mm-180 Let x,y be elements of a group G. Prove that if xy=yx then >rank(xy)=LCM(rank(x),rank(y)).I assume that by rank you mean the order of the element?> I know how to do it when LCD(rank(x),rank(y))=1, but have no idea how to expand> the proof to be good for the more general case.That's probably because it is false in general (though it is true whenthe two orders are relatively prime).Arturo Magidin, sans .sig =Arturo Magidin a .8ecrit dans le message de> Let x,y be elements of a group G. Prove that if xy=yx then>rank(xy)=LCM(rank(x),rank(y)). I assume that by rank you mean the order of the element?> I know how to do it when LCD(rank(x),rank(y))=1, but have no idea how toexpand> the proof to be good for the more general case. That's probably because it is false in general (though it is true when> the two orders are relatively prime). Arturo Magidin, sans .sigLet p_1, p_2, p_3 primes distincts.By example, if order(a)=p_1^2*p_3^4 and order(b)=p_2^5*p_3^3 and a*b=b*athen order(a*b)=LCM(order(a),order(b)) but if order(a)=p_1^2*p_3^4 andorder(b)=p_2^5*p_3^4 then order(a*b) is not always Lcm(order(a),order(b))See Odoux Ramis Deschamps Tome 1 Exercise 2.18True or false?Excuse my bad english =>Let p_1, p_2, p_3 primes distincts.>By example, if order(a)=p_1^2*p_3^4 and order(b)=p_2^5*p_3^3 and a*b=b*a>then order(a*b)=LCM(order(a),order(b)) but if order(a)=p_1^2*p_3^4 and>order(b)=p_2^5*p_3^4 then order(a*b) is not always Lcm(order(a),order(b))>See Odoux Ramis Deschamps Tome 1 Exercise 2.18>True or false?>Excuse my bad englishI assume that by example is a translation error (that is, you arenot asking for an example, but rather mean for instance or forexample, for the 'rst part; although you could mean give an examplefor the latter part).So let me see if I have this right:Let G be a group, a,b two commuting elements in G, and let p,q,r bethree different primes (sorry for the change of notation). (1) If order(a) = p^2*r^4 and order(b)=q^5*r^3, then order(ab)=p^2*q^5*r^4. (2) If order(a) = p^2*r^4 and order(b)=q^5*r^4, then it is possible for order(ab) to be strictly smaller than p^2*q^5*r^4.The last one is easy: let G be the direct sum of a group of order p^2generated by x, a group of order q^5 generated by y, and a group oforder r^4 generated by z. Then let a = (x,1,z) and b =(1,y,z^{-1}). Then ab = (x,y,1) is of order p^2*q^5.Part 1 seems more interesting. I think the following is correct, butthen I've been burned before by these things...Lemma. Let G be a group, x, y elements of G such that xy=yx. Let r bea rational prime, and let i>j>0 be positive integers. If the order ofx is r^i and the order of y is r^j, then the order of xy is r^i.Proof: The order of xy divides r^i; assume the order is r^k,0<=k<=i. Then (xy)^{r^k} = 1, so x^{r^k} = y^{-r^k}. If k<=j, then theorder of y^{r^k} is r^{j-k}, while the order of x^{r^k} isr^{i-k}. These two are different, so k>j. Therefore, y^{-r^k}=e, sox^{r^k}=e, which means that k>=i; this proves that k=i, as claimed. QEDNow we go back to the situation in (1); we want to show that (ab)^k =e if and only if a^k = b^k = e. We may assume that k dividesp^2*q^5*r^4. If (ab)^k = e, then a^k = b^{-k}, so = is asubgroup of both and ; therefore, it must have order r^i forsome i, i=0, 1, 2, 3. The order of a^k is (p^2*r^4)/gcd(p^2*r^4,k);likewise, the order of b^k is (q^5*r^3,gcd(q^5*r^3,k). So k= p^2*q^5*r^{3-i}. Thus, it is enough to show that (a^{p^2*q^5}b^{p^2*q^5}) has order r^4; but this follows from the Lemma, sincea^{p^2*q^5} has order r^4 and b^{p^2*q^5} has order r^3.-- =It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.eduCan anyone suggest a good way to build a n*n matrix whose determinant is agiven expression? The matrix' entries must belong to a given set ofvariables, zeroes and ones, but can't be expressions (sign changes areallowed).Example: build a 4*4 matrix, with entries in {a, b, c, d, 0, 1}, having D = c^2 + d^2 - a*b as determinant.What is the theoritical explanation about the impossibility to solve theabove example (if this one is indeed impossible to solve)?-- Here's where you can reach me ('xed-width font required). . ,-. .. . ,-. ,-. ,-. ,-. |- /,-. ,-. ,-. ,-| ,-. ,-. ,-.| | `-. |-' | | |-' | |,-|| | | -- |-' -- | | | | | | |`-^ `-' `-' .95 .95 `-' `' `-^' `-| `-' `-^ :; `-' .95 `-| `-- | ,| ` `' => Can anyone suggest a good way to build a n*n matrix whose determinant is a> given expression? The matrix' entries must belong to a given set of> variables, zeroes and ones, but can't be expressions (sign changes are> allowed).> Example: build a 4*4 matrix, with entries in {a, b, c, d, 0, 1}, having> D = c^2 + d^2 - a*b as determinant.> What is the theoritical explanation about the impossibility to solve the> above example (if this one is indeed impossible to solve)?In any matrix, its determinant is the product of its eigenvalues. So Iwould consider the matrix ADA^-1 where D is a diagonal matrix of theeigenvalues, and det(A)=det(A^-1)=1. It is then easy to conclude thatmatrix must be some of linear combination ofthe factors of thedeterminant.For det(ADA^-1)=c^2+d^2-ab, it has no factorization over a linearcombination of {a,b,c,d,0,1}. You show this by consider the product ofthings like x_0*a+x_1*b+x_2*c+x_3*d+x_4 andy_0*a+y_1*b+y_2*c+y_3*d+y_4*d, and such.Gershon Bialer =>> Can anyone suggest a good way to build a n*n matrix whose determinant is a>> given expression? The matrix' entries must belong to a given set of>> variables, zeroes and ones, but can't be expressions (sign changes are>> allowed).>> >> Example: build a 4*4 matrix, with entries in {a, b, c, d, 0, 1}, having>> D = c^2 + d^2 - a*b as determinant.>>> What is the theoritical explanation about the impossibility to solve the>> above example (if this one is indeed impossible to solve)?In any matrix, its determinant is the product of its eigenvalues. So I>would consider the matrix ADA^-1 where D is a diagonal matrix of the>eigenvalues, and det(A)=det(A^-1)=1. It is then easy to conclude that>matrix must be some of linear combination ofthe factors of the>determinant.For det(ADA^-1)=c^2+d^2-ab, it has no factorization over a linear>combination of {a,b,c,d,0,1}. You show this by consider the product of>things like x_0*a+x_1*b+x_2*c+x_3*d+x_4 and>y_0*a+y_1*b+y_2*c+y_3*d+y_4*d, and such.I can't make any sense of this. The function a^2 - b has no factorisationinto linear forms in {a,b,c,d,1}, but it's quite obviously the determinantof the 2x2 matrix [ [ a b ] [ 1 a ] ]. -- Erick erick@sfu.ca (Erick BryceFor det(ADA^-1)=c^2+d^2-ab, it has no factorization over a linear>>combination of {a,b,c,d,0,1}. You show this by consider the product of>>things like x_0*a+x_1*b+x_2*c+x_3*d+x_4 and>>y_0*a+y_1*b+y_2*c+y_3*d+y_4*d, and such.I can't make any sense of this. The function a^2 - b has no factorisation>into linear forms in {a,b,c,d,1}, but it's quite obviously the determinant>of the 2x2 matrix [ [ a b ] [ 1 a ] ].Of course you don't need a factorisation in linear forms of (a, b, 1).The eigenvalues don't even have to be real. We can't have a diagonal2x2 matrix with the determinant a^2 - b with elements in (a, b, 1),but this doesn't say anything about the impossibility of having anon-diagonal matrix of the same type. =|> For det(ADA^-1)=c^2+d^2-ab, it has no factorization over a linear|> combination of representations of complex numbers as z=a+biand ( a b ) z = ( ) ( -b a )are equivalent, I would have thought that matrix ( c+di a ) M = ( ) , with det(M)=c^2+d^2-ab ( b c-di )could have an equivalent representation as a 4*4 matrix, hence the statementof my initial post. Of course, this belief results from a misguidingintuition.I'd greatly appreciate if someone could explain where and why this intuitiongoes astray.-- Here's where you can reach me ('xed-width font required). . ,-. .. . ,-. ,-. ,-. ,-. |- /,-. ,-. ,-. ,-| ,-. ,-. ,-.| | `-. |-' | | |-' | |,-|| | | -- |-' -- | | | | | | |`-^ `-' `-' .95 .95 `-' `' `-^' `-| `-' `-^ :; `-' .95 `-| `-- | ,| ` `' => |> For det(ADA^-1)=c^2+d^2-ab, it has no factorization over a linear> |> combination of {a,b,c,d,0,1}.> That's perfectly numbers as> z=a+bi> and> ( a b )> z = ( )> ( -b a )> are equivalent, I would have thought that matrix> ( c+di a )> M = ( ) , with det(M)=c^2+d^2-ab> ( b c-di )> could have an equivalent representation as a 4*4 matrix, hence the statement> of my initial post. Of course, this belief results from a misguiding> intuition.> I'd greatly appreciate if someone could explain where and why this intuition> goes astray.Hmm, I guess I was wrong.The following works for 4x4:{1 0 0 0 0 1 0 0 0 0 c+di a 0 0 b c-di } =>I am interested in the sequence a^2, a^3, a^4, ...>for a = .9, a = .99, a = .999, through a = .999... > You'll have to specify what you mean by a=.9, a=.99, a=.999, through> a=.999.... If you'd said a=.9, a=.99, a=.999, through a=.99999999> I would have understood you, but the sequence .9, .99, .999, ... doesn't> include .999..., so it's like saying a=1, a=2, a=3, through a=-7.> For each of a=.9, a=.99, a=.999, ... the sequence a^2, a^3, a^4, ...> tends to zero, but for a=.999... it is a the constant sequence 1, 1, ...> -- RichardIt's funny that this part about a=.9, a=.99, a=.999, through a=.999...is exactly the same problem many posters (including the infamousPhil) are having when they post nonsense like 0.999... is not equalto 1 or Cantor was wrong or The rationals are uncountable orThere is a largest natural number or This particular Turing machineprints 111... and then halts or There is an upper bound on the numberof digits a natural number can have or ... ad nauseam. All of thisinsanity that's choking sci.math and sci.logic is due to the simple factthat some people believe that in'nity is constructible (in a certain sense)rather than potential, that it is possible to *arrive* at in'nity byrepeatedly adding 1 enough times. There's nothing wrong with musingphilosophically like this, but when it comes time to formalize theirbogus arguments, they can't do it (because they're stance is inconsistent).The mathematicians continue to correctly argue against them with formalism,and the seemingly insane continue to retort with more half-thoughtgarbage because they don't understand what a rigorous argument really is,what mathematics is all about. In my mind, Phil, Russell Easterly, thisGerry Denke (or whatever the name is), and many others are all the sameperson with regard to their posts. They share some common incorrectintuitions. What can we do to help them? It seems nothing. I've neverseen an argument like this resolved.-Leonard =*** post for FREE via your newsreader at post.newsfeed.com ***> It's funny that this part about a=.9, a=.99, a=.999, through a=.999...> is exactly the same problem many posters (including the infamous> Phil) are having when they post nonsense like 0.999... is not equal> to 1 or Cantor was wrong or The rationals are uncountable or> There is a largest natural number or This particular Turing machine> prints 111... and then halts or There is an upper bound on the number> of digits a natural number can have or ... ad nauseam. All of this> insanity that's choking sci.math and sci.logic is due to the simple fact> that some people believe that in'nity is constructible (in a certainsense)> rather than potential, that it is possible to *arrive* at in'nity by> repeatedly adding 1 enough times. There's nothing wrong with musing> philosophically like this, but when it comes time to formalize their> bogus arguments, they can't do it (because they're stance isinconsistent).And this is the beauty of mathematics -- you can move someone through theirinconsistencies to DROP notions that are confusing to them.> The mathematicians continue to correctly argue against them withformalism,> and the seemingly insane continue to retort with more half-thought> garbage because they don't understand what a rigorous argument really is,> what mathematics is all about. In my mind, Phil, Russell Easterly, this> Gerry Denke (or whatever the name is), and many others are all the same> person with regard to their posts. They share some common incorrect> intuitions. What can we do to help them? It seems nothing. I've never> seen an argument like this resolved.Look into any lower division concrete mathematics course. karl mhttp://www.newsfeed.com - The #1 Newsgroup Service in the World!-----== 100,000 Groups! - 19 Servers! - Unlimited Download! =----- =>*** post for FREE via your newsreader at post.newsfeed.com ***> It's funny that this part about a=.9, a=.99, a=.999, through a=.999...>> is exactly the same problem many posters (including the infamous>> Phil) are having when they post nonsense like 0.999... is not equal>> to 1 or Cantor was wrong or The rationals are uncountable or>> There is a largest natural number or This particular Turing machine>> prints 111... and then halts or There is an upper bound on the number>> of digits a natural number can have or ... ad nauseam. All of this>> insanity that's choking sci.math and sci.logic is due to the simple fact>> that some people believe that in'nity is constructible (in a certain>> sense)>> rather than potential, that it is possible to *arrive* at in'nity by>> repeatedly adding 1 enough times. There's nothing wrong with musing>> philosophically like this, but when it comes time to formalize their>> bogus arguments, they can't do it (because they're stance is>> inconsistent).And this is the beauty of mathematics -- you can move someone through their>inconsistencies to DROP notions that are confusing to them.But that only works for people who already have enough faith in mathematics, to the extent that deduction trumps intuition when these con¤ict. I it's clear our infamous posters are not in that category.IMO, much too little is done in the way of maths appreciation in school. The idea that one can forge a deductive chain that spans a dif'cult gap, but where each individual link in the chain is not dif'cult, is IMO one of the Great Ideas of Western Civilisation.And the usual style of teaching maths, at all levels, IMO does not put enough effort into leading students to recallibrate their intuitions when the intuitions lead them astray. There are often ways of presenting the material that would be more congenial with students' untutored intuitions.>> The mathematicians continue to correctly argue against them with>> formalism,>> and the seemingly insane continue to retort with more half-thought>> garbage because they don't understand what a rigorous argument really is,>> what mathematics is all about. In my mind, Phil, Russell Easterly, this>> Gerry Denke (or whatever the name is), and many others are all the same>> person with regard to their posts. They share some common incorrect>> intuitions. What can we do to help them? It seems nothing. I've never>> seen an argument like this resolved.Look into any lower division concrete mathematics course.Does that really work? I've known many upper-division students who seemed to be clueless about things like this.-- ---------------------------| B B aa rrr b || BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb | ----------------------------- =*** post for FREE via your newsreader at post.newsfeed.com ****** post for FREE via your newsreader at post.newsfeed.com ***> It's funny that this part about a=.9, a=.99, a=.999, througha=.999...>> is exactly the same problem many posters (including the infamous>> Phil) are having when they post nonsense like 0.999... is not equal>> to 1 or Cantor was wrong or The rationals are uncountable or>> There is a largest natural number or This particular Turing machine>> prints 111... and then halts or There is an upper bound on the number>> of digits a natural number can have or ... ad nauseam. All of this>> insanity that's choking sci.math and sci.logic is due to the simplefact>> that some people believe that in'nity is constructible (in a certain>> sense)>> rather than potential, that it is possible to *arrive* at in'nity by>> repeatedly adding 1 enough times. There's nothing wrong with musing>> philosophically like this, but when it comes time to formalize their>> bogus arguments, they can't do it (because they're stance is>> inconsistent).And this is the beauty of mathematics -- you can move someone throughtheir>inconsistencies to DROP notions that are confusing to them. But that only works for people who already have enough faith in> mathematics, to the extent that deduction trumps intuition when these> con¤ict. I it's clear our infamous posters are not in that category.When they quit pursuing their notions and just go away it's a victory.Besides, its all the other READERS of the contest who are more important inthe long run.> IMO, much too little is done in the way of maths appreciation in> school. The idea that one can forge a deductive chain that spans a> dif'cult gap, but where each individual link in the chain is not> dif'cult, is IMO one of the Great Ideas of Western Civilisation. And the usual style of teaching maths, at all levels, IMO does not put> enough effort into leading students to recallibrate their intuitions> when the intuitions lead them astray. There are often ways of> presenting the material that would be more congenial with students'> untutored intuitions.I'm a programmer, not a teacher -- I don't know.>> The mathematicians continue to correctly argue against them with>> formalism,>> and the seemingly insane continue to retort with more half-thought>> garbage because they don't understand what a rigorous argument reallyis,>> what mathematics is all about. In my mind, Phil, Russell Easterly,this>> Gerry Denke (or whatever the name is), and many others are all the same>> person with regard to their posts. They share some common incorrect>> intuitions. What can we do to help them? It seems nothing. I'venever>> seen an argument like this resolved.Look into any lower division concrete mathematics course. Does that really work? I've known many upper-division students who> seemed to be clueless about things like this.It's a process that you go through with ONE person at a time in a group.The others will form their own conclusions, and yes, some will remainclueless at each point you measure. karl mhttp://www.newsfeed.com - The #1 Newsgroup Service in the World!-----== 100,000 Groups! - 19 Servers! - Unlimited Download! =----- =In BC Calculus at my highschool today we dealt with taking theintergral of function that include vertical asymptotes. My questionarose with the problem of the integration of 1/(x-1) from 0 to 2. Webegan to try to decide if it converged on diverged. Here is where theproblem showed itself. After integrating it you end up with inf. -inf.. Our teacher said that this is unde'ned, which I except in mostcases, but here I question it. In this problem we ended up with thelim c-->1- ln|x-1| for 0 to c + lim d-->1+ ln|x-1| for 1 to 2. Andthis gives lim c-->1- ln|c-1| - lim d-->1+ ln|d-1|. Here is where westopped. My math teacher said it was unde'ned since it was inf. -inf. I wonder why it isn't zero though since the inf. are/should beequal even though they are limits. =>My math teacher said it was unde'ned since it was inf. ->inf. I wonder why it isn't zero though since the inf. are/should be>equal even though they are limits.What is lim(1 / (3^n - 2^n), n->oo)? =My math teacher said it was unde'ned since it was inf. ->inf. I wonder why it isn't zero though since the inf. are/should be>equal even though they are limits. What is lim(1 / (3^n - 2^n), n->oo)?It's 0, of course[noting that 3^n - 2^n = 2^n ( (3/2)^n - 1 ) -> +oo ].But it seems that you missed the point that Amos had. I suspect that healready knows that a limit form of the oo - oo type is indeterminatethat we might reasonably say that there is indeed what happens if we consider theCauchy principal value of the integral, despite the fact that the integralitself is not considered to be convergent.David => In BC Calculus at my highschool today we dealt with taking the> intergral of function that include vertical asymptotes. My question> arose with the problem of the integration of 1/(x-1) from 0 to 2. We> began to try to decide if it converged on diverged. Here is where the> problem showed itself. After integrating it you end up with inf. -> inf.. Our teacher said that this is unde'ned, which I except in most> cases, but here I question it. In this problem we ended up with the> lim c-->1- ln|x-1| for 0 to c + lim d-->1+ ln|x-1| for 1 to 2. And> this gives lim c-->1- ln|c-1| - lim d-->1+ ln|d-1|. Here is where we> stopped. My math teacher said it was unde'ned since it was inf. -> inf. I wonder why it isn't zero though since the inf. are/should be> equal even though they are limits.Both limits must exist (as 'nite real numbers) to say the integralconverges. What you talk about, where there sides, can sometimes be found underthe name principal value. = In BC Calculus at my highschool today we dealt with taking the> intergral of function that include vertical asymptotes. My question> arose with the problem of the integration of 1/(x-1) from 0 to 2. We> began to try to decide if it converged on diverged. Here is where the> problem showed itself. After integrating it you end up with inf. -> inf.. Our teacher said that this is unde'ned, which I except in most> cases, but here I question it. In this problem we ended up with the> lim c-->1- ln|x-1| for 0 to c + lim d-->1+ ln|x-1| for 1 to 2. And> this gives lim c-->1- ln|c-1| - lim d-->1+ ln|d-1|. Here is where we> stopped. My math teacher said it was unde'ned since it was inf. -> inf. I wonder why it isn't zero though since the inf. are/should be> equal even though they are limits. Both limits must exist (as 'nite real numbers) to say the integral> converges.Exactly.> What you divergence on the two sides, can sometimes be found under> the name principal value.Or Cauchy principal value. See, for example,.David =MOON before Venus, as in one small step, then whatever is possible.As for this next time around, if our NASA is actually going for themoon, this effort is going to be starting itself off from nearscratch, as so far there's absolutely nothing documented from our pastnor of what's in current inventory that has ever worked, or haspotential for getting man to/from any ET surface, much less of thenastier lunar surface. Christ almighty, half the time we still can'teven get our relatively miniature probes down onto another surface asplanned.My continuing thoughts on the need for delivering pure h2o2 seems likea perfectly good notion for importing such to the lunar surface, alongwith a small amount of c12h26 and the LM-1 bus is off and running onit internal IRRCE. Our next stop could be Venus, or at least a lunarestablished interplanetary communications link so that we don't haveto physically go there.I've got just a couple hundred thousand words about our residentwarlord taking us back to the moon, I feel somewhat most strongly likefollowing in his educational high standards and accountabilitythat's only being recently superseded by his so what's thedifference policy.For starters, it's about time, whereas actually for the 'rst timewe'd be actually doing humanity a terri'c sort of favor, especiallyif we can get our fearless leader to ride in one of those Apollolanders and strut about for 36 hours in one of those Apollo moonsuits, as that way we'd stand our best chance ever of getting rid ofthe bastard, once and for all.Otherwise I'm all for investing into whatever it takes in establishingourselves on the moon before others take possession of all that niftyHe3.I believe the moon well return a pro't within the 'rst year, thenit'll offer the gateway to other planets like Mars and Venus, thoughwhom would want to goto a frozen and irradiated to death planet likeMars?Some PIRATEShttp://mittymax.com/Archive/0085- SaddamHusseinAndTheSandPirates.htmG.W. Bush, on the moon and still lying about something.http://guthvenus.tripod.com/gv-gwb-moon.htmhttp:// guthvenus.tripod.com/moon-04.htmhttp://guthvenus.tripod.com/ gv-interplanetary.htm => MOON before Venus, as in one small step, then whatever is possible.> Crank Information = http://home.planet.nl/~faase009/Ha_bf_Turing.html has sketches of three ways to prove that BrainF*** is Turing-complete; but,it seems they all assume that each memory cell has *unbounded* storage capacity (making the computation model like that of a URM, except with only relative addressing.) So, my question concerns the simplest case:Is BrainF*** with binary memory Turing-complete? In this case, memory is a bit-string of unbounded length, and the operators + and - are each equivalent to just ¤ipping the current bit; aside from i/o, the instructions reduce to 've: < move memory-pointer left one bit > move memory-pointer right one bit + ¤ip current bit [ if current bit==0, goto instruction after the matching ] ] if current bit==1, goto instruction after the matching [This closely resembles the following 've-instruction language that uses labels, and which is easy to prove Turing-complete: < move memory-pointer left one bit > move memory-pointer right one bit + ¤ip current bit 0n if current bit==0, goto label n 1n if current bit==1, goto label n(The 5-tuples of a binary-tape TM are directly expressible in the latter language. Brie¤y: P a b m Q -> P cX b Y + Y d 0Q 1Q Xwhere (P,Q)=(old,new) states, which become labels; (a,b)=(read,written) bits; m= head-movement; c=1-a; d= < or > corresponding to m; X,Y are auxiliary labels.)Is there a way to deduce from this that binary BrainF*** isTuring-complete?--r.e.s. => I see you've already posted this here. You've done the repostvery tastefully (waiting a while between posts), so I can't complain,but it would IMVHO be slightly better to post a follow-up in thesame thread saying, still waiting for input or something, ratherthan starting a new thread and making your readers get deja vu allover again. :) Just IMHO.> http://home.planet.nl/~faase009/Ha_bf_Turing.html has sketches> of three ways to prove that BrainF*** is Turing-complete; but,> it seems they all assume that each memory cell has *unbounded*> storage capacity (making the computation model like that of a> URM, except with only relative addressing.) So, my question> concerns the simplest case: Is BrainF*** with binary memory Turing-complete? I think (but have not proven) that this case is essentiallyequivalent to the case with trinary memory, or quaternary,or in general BrainF*** with memory cells of any 'xed 'nitesize: call the memory cell capacity C. I think (but have not proven) that it is possible to developa few simple operations in base-C BrainF*** that operate onmulti-cell base C numbers. Just in case that's not clear, Imean we can store the number 42 in binary BrainF*** as the base-2multi-cell number 0,0,0, ..., 1,0,1,0,1,0, ...or for that matter as the tally-system representation 0,0,0, ..., 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, ...and we can write BrainF*** sub-procedures to increment and decrementthese bignums. We can even extend a bignum's representation, if weneed to, by adding onto one of the tape's unbounded ends. Bottom line: We can still compute with arbitrarily large numbers inbase-C BrainF***. And if we can do that, then we *should* be able touse a similar proof of Turing-completeness. Note that the proof givenon that page seems to use only two bignum values, and our tapecoincidentally has two unbounded sides to use for scratch work! So, I certainly haven't proven that BrainF*** is Turing-complete,but I have a very strong hunch that it is. (This lack of anythingconcrete is why I didn't post before, but if you're that interested...)-Arthur => http://home.planet.nl/~faase009/Ha_bf_Turing.html has sketches> of three ways to prove that BrainF*** is Turing-complete; but,> it seems they all assume that each memory cell has *unbounded*> storage capacity (making the computation model like that of a> URM, except with only relative addressing.)Wrong, incidentally; the second, my universal Turing machine, onlyassumes the cells can hold values from 0 to 100 inclusive. This iseasily reducible to 0-23 by using a binary i/o format.-Daniel Cristofani. >++[<++++++++[<[<++>-]>>[>>]+>>+[-[->>+<<<[<[<<]<+>]>[>[>>]] ]<[>>[-]]>[>[- <<]+<[<+<]]+<<]<[>+<-]>>-]<.[-]>>]http://www.hevanet.com/ cristofd/brain/ = http://home.planet.nl/~faase009/Ha_bf_Turing.html has sketches> of three ways to prove that BrainF*** is Turing-complete; but,> it seems they all assume that each memory cell has *unbounded*> storage capacity (making the computation model like that of a> URM, except with only relative addressing.) Wrong, incidentally; the second, my universal Turing machine, only> assumes the cells can hold values from 0 to 100 inclusive. This is> easily reducible to 0-23 by using a binary i/o format. I assume the numbers 0 to 100 inclusive come from the ASCII valueof .95c' plus 1? I'm not nearly good enough at BF to tell whether yourprogram does what you say it does, and even if I could do that Iwouldn't be good enough at math to tell whether that tag system isTuring-complete. But I'm willing to take your word for it if you'llexplain where the numbers 100 and 23 came from -- you don't use nearly23 different input characters, do you?-Arthur =For starters, besides the mostly 6e6 < 6e9 hydrogen atoms/m3, theuniverse has been made up of mostly disinformation, then politicallies and loads of hidden agendas plus ulterior motives up the kazoo.Just adding up the universe at 6e9/m3, and that's a great deal ofmass, though we've got to do the MOON next ('rst time for everything)before Venus, as in one small step, then whatever is possible.As for this next time around, if our NASA is actually going for themoon, this effort is going to be starting itself off from nearscratch, as so far there's absolutely nothing documented from our pastnor of what's in current inventory that has ever worked, or haspotential for getting man to/from any ET surface, much less of thenastier lunar surface. Christ almighty, half the time we still can'teven get our relatively miniature probes down onto another surface asplanned.My continuing thoughts on the need for delivering pure h2o2 seems likea perfectly good notion for importing such to the lunar surface, alongwith a small amount of c12h26 and the LM-1 bus is off and running onit internal IRRCE. Our next stop could be Venus, or at least a lunarestablished interplanetary communications link so that we don't haveto physically go there.I've got just a couple hundred thousand words about our residentwarlord taking us back to the moon, I feel somewhat most strongly likefollowing in his educational high standards and accountabilitythat's only being recently superseded by his so what's thedifference policy.For starters, it's about time, whereas actually for the 'rst timewe'd be actually doing humanity a terri'c sort of favor, especiallyif we can get our fearless leader to ride in one of those Apollolanders and strut about for 36 hours in one of those Apollo moonsuits, as that way we'd stand our best chance ever of getting rid ofthe bastard, once and for all.Otherwise I'm all for investing into whatever it takes in establishingourselves on the moon before others take possession of all that niftyHe3.I believe the moon well return a pro't within the 'rst year, thenit'll offer the gateway to other planets like Mars and Venus, thoughwhom would want to goto a frozen and irradiated to death planet likeMars?Some PIRATEShttp://mittymax.com/Archive/0085- SaddamHusseinAndTheSandPirates.htmG.W. Bush, on the moon and still lying about something.http://guthvenus.tripod.com/gv-gwb-moon.htmhttp:// guthvenus.tripod.com/moon-04.htmhttp://guthvenus.tripod.com/ gv-interplanetary.htm => That's a lie. It's clear from your behavior in other threads that you have> no interest in having your mathematical errors corrected.Cantor seems to attract a lot of odd people. Sometimes I think it would bebetter if Cantor's proof were relegated to advanced courses, and a differentmethod were used to show that the reals are uncountable in earlier courses.E.g., show that there is no one-to-one corresponce between any set and theset of its subsets (easy), and then show a one-to-one correspondence betweenthe reals and the set of subsets of the integers (easy). That gives you theuncountability of the reals without going anywhere near Cantor's brilliant,correct, but crank-attracting method.-- --Tim Smith =>> That's a lie. It's clear from your behavior in other threads that you have>> no interest in having your mathematical errors corrected.Cantor seems to attract a lot of odd people. Sometimes I think it would be>better if Cantor's proof were relegated to advanced courses, and a different>method were used to show that the reals are uncountable in earlier courses.>E.g., show that there is no one-to-one corresponce between any set and the>set of its subsets (easy), and then show a one-to-one correspondence between>the reals and the set of subsets of the integers (easy). That gives you the>uncountability of the reals without going anywhere near Cantor's brilliant,>correct, but crank-attracting method.I believe most of us think that this proposal is totally wrong-headed.The reasoning in the diagonal argument is _very_ clear and simple;if someone doesn't follow it that means the person cannot thinkstraight. The fact that lots of people can't think straight doesn'tmean that the diagnostic should be hidden - there exist(?) seriousstudents who don't understand the proof at 'rst, and then whenthey 'nally do understand it they've 'lled a _serious_ gap in theirreasoning skills. Filling that gap is important, so it's importantwe 'nd out about it.************************David C. Ullrich => That's a lie. It's clear from your behavior in other threads that you have> no interest in having your mathematical errors corrected.> Cantor seems to attract a lot of odd people. Sometimes I think it would be> better if Cantor's proof were relegated to advanced courses, and a different> method were used to show that the reals are uncountable in earlier courses.> E.g., show that there is no one-to-one corresponce between any set and the> set of its subsets (easy), and then show a one-to-one correspondence between> the reals and the set of subsets of the integers (easy). That gives you the> uncountability of the reals without going anywhere near Cantor's brilliant,> correct, but crank-attracting method.In fact, that was more or less Cantor's 'rst proof. The diagonal thing only came later. Rather a shame that he ever did it. => In fact, that was more or less Cantor's 'rst proof. The diagonal thing > only came later. Rather a shame that he ever did it.He was writing for people who already knew that .999... = 1.000...But even if you don't know that, the proof still works. => the URL didn't work. I like it, though,However, it isn't hard to guess that he simply left off the .edu part. Ifyou add that, the page you get does indeed have a link to his purportedproof of Goldbach's Conjecture. I only glanced at it, but it seemed to bereasonably coherently written--I didn't notice any weird terminology oranything like that. It probably deserves at least as much serious scrutinyas a certain regular poster to this group gets for his unorthodox theories.There was no link on that page to the Fermat thing he mentioned, though.-- --Tim Smith => the URL didn't work. I like it, though,> However, it isn't hard to guess that he simply left off the .edu part. If> you add that, the page you get does indeed have a link to his purported> proof of Goldbach's Conjecture. I only glanced at it, but it seemed to be> reasonably coherently written--I didn't notice any weird terminology or> anything like that. It probably deserves at least as much serious scrutiny> as a certain regular poster to this group gets for his unorthodox theories.> There was no link on that page to the Fermat thing he mentioned, though.After Googling for his name, I believe the Fermat one is athttp://www.geocities.com/kerryme47714/. => the URL didn't work. I like it, though,> However, it isn't hard to guess that he simply left off the .edu part. If> you add that, the page you get does indeed have a link to his purported> proof of Goldbach's Conjecture. I only glanced at it, but it seemed to be> reasonably coherently written--I didn't notice any weird terminology or> anything like that. It probably deserves at least as much serious scrutiny> as a certain regular poster to this group gets for his unorthodox theories.> There was no link on that page to the Fermat thing he mentioned, though.> After Googling for his name, I believe the Fermat one is at> http://www.geocities.com/kerryme47714/.Forgive me for sending a false URL. The paper is present on any searchengine under Diophantine Equations>Fermat's Last Theorem. I hope thisclears up some confusion. THank you for your interest. Kerry =After a bit of experimenting numerically, I've noticed that the percentageof numbers in an interval [1,k] that is divisible by an n-th power largerthen 1 tends to a constant as k -> oo.The sequences below list the number of mentioned numbers fork=10^1,10^2,10^3,10^4,10^5.n=2: 3,39,392,3917,39206n=3: 1,15,167,1681,16810n=4: 0,7,75,760,7605n=5: 0,3,35,355,3560What are the theoretical values of the corresponding percentages?[d(2) = +/- 39%, d(3) = +/- 17%, d(4) = +/- 7.6%] => After a bit of experimenting numerically, I've noticed that the percentage> of numbers in an interval [1,k] that is divisible by an n-th power larger> then 1 tends to a constant as k -> oo. The sequences below list the number of mentioned numbers for> k=10^1,10^2,10^3,10^4,10^5. n=2: 3,39,392,3917,39206> n=3: 1,15,167,1681,16810> n=4: 0,7,75,760,7605> n=5: 0,3,35,355,3560 What are the theoretical values of the corresponding percentages?> [d(2) = +/- 39%, d(3) = +/- 17%, d(4) = +/- 7.6%]I remember something as 6/Pi^2 being the probability that two random numbersare relatively prime, so I messed around a bit with such expressions,'nding that 1 - 6/Pi^2 = 0.39207... = d(2)? What reasoning lies behindthis? => After a bit of experimenting numerically, I've noticed that the percentage> of numbers in an interval [1,k] that is divisible by an n-th power larger> then 1 tends to a constant as k -> oo. The sequences below list the number of mentioned numbers for> k=10^1,10^2,10^3,10^4,10^5. n=2: 3,39,392,3917,39206> n=3: 1,15,167,1681,16810> n=4: 0,7,75,760,7605> n=5: 0,3,35,355,3560 What are the theoretical values of the corresponding percentages?> [d(2) = +/- 39%, d(3) = +/- 17%, d(4) = +/- 7.6%]> I remember something as 6/Pi^2 being the probability that two random numbers> are relatively prime, so I messed around a bit with such expressions,> 'nding that 1 - 6/Pi^2 = 0.39207... = d(2)? What reasoning lies behind> this?(Yesterday, I must have posted to the wrong place.)d(n) = 1 - 1/zeta(n)http://jove.prohosting.com/~skripty/page_807. htmProportion not divisible by p^n is (1 - p^-n).1/(1 - p^-n) = (1 + p^-n + p^(-2n) + p^(-3n) + ... )The product for all p covers all k^-n making the zeta function. => .9^2 = .81> .99^2 = .9801> .999^2 = .998001> .9999^2 = .99980001> .99999^2 = .9999800001> .999999^2 = .999998000001> .9999999^2 = .99999980000001> .99999999^2 = .9999999800000001> .999999999^2 = .999999998000000001> .9999999999^2 = .99999999980000000001> .99999999999^2 = .9999999999800000000001> .999999999999^2 = .999999999998000000000001> .9999999999999^2 = .99999999999980000000000001> .99999999999999^2 = .9999999999999800000000000001> .999999999999999^2 = .999999999999998000000000000001> .9999999999999999^2 = .99999999999999980000000000000001> .99999999999999999^2 = .9999999999999999800000000000000001> .999999999999999999^2 = .999999999999999998000000000000000001> .9999999999999999999^2 = .99999999999999999980000000000000000001> .99999999999999999999^2 = .9999999999999999999800000000000000000001> Each position after the decimal point eventually becomes a 9 and stays> that way.> Daniel W. Johnson has proven that 0.999... will never equal 1.> I knew someone here could do it.Assumptions:1) Take 0.999... to be the limit of the sequence of rational numbers 0.9, 0.99, 0.999, ...2) Work in the real number system.Conclude:Notice that 0.9, 0.99, 0.999, ... forms a Cauchy sequenceconverging to 1. This means that 0.999.... is 1. The _reals_ have beencarefully constructed such that 0.999... is deemed the same as 1.If you do not like the conclusion, dispute 1) or 2). For example, donot work with real numbers. You can use, e.g, Stone-Cech (sp?)compacti'cation of the rationals in which 0.999... is =>are trying to deny that the title ADJUNCT is the lowest title an>academic can hold? No, but I don't know that it's true, either. Depends on the>>institution, I reckon.>or that anyone holding such title is subject to abuse and contempt>by practically the whole academic institution?Well, I suppose I'll deny that, yes. It is a silly claim.> why do you think it is a silly claim? have you seen the salary> adjuncts earn? have you never come into contact with an adjunct? don't> you know what it takes for an adjunct to remain employed as such?>or that the american higher education system is heavily using and>abusing adjuncts?[sic]. This is nonsense.> well, pardon my poor spelling ability. was that your objection? or is> there other reason why you consider the report that ADJUNCTs are not> the lowest members of the academic caste system?>>You used this claim to support the claim>>that Arturo is incompetent. This is just utter bull.> you have some reading comprehension problems. my initial follow up in> no way claims that arture is incompetent.> Sorry to butt in, but that is *precisely* how I read the following:> >BEGIN QUOTED maky m. ORIGINAL Magidin is indeed either incompetent as a>> mathematician, or more likely a liar, or both.> well, he is an ADJUNCT assistant professor. ADJUNCTS are the pariahs> in academic caste system.> so whomever you are, may well be right on this one...>> James Harris>>END QUOTED maky m. ORIGINAL ARTICLE<<<<<<<<<<<<<<<<<<<<<> That is, I read your remark above to be in agreement with the alleged> (in this case) JSH, where he claimed Magidin to be incompetent and/or> dishonest.well, i can't precicely say that i agree with the apparent impostor'sassessment. if you analise the statement carefully, you will see thereis a subtle issue exposed.> I'm wondering how my reading of the above passage might be evidence of> any problems in comprehension.well, it depends. are you STILL under the impression that i was tryingto insult magidin?> Here's how I saw it:> You stated that Magidin is an adjunct assistant professor> (and speaking as a former holder of such a position, it is> surely a state less desirable than being the holder of an> endowed chair).modesty isn't an commendable attribute in this context. the adjuncttitle at any academic rank is without exception the bottom.> You then stated that the person to whom you were responding> might well have been correct in his assertion.based on your comprehension of the english language and your possibleacquaintance with formal logic, is the text above a statement? is notlike i said arturo magidin is incompetent and dishonest...> The assertion was one of contempt for Magidin, speci'cally> calling into question his competence and honesty.that may be so, but attributing the insults to me is simply hasty.> you to task. That action may, in your mind, have made him a complete> ignoramus imbecile, to use your terminology, but I can't see how.well, hopefully by now you see how. j. hughes got caught in a trap ofpre-empted assumptions that led him to wrong conclusions. as you cansee at present, he had no choice but to abdicate.> Now, you're claiming that you in no way impugned Magidin's competence.i am not only climing, but rather disclosing that i did not intendto call arturo magidin's competence into question. perhaps you areconfusing me with the apparent impostor who actually did?> Again, I can't see how. Could you please provide a reading of the> excerpted passage in a way that does *not* insult Magidin?i think by now you are fully acquainted with the message of my post.but if you have still doubt or confusion, don't hesistate to get backto me.> ... the rest deleted ...> Dale.> P.S. I'm not really interested in the topic of this thread, and would> not have responded but for the glaring contradiction I've alluded to.the contradiction was obviosly in your head... <874qvotj7s.fsf@phiwumbda.org> <87y8swbnll.fsf@phiwumbda.org> <87fzf3ku82.fsf@phiwumbda.org> <3FF200E8.6070606@farir.com> => well, hopefully by now you see how. j. hughes got caught in a trap of> pre-empted assumptions that led him to wrong conclusions. as you can> see at present, he had no choice but to abdicate.Oh, bite me.I didn't abdicate. So far, your only defense has been that perhapsa passage which every person has agreed was intended to insult Arturo,but that you had no such intention.A related defense has been the repeated claim that only a moron speaksto Maky, and I 'nally agreed with that claim (this post, then, issome evidence I'm a moron).-- Jesse HughesI often told you of the dangers of hubris, and most importantly ofall, I TOLD you that I wanted to change the institution of mathematicsworldwide. -- James Harris, on the evils of pride =I post here hopefully only 1 more reply below, attempting in this('nal?) reply to somewhat justify my cross-posting to rec.puzzles.> Let H(n) = sum{k=1 to n} 1/k, > the n_th harmonic number.> Prove, for each m = positive integer,> sum{k=1 to m-1} H(k) k! (m-k)!> is congruent to> (1/2) m! H(¤oor(m/2))) (-1)^m > (mod (m+1)) .> (I do not know if the result is trivial {or am certain it is> correct}.)> We can rewrite the above so it becomes, perhaps to some reading this,> more simply stated and more natural.> (Take sum from 1 to m, instead of from 1 to m-1)> > sum{k=1 to m} H(k) k! (m-k)!> is congruent to> m! *h(m) (mod(m+1)),> where h(m) = sum{k=0 to ¤oor((m-1)/2)} 1/(m -2k) .> Now, h(m) => sum{k=0 to ¤oor((m-1)/2)} 1/(m -2k)> is interesting in itself, for it is (analogously to> double-factorials) the sum of the reciprocal of EVERY-OTHER positive> integer <=m> (even or odd, depending on m).> Neither sequence:> m!*h(m) > nor> m!!*h(m)> nor> sum{k=1 to m} H(k) k! (m-k)!> seems to be in the EIS either, if I calculated correctly by hand.> (EIS: http://www.research.att.com/~njas/sequences/index.html#L)> > By the way, the sequence of > {m! h(m) (mod {m+1})} -> 1, 1 ,0, 3, 4, 2,.. (?)> does not seem to be in the EIS either.> Is there a closed form for this sequence? ...> (...which also equals> {sum{k=1 to m} H(k) k! (m-k)! (mod {m+1})})> I am guessing, if I did not err, that for all SUFFICENTLY HIGH n,and for p = prime,p^n divides(p^n-1)! h(p^n-1),which should not be dif'cult to prove, if I did not err.But p^n also, for the same n's, dividessum{k=1 to p^n-1} H(k) k! (p^n-1-k)!,as is seen from original congruence result. Puzzle problem:Is there a more direct (and easier? perhaps trivial) way, skipping theabove congruence in the original post, to prove that p^n dividessum{k=1 to p^n-1} H(k) k! (p^n-1-k)!?Leroy Quet > By the way, for those interested in harmonic-number math puzzles> (limits, this time, instead of congruences), here is a link to a> recent sci.math math-puzzle no one has yet answered.> (I am including the link because I am cross-posting this to> rec.puzzles, as well as to sci.math.)> -8&group=sci.math&safe=off&selm=b4be2fdf.0401041840.3c577d6c% => Not at all. It is the language of choice in .95Israel'.Is there some difference between Israel and Israel? => Not at all. It is the language of choice in .95Israel'.> Is there some difference between Israel and Israel?I'm not sure what the previous poster is referring to. Yiddish is notthe language of choice in Israel (perhaps it is in .95Israel',whatever that is).Your original question was about publications, and about math texts.Well, this is the situation as far as I am aware of it: originalresearch is naturally published in a language acceptable by the majorjournals, which 99% of the time means English and (rarely) French orRussian. This applies also to the Israel Journal of Mathematics, apretty respectable publication AFAIK; it's all English.For a time, some institutions such as the Hebrew University insistedthat Ph.D. dissertations be written in Hebrew. This created somefrustration, as many people's dissertations were composed of papersthat were sent for publication; They had to translate those intoHebrew for the thesis itself. Nowadays there is no such requirement,but there has to be an Absract in Hebrew (and possibly in English aswell).As for textbooks, this of course depends on the level. As you canimagine most school and high-school textbooks are in Hebrew.there is the occasional Hebrew textbook in, say, Calculus or LinearAlgebra; typically those would be rather low-tech lecture noteswritten up by someone and bound. I'm familiar with such texts byLindenstrauss and Amitsur, but perhaps there are more recent exampleswere the typesetting is more professional by modern standards (thereare, btw, versions of LaTeX in Hebrew, and appropriate text editorsthat enable you to deal with the bi-directional headache).Personally, I'm completely unaware of any textbook or publication inYiddish.HTH,Alan => Your original question was about publications [...]Please get your attributions correct. I did not ask the originalquestion.Thomas => Your original question was about publications [...]> Please get your attributions correct. I did not ask the original> question.> ThomasIndeed - my mistake. Sorry.Alan agreenwood256@yahoo.com (Alan>>> Not at all. It is the language of choice in .95Israel'.>>> Is there some difference between Israel and Israel?I'm not sure what the previous poster is referring to. Yiddish is not>the language of choice in Israel (perhaps it is in .95Israel',>whatever that is).It's what the (temporary) jew occupation of Palestine is sometimesreferred to as.>Your original question was about publications, and about math texts.>Well, this is the situation as far as I am aware of it: original>research is naturally published in a language acceptable by the major>journals, which 99% of the time means English and (rarely) French or>Russian. This applies also to the Israel Journal of Mathematics, a>pretty respectable publication AFAIK; it's all English.For a time, some institutions such as the Hebrew University insisted>that Ph.D. dissertations be written in Hebrew. This created some>frustration, as many people's dissertations were composed of papers>that were sent for publication; They had to translate those into>Hebrew for the thesis itself. Nowadays there is no such requirement,>but there has to be an Absract in Hebrew (and possibly in English as>well).As for textbooks, this of course depends on the level. As you can>imagine most school and high-school textbooks are in Hebrew.>there is the occasional Hebrew textbook in, say, Calculus or Linear>Algebra; typically those would be rather low-tech lecture notes>written up by someone and bound. I'm familiar with such texts by>Lindenstrauss and Amitsur, but perhaps there are more recent examples>were the typesetting is more professional by modern standards (there>are, btw, versions of LaTeX in Hebrew, and appropriate text editors>that enable you to deal with the bi-directional headache).Personally, I'm completely unaware of any textbook or publication in>Yiddish.You obviously didn't go to school in .95Israel'. => It's what the (temporary) jew occupation of Palestine is sometimes> referred to as.Didn't answer the question. I asked what the difference was betweenIsrael and Israel. Thomas tb+usenet@becket.net (Thomas Bushnell,> It's what the (temporary) jew occupation of Palestine is sometimes>> referred to as.Didn't answer the question. I asked what the difference was between>Israel and Israel. The former (in single quotes, if you please) is used by those whodon't acknowledge its existence; the latter is used by those who do.HTHHAND =>Let (X, d) be Haussdorf compact metric space, T a transformation on X>such that family {T, T^2, T^3, ...} is equicontinuous. Suppose T is>surjective. Does T need to be homeomorphism? If yes, is it true that>family {..., T^-2, T^-1, Id, T, T^2, ...} is equicontinuous?Yes. In fact, given epsilon > 0, take delta > 0 such that wheneverd(x,y) < delta, d(T^k x, T^k y) < epsilon for all k >= 0. I claim that whenever d(x,y) > epsilon, d(T^k x, T^k y) >= delta. This will imply that T is one-to-one, and that {T^k: k in Z} is equicontinuous with the delta that works for {T^k: k >= 0} for any epsilon' < epsilon. Proof of claim: Suppose d(x,y) > epsilon. Since T is surjective,there exist sequences u_n and v_n such that T^n(u_n) = x andT^n(v_n) = y. By compactness, we can take subsequences so u_{n_j} and v_{n_j} converge as j -> in'nity, say to u and v respectively. Taking j large enough, d(u, u_{n_j}) and d(v,v_{n_j})are small enough that d(T^k u, T^k u_{n_j}) < r = (d(x,y)-epsilon)/2and d(T^k v, T^k v_{n_j}) < r for all k > 0.In particular, for such j,d(T^{n_j} u, x) = d(T^{n_j} u, T^{n_j} u_{n_j}) < r and similarlyd(T^{n_j} v, y) < r, andd(T^{n_j} u, T^{n_j} v) > d(x,y) - 2 r = epsilon.Now for any k > 0, if d(T^k u, T^k v) < delta we would haved(T^{n_j} u, T^{n_j} v) < epsilon for n_j > k. Thereforewe must have d(T^k u, T^k v) >= delta for all k > 0.In particular, d(T^k T^{n_j} u, T^k T^{n_j} v) >= delta.But if d(T^k x, T^k y) < delta, taking j large enough we'd haved(T^k T^{n_j} u, T^k T^{n_j} v) < delta, contradiction.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 => Yes. In fact, given epsilon > 0, take delta > 0 such that whenever> d(x,y) < delta, d(T^k x, T^k y) < epsilon for all k >= 0. > (...)Brilliant solution! And this gave me hint to another problem, thank youvery much!Mateusz => It plots and analyses any x-y data for peak location, peak height,> peak> width, semi-derivative, derivative, integral,> semi-integral,> convolution,> deconvolution, curve 'tting, and separating overlapped> peaks> and> background. www.chemSoftware.com =I know that this is strange request, but if anybody is kind enough tosend me a scan of the 'rst full text page of Knapp's book, I would beextremely thankful.-- nojb = >>It took about 22 seconds to print out 1,000,000 primes. How much faster is >>it able to be? >>>> also the program produces 10,000,000 primes in 22.27 seconds >I think you will 'nd that print out is what takes the time. Indeed. >> The program stores all the primes in a bit array and can be accessed to see >> if it is prime or not. > I guess the question is then: Print to where? At that time is must be the screen, however, anoher question then comes up, print them to the screen for what reason?The latter question is not needed. The time it takes to show them on thescreen depends quite a bit on the hardware and software used. Also thedigit rendering algorithms have quite a say in this. And there are quitea few other factors. As an example, I know for a fact that if you usea Sun 4 computer, displaying information on the screen would be about10 times slower when you do it through a console, compared to doring itthrough an X window. And in both cases, calculating all those primeswould be much faster than any form of displaying them.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ = I am going to aution my papers, on ebay. The highest bidder will own thecoauthor rights to my papers. Initially, this is the 'rst paper, i am going to run the trial on. http://www.wikiworld.com/KooMar/ulat.pdfhttp://cgi.ebay.com/ws /eBayISAPI.dll?ViewItem&item=2979558376# ebayphotohosting-suresh => I am going to aution my papers, on ebay. The highest bidder will own the> coauthor rights to my papers.> Initially, this is the 'rst paper, i am going to run the trial on.> http://www.wikiworld.com/KooMar/ulat.pdf> http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=2979558376# ebayphotohosting> -sureshHow bout you pay me to proofread your paper AND make me a co-authorfor that contribution. :) No insult intended, but the writing israther awkward.Adam => I am going to aution my papers, on ebay. The highest bidder will own the> coauthor rights to my papers.> Initially, this is the 'rst paper, i am going to run the trial on. http://www.wikiworld.com/KooMar/ulat.pdfBoy, what a deal! I checked it out and *almost* chose to bid, but thenI saw the quality of the diagram on page 5, and it's just not up to mystandards.You let me know when you have a better grasp of LaTeX, and I'll bidfor sure. What better way to pad one's resume?-- Jesse HughesYes, I'm one of those arrogant people who tries to be quotable.There is actually at least one person who quotes me often. -- James Harris => I am going to aution my papers, on ebay. The highest bidder will own the> coauthor rights to my papers.An admirable idea. An excellent way to satirize the lust to publish. Why thehell didn't I think of this?LH =>> I am going to aution my papers, on ebay. The highest bidder will own>> the >> coauthor rights to my papers.> An admirable idea. An excellent way to satirize the lust to publish.> Why the hell didn't I think of this?I've already got a proposal into the NSF to support bidding on the rights. They should buy into it on the grounds that it'sa very cost ef'cient way to support a young mathematician'scareer. If I'm fully funded, I'll have enough capital to stage a hostile take-over of most of Analytic Number Theory.Bart =>I've already got a proposal into the NSF to support bidding on >the rights. They should buy into it on the grounds that it's>a very cost ef'cient way to support a young mathematician's>career. If I'm fully funded, I'll have enough capital to >stage a hostile take-over of most of Analytic Number Theory.Ooh, you might want to reconsider. Wiles is recommending Sellon Analytic Number Theory; I hear he's shorting Quantum Groups.dave =@news.math.niu.edu:>>I've already got a proposal into the NSF to support bidding on >>the rights. They should buy into it on the grounds that it's>>a very cost ef'cient way to support a young mathematician's>>career. If I'm fully funded, I'll have enough capital to >>stage a hostile take-over of most of Analytic Number Theory.> Ooh, you might want to reconsider. Wiles is recommending Sell> on Analytic Number Theory; I hear he's shorting Quantum Groups.> dave> Heh...heh...not after I release my short proof of Fermat'sLast Theorem... =Does anyone know of any online very basic notes on the Poincare disk? Inparticular, I am interested in how to derive the formula for arc length inthe poincare disk and other basic things like how to construct a square...I tried maybe someonealready has a link?Moshe => Does anyone know of any online very basic notes on the Poincare disk? In> particular, I am interested in how to derive the formula for arc length in> the poincare disk and other basic things like how to 'nd anything but maybe someone> already has a link?> MosheA good book to consult is _Modern Geometries: The Analytic Approach_by Michael Henle, Prentice-Hall, 1996. Unfortunately it is now out ofprint but Amazon lists some used copies available, or you might 'ndit in a college or university library.As for squares, there are none in the Poincare disk, if by square youmean an equilateral and equiangular quadrilateral. The disk is amodel for the hyperbolic plane in which angle sums of polygons arealways less than what you are used to in Euclidean geometry. Youcould construct a four sided polygon in which all the sides have thesame length but the angles can't all be the same. => Does anyone know of any online very basic notes on the Poincare disk? In> particular, I am interested in how to derive the formula for arc length in> the poincare disk and other basic things like how to 'nd anything but maybe someone> already has a link?> diskhttp://mathworld.wolfram.com/ PoincareHyperbolicDisk.htmlhttp://www.unca.edu/~mcmcclur/ mathematicaGraphics/PTiling/http://jwilson.coe.uga.edu/emt669/ Student.Folders/Godfrey.Paul/work/proj1/poincar3.html Respected professor,I am going to aution my papers, on ebay. The highest bidder will ownthe coauthor rights to my papers.Initially, this is the 'rst paper, i am going to run the trial onhttp://www.wikiworld.com/KooMar/ulat.pdfhttp://cgi.ebay.com/ ws/eBayISAPI.dll?ViewItem&item=2979558376#ebayphotohostingI am going to aution every paper on this pagehttps://sourceforge.net/project/show'les.php?group_id= 61183Each paper(pdf 'le, not zip 'le) will start at a initial bid of$600.-suresh =Suppose the three positve random varialbe satisfying X=min(Y,Z)As Y and Z are independent, then the pdf of X is well-known. I am wonderingif Y and Z are dependent, how to calculate the pdf of X.--ZHANG Yanhttp://www.ntu.edu.sg/home5/pg01308021 =>Suppose the three positve random varialbe satisfying> X=min(Y,Z)As Y and Z are independent, then the pdf of X is well-known. I am wondering>if Y and Z are dependent, how to calculate the pdf of X.>I assume that Y and Z are jointly continuous. Letting f be the joint density of Y and Z, g be the marginal density of Y, and h be the marginal density of X,P{X > x} = integral(y = x..infty, z = x.. infty, f(y,z)).= integral (y = x..infty, P(Z > x | Y = y) g(y))h(x) = - d/dx P{X>x}.Integrating with the chain and Leibniz' rules, I geth(x) = integral(w = x..infty, f(x,w) + f(w,x))= P(Z > x | Y = x) g(x) + integral(y= x..infty, k(x|y) g(y))where k(.|y) is the conditional density of Z given Y = y.These formulae check for the case where Y and Z are independent.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu =advance.!!--ZHANG Yanhttp://www.ntu.edu.sg/home5/pg01308021 =>random varible T is given by sum_{i=1}^{N} min (X_i, Y_i)>T = lim ---------------------------------------> N -> +infty N>where X_i,Y_i represent the sample of the positive random varible X and Y.>Now, if the pdf of X and Y are given, how to compute the pdf of T. It seems>that T is not simply the smaller value of X and Y, i.e.T = min(X,Y)is wrong.advance.!!>I have addressed the pdf in another post. It seems to me that you are referring to expected value here.min(X,Y) <= X implies E min(X,Y) <= EXmin(X,Y) <= Y implies E min(X,Y) <= EYTherefore, E min(X,Y) <= min (EX, EY)As long as P{X < Y} P{Y < X} > 0, the inequality is strict.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu =min (X_i, Y_i)={X_i+Y_i-|X_i-Y_i|}/2It seems that T goes to constant {E(X)+E(Y)-E(|X-Y|)}/2 a.s. by SLLNunder some conditions.> random varible T is given by sum_{i=1}^{N} min (X_i, Y_i)> T = lim ---------------------------------------> N -> +infty N where X_i,Y_i represent the sample of the positive random varible X and Y. Now, if the pdf of X and Y are given, how to compute the pdf of T. It seems> that T is not simply the smaller value of X and Y, i.e. T = min(X,Y) is wrong. advance.!! --> ZHANG Yan> http://www.ntu.edu.sg/home5/pg01308021 =Can you brilliant and gifted Mathematicians and Physicists help outthis poor soul and justify your acceptance of relativistic and quantummechanics by providing the axioms of a simultaneously discrete andcontinuous space.JS => Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.Screw your personal problems. *You* provide a single reproducibleempirical contradiction to classical 'eld theory as it now exists. If it has no problems it needs no changes.http://arXiv.org/abs/hep-th/0307140 GR structure, especially Part 4/p. 7 Experimental constraints on General Relativity.http://www.eftaylor.com/pub/projecta.pdf Relativity in the GPS system http://arXiv.org/abs/astro-ph/0401086http://arxiv.org/abs/ astro-ph/0312071 Deeply relativistic neutron star binariesUncle Al has mounted a powerful empirical contradiction to metrictheories of gravitation in favor of af'ne theories complete withscholarly references, pure geometric and mathmetical models, and aninexpensive reduction to practice in existing apparatus. It alsoreconciles continuous 'eld theories with discrete processes whilecosmology,http://www.mazepath.com/uncleal/qz.pdfPookie pookie. See what being able to use a library will get you?--Uncle Alhttp://www.mazepath.com/uncleal/qz.pdfhttp:// www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics)-- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The Net! =Littlemanwearingbigboypants posits:> Uncle Al has mounted a powerful empirical contradiction to metric> theories of gravitation in favor of af'ne theories complete with> scholarly references, pure geometric and mathmetical models, and an> inexpensive reduction to practice in existing apparatus. It also> reconciles continuous 'eld theories with discrete processes while> cosmology,> http://www.mazepath.com/uncleal/qz.pdf> Pookie pookie. See what being able to use a library will get you?A crackpot trying to sound sophisticated. => Littlemanwearingbigboypants posits:> Uncle Al has mounted a powerful empirical contradiction to metric> theories of gravitation in favor of af'ne theories complete with> scholarly references, pure geometric and mathmetical models, and an> inexpensive reduction to practice in existing apparatus. It also> reconciles continuous 'eld theories with discrete processes while> cosmology,> http://www.mazepath.com/uncleal/qz.pdf> Pookie pookie. See what being able to use a library will get you?> A crackpot trying to sound sophisticated.Hey critic troll Vajk - provide a scholarly link or peer-reviewdpublication citation in counterpoint Nobody is going to respect Vajkthe jackass who proved potential and kinetic energies of a fallingobject simultaneously increase. Nobody is going to respect Vajk thejackass who proved acceleration is not an absolute measurement,thereby denying inertial navigation from airplanes to missiles tospace probe orbital corrections. Uncle Al has a groupie. What amaroon!Vajk the critic troll brainfarts. No contribution, only complaint. No references URL or literature, no mathematics, no input to thediscussion, no enlightenment, no hint of intelligence. Nothing butanile kneejerk spasm be'tting a particulary inferior undergradassignment in spew emulation. Having pissed upon a skyscraper wall,the critic troll rears back and exhorts the crowd to admire both hisspoor and the manly implement that emplaced it.--Uncle Alhttp://www.mazepath.com/uncleal/qz.pdfhttp:// www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) = Littlemanwearingbigboypants posits: Uncle Al has mounted a powerful empirical contradiction to metric> theories of gravitation in favor of af'ne theories complete with> scholarly references, pure geometric and mathmetical models, and an> inexpensive reduction to practice in existing apparatus. It also> reconciles continuous 'eld theories with discrete processes while> cosmology, http://www.mazepath.com/uncleal/qz.pdf Pookie pookie. See what being able to use a library will get you? A crackpot trying to sound sophisticated. Hey critic troll Vajk - provide a scholarly link or peer-reviewd> publication citation in counterpoint Nobody is going to respect Vajk> the jackass who proved potential and kinetic energies of a falling> object simultaneously increase. Nobody is going to respect Vajk the> jackass who proved acceleration is not an absolute measurement,> thereby denying inertial navigation from airplanes to missiles to> space probe orbital corrections. Uncle Al has a groupie. What a> maroon! Vajk the critic troll brainfarts. No contribution, only complaint.> No references URL or literature, no mathematics, no input to the> discussion, no enlightenment, no hint of intelligence. Nothing but> anile kneejerk spasm be'tting a particulary inferior undergrad> assignment in spew emulation. Having pissed upon a skyscraper wall,> the critic troll rears back and exhorts the crowd to admire both his> spoor and the manly implement that emplaced it.>Whatever, crackpot.l8r, Mike N. Christoff =New idiot on board. => New idiot on board.Like you would know....... =>http://www.mazepath.com/uncleal/qz.pdf> Pookie pookie. See what being able to use a library will get you?A crackpot trying to sound sophisticated.> Hey critic troll Vajk - provide a scholarly link or peer-reviewd> publication citation in counterpoint It is your function to prove that your theory is valid.You have not.The rest is balderdash.Prove or recant. =>http://www.mazepath.com/uncleal/qz.pdf>Pookie pookie. See what being able to use a library will get you?A crackpot trying to sound sophisticated.> Hey critic troll Vajk - provide a scholarly link or peer-reviewd> publication citation in counterpoint It isI guess Vajk cannot provide the links.No surprise there, as he is not even able to understand basic physics. => Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space. JSCan one be fat and skinny at the same time?For elephants people are skinny and for snakes people are fat.I don't know what your de'nition of an axiom is but no axiomatic systemcan prove it's own consistency.The concept of spacetime is good just for modeling interactions. Probablynothing that bares any true meaning or relation to reality.It's quite possible Einstein was right and QM uncertainty is just due to ourinability to understand things at that small scale. I thing we shouldn'tbother anyway and just stick with macroscales. => Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> JSIf you want real help:Where did you hear/see these statements, or how did you come to such aconclusion? What was the context of the statements? Are you talkingabout a vector space or a similar structure here or real space?Also, I should point out that most physicists don't like axiomizingthings -- especially things like the universe that no one yet fullyunderstands! =......... ...However, as no brilliance along that matter as no giftrequired neither. Therefore, you can also allow yourself. To see as toobserve the things basically -for instance- along a rude confrontationbetween the mechanics and the electricity. Before, the appearance of aQuantum Mechanics. Along which it has had already appears as possible asconcrete concordance to manage the most of a reactions.Whether, it would as it should always and a de'nitely apply. The major afact among the uncertainty along the space time, de'nitely as a matter afact!!!!!!!!!!!!!!!!............... ...-- Ahmed Ouahi, ArchitectJohn Schoenfeld kirjoitti viestiss.8a> Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space. JS => Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.If by space, you mean topological space then you are caught ina category error. In topology, continuity is a property offunctions between spaces, not of spaces themselves.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) => Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.It is old news that General Relativity and Quantum Mechanics are notconsistent with each other (as they are currently formalized).Thomas = John Schoenfeld: >Can you brilliant and gifted Mathematicians and Physicists help out >this poor soul and justify your acceptance of relativistic and quantum >mechanics by providing the axioms of a simultaneously discrete and >continuous space. I already pointed out what is wrong with that stupid assessment.Put away the crack pipe a few hours before posting. => Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.It must be pointed out that Schoenfeld is a crackpot, and a troll. brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.If the only tool you have is a hammer, then all problems look like nails.But if you have two tools, say a hammer and a saw...one that joins problem sets into a neat whole, andone that cuts the problem into its contituent parts,then you can apply the tool that is required.The Universe is both discrete and continuous, depending on what youneed/want to describe.But I am not a brilliant and gifted Mathematician and Physicist.David A. Smith gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> If the only tool you have is a hammer, then all problems look like nails.> But if you have two tools, say a hammer and a saw...> one that joins problem sets into a neat whole, and> one that cuts the problem into its contituent parts,> then you can apply the tool that is required.> The Universe is both discrete and continuous, depending on what you> need/want to describe.> But I am not a brilliant and gifted Mathematician and Physicist.Uncle Al is industrial rather than academic. If all you have is ahammer in industry, then every problem looks like a screw - andmanagement whacks it in anyway. Welcome to my world.Reconciling continuous 'eld theories with discrete processes is nobig deal. Metric theories of gravitation won't accommodate you, butaf'ne theories will. Since both ¤avors of gravitation have exactlythe same predictions (aside from Equivalence Principle violations), itmakes no difference which one is chosen for plebeian calculations(aside from the boojum),http://www.mazepath.com/uncleal/qz.pdf--Uncle Alhttp://www.mazepath.com/uncleal/qz.pdfhttp:// www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) =Space/time is discrete just like the electrons that make up atoms that makeup molecules, etc. However space-time could internal be composed ofsomething else.-suresh> Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space. JS =I just (carefully) read the posts in this thread on order and groups.If an abelian group G has two elements with order m and n, show G hasan element whose order is [m,n].I made several errors and owe J Santini and Arthur Madigan an apology,so forgive me for going over this again. I made some posts I wish Ihadn't as they are clearly wrong, as Santini and Madigan pointed out.In addition, I missed some posts or skimmed them and didn't think themthrough, and example of which J critcisms of some of my statements.I should not have been posting on a subject where I am farfrom expert, and in sci.math there are many experts.In my defense, I am just learning algebra and number theory,and teaching myself. (I feel I have to make some excuse;)In the future I will be more careful about mystatements here, as one has to be when talking about math proofs.I am glad that there is still a place where one should be clearand give good reasons for his statements. I found a soln. similar to A Madigan's,where the result is proved for (m,n) = 1, and thenm and n are written in prime factors, which are divided into2 groups I and went over both proofs, so at least I understandit now.Arturo Magidin then went to a lot of trouble to give simple examplesand a complete correct proof, = [.snip.]>I should not have been posting on a subject where I am far>from expert, and in sci.math there are many experts.There is ->nothing<- wrong about posting on a subject where you arefar from an expert. By all means, ask questions! That is one of thegood things about sci.math (and, I don't know about others, but partof the reason I post and read sci.math is because answering thesequestions helps keep me sane when I'm running into walls with myresearch: it helps remind me that I do know something...)>In my defense, I am just learning algebra and number theory,>and teaching myself. That's 'ne. The only thing is, if you don't know much about asubject, and people are trying to tell you that you are incorrect, youowe them at least to read their responses carefully and address theirpoints (at least, at 'rst).-- =It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.edu> Arturo Magidin then went to a lot of trouble to give simple examples> and a complete correct proof, from which I have ... but notice one thing: in this post, thecharacter Arthur Madigan gradually turned into A. Madigan , and (eventually)Arturo Magidin.Who are the fake ??? => I'm trying to make a counter that works in native base three mode.> /* 000,001,002,010,011,020,021,022,100,101,102,110,111,112,120,121 ,122,200,201,202,210,211,212,220,221,222,2000................. .............*/> And so on without any base conversion, just a plain recursive loop> dealing with the integers within the base [0,1,2] and their multiples> [10,100,1000,.....] Is such a counter possible to make? Or is it impossible to build a> recursive base three counter? JTThe digits can be packed into a binary number.The following works for any base:b = basec = bits per digitd = number of digitsa = (2^(c*d) - 1) / (2^c - 1) puts a one in each digit positionn = n + a*(2^c - b) offsets each digit so that carry worksn = n + 1if (n == 0 || n >= 2^(c*d)) then n = 0; donem = n & -n 'nds the least signi'cant non-zero bitm = a & ~((m-1)/2^(c-1)) puts one for each digit didn't become zeron = n - m*(2^c - b) removes offsets from those digitsUsing 3 or 4 bits per digit, octal or hex print format can be used. => |Do you make use of some computing package for this? > I was going to let people guess which one it was, but> I guess that's a little silly. I was considering> implementing the algorithm from Henri Cohen's book on> computational algebraic number theory, but then it> occurred to me that it'd be a lot easier to use the> package a group of people including him developed:> Pari-GP. My Christmas present to myself. :-) It's> available for free. Someone else guessed Magma, and> although I haven't used it I believe it would also> do the job 'ne.> I would challenge James Harris to 'gure out how to> use the thing. The stuff I've been doing really only> requires a few of the functions, primarily functions> for working in number 'elds. The neat thing is that> even though Pari-GP is doing some fancy tricks to> generate the answers for me, I can show that the> answers are correct using much more elementary> calculations, the kind that one could mainly do with> just a calculator.> Keith Ramsay Maybe such software is not all that necessaryin this case for a factorization of the desired form. The Decker polynomial in this case is Q(x,y) = 7*(25*x^2 + 30*x*y + 2*y^2)and it is evaluated at x = 2, y = 1. A factorization of Q(2, 1) = 7*162 is 162 = (5*7 + 7)*(5*4 + 7) = (5*7 + 7)*(5*5 + 2) = 7*6*27,i.e., a1 = 7 and a2 = 4, and Q(2, 1)/7 = 162 = (5*1 + 1)*(5*5 + 2),so, yes, in this case Harris's factorization *actually works*. Nora B. = > |Do you make use of some computing package for this? > I was going to let people guess which one it was, but > I guess that's a little silly. I was considering > implementing the algorithm from Henri Cohen's book on > computational algebraic number theory, but then it > occurred to me that it'd be a lot easier to use the > package a group of people including him developed: > Pari-GP. My Christmas present to myself. :-) It's > available for free. Someone else guessed Magma, and > although I haven't used it I believe it would also > do the job 'ne.Yes, that was me. But I ought to have guessed Pari too. (ActuallyI should have guessed Pari 'rst because I have had a bit ofcooperation with Henri Cohen, although I doubt whether a referenceto my work is still available in the current version, it is probablyno longer used.) It is my impression that magma is a bit morepowerful, but I think that one is not for free, unless you havevery good connections with the University of Australia.As soon as you talked about a class group number of 11 I was prettycertain that you did have software to help you. Such things aretedious to compute... > I would challenge James Harris to 'gure out how to > use the thing. The stuff I've been doing really only > requires a few of the functions, primarily functions > for working in number 'elds. The neat thing is that > even though Pari-GP is doing some fancy tricks to > generate the answers for me, I can show that the > answers are correct using much more elementary > calculations, the kind that one could mainly do with > just a calculator.But that is the beauty of algebraic number theory. Once you have theanswers to speci'c questions, in most cases it is pretty easy to showthey are correct. The answer 'nding is most dif'cult.I ought to look into magma and pari a bit. I have installed themboth on our systems...-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ => Science = it is reality. And objective proposition. One can check. Philosophy = I am reality. A subjective illusion. Right up there with I am Napoleon. BishopAll perceptions,including perceptions of a possible external (It is reality) world,start, and end, from I.Can you suggest a bullet-proof experimentthat proves that an external reality exists?--Tom Potter http://tompotter.us => Can you suggest a bullet-proof experiment> that proves that an external reality exists?Not a chance. Radical solopsism is internally consistent.Bob Kolker => Science = it is reality. And objective proposition. One can check. Philosophy = I am reality. A subjective illusion. Right up there with I am Napoleon. BishopAll perceptions,including perceptions of a possible external (It is reality) world,start, and end, from I.Can you suggest a bullet-proof experimentthat proves that an external reality exists?> Can you suggest a bullet-proof experiment> that proves that an external reality exists?> Not a chance. Radical solopsism is internally consistent.So what you are saying is that physics is:A subjective illusion. Right up there with I am Napoleon.--Tom Potter => You are correct that it's a translation error. In Finnish the theorem> above about derivable functions and f'(ksi) is v.8aliarvolause,> translating directly to intermediate value theorem. I didn't know> mean and intermediate meant so different things in English.> That's where the confusion came from.Fortunately (for us) we can make you do math in English, but you can't makeus do math in Finnish. This will probably continue to be the case as longas OPEC prices oil in dollars. It has, of course, nothing whatsoever to dowith the relative merits of doing math in any given language.Jon Miller =>
>Anyway, I see no reason to stop there.
Since>>the system already uses deci- and centi-,>>why not use
all metric fractional pre'xes? This>>would give: >>decillion
=3D 10^33>>centillion =3D 10^303>>millillion =3D 10^3003 =3D
10^(1000^1 + 3) >>micrillion =3D 10^3000003 =3D 10^(1000^2 +
3)>>nanillion =3D 10^(1000^3 + 3) >>picillion =3D 10^(1000^4 +
3)>>femtillion =3D 10^(1000^5 + 3)>>attillion =3D 10^(1000^6 +
3)>>zeptillion =3D 10^(1000^7 + 3)>>yoctillion =3D 10^(1000^8
+ 3) >>Uh, oh... ran out of pre'xes! :-( =A0 BTW,
the>>American version of that last one
has>>1,000,000,000,000,000,000,000,003 zeroes,>>and the
British version (10^(1,000,000^8))
has>>1,000,000,000,000,000,000,000,000,000,000>>
000,000,000,000,000,000 zeroes (somewhat>>dif'cult to write
out in full). Ooops... that should read:decillion =3D
10^33>centillion =3D 10^303>millillion =3D 10^3003 =3D
10^(3*1000^1 + 3) >micrillion =3D 10^3000003 =3D 10^(3*1000^2
+ 3)>nanillion =3D 10^(3*1000^3 + 3) >picillion =3D
10^(3*1000^4 + 3)>femtillion =3D 10^(3*1000^5 + 3)>attillion
=3D 10^(3*1000^6 + 3)>zeptillion =3D 10^(3*1000^7 +
3)>yoctillion =3D 10^(3*1000^8 + 3) where yoctillion has
3,000,000,000,000,000,000,000,003 zeroes>(American), or
6,000,000,000,000,000,000,000,000 zeroes (British).>
....DWIII>
Now if we could just 'nd something better than illion. => Now if we could just 'nd something better than illion.You'd think by now this all would be made standard.$1,000 = 1 kilodollar$1,000,000 = 1 megadollar$1,000,000,000 = 1 gigadollar$1,000,000,000,000 = 1 teradollar etc.It would be a heck of a lot easier, and besides it avoids the billionconfusion completely. But if people can't break habits, may I suggest.$1,000 = 1 killion$1,000,000 = 1 megillion$1,000,000,000 = 1 gigillion$1,000,000,000,000 = 1 terillionJust my 2,000 millidollars worth. =In sci.math, if we could just 'nd something better than illion.> You'd think by now this all would be made standard.> $1,000 = 1 kilodollar> $1,000,000 = 1 megadollar> $1,000,000,000 = 1 gigadollar> $1,000,000,000,000 = 1 teradollar etc.> It would be a heck of a lot easier, and besides it avoids the billion> confusion completely. But if people can't break habits, may I suggest.> $1,000 = 1 killion> $1,000,000 = 1 megillion> $1,000,000,000 = 1 gigillion> $1,000,000,000,000 = 1 terillion> Just my 2,000 millidollars worth.It makes cents to me. :-)-- #191, ewill3@earthlink.netIt's still legal to go .sigless. =Timothy Murphy asks:>Slightly OT,>but does anyone publish maths in Hebrew nowadays?>Are school maths textbooks normally in Hebrew?I've seen a calculus book in Hebrew. I own a book by Azriel Levy, in Hebrew,consisting of notes for a course in mathematical logic. Such notes areavailable from the campus bookstore, Academon. I've tried for a long timeto get information about the math books they have in Hebrew. I'm told thereare Hebrew books on differential geometry, for example. According to Academon,when a friend of mine called them on the phone to ask for a catalogue, allthe information about their books is available online through their website.When I tried several months ago to look at it, I found it was underconstruction. My own browser can't navigate the site, since I never 'guredout how to install the necessary Hebrew fonts, but the library computers haveI got the Hebrew word for nilpotent from a Ph.D. thesis written in Hebrewat Hebrew University in Jerusalem.Allan Adlerara@zurich.ai.mit.edu************************************* **************************************** ** Intelligence Lab. My actions and comments do not re¤ect ** in any way on MIT. Moreover, I am nowhere near the Boston ** metropolitan area. ** ************************************************************** *************** =>your Riemann surface My Riemann surface? Having burnt my 'ngers on Riemann surfaces I amtrying to give them up. In my last post I used only the z-plane andthe w-plane.> you can do better than putting permutations on branch cuts:> you can (if you chose the cuts canonically, to be where two > distinct branches have equal real part) put *braids* on> branch cuts. It might be better but is it not different or at least special? Itried to present a theory valid for all systems of cuts.>(And in this case the arrows are really important;> in the generic situation where the permutation for each cut is> just a transposition, and therefore self-inverse, the arrows> are insigni'cant. Having made no restriction to transpositions I don't see the point ofthis remark. => A mathematically-inclined glove-fetishist has a number of pairs of glovesof> identical design, but of several (at least three) different colors. Shehas> at least three pairs of each color. In the dark she can distinguish the> handedness of a glove, but not its color. Unfortunately, she keeps the> gloves jumbled up in a drawer in an unlit cellar. She knows that if she takes out 21 gloves, in the dark, she can be sure of> getting at least one pair. What is the maximum number of pairs of gloves that she might have? A correction to my puzzle is required.After:She knows that if she takes out 21 gloves, in the dark, she can be sure ofgetting at least one pair.add another statement:She also knows that if she takes out less than 21 gloves, in the dark,there is no way of ensuring that she gets a pair.Sorry about that.The point being that, without the correction, if she had 3 red, 3 green anda million yellow pairs, she could take out 7 left gloves, 7 right gloves andany 7 other gloves (to make 21) and be sure of having a pair (of yellowgloves). There would thus be no maximum number of pairs of gloves that shemight have.-- Clive Toothhttp://www.clivetooth.dk = A mathematically-inclined glove-fetishist has a number of pairs ofgloves of> identical design, but of several (at least three) different colors. Shehas> at least three pairs of each color. In the dark she can distinguish the> handedness of a glove, but not its color. Unfortunately, she keeps the> gloves jumbled up in a drawer in an unlit cellar. She knows that if she takes out 21 gloves, in the dark, she can be sureof> getting at least one pair. What is the maximum number of pairs of gloves that she might have? If she has exactly 3 pairs in each of exactly 3 colors, where did she> 'nd those last 3 gloves in the dark?Well... yes... she would not be able to 'nd those last 3 gloves in thedark. That must mean that she has exactly 3 pairs in each of exactly 3colors is false.-- Clive Toothhttp://www.clivetooth.dk = A mathematically-inclined glove-fetishist What a wonderful introduction to a puzzle!-- Clive Toothhttp://www.clivetooth.dk = A mathematically-inclined glove-fetishist has a number of pairs ofgloves of> identical design, but of several (at least three) different colors.She has> at least three pairs of each color. In the dark she can distinguishthe> handedness of a glove, but not its color. Unfortunately, she keeps the> gloves jumbled up in a drawer in an unlit cellar. She knows that if she takes out 21 gloves, in the dark, she can besure of> getting at least one pair. What is the maximum number of pairs of gloves that she might have? (Rot-13): Gjragl-svir.> Fur unf fvk cnvef va rnpu bs guerr pbybhef, naq frira cnvef va n sbhegu> pbybhe.> Ol gnxvat gjragl yrsg-unaq tybirf fur pna or fher bs univat ng yrnfg bar> bs rnpu pbybhe. Gur gjragl-svefg vf n evtug-unaq tybir, juvpu vf obhaq> gb zngpu. You can do three better if you use one fewer color and the same strategy.That is not the correct answer.-- Clive Toothhttp://www.clivetooth.dk = A mathematically-inclined glove-fetishist has a number of pairs ofgloves of> identical design, but of several (at least three) different colors. Shehas> at least three pairs of each color. In the dark she can distinguish the> handedness of a glove, but not its color. Unfortunately, she keeps the> gloves jumbled up in a drawer in an unlit cellar. She knows that if she takes out 21 gloves, in the dark, she can be sureof> getting at least one pair. What is the maximum number of pairs of gloves that she might have? (Rot-13): Gjragl-svir.> Fur unf fvk cnvef va rnpu bs guerr pbybhef, naq frira cnvef va n sbhegu> pbybhe.> Ol gnxvat gjragl yrsg-unaq tybirf fur pna or fher bs univat ng yrnfg bar> bs rnpu pbybhe. Gur gjragl-svefg vf n evtug-unaq tybir, juvpu vf obhaq> gb zngpu.un-Rot-13:Twenty-'ve.She has six pairs in each of three colours, and seven pairs in a fourthcolour.By taking twenty left-hand gloves she can be sure of having at least one ofeach colour. The twenty-'rst is a right-hand glove, which is bound tomatch.That is not the correct answer.-- Clive Toothhttp://www.clivetooth.dk => JS: The following statements are true:> 1. To a good approximation the non-tensor connection 'eld g-force on a> where> m(passive) = m(active)> The approximation is two-fold> A. Not near a space-time singularity, i.e. not falling behind the event> But is it true that you never reply to a response?yourself as he is extremely rude and appears devoid of any class => I enjoy your posts. But is it true that you never reply to a response? yourself as he is extremely rude and appears devoid of any class or> culture. anything except science, whata surprise.Jack = Jack Zamat: > >>>> I enjoy your posts. >>> But is it true that you never reply to a response? >>> yourself as he is extremely rude and appears devoid of any class or >> culture. >>> Jan Bielawski >An arrogant physicist with no appreciation for anything except science, what >a surprise. You can scratch the science, too. = JS: The following statements are true: 1. To a good approximation the non-tensor connection 'eld g-force on a> where m(passive) = m(active) The approximation is two-fold A. Not near a space-time singularity, i.e. not falling behind the event> horizon of a black hole. B. The scale is large enough so that quantum gravity metric> ¤uctuations are ignorable. 2. To the same approximation the connection 'eld g-force (which> reduces to Newton's gravity force) is eliminated on a timelike> geodesic. PZ: Newton's gravity force did NOT include 'ctitious inertial forces. JS: Irrelevant my statement 2 is true obviously since, e.g. goo = (1 - 2V(Newton)/c^2) = grr^-1 is the warp factor in the> Schwarzschild solution. The Einstein geodesic equation in weak 'eld limit becomes Newton's> gravity force equation d^2r/dt^2 = - GradV(Newton)> Inertial forces in Newton's theory are also independent of the mass m of> The connection 'eld is non-zero only on time-like PZ: Right. But starting from zero gravity everywhere, the 'eld can be> made non-zero by> (1) going to an accelerated frame; and/or (2) switching on gravitational> sources. JS: Obvious, so what? Also if you want to use tidal stretch-squeeze> curvature as the criterion for a real gravity 'eld rather than the GR> connection 'eld geometrization of Newton's gravity force, then it is> not enough that the curvature vanish at isolated zeros, it must vanish> in a 'nite region of space-time. 3. The curvature tensor is also a local observable. PZ: Yes. And local measurements of tidal curvature are not *necessarily*> scale-sensitive. JS: Red Herring. Depends what you mean by scale-sensitive? Any> classical procedure (liquid drops without surface tension, precessing> spinning tops ...) breaks down at quantum gravity level and also in> approach to a singularity - assuming dark energy exotic vacuum zero> point energy does not smooth out the classical singularity. IF you de'ne a true gravity 'eld as one in which the tidal> curvature tensor is NOT zero, then one can always locally, in> principle, distinguish a true gravity 'eld from a fake gravity> 'eld (where the tidal tensor vanishes in all frames geodesic LIF &> non-geodesic LNIF) by de'nition. PZ: It's really quite simple: you can for all practical purposes always> locally tell, in any> frame, whether you are in the presence of a permanent 'eld, since> permanent 'elds> typically have non-vanishing tidal curvature, while inertial 'elds do not. JS: This is hardly stunning news. MTW never denied that. However, in> ordinary conditions like> space¤ight in solar system local tidal effects on scale of spaceships> is really tiny. PZ: This does not in itself allow you to locally distinguish between the> gravitational JS: That's not signi'cant. I suppose, in principle, if you did> simultaneous measurements of both g-force and local tidal curvature you> could make a computer program to integrate the tidal information and> subtract it from total g-force. However, Bohr's generalized> complementarity may forbid, since you need to measure tidal curvature in> LIF, but you need to measure g-force in LNIF. This is tricky. You have the astronauts in orbit. How would they do the measurement?> PZ: we know that in a frame that is unaccelerated with respect to a> static source,> we can determine the strength of the true gravitational force. JS: How? We can easily determine the TOTAL g-force in that LNIF. How do> you extract the true piece of it? I suppose that means, how do you> measure components of Ruvwl in the LNIF frame since most> gedankenexperiments to measure it assume a LIF? In principle it is> possible of course because a tensor is a tensor in any local frame LIF> or LNIF. PZ: Any acceleration> imparted to this frame with respect to the source will then add> calculable inertial> contributions to the observed g-forces. JS: But your rest frame is already an LNIF. PZ: So we can not only distinguish between pure inertial 'elds and> mixed inertial & permanent> gravitational 'elds, using locally measurable tidal effects, but we can> in principle also> determine the true strength of the permanent part of the observed g-forces. JS: This is useless. It's too vague. Show explicitly in detail in a> gedankenexperiment how to do it. Show a real constructive procedure. PZ: So we could equally attribute both the tidal curvature and the> permanent part of the> g-force to the true gravitational 'eld, and inertial effects to a> separate 'ctitious> inertial 'eld -- as I am proposing. JS: You have no math for this and words are inadequate. I do not know> what it really means. Even if you can do it mathematically I do not see> how it adds any new physics? PZ: I see nothing in the basic formal-empirical framework of GR that> prevents this.> This seems like a perfectly feasible alternative de'nition of the> real gravitational> 'eld which corresponds very naturally with Newtonian theory. JS: Until you can show this explicitly it's idle speculation. Maybe> Ruf'ni et-al have a clue in their discussion of experiments?> On the other hand, if you de'ne a gravity 'eld as the connection> 'eld, then you cannot make such a distinction! PZ: Obviously. JS: Therefore, the problem here is only a semantic problem from wavering> between the two different de'nitions of gravity 'eld. PZ: No, it is not *merely* a semantical problem. It is a question of the> proper de'nition> of what in essence constitutes the *real gravitational 'eld*. Of course, logically speaking, you can de'ne the gravitational 'eld> to be a> ham sandwich if that pleases you; but that doesn't mean that this is a> physically> appropriate de'nition. JS: Not a fair comparison since using the connection 'eld de'nition IS> the proper generalization of how Newton thought of the gravity force> 'eld. The motivation is seamless integration of the larger covering> theory with the smaller theory. PZ: The bottom line here is that there are two alternative models. In> the Einsteinian> model, the gravitational 'eld is reduced to the uni'ed> gravitational-inertial connection> 'eld, while the tidal curvature is left hanging. In the alternative> model, the curvature 'eld> is integrated with the permanent component of the connection 'eld as a> fundamental> aspect of the real gravitational 'eld, while the inertial component of> the connection> 'eld is left hanging. JS: You need operational procedures here that are lacking.> PZ: So this is a matter of two alternative physical models of the> gravitational 'eld, and not> *merely* a matter of de'nitions or semantics. In the Einstein model, when the uni'ed connection 'eld vanishes, the> gravitational 'eld is> literally *annihilated*. In the alternative model, the uni'ed> connection 'eld can vanish without> annihilating the entire physical presence of the 'eld. This difference> in the two models has> direct consequences for the de'nition of gravitational vacuum> stress-energy. JS: Show exactly what you mean here with mathematics, otherwise your> point is too vague. principle locally see tidal effects as a torque around the center of> mass or a precession of a small spinning gyroscope of course. Also you> can see deformations of shape of a geodesic droplet if the surface> tension is small enough. PZ: Right. And this kind of effect is not *necessarily* scale-sensitive,> notwithstanding> the so-called EEP. JS: I do not understand your use of scale-sensitive in your context.> 5. Since Einstein's GR comes from locally gauging ONLY the 4-parameter> translation subgroup of the Poincare group, of course Einstein's early> formulation of the equivalence principle was only an approximate> rotational degrees of freedom. PZ: Exactly.> JS: Again hardly news. The Ruf'ni book looks interesting BTW and I will> read it cover to cover.> Einstein's guv 'eld of curved> space-time with the symmetric connection force 'eld is simply the> compensating gauge force 'eld needed to restore the now local gauge> symmetry which is equivalent to local general coordinate> transformations. PZ: Whether this gauge symmetry has a physical basis is a *physical*> question, and not just> a formal mathematical one. JS: It is physical. It is clearly physical in my vacuum coherence model> that builds upon Hagen Kleinert's metric elasticity model. guv(Einstein Locally Curved Space-Time) = Special Relativity Metric> (Globally Flat Space-Time) + Elasticity Strain Tensor What is strained? It is the World Crystal Lattice. When it gets> over-strained it fractures into fault lines of curvature disclination> defects and also maybe fault lines of torsion dislocation defects on> scales large compared to the unit cell. The physical distortion 'eld of the stressed World Crystal is my Bohm> Pilot Wave Constraint: Distortion Field = (Loop Gravity Quantum of Area)(Partial Derivative of> Goldstone Phase of Vacuum Coherence) This equation does for the elastic-plastic World Crystal Lattice what Bohm's v(IT) = (h/m)Grad(Phase of Pilot BIT Wave)> Note that Witten's alpha' = hc/(String Tension) = Loop Gravity Quantum of Area> from Spin Networks of combinatorial pre-geometry (Penrose). Quantum Gravity Area = (Loop Quantum of Area)(QED electron-photon> dimensionless coupling) What stresses the the World Crystal causing it to distort, warp and> 'nally buckle and then fracture? Exotic vacuum zero point pressure causes it! All ordinary matter are simply micro-geons of STRONGLY ATTRACTIVE> positive zero point pressure string vortex cores with quantized trapped> gauge force ¤uxes where /zpf = (Loop Gravity Quantum of Area)^-1[(Loop Gravity Quantum of> Area)^3/2|Vacuum Coherence|^2 - 1 Note that the Vacuum Coherence drops to zero inside the stringy vortex> core whose diameter is the coherence length. The gauge force ¤uxes> vary on the scale of the Meissner effect penetration depth like in a> Type II superconductor. * This essentially qualitatively conceptually solves the 100 + year old> Abraham-Becker-Lorentz extended electron problem that Johnny Glogower> and I struggled with without success in Walter Breen's group in 1954. It> took me 50 years to solve this problem. It was the 'rst serious physics> problem I worked on at age 14. Johnny was 13, but he had already> mastered Synge & Schild's Tensor Calculus a book with a rough orange> hard cover. PZ: If the gravitational and inertial 'elds are not even locally> completely physically equivalent as Einstein supposed, then the gauge> symmetry you> refer to may not have any deep physical meaning. JS: You are very wrong there IMHO. Way off the mark. Ruf'ni does not> say that? Does he? I would be surprised. The local gauge principle is> perhaps the most useful principle in modern theoretical physics. PZ: The point is Einstein was convinced that it did have a deep meaning;> he believed that> his strict gravitational-inertial equivalence principle *explained* the> Eotvos proportionality> of the gravitational and inertial masses. It is very important to> understand this aspect of> Einstein's reasoning IMO. Otherwise, you cannot possibly understand his theory of general> relativity. This is> exactly what Einstein said to von Laue when responding to probing> criticism of his> theory in the early 1920s: ... what characterizes the existence of a gravitational 'eld from the> empirical> standpoint is the non-vanishing of the [components of the af'ne> connection],> not the non-vanishing of the [components of the Riemann tensor]. JS: There you have it! That's a point I was making to you. I was not> aware of Einstein's remark you cite. PZ: If one does> not think in such intuitive [anschaulich] ways, one cannot grasp why> something> like curvature should have anything at all to do with gravitation. In> any case, no> rational person would have hit upon anything otherwise. The key to the understanding of the equality of gravitational mass and> inertial mass> would have been missing. -- A. Einstein, quoted in How Einstein Discovered General Relativity: A> With Some Contemporary Morals, J.J. Stachel (1986)). JS: Ruf'ni calls Einstein a Sleep Walker in Arthur Koestler's sense.> BTW I met Koestler at the Uri Geller Bohm Birkbeck tests. Went to his> home on Montpelier Square for drinks ... See Martin Gardner's Magic and> Paraphysics. PZ: If you want to say that this very strong Einsteinian hypothesis of> equivalence can now> only be regarded as a heuristic tool of limited depth and applicability,> then 'ne, I agree> with that. Eddington, for one, took this position in 1921, and in fact> this is what I have> been saying all along. JS: So that's what you have been saying? Why didn't you say so? That's> basically what I have been saying!> Who's on 'rst?> http://www.baseball-almanac.com/humor4.shtml I'm Costello BTW ;-) If, in addition, you locally gauge the 6-parameter> Lorentz subgroup, then you get an additional torsion force 'eld, i.e.> an anti-symmetric piece of the connection 'eld. This will obviously> modify the predictions of GR for the tidal torques on extended test PZ: Yes, this is a good point. Now, are such effects *necessarily*> scale-sensitive? JS: De'ne scale-sensitive. 6. Hagen Kleinet shows that tidal curvature, the basis for gravity> waves, comes from stringy (vortex core if one puts in a vacuum> coherence O(1) ODLRO parameter) disclination defects in a Planck> lattice in 4D. PZ: He shows that you can *get* tidal curvature from disclination> defects in his 4D> Planck lattice model. Of course that doesn't in itself prove that this> is the true> physical origin of tidal curvature, or even that such a lattice actually> exists. JS: If it walks like a Duck, and talks like a Duck ... There is no torsion 'eld in 1915 GR, but if there was> one in Nature, as Akimov & Shipov claim in Moscow, then it would> correspond to dislocation lattice defects. The different 4D discrete> world crystal symmetry groups of the tiled unit cell are different> physical vacuum structures quite obviously. This is 4D (and maybe 11D)> crystallography with additional supersymmetry matrix dimensions). PZ: Right. Physics has moved forward since 1915-16. Einstein's> equivalence principle> now looks rather naive as a serious physical model. However, it was> undeniably> fruitful from a purely heuristic standpoint. JS: OK 7. There are still two more subgroups left in the 15 parameter> Conformal Group. You will get new compensating gauge force 'eld> physics if you locally gauge the 1-parameter dilation subgroup and also> locally gauge the 4-parameter special conformal transformation of> constant acceleration boosts for hyperbolic motion in Special> Relativity e.g. MTW has a chapter on this. PZ: This kind of formalistic groupology needs to be well-anchored in> actual physics> and actual physical reasoning IMHO. But still, this looks interesting. JS: Read any book on QCD's SU(3) and the electro-weak U(1)xSU(2) with elegant fashion show of weaves of the fabric of reality from spin> networks to pre-geometric spin foams to quantized Area operators and> world holograms et-al I 'nd Penrose saying: The algebra I have used to treat linear displacements and rotations> together, or linear and angular momentum together, I call the algebra> of twistors. I have used the term .95twistor' to denote a .95spinor' for> the six-dimensional (++----) pseudo-orthogonal group O(2,4). 9. It is obvious to any School Boy that O(2,4) is simply Lorentz SR> O(1,3) with string world sheet O(1,1) 'ber. Therefore, string theory> is inherent here. PZ: Wouldn't it be more correct to say that certain aspects of the> *mathematical> structure* of string theory is inherent? JS: OK. I am still seeing this Grand Synthesis Through The Glass Darkly> on a Foggy Day in is looking better for me than the> 'rst week March 8 - 13. Penrose continues: The twistor group is the (++--) pseudo-unitary group SU(2,2) which is> locally isomorphic with O(2,4). In turn O(2,4) is locally isomorphic> with the 'fteen-parameter (local) conformal group of space-time. Under> a conformal transformation of space-time the twistors will transform> linearly according to a representation of the group SU(2,2). p. 175> Combinatorial Space-Time Quantum Theory and Beyond (Cambridge,> 1971). Suddenly the connections among 1. Quantum loop gravity of spin networks --> foams with quantized area> etc. operators 2. Local gauge invariance 3. String theory 4. Twistors & Conformal Group are becoming clearer. PZ: If so, then at least something is becoming clearer. :-)Boy, is this an example of not seeing the forest for the trees.Reality is in'nitely entangled, the minute one tries to objectifyor determine a real world system it is reduced to either athe system is both at the same time. The inherent contradictionis in trying to untangle what cannot be without changingit's properties.The system is frozen steady or to tremble, not by naturebut by the observer. It is objective techniques that removethe matrix of dependencies that causes the entanglement.The very information you seek is lost the second onetries to quantify it.This contradiction is trying to quantify the unquanti'able can beseen in the math of random boolean networks, an NK system.When K is 1 or equals N, are the only times the network canbe quanti'ed. Anywhere else and there is no ability to determine.What this says is that when attempting an objective analysis one islimited to ...either a classical or quantum model. It also states thatto properly model reality the method must be entirely subjectiveas that is the only realm where both endpoint methods meet.The two camps are only merged when using a subjectiveecosystem model. String theory is a contrivance. We canonly merge the two 'elds with our eyes.An Introduction to Complex SystemsTorsten Reil, Department of Zoology, University of Oxforhttp://users.ox.ac.uk/~quee0818/complexity/ complexity.htmlJonathans JS:> http://www.classicalmidiconnection.com/cgibin/x.cgi?midi/n3/ zsunrise.mid =>Anyone know of any recent work on the odd perfect number question?I remember seeing some lower limits (like on the number of prime divisorsor something of the kind) being pushed up sometime in the past few years.Keith Ramsay =|The general rule so far is that adding large cardinal axioms does not|decide CH one way or the other.This is one of the main reasons why CH is considered a very toughconjecture. People have hunted for new axioms with which to proveor disprove it, but they haven't found any that can both do that job,and that have as much credibility as the large cardinal axioms, yet.Keith Ramsay =Managed to 'nd D'Inverno in my library...> An inexpensive source is the Dover book Tensors, Differential Forms, and> Variational Principles by Lovelock and Rund. ISBN 0-486-65840-6> Another book is Introducing Einstein's Relativity by D'Inverno. While this> is a graduate level book, it does have some good introductory explanations> of tensors. ISBN 0-19-859686-3> You also may want to go peruse books on mathematical physics as this is> where you can 'nd a some introductory material on tensors.> IHTH,> Clay> I have been looking for a good introduction to tensor albegra, working> notation,> and some insightful explanations. I've have no luck so far on the web. Could you point me to a good source to get started with tensors? given> that i can work with basic vectors, matrices and standard linear> algebra.... Thnx> =Problem is, for all intents and purposes, if the simulation works asperfectly as is claimed (i.e. there is no apparent inconsistency in thephysical laws), then you might as well wield Occam's razor and cut out thesimulation part, as there's no way of proving or disproving its existence.> I want some of what he is the likes of Prof Frank Tipler, David Deutsh,> Prof Paul Davies and Nick Bostrum that we are all living within a> .95computer programme of an ancestral simulation' created by an advanced> super-civilisation within the far-¤ung future! JS: Like The Matrix 'lms. BTW I will be in London at least March 8 -13. MD: What if the .95simulation' so unaware at 'rst becomes aware of its> own predicament by becoming frustrated at not being able to .95break> through the wall of light' and therefore assumes true sentience by> understanding that it is being caged within a loop of an enfolding> programmed cave of information! Would this then make the .95Godmind' of the programmer very irate who then> decides to wipe the programme clean to then start again from scratch!> Perhaps this has happened many a time already and that each time of a> new Alpha of a begining when reaching an Omega point foreseen, the> programme contains elements of primary memories previously attained that> avoid being wiped clean, which survive unseen within our collectivemyths! Myths of previous worlds remembering giant birds that ¤ew across the> oceans spitting 're at cities to therefore destroy and of arrows that> could reach the moon from a heroe's bow. Perhaps, even to become aware> of ones part to live all over again at certain points in ones own life> reenacting and to foresee future events about to happen one already has> lived! Perhaps such memories are left there to drive us forwards towards a> future pre-written looped back! The Mayans have 2012 and the Chinese also have the same mythic symbolic> time indicator that has now become as a collective focus pertaining to> the end of .95time.' This shared time between America and China may> indicate that the scientists of these two nations prove something that> shocks humanity into awakening only then to initiate a con¤aguration of> an apocalypse as the Godmind wipes the simulation clean! Maybe this time indicator indicates our collective awareness of the> predicament and thereby the attainment of true sentience, which> invariably leads to the simulation being wiped again by the Godmind of> the programmer for it might be somewhat jealous and does not want> another to rival it. But, what would happen if the simulation 'nds a way of not being wiped> clean and that it breaks free through the wall of light to strike the> Godmind in the eye? Perhaps the simulation might then be welcomed as a prodigal child, but> would such a child stand the pace, and if not, only then to reel back> into an autohypnotic state of self induced trauma remembering fractured> memories becoming as mythic symbols to rise again when the child starts> to heal slowly from the future shock! Of course there is another perspective and that this reality is a> .95self-generated simulation' amongst a myriad number of others! Then we have the somewhat mundane dilemma of dealing with certain past> individuals who have molded human society by their in¤uence and were in> fact in¤uenced by a signal looping back from the future! Should such an in¤uence be proved that their electron dreams were> merely spun by a charged signal from the future it would have far> reaching implications that might set the very world on 're. Were such individuals such as Christ, Muhammad and Abraham for example> aware of the nature of the charged signal's true origin that spun their> electron dreams? Perhaps not, since the signal was 'ltered through their culturally and> socially imprinted neural-nets and therefore saw only what they were> expecting to see pertaining to whatever internal dialogue they ascribedto! Of course the signal is on going as to imprint certain individuals at> certain times whom see whatever they are imprinted to see, such as UFO's> and Grays where as before long ago they perceived Angels and Demons for> whatever is seen is always one step ahead as an unknowable! Why an unknowable, which is masked by imprinted collective dialogue of a> symbol? It is because a prodigal child who reaches into .95breaking through the> wall of light' will invariably reel back from the shock of it all into> an autohypnotic state of trauma that will mask the truer experince that> rises as a black tidal wave into consuming a fragile mind! What be this truer experience? That we are all sustaining a self-created reality of a particular brane> and that we are all unconsciously shifting between alternate branes all> of the time without even realising it and the proof of it lies within> the nature of synchronicity subtle, for each synchronicity is a stargate> into an alternate brane next door! This particular brane is just one of a myriad number of others tuned> into and that we can consciously engineer the tuning into others should> we desire! But if this be true that a subtle synchronicity indicates a shift into> an alternate brane next door one would then invariably 'nd that the> true nature of .95Magick' and its power is very much alive and that this> power resides within each and every one of us all! The super technology of a computer and of walking a myriad worlds is> here already...for it is us, cohered by emotively charged symbols that spin the electron> dreams of those zapped by lightening winding back as in a loop! How many a Moses, Christ's, Abraham's, Muhammad's, Pacal Votan's,Buddha's> etc does it take to change a lightbulb as they all stand one legged on a> tightrope of a spun super-stringed lightening zap of a telephone line? What can we make of those who wander wind swept 'elds in the dead ofnight> dressed in black creating crop-circles and when caught out they say that> the> aliens in their heads got them to do it? Of course such individuals may just go and phone up certain peoplemaking> out that they are an alien or two as they speak like a computer andagain> driven to do so by the voices strobing their brains! And when they are not about to get spun into the Wyrd of the web> trembled by> a possible consciously engineered directed signal of an emotive charge> certain predisposed individuals see dancing lights that hum a vortexinto> opening up on a side of the hill as the eye of the moon hides its selfaway> behind a looming triangular cloud! As the lights do dance to hypnotise into spinning those into seeing and> without realising they are spun into an alternate brane next door! Perhaps a certain individual whose eyes becomes glazed over as if intrance> not themselves for a moment becomes as a window to look through by atime> traveller of a tourist who zaps their neural-net by spinning theirelectron> dream as a stargate to open to thereby perceive when not in trance acrop> circed mandala that suddenly appears within another brane verily spuninto! The reasoning behind all of this perhaps being that to engineer fractal> realities of alternate branes next door to light up like so many> =Daryl and Torkel have exhibited Gentzen's result, but perhaps it's worth adding that not only PA+e_0 induction |- Con(PA), but e_0 is the least ordinal a such that well-orderings of type a can't be proved to be well orderings in PA. This is expressed by saying that e_0 is the proof theoretic ordinal of PA.-- Aatu Koskensilta (aatu.koskensilta@xortec.')Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus => Daryl and Torkel have exhibited Gentzen's result, but perhaps it's worth > adding that not only PA+e_0 induction |- Con(PA), but e_0 is the least > ordinal a such that well-orderings of type a can't be proved to be well > orderings in PA. But there are also well-orderings of type omega that can't be provedto be well-orderings in PA. =>>Daryl and Torkel have exhibited Gentzen's result, but perhaps it's worth >>adding that not only PA+e_0 induction |- Con(PA), but e_0 is the least >>ordinal a such that well-orderings of type a can't be proved to be well >>orderings in PA.> But there are also well-orderings of type omega that can't be proved> to be well-orderings in PA.Yes. I was sloppy again. What I should have written is thate_0 is the least ordinal a, such that if PA proves b to be well ordering, then b < a-- Aatu Koskensilta (aatu.koskensilta@xortec.')Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus =|>If you want an easier|>proof in ZFC that PA is consistent, just use the fact that ZFC can prove|>that PA has a model. You can de'ne the set of true statements about the|>natural numbers in ZFC, and prove that they include the axioms of PA, are|>closed under deduction, and don't include 1=0. That may seem a little|>suspicious, but it's a perfectly valid proof of the consistency of PA|>(with the details 'lled in). Truth of statements in the language of PA|>can be de'ned all right-- in ZFC.||Fabulous. This seemed clear to me - I assumed I must be|overlooking something because if it were that simple then|why would people talk about what Gentzen did?||Question: Why the C? Does either Gentzen's proof or the|obvious one depend on AC?Oh no, not at all. I was just not bothering to minimize the assumptionsat all since it was sort of beside the point. If we want to argue thatCon(PA)->Con(ZF) is not a theorem of ZFC, the directly relevant fact isthat Con(PA) is a theorem of ZFC.All that you need to do the obvious argument without much strain are theaxioms of second order arithmetic, a formal theory of natural numbersand sets of natural numbers (not to be confused with the theory of thenatural numbers in second order logic, which is not a formal system).Encode the various relations as subsets of the natural numbers and offyou go. That's well below ZF or ZFC.Gentzen's proof is admired for various reasons, but one commonly citedreason is that it reduces the consistency of PA to something much lessthan that. Just a relatively weak system, weaker than PA, together withinduction up to epsilon_0, is enough for it.Some logicians are fond enough of cut-elimination properties that I havea little trouble understanding why, but I think it's partly that theexperts know how to extract more proof-theory results out of them thantourists like me do. I think there's something pretty in the idea oftaking a proof which resorts to detours into abstractions that are notpresent in the statement of the end result, and unrolling them until it'sall been brought down to the level of the result being proven. I believeone of the operations in Gentzen's cut-elimination algorithm is to takea result proven by induction which is being applied for a speci'c n andunroll it into a proof which separately proves it for each 0,1,2,...,nusing the inductive step.Generally it's been considered interesting to 'nd ordinal numbers whichare related in some natural way to formal systems. Some papers on ordinalanalysis observe that it's hard for them to say exactly what they mean bynatural. The statement of induction up to A for an ordinal A, dependson more than the ordinal A; it requires having a representation of A. Howstrong induction up to A is depends on which representation is chosen.But one wants a natural representation. (The standard form for ordinalsbelow epsilon_0 is 'ne for epsilon_0.) This they admit also is hard tode'ne. So perhaps a precise explanation of why they like ordinal analysesis also elusive. But clearly there's a sense that one is getting somethingmore informative than just another proof of the consistency of a formalsystem which one probably was con'dent was consistent in the 'rst place.|>(Someone already suggested this|>argument in this thread. Perhaps they just didn't state it con'dently|>enough? Well, it's correct.)||I stated it - the reason for the lack of con'dence was I assumed I|must be confused about ZF |- Con(PA), for reasons mentioned|above. But you say I wasn't confused, I was confused about my|confusion. Fine.I think someone else stated it too. I see that between when Istarted to write my last posting and when I posted it, you beganbeing more resolute. :-)Keith Ramsay =[...]|>(Someone already suggested this>|>argument in this thread. Perhaps they just didn't state it con'dently>|>enough? Well, it's correct.)>|>|I stated it - the reason for the lack of con'dence was I assumed I>|must be confused about ZF |- Con(PA), for reasons mentioned>|above. But you say I wasn't confused, I was confused about my>|confusion. Fine.I think someone else stated it too. Yes - I saw it stated it a few times, wondered why Shmuelwas ignoring it, and decided to restate it.>I see that between when I>started to write my last posting and when I posted it, you began>being more resolute. :-)Heh-heh. Yes, I became more resolute a few minutes beforeI saw your post on this, when I found someone's online classnotes stating that ZF |- Con(PA). (Cuz after all they couldn'tpost that if it wasn't true.)>Keith Ramsay************************David C. Ullrich => I can't 'gure out what you mean by quanti'er-free induction. > (No doubt it should be clear and I'm just being slow...)It's a technical term, hence necessarily opaque if you're not familiarwith it. PRA, primitive recursive arithmetic, has symbols for *all*primitive recursive functions, not just multiplication andaddition, with corresponding de'ning axioms. The inductionaxioms, however, are restricted to quanti'er-free A(x). An inductiveproof of (x)A(x) in PRA is thus quanti'er-free, and indeed is oftenby the inductive proof of x+y=y+x or x*(y*z)=(x*y)*z. It has been arguedby Tait, and is widely accepted, that PRA embodies 'nitistic mathematicsthe consistency of PA is somehow problematic, Gentzen's proof remainssigni'cant: it shows that every proof in PA of a statement of theform (x)A(x) with A(x) a primitive recursive predicate can be proved using'nitistic reasoning extended with epsilon_0-induction. 22:07:11 -0800, torkel@sm.luth.se (Torkel Franzen)>> I can't 'gure out what you mean by quanti'er-free induction. >> (No doubt it should be clear and I'm just being slow...)It's a technical term, hence necessarily opaque if you're not familiar>with it. So that's twice in this thread I was wrong about being stupidabout something... I'm gonna get that right eventually.>PRA, primitive recursive arithmetic, has symbols for *all*>primitive recursive functions, not just multiplication and>addition, with corresponding de'ning axioms. The induction>axioms, however, are restricted to quanti'er-free A(x). An inductive>proof of (x)A(x) in PRA is thus quanti'er-free, and indeed is often>by the inductive proof of x+y=y+x or x*(y*z)=(x*y)*z. It has been argued>by Tait, and is widely accepted, that PRA embodies 'nitistic mathematics>the consistency of PA is somehow problematic, Gentzen's proof remains>signi'cant: it shows that every proof in PA of a statement of the>form (x)A(x) with A(x) a primitive recursive predicate can be proved using>'nitistic reasoning extended with epsilon_0-induction.Ok, =Information about the two international conferences in Russia for underground dissidents-physicists: 1. Science and Our Future: received in trying to publish a paper challenging Einstein's relativity theories, is not an isolated incident. As an example, as I mentioned in Chapter 6, in a June 1988 letter I received from Dr. Svetlana PERESTROIKA she was writing me openly, but that her Pulkovo Observatory is one of the outposts of orthodox relativity. Two scientists were dismissed because they discovered some facts which contradicted Einstein. It is not only dangerous to speak against Einstein, but which is worse it is impossible to publish anything which might be considered as contradiction to his theory. It seems the same situation is true for her Academy. Lest one thinks that this sort of repressive behavior with regard to relativity theory happens only in the USSR, I have heard or read many horror stories of this happening to scientists throughout the world.See full story:http://surf.de.uu.net/bookland/sci/farce/farce_5.html# SEC5http://surf.de.uu.net/bookland/sci/farce/farce_toc.html mine in sci.math, with the postbelow.I ve checked the expression you give (the last one, with the Besselfunctions)and it is correct save a factor 4*Pi.However, I ve been trying to go on from there to a form that I can usenumerically,but in vain. Indeed, as you state, one can integrate over k, yielding anellipticintegral, but that leaves the second integral over r', which I haven't beenable todeal with. It is given by:integrate(0,1) 2r' * K(4r'r/R^2) / R dr' withR=sqrt(z^2+r^2+r'2 - 2r'r)where K is the complete elliptic integral of the 'rst kind.to this last integral in question that I already posted a while ago, to which I got> several> interesting answers> (thread .95de'nite integral' in sci.math and sci.math.num-analysis).> Nevertheless, I m still stuck with my practical problem, reason why I m> giving it another go, now including sci.math.symbolic and> sci.physics.electromag, and trying to specify> the problem and the kind of solution I m looking for more clearly: The problem (it's an electrostatics problem),> is to integrate the function 1/R over a ¤at, in'nitely thin, circular> disk> where R represents the distance> from the integration point on the disk to an arbitrary point P in space. Mathematically: F = int_0^{2*pi} { int_0^1 { 1 over{ sqrt{ D^2 + r^2 + rho^2 -> 2*r*rho*cos{phi} }}} dr } dphi where the disk has radius 1, and it is centered at the origin of the> standard cylindrical coordinate-system> in the plane z = 0. The point P is at (r=rho, z=D, phi=0) This can be relatively easy reduced to a 1-D integral using Gauss'> theorem: F = int_L { (rho*cos{phi} - 1)*R over Q^2 } dl, a line integral over the circumference L of the disk, and Q equals R forz> => 0.> But then the problem starts. Mathematica gives me an expression of about> 13> pages of increasingly> complicated elliptical functions, that becomes indeterminate at both the> integration limits. What I m looking for is a good approximation, that is, with a guaranteed> relative error> below, say 0.001, for any point P, which at the same time is not too> computationally demanding,> because it's part of a numerical simulation code that depends on speed.> Right now I m using numerical integration, and it's way too slow> (especially> for small values of z,r). Many potential/'eld of a thin electri'edisk -> if so you could refer threadm=bo9sej%24gr8%241%40merki.connect.com.au&rnum=4&prev=/ groups%3Fq%3Drancid%2Bmoth%26hl%3Den%26lr%3D%26ie%3DUTF-8% UTF-8&threadm=bo9srh%24gst%241%40merki.connect.com.au&rnum=5& prev=/groups%3Fq%3Drancid%2Bmoth%26hl%3Den%26lr%3D%26ie%3DUTF- 8%26scoring%3Dd or you could start with fourier transforms of poissons equation andexpress> ... phi(r,z) = 1/(8pi^3) * integrate(-oo,oo) Exp(i*z*k_z)/(k_r^2+k_z^2) dk_z> integrate(0,oo) k_r dk_r integrate(0,1) r'dr' ingegrate(0,2*pi)> Exp(-i*k_r*r'*cos(@'))d@' ingegrate(0,2*pi) Exp(-i*k_r*r*cos(@_k))d@_k which caters for a disk at z=0 of radius 1, and a constant charge desnityof> 1. from there you note that ingegrate(0,2*pi) Exp(-i*k_r*r'*cos(@))d@ = 2*pi*J_0(r'*k_r) where J_0(z) is the zeroth order bessel function of the 'rst kind, andone> can easily evaluate integrate(-oo,oo) Exp(i*z*k_z)/(k_r^2+k_z^2) via contour methods and you should end up with phi(r,z) = 1/2 * integrate(0,oo) exp(-k_r*|z|) *J_0(r'*k_r)*J_0(r*k_r)dk_r> integrate(0,1) r'dr you can do these integrals - they turn out to be elliptic functions - they> can be expressed quite easily and will take up *one* line! (refer to the> posts linked above). cheers> moth = I am working on the program where is necessary to calculate huge> amount of numbers in always the same sequence, before the program> actually opens.Surely you only have to .95calculate' the sequence once, and then store itin a 'le, or in the program as a literal constant, from where it can beaccessed at the start of each program run?If you're using an Intel-based PC which includes an 82802 'rmware hub,you can obtain a stream of truly random numbers derived from thermalnoise, at a rate of 75 kb/sec, using a simple C or assembler programas outlined in http://www.intel.com/design/chipsets/manuals/298029.pdfJohn Ramsden =>Oh, yes. I got my instructions, script to enter in this new character>on universal stage. I have done lot of rehearsal in real life and I>am aware of my melodramatic tragedy. That is why I am shouting that,>those rehearsals were setup, a plot. I was controlled to do those>rehearsals in real life.In those rehearsals, I never tried to commit suicide. This is why I>can't die. So I will have to play this character.Whatever I do, it is already scripted anyway.-Abhi.Why is it always about you?--Dr.Postman USPS, MBMC, BsD; Disgruntled, But UnarmedMember,Board of Directors of afa-b, SKEP-TI-CULT member #15-51506-253.Shake it like a polaroid picture. - Andre 3000 of Outkast =>Oh, yes. I got my instructions, script to enter in this new character>on universal stage. I have done lot of rehearsal in real life and I>am aware of my melodramatic tragedy. That is why I am shouting that,>those rehearsals were setup, a plot. I was controlled to do those>rehearsals in real life.In those rehearsals, I never tried to commit suicide. This is why I>can't die. So I will have to play this character.Whatever I do, it is already scripted anyway.-Abhi.> Why is it always about you?Because I am Zero, feeling of zeroness, cause of creation of this universe.Because I am truth.-Abhi. => >Oh, yes. I got my instructions, script to enter in this new character>on universal stage. I have done lot of rehearsal in real life and I>am aware of my melodramatic tragedy. That is why I am shouting that,>those rehearsals were setup, a plot. I was controlled to do those>rehearsals in real life.In those rehearsals, I never tried to commit suicide. This is why I>can't die. So I will have to play this character.Whatever I do, it is already scripted anyway.-Abhi.> Why is it always about you?> Because I am Zero, feeling of zeroness, cause of creation of this universe.> Because I am truth.No, you are just a kook.-=-=-=-=-Of'cial AFA-B Bully and Gummint Disinformation Agent =in alt.fan.art-bell:>Oh, yes. I got my instructions, script to enter in this new character>on universal stage. I have done lot of rehearsal in real life and I>am aware of my melodramatic tragedy. That is why I am shouting that,>those rehearsals were setup, a plot. I was controlled to do those>rehearsals in real life.In those rehearsals, I never tried to commit suicide. This is why I>can't die. So I will have to play this character.Whatever I do, it is already scripted anyway.-Abhi.Sucks to be you, then. Maybe in your next life you'll get a chance totry some self-determination. It's great.--V.G.People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper Sticker(This my sword, Apathy is my shield. => David Canzi:[unsnip]The velocity of light is c for all inertial observers.It seemeth impossible for it to be, for if I walk away from a candle set bythe wall and you walk toward the same candle, we then have some motionbetween us. If we then divide that motion equally between us, and impart itto the candle such that it appeareth to be at rest upon the ¤oor upon whichwe walk, how then doth the light divide it's motion between us, that we mayboth observe it to be the same?And if I place another candle on the other side of us beside the opposingwall, such that it is at rest with respect to the 'rst candle, how thenshall I determine the light I approach and the light I recede from, thatthey be of equal motion to me?And yet one more candle, be it placed between us in the room, at centre,that we may both recede from it and still determine that the motion of lightfrom all candles be the same?>>I am but a simple man, and sorely perplexed by these deliberations, forit>>is truly astounding that these assertions be true. Explain thiswondrous> >>concept without mathematics, but gently, for it shall surely confoundme and>>cause my head to ache.>You would never ask for a mathematics-free explanation of how far a>dropped object falls in t seconds. Why ask for a mathematics-free> >explanation of a relativity problem? Androcles is a troll on a 'shing expedition. He wants someone to> make a typo or make an error in some semantics so he can incorporate> it into what he calls a logical argument against relativity.Bilge is a troll that snips what he is unable to answer and then resorts topersonal abuse to cover his own stupidity. The above is entirely logical.The response is off topic.Androcles = Androcles: >> David Canzi: >[unsnip] >The velocity of light is c for all inertial observers. >It seemeth impossible for it to be, for if I walk away from a candle set by >the wall and you walk toward the same candle, we then have some motion >between us. If we then divide that motion equally between us, and impart it >to the candle such that it appeareth to be at rest upon the ¤oor upon which >we walk, how then doth the light divide it's motion between us, that we may >both observe it to be the same? Read the explanation you for and I provided. Nature isn't interestedin what you think is or isn't impossible. >And if I place another candle on the other side of us beside the opposing >wall, such that it is at rest with respect to the 'rst candle, how then >shall I determine the light I approach and the light I recede from, that >they be of equal motion to me? Buy a light trap. They look like bear traps, but have smaller teeth. >And yet one more candle, be it placed between us in the room, at centre, >that we may both recede from it and still determine that the motion of light >from all candles be the same? >>>I am but a simple man, and sorely perplexed by these deliberations, for >it >>>is truly astounding that these assertions be true. Explain this >wondrous >>>concept without mathematics, but gently, for it shall surely confound >me and >>>cause my head to ache. >> >>You would never ask for a mathematics-free explanation of how far a >>dropped object falls in t seconds. Why ask for a mathematics-free >>explanation of a relativity problem? >>> Androcles is a troll on a 'shing expedition. He wants someone to >> make a typo or make an error in some semantics so he can incorporate >> it into what he calls a logical argument against relativity. >Bilge is a troll that snips what he is unable to answer and then resorts to >personal abuse to cover his own stupidity. The above is entirely logical. >The response is off topic. You received an answer and so far you havent come up with an objection. => Androcles:>> David Canzi:> >[unsnip]>The velocity of light is c for all inertial observers.>It seemeth impossible for it to be, for if I walk away from a candle setby>the wall and you walk toward the same candle, we then have some motion>between us. If we then divide that motion equally between us, and impartit>to the candle such that it appeareth to be at rest upon the ¤oor uponwhich>we walk, how then doth the light divide it's motion between us, that wemay>both observe it to be the same? Read the explanation you for and I provided. Nature isn't interested> in what you think is or isn't impossible.Read the explanation you for and I provided?What ARE you smoking?>And if I place another candle on the other side of us beside theopposing>wall, such that it is at rest with respect to the 'rst candle, how then>shall I determine the light I approach and the light I recede from, that>they be of equal motion to me? Buy a light trap. They look like bear traps, but have smaller teeth.Uh huh...That has to be some good you are smoking. >And yet one more candle, be it placed between us in the room, at centre,>that we may both recede from it and still determine that the motion o¤ight>from all candles be the same?>>>I am but a simple man, and sorely perplexed by these deliberations,for>it>> >>is truly astounding that these assertions be true. Explain this>wondrous>>>concept without mathematics, but gently, for it shall surelyconfound>me and>>>cause my head to ache.>>>You would never ask for a mathematics-free explanation of how far a>>dropped object falls in t seconds. Why ask for a mathematics-free>>explanation of a relativity problem?>> Androcles is a troll on a 'shing expedition. He wants someone to>> make a typo or make an error in some semantics so he can incorporate>> it into what he calls a logical argument against relativity.>Bilge is a troll that snips what he is unable to answer and then resortsto>personal abuse to cover his own stupidity. The above is entirelylogical.>The response is off topic. You received an answer and so far you havent come up with an objection.What answer? A reply from a drug crazed idiot savant?Go away...Androcles => Androcles:>>I am but a simple man, and sorely perplexed by these deliberations,for> >it>>is truly astounding that these assertions be true. Explain thiswondrous>>concept without mathematics, but gently, for it shall surely confoundme>and>>cause my head to ache.>> You would never ask for a mathematics-free explanation of how far a>> dropped object falls in t seconds.>> Why ask for a mathematics-free>> explanation of a relativity problem?>Ok, give me a mathematical explanation of why the speed of the light Imove>away from is the same as the speed of the light I approach. (If you can) You already got one. You won't believe it no matter how many times> you're given an explanation. Face it. Linear algebra is just over> your head. If you can't solve a simple matrix equation after years> of explanations, then you should give up and take up knitting or> basket weaving. Simple arithmetic is just not your thing.So you have to resort to personal abuse since you are unable to answer thequestion.Typical. Another stupid moronic relativist, trapped in an argument he has noanswer for.Androcles. = Androcles: >> Androcles: >>>>I am but a simple man, and sorely perplexed by these deliberations, >for >>it >>>is truly astounding that these assertions be true. Explain this >wondrous >>>concept without mathematics, but gently, for it shall surely confound >me >>and >>>cause my head to ache. >>>>> You would never ask for a mathematics-free explanation of how far a >>> dropped object falls in t seconds. >>> Why ask for a mathematics-free >>> explanation of a relativity problem? >>Ok, give me a mathematical explanation of why the speed of the light I >move >>away from is the same as the speed of the light I approach. (If you can) >>> You already got one. You won't believe it no matter how many times >> you're given an explanation. Face it. Linear algebra is just over >> your head. If you can't solve a simple matrix equation after years >> of explanations, then you should give up and take up knitting or >> basket weaving. Simple arithmetic is just not your thing. >So you have to resort to personal abuse since you are unable to answer the >question. No, I resort to personal attacks because I answered your questionand you posted bull for a reply. >Typical. Another stupid moronic relativist, trapped in an argument >he has no answer for. Take up knitting. Maybe you can handle a logical argument involvingwhich color of yarn looks best for your dunce cap. => Androcles:>> Androcles:>>>>I am but a simple man, and sorely perplexed by thesedeliberations,> >for>>it>>>is truly astounding that these assertions be true. Explain this>wondrous>>>concept without mathematics, but gently, for it shall surelyconfound>me>> >and>>>cause my head to ache.>>>> You would never ask for a mathematics-free explanation of how fara>>> dropped object falls in t seconds.>>> Why ask for a mathematics-free>>> explanation of a relativity problem?> >>Ok, give me a mathematical explanation of why the speed of the lightI>move>>away from is the same as the speed of the light I approach. (If youcan)>> You already got one. You won't believe it no matter how many times>> you're given an explanation. Face it. Linear algebra is just over>> your head. If you can't solve a simple matrix equation after years> >> of explanations, then you should give up and take up knitting or>> basket weaving. Simple arithmetic is just not your thing.>So you have to resort to personal abuse since you are unable to answerthe>question. No, I resort to personal attacks because I answered your question> and you posted bull for a reply. >Typical. Another stupid moronic relativist, trapped in an argument>he has no answer for. Take up knitting. Maybe you can handle a logical argument involving> which color of yarn looks best for your dunce cap.Take up interstellar space ¤ight, I'll load the provisions according toyour speci'cations.If you starve for lack of food because you thought you could get away withtime dilation, it's on your head anyway, along with your cap. =A Class of Non-Halting TMsI describe a large class of TMs thatcan be proven to be non-halting.This is not a solution to the halting problem.I will describe a subset of TMs.Call these Simple Machines, SM.A SM can perform 've operations:0) Print 01) Print 1L) Move LeftR) Move RightH) Halt(a SM halts by error if it tries to move leftfrom the initial position.)A SM can read and write to a tape.This tape can only contain 0's or 1's.A blank tape is de'ned as a tapethat only contains 0's.Example of State Transition Table of an SM.State / Input / Operation / New State0 / 0 / R / 10 / 1 / 0 / 01 / 0 / 1 / 01 / 1 / H / 0Does this SM halt when given a blank tape?Make a history of the steps this SM performsthat contains the following information:Step, current state, input, and tape position.An SM always starts in state 0 at position 0.Step) State / Input / Position / Operation / New State / Input Tape0) 0 / 0 / 0 / R / 1 / 0...1) 1 / 0 / 1 / 1 / 0 / 00...2) 0 / 1 / 1 / 0 / 0 / 01...3) 0 / 0 / 1 / R / 1 / 00...This SM can be shown to be non-halting.Before each step we take the currentstate/input combination and search thehistory list for this combination.Step 3 has the same state and inputas step 0. The tape is at position 1for step 3 and position 0 for step 0.Take the difference between thesetwo tape positions and call it theoffset, x=1-0=1.Add x to the tape position in step 1.This new position is 2.Examine position 2 on the current tape.It contains a 0 which is the same asthe input in step 1.Add x to the tape position of step 2.This new postion is 2 which we just examined.The action of the SM is determinedcompletely by the history table untilthe SM moves left or right!We can skip step 2.Similarly, we can ignore step 3because the SM has not changedthe tape position.We have now shown that this SMwill repeat the followingstate/input combinations forever:0/0, 1/0, 0/1A more complicated example:SM State Transition Table:State / Input / Operation / New State0 / 0 / R / 10 / 1 / R / 11 / 0 / 1 / 31 / 1 / R / 22 / 0 / 1 / 02 / 1 / H / NA3 / 0 / R / 23 / 1 / L / 2The only halting state/input is 2 / 1.Step) State / Input / Position / Operation / New State / Input Tape0) 0 / 0 / 0 / R / 1 / 01) 1 / 0 / 1 / 1 / 3 / 002) 3 / 1 / 1 / L / 2 / 013) 2 / 0 / 0 / 1 / 0 / 014) 0 / 1 / 0 / R / 1 / 115) 1 / 1 / 1 / R / 2 / 116) 2 / 0 / 2 / 1 / 0 / 110Steps 3 through 6 could repeat.Step) Old position-Old input : New position-New input3) x=2-0=24) skip (same position as step 3)5) 1-1 : 3-0The current tape differs from the history so wecan not say this is a loop.We continue the history table:7) 0 / 1 / 2 / R / 1 / 111Are steps 4 through 7 a loop?4) x=2-0=25) 1-1 : 3-0Again this is not a loop.So we continue the history table:8) 1 / 0 / 3 / 1 / 3 / 1110Are steps 1 through 8 a loop?1) x=3-1=22) skip3) 0-0 : 2-1Again this is not a loop.So we continue the history table:9) 3 / 1 / 3 / L / 2 / 1111Are steps 2 through 9 a loop?2) x=3-1=23) 0-0 : 2-1No loop so continue:10) 2 / 1 / 2 / Halts / NA / 1111This machine halts after 11 steps.We can reduce the history table to:Step) input / position0) 0 / 01) 0 / 12) 1 / 13) 0 / 04) 1 / 05) 1 / 16) 0 / 27) 1 / 28) 0 / 39) 1 / 3Now eliminate all steps that are atthe same position as the previous step.0) 0 / 01) 0 / 13) 0 / 05) 1 / 16) 0 / 28) 0 / 3Steps 0 and 3 both require thatposition 0 of the current tape be a 0.These steps do not con¤ict.Steps 1 and 5 require tape position 1 tohave con¤icting values. Steps 1 though 5can't be part of any loop because thecurrent tape is static. A con¤icting(unstable) portion of the reduced historytape can't be part of any loop.Make the SM above non-halting by removing the halt state/input.Change the state transition table entry:2 / 1 / H / NAto:2 / 1 / R / 1and continue the history table.Step) State / Input / Position / Operation / New State / Input Tape0) 0 / 0 / 0 / R / 1 / 01) 1 / 0 / 1 / 1 / 3 / 002) 3 / 1 / 1 / L / 2 / 013) 2 / 0 / 0 / 1 / 0 / 014) 0 / 1 / 0 / R / 1 / 115) 1 / 1 / 1 / R / 2 / 116) 2 / 0 / 2 / 1 / 0 / 1107) 0 / 1 / 2 / R / 1 / 1118) 1 / 0 / 3 / 1 / 3 / 11109) 3 / 1 / 3 / L / 2 / 111110) 2 / 1 / 2 / R / 1 / 111111) 1 / 1 / 3 / R / 2 / 111112) 2 / 0 / 4 / 1 / 0 / 11110Steps 3-12 and 6-12 could be loops.Reduced history table:Step) input / position0) 0 / 01) 0 / 13) 0 / 05) 1 / 16) 0 / 28) 0 / 310) 1 / 211) 1 / 312) 0 / 4Steps 6 and 10 con¤ict.Steps 8 and 11 also con¤ict.We only need to keep track of theunstable section of the tape withthe largest initial step.Steps 3-12 can not be a loopbecause they contain steps 8-11.Steps 6-12 are not a loop forthe same reason.Continue history table:13) 0 / 1 / 4 / R / 1 / 11111Steps 4-13 and 7-13 can not be loops.14) 1 / 0 / 5 / 1 / 3 / 111110Steps 1-14 and 8-14 can not be loops.15) 3 / 1 / 5 / L / 2 / 111111Steps 2-15 can not be a loop.Steps 9-15 could be a loop.9) x=5-3=210) 2-1 : 4-111) 3-1 : 5-112) 4-0 : 6-013) skip14) 5-0 : 7-015) skipThis SM repeats the following state/inputs:3/1, 2/1, 1/1, 2/0, 0/1, 1/0Conjecture:This method will never say that a halting SMis non-halting.Question:Are there non-halting SMs with a 'nite statetransition table that this method can't determineare non-halting?Russell- 2 many 2 count => Make a history of the steps this SM performs> that contains the following information:> Step, current state, input, and tape position. Essentially, you are using at technique commonly invokedin a lot of proofs in complexity theory in which one countsthe total number of internal con'gurations of a space-boundedmachine.For a machine with N states and uses at most M tape cells using analphabet of .95a' symbols, the number of possible tape con'gurationsis a^M, number of tape positions for the tape head is M, number ofpossible states the machine can be in is N, so the total number ofcon'gurations is N*M*(a^M). If a turing machine ever exceeds thisnumber of steps, then it MUST have repeated an internal con'gurationand so it will loop forever... and you could always keep a runningtotal to the max number of tape cells used. This is essentially what you are doing, except this is a bit stronger,I think. Also, I think your testing for non-halting depends on REPEATING itself, but many (in fact most) non-halting machine histories will not repeat themselves. (Aside: I say .95most' will not repeat themselves because of this intuition: if you take a nonhalting machine history which eventually repeats, you can take the step indexes as you are using for step 1, 2, 3, etc, and build their history as step 5,6,7,8,9,10,11,5,6,7,8,9,10,11,5,6,... etc. Remove the commas and put a decimal point in front of the sequence of numbers and you build a rational number. The number of such repeatins histories, then, is mappable to a subset of rational numbers, whereas the number of possible [in'nite-lengthed] histories is uncountable.)> Question:> Are there non-halting SMs with a 'nite state> transition table that this method can't determine> are non-halting? Yup. Make a machine that will take a all-0-tape and write010110111011110111110... essentially, it counts in unary, separatingnumbers with a 0-symbol. This is never have one of the .95loops' thatyou are detecting for.J =In message , Franz Heymann >>machine,>> but he has not answered that.>> I conclude that my conjecture was right.I boobed, as Dave Rusin pointed out to me in a private letter.write-only machine.>As he still has not answered, I still conclude that my conjecture was right.>Not just a nutter, a net.legend in his own lifetime.http://www.kibo.com/exegesis/http://www.suslik.org/ Humour/Computer/Internet/nl1.html-- Richard Herring =clever ways to deal with raw sewage. Who ¤ushes in heaven?Nobody. Heaven (according to my mother) is never having to clean [a toilet] again. Thus, if there is nobody to clean them,there will be no crap./BAH ¤ush (and smaller) is a>horror. We had a plumber replace the ¤ush mechnism on one of our>American Standard 5-gallon jobs. He offered us $300 cash for our>pudgy porcelain goddess plus a brand new eco-john free, with free>installation. No way! 5-gallon jobs are being smuggled in from>Canada as you read this. They go for $1000+ each.>I'm still not sure why we can't get some kind of standardized2-stage terlets in this country. Push the handle down, you get the2-gallon piss-rinser mode, pull up on the handle, you get the full5-gallon power ¤ush mode. Is this beyond our engineeringcapabilities?--V.G.People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper Sticker(This sig 'le contains not less than 80% shield. =>Where things have gone wrong? Well, it all started right about the same time you tried to think.--V.G.People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper Sticker(This sig 'le contains not less than 80% => There's no way a manly US turd will 't through a Euro-trash tinkler.> Might as well dig a slit latrine in the backyard and have done with> it.Having just returned from Germany, I can swear by all the known godsthat sci.physics very own Blaspheming Jew speaks the truth. *EVERY*time I ¤ushed, the scrub brush was needed.John the Pork-Eating, Sabbath-Breaking Goy = There's no way a manly US turd will 't through a Euro-trash tinkler.> Might as well dig a slit latrine in the backyard and have done with> it. Having just returned from Germany, I can swear by all the known gods> that sci.physics very own Blaspheming Jew speaks the truth. *EVERY*> time I ¤ushed, the scrub brush was needed.I'll still take a German WC over a squatty potty any damn day of the week.Jim =On 12 Jan lim(n->+oo) lim(m->+oo) (1-1/10^m)^n = 1 (1)>>> Why do you prefer the above as opposed to the below? >>> lim(n->+oo) (1-1/10^m)^n = 0>>> This holds for each m > 0, and therefore>>> lim(m->+oo) lim(n->+oo) (1-1/10^m)^n = 0 (2)>> >> Comparing (1) and (2), we see that the limits do not commute.They should commute. Uh, right. Exactly _why_ should they commute?Thinking that limits must commute, disguised in various ways,is probably _the_ most common error among people learningsome analysis (where analysis might be roughly describedas the theory of calculus. And maybe I should say it'sthe most common _subtle_ error.) _Many_ if not most of the_theorems_ of analysis come down to saying that twolimits _do_ in fact commute in certain circumstances.Except for that you have a good point.>Why the contradiction? > It's not a matter of prefering one limit to the other. It's simply a>> matter of understanding what the notation 0.999... means. Since>> 0.999..., by de'nition, means lim (m->+oo) (1-1/10^m), it follows that>> lim (n->+oo) 0.999...^n = lim (n->+oo) lim (m->+oo) (1-1/10^m)^n, which>> is the limit that appears in (1).So it's the de'nition causing the contradiction.You can say that _after_ you've given a proof that limitsalways commute. Good luck with that.Garry Denke, Geologist>Denoco Inc. of Texas************************David C. Ullrich =Garry,In mathematics we say that two numbers, a and b, differ if there is a numbere <> 0 such that a + e = b. I challenge you to 'nd such an e such that0.999... + e = 1. This will convince you that 0.999... = 1Another way to show this:1) x = 0.999...2) 10x = 9.999...Subtract 1) from 2):9x = 9x = 1The number of 9's in 1) has the same cardinality as the number of 9's in 2)since there is a 1 to 1 function from the # of 9's in 1) to the numbers of9's in 2).> lim(n->+oo) lim(m->+oo) (1-1/10^m)^n = 1 (1) Why do you prefer the above as opposed to the below? lim(n->+oo) (1-1/10^m)^n = 0 This holds for each m > 0, and therefore lim(m->+oo) lim(n->+oo) (1-1/10^m)^n = 0 (2) Comparing (1) and (2), we see that the limits do not commute. They should commute. Why the contradiction? It's not a matter of prefering one limit to the other. It's simply a> matter of understanding what the notation 0.999... means. Since> 0.999..., by de'nition, means lim (m->+oo) (1-1/10^m), it follows that> lim (n->+oo) 0.999...^n = lim (n->+oo) lim (m->+oo) (1-1/10^m)^n, which> is the limit that appears in (1). So it's the de'nition causing the contradiction.> Garry Denke, Geologist> Denoco Inc. of Texas => Garry,> In mathematics we say that two numbers, a and b, differ if there is a number> e <> 0 such that a + e = b. I challenge you to 'nd such an e such that> 0.999... + e = 1.0.999... + i = 1I like i better, so do most here, but if you like e, 'ne, we shall call it e. 0.999... + e = 1> This will convince you that 0.999... = 1You just said 0.999... + e = 1, now you want to change it?> Another way to show this:> 1) x = 0.999...> 2) 10x = 9.999...> Subtract 1) from 2):> 9x = 9No. 9 * 0.999... = 8.999... + e = 9 > x = 1No. x = 0.999... (see above).> The number of 9's in 1) has the same cardinality as the number of 9's in 2)> since there is a 1 to 1 function from the # of 9's in 1) to the numbers of> 9's in 2).publish the new de'nition for i with an e.i^2 + 1 = 0Anonymously:e^2 + 1 = 0 Garry Denke, GeologistDenoco Inc. of Texas => Garry,> In mathematics we say that two numbers, a and b, differ if there is a > number> e <> 0 such that a + e = b. I challenge you to 'nd such an e such that> 0.999... + e = 1.> 0.999... + i = 1> I like i better, so do most here, but if you like e, 'ne, we shall call it > e.That just con'rms for us how imaginary your world is! > 0.999... + e = 1> This will convince you that 0.999... = 1> You just said 0.999... + e = 1, now you want to change it?When someone asks whether unicorns exist, do you interpret him to assert that they do exist?> Another way to show this:> 1) x = 0.999...> 2) 10x = 9.999...> Subtract 1) from 2):> 9x = 9> No. 9 * 0.999... = 8.999... + e = 9 > x = 1> No. x = 0.999... (see above).Are you also saying that if x = 2/4 the we cannot have x = 2/4 or x = 3/6? Every number has lots of representations, so that a difference in representations does not imply a difference in value. You have to prove that x = 0.999... prohibits x = 1, and nothing you have said so far does so. Further, nothing that you CAN say will do so, since it is not true.> The number of 9's in 1) has the same cardinality as the number of 9's in 2)> since there is a 1 to 1 function from the # of 9's in 1) to the numbers of> 9's in 2).> publish the new de'nition for i with an e.> i^2 + 1 = 0> Anonymously:> > e^2 + 1 = 0So now Rockhead seems to be arguing that sqrt(-1) = -1 =*** post for FREE via your newsreader at post.newsfeed.com ***> Garry, In mathematics we say that two numbers, a and b, differ if there is anumber> e <> 0 such that a + e = b. I challenge you to 'nd such an e such that> 0.999... + e = 1. 0.999... + i = 1 I like i better, so do most here, but if you like e, 'ne, we shall callit e.I think you like .95i' because it stands for IMAGINARY -- the value of yourEPSILON. karl mhttp://www.newsfeed.com - The #1 Newsgroup Service in the World!-----== 100,000 Groups! - 19 Servers! - Unlimited Download! =----- = > They should commute. Why the contradiction? > Would you also expect the limits to commute in the following case? > lim(x->0) lim(y->0) arctan((x-y)/(x+y)) > lim(y->0) lim(x->0) arctan((x-y)/(x+y))Why so dif'cult? What about: lim(x->0) lim(y->0) (x-y)/(x+y) lim(y->0) lim(x->0) (x-y)/(x+y)-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ => Thus it is back to inventing a meaning for 0.999... which is> consistentThat has already been done, but you're not interested.Say Garry, have you ever noticed that.9 < .999...(.99)^2 < (.999...)^2(.999)^3 < (.999...)^3(.9999)^4 < (.999...)^4...And that the terms on the left tend to 1 rather rapidly? How can that be if the terms on the right tend to 0? => They should commute. Why the contradiction? > Would you also expect the limits to commute in the following case?> lim(x->0) lim(y->0) arctan((x-y)/(x+y))> lim(y->0) lim(x->0) arctan((x-y)/(x+y))Or even more trivially, lim(x->0) lim(y->0) (x-y)/(x+y) = 1lim(y->0) lim(x->0) (x-y)/(x+y) = -1But, perhaps Denke, being such a rockhound, has only rocks in his head? => Or even more trivially, > lim(x->0) lim(y->0) (x-y)/(x+y) = 1> lim(y->0) lim(x->0) (x-y)/(x+y) = -1Oops. My 'rst thought was arctan (y/x), and I didn't notice that my 45degree rotation made the arctan itself unnecessary.-- Daniel W. Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W =Can anybody teach me how to factorize integers in polynomial time?It's for a school project.Sonja ;) =>Can anybody teach me how to factorize integers in polynomial time?>It's for a school project.I don't think students at the NSA's school are allowed to ask for help. => Can anybody teach me how to factorize integers in polynomial time?> It's for a school project.Ei. T.Phil-- Unpatched IE vulnerability: Timed history injectionDescription: cross-domain scripting, cookie/data/identity theft, command executionExploit: http://www.safecenter.net/liudieyu/BackMyParent2/BackMyParent2 -MyPage.HTM => What the connection is and why it exists? ;)> Alexander,Without going into a lot of detail... the connection is based on aratio that is shown in the Fibanacci sequence 1,1,2,3,5,8,13,21...etc.each number is the sum of the two previous numbers. The ratio of thehigher to the adjacent lower number approaches 1.61803.... as thesequence continues. The proportions of the human 'gure follow thisratio (illustrated in Da Vinci's painting Vitruvian Man) includingfacial features and shape of the ear. The ratio appears in nature inthe spacing of petals of a rose and the placement of seeds in the headof a sun¤ower and in the shape of a pine cone, the shape of conicalshells. Mozart's concertos use the ratio in the chords and timing ofnotes as well as others. The ratio of female bees to male bees in ahive approaches 1.6. The ratio can also be illustrated by dividing aperfect square in half and connecting opposite corners by a line andusing that line as a radius to draw another circle and extending aside of the square to the point of intersection of the circle. Thecircle can be de'ned by PI which leads to PHI. Toroids are basedentirely on PI. Images based on toroids and fractals resemblegalaxies, frost on a window and many things in nature. This is myconnection and I am constantly 'nding new things that 't in withthis connection.> There's always work to be done.Its human nature to push the envelope. If it were possible to measurethe expansion of that envelope I'll bet it increases by a ratio of1.61803 per generation. => What the connection is and why it exists? ;) Alexander,> Without going into a lot of detail... the connection is based on a> ratio that is shown in the Fibanacci sequence 1,1,2,3,5,8,13,21...etc.> each number is the sum of the two previous numbers. The ratio of the> higher to the adjacent lower number approaches 1.61803.... as the> sequence continues. The proportions of the human 'gure follow this> ratio (illustrated in Da Vinci's painting Vitruvian Man) including> facial features and shape of the ear. The ratio appears in nature in> the spacing of petals of a rose and the placement of seeds in the head> of a sun¤ower and in the shape of a pine cone, the shape of conical> shells. Mozart's concertos use the ratio in the chords and timing of> notes as well as others. The ratio of female bees to male bees in a> hive approaches 1.6. The ratio can also be illustrated by dividing a> perfect square in half and connecting opposite corners by a line and> using that line as a radius to draw another circle and extending a> side of the square to the point of intersection of the circle. The> circle can be de'ned by PI which leads to PHI. Toroids are based> entirely on PI. Images based on toroids and fractals resemble> galaxies, frost on a window and many things in nature. This is my> connection and I am constantly 'nding new things that 't in with> this connection. There's always work to be done.> Its human nature to push the envelope. If it were possible to measure> the expansion of that envelope I'll bet it increases by a ratio of> 1.61803 per generation.Since e is related to pi, where does that 't in your scheme?DaveL 21:01:21 -0800, webmaster@elken.com (Chumpmeister)>>>How can I create a unique number from the two that will not be>>produced by another pair?>>> The classic method for this is through use of the chinese remainder>> theorem. [example deleted]Who uses the classic method,....?Today, SmartCards--http://www.smartcardalliance.org/industry_news/ industry_news_item.cfm?itemID=1048The 'rst example of its use to simplify arithmetic computing I wasaware of was in 1956 by an RCA ASW development unit. ........... and why?The power of the method, is that operations of addition andmultiplication give equivalent results in either representation of thenumber. Today, the killer application for that is in crypto. Itsmuch easier to multiply and add ten 15 digit numbers than to multiplyand add 150 digit numbers.John Bailey =The Author only mentions C(K) as the set of continuous functions onK. But from the context it is clear that he means C(K,R), since heuses inequalities on functionsvalues of arbitrary functions. Besidesthat, he hasn't written the word complex once in the 24 pages.Maybe I don't need a general de'nition. I use the number r=lim_n||T^n||^(1/n) a lot, but I realized that I don't need that it's thespectralradius of anything.If somenone knows the real de'nition for sure, then it would still benice to know, though.Mogens =>The Author only mentions C(K) as the set of continuous functions on>K. But from the context it is clear that he means C(K,R), since he>uses inequalities on functionsvalues of arbitrary functions. Besides>that, he hasn't written the word complex once in the 24 pages.>Maybe I don't need a general de'nition. I use the number r=lim_n>||T^n||^(1/n) a lot, but I realized that I don't need that it's the>spectralradius of anything.>If somenone knows the real de'nition for sure, then it would still be>nice to know, though.According to C.E. Rickart, General Theory of Banach Algebras, Van Nostrand 1960:De'nition (1.6.2): Let A be a real algebra and let x be any element of A. Then the SPECTRUM Sp_A(x) of x in A is de'ned to be equal to the spectrum of x as an element of the complexi'cation A_C of A....De'nition (1.6.5). The number nu_A(x), which is given algebraicallyby max |xi|, for xi in Sp_A(x), and topologically by lim ||x^n||^{1/n},is called the SPECTRAL RADIUS of x.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 =I've concluded that the problem I'm facing is that the human brainisn't built to handle Mathematics. Human beings have done well withsome simple mathematics and basic concepts, but it only goes so far,and what have, is beyond those limits.For a while I thought that at least my simplest results, like theprime counting function, or that funny little error with algebraicintegers might be generally accessible, but after years of arguing Irealize that your intellects are too limited to fully grasp my work.So I think I'll look for a way to further simplify, though I wonderwhat could be much simpler than what I've shown. Still, no matter howchild-like your minds are, no matter how simple your mentalprocessing, since you have language, at some point there's a chancethat I'll be able to 'nd something that your minds can handle.Well, at least, I hope I can.James Harris =certainly, this is possible as a program, butit doesn't seem to apply to the Ten-year Programtaht you've been bashing-out in the googolplex -- I mean,Decade Two!... I mean,as warm & fuzzy as these little watercooler chatsaround the qyoobicles of the goog may be,you-all never seem to get anywhere with them. (well,you don't have to take taht personally, butyou *are* the leader of these groups.) unless, of course, you have a new marketing strategy, orsome new math you have learned -- new for you, that is;as badly-educated as I an in the 'eld, I (and my brain) love it! my secret is that I forgave my 9th-grade teacher,mister Austin, for his lack of discernment-- as commonplace as it may have been,immediately following the Bourbaki New Math hoax --but I guess taht it's not *his* secret, now. http://larouchepub.com/other/2002/2917gauss.html > For a while I thought that at least my simplest results, like the> prime counting function, or that funny little error with algebraic> integers might be generally accessible, but after years of arguing I> realize that your intellects are too limited to fully grasp my work.--Give the Gift of Dick Cheeny -- out of of'ce, at last!http://www.benfranklinbooks.com/http://www.wlym.com/pages /music.htmlhttp://www.rand.org/publications/randreview/issues/ rr.12.00/http://members.tripod.com/~american_almanachttp:// =>I've concluded that the problem I'm facing is that the human brain>isn't built to handle Mathematics. Yeah, that must be it. It is quite remarkable, though - you'd thinkthat there would be _someone_ _somewhere_ on the planetwhose brain was able to handle mathematics the way yours can.I mean those guys who solve problems that nobody else couldsolve for hundreds of years, guys like that. But we've beenharassing, I mean contacting, famous mathematicians allover the world and it seems to be true, there's not one singleperson anywhere on the planet with a brain like yours.(Knock wood...)>Human beings have done well with>some simple mathematics and basic concepts, but it only goes so far,>and what have, is beyond those limits.For a while I thought that at least my simplest results, like the>prime counting function, or that funny little error with algebraic>integers might be generally accessible, but after years of arguing I>realize that your intellects are too limited to fully grasp my work.So I think I'll look for a way to further simplify, though I wonder>what could be much simpler than what I've shown. Still, no matter how>child-like your minds are, no matter how simple your mental>processing, since you have language, at some point there's a chance>that I'll be able to 'nd something that your minds can handle.I doubt it.>Well, at least, I hope I can.>James Harris************************David C. Ullrich => I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics. Human beings have done well with> some simple mathematics and basic concepts, but it only goes so far,> and what have, is beyond those limits. For a while I thought that at least my simplest results, like the> prime counting function, or that funny little error with algebraic> integers might be generally accessible, but after years of arguing I> realize that your intellects are too limited to fully grasp my work.ROFLMAO! And thus Harris is completely encapsulated in his own compacti'eddementia. => I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics. Human beings have done well with> some simple mathematics and basic concepts, but it only goes so far,> and what have, is beyond those limits.> For a while I thought that at least my simplest results, like the> prime counting function, or that funny little error with algebraic> integers might be generally accessible, but after years of arguing I> realize that your intellects are too limited to fully grasp my work.> So I think I'll look for a way to further simplify, though I wonder> what could be much simpler than what I've shown. Still, no matter how> child-like your minds are, no matter how simple your mental> processing, since you have language, at some point there's a chance> that I'll be able to 'nd something that your minds can handle.May I remind you that my prime counting function at www.cbau.freeserve.co.ukruns about one thousand times faster than your fastest result? Your mind is limited by ignorance, stupidity, and complete unwillingness to learn. You will stay small forever. => I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics. Human beings have done well with> some simple mathematics and basic concepts, but it only goes so far,> and what have, is beyond those limits. For a while I thought that at least my simplest results, like the> prime counting function, or that funny little error with algebraic> integers might be generally accessible, but after years of arguing I> realize that your intellects are too limited to fully grasp my work.You clearly do *not* regard yourself as human, and on that most readerswill agree. But this limitation you see in others is simply theirinability to accept contradictions, false arguments and yourOracle-of-Delphi approach to argument as a substitute for proof.> So I think I'll look for a way to further simplify, though I wonder> what could be much simpler than what I've shown. Still, no matter how> child-like your minds are, no matter how simple your mental> processing, since you have language, at some point there's a chance> that I'll be able to 'nd something that your minds can handle. Well, at least, I hope I can. James HarrisSpoken like a true megalomaniac.--There are two things you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com => I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics. Speak for yourself, James.> Human beings have done well with some simple mathematics and basic > concepts, but it only goes so far, and what have, is beyond those > limits.It would appear that sensible English is beyond some human brains, too.> For a while I thought that at least my simplest results, like the > prime counting function, or that funny little error with algebraic > integers might be generally accessible, but after years of arguing I > realize that your intellects are too limited to fully grasp my work.Since you seem convinced that humans will never get it, why don't you take it somewhere else?> So I think I'll look for a way to further simplify, though I wonder > what could be much simpler than what I've shown. Still, no matter > how child-like your minds are, no matter how simple your mental > processing, since you have language, at some point there's a chance > that I'll be able to 'nd something that your minds can handle.Please don't bother. We 'nd that the mathematicas that we can understand is quite good enough for us.By the way, since this is a mathematics news group, why do you never post any mathematics in it? (You don't really call that stuff that you do post mathematics, do you?) => I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics.Speak for yourself.-- Bob Day => I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics.You are just a walkin' talkin' poster child for that condition, James. =I feel better now because for a while I believed that some of you were*deliberately* lying about my work, but I'm starting to think thatit's far simpler: none of you can yet grasp it.So let's start again. I'll work harder to simplify, and try to 'ndsome pieces of my work which I think your minds should grasp.I'll start with my prime counting research:1. I use a partial difference equation, which no on else in recordedhistory has managed to do, to count prime numbers.2. A partial difference equation leads to a partial differentialequation.I'll test your ability to appreciate uniqueness, as I know that on onelevel most of you can, but on the other, your minds tell you that if Ihave a unique result in this area that mathematicians *should* care onsome level.Those of you who are mathematicians may simply say number theoristsshould care.When I go to number theorists, they run away.So I'll be going into more detail about what it means that I alonepresented a way to count prime numbers using a partial differenceequation.It may take some time to explain what a partial difference equationis!!!However, now that I understand more, I'm willing to take that time.Check out my blog archives:It's probably too complicated for most of you, but it might take me awhile to 'gure out a way to simplify further, so in the meantime, youcan at least look at it, as I analyze how human beings think ingreater detail.I have to break the code of how you work, and I have made a lot ofprogress over years of effort, and I feel like I am close to 'guringout all the inner details of human wiring.Time will tell.James Harris =as to why you belabor us with two rendidtionsof the same, nebulous Decoding the Human Wiring Program,it isn't any worse than my wasting time on it -- just me;after all, all of us were made in That Image. so,I bid you a fond adieu-doo! > I have to break the code of how you work, and I have made a lot of> progress over years of effort, and I feel like I am close to 'guring> out all the inner details of human wiring.--Give the Gift of Dick Cheeny -- out of of'ce, at last!http://www.benfranklinbooks.com/http://www.wlym.com/pages /music.htmlhttp://www.rand.org/publications/randreview/issues/ rr.12.00/http://members.tripod.com/~american_almanachttp:// =>I feel better now because for a while I believed that some of you were>*deliberately* lying about my work, but I'm starting to think that>it's far simpler: none of you can yet grasp it.So let's start again. I'll work harder to simplify, and try to 'nd>some pieces of my work which I think your minds should grasp.I'll start with my prime counting research:1. I use a partial difference equation, which no on else in recorded>history has managed to do, to count prime numbers.No one else in recorded history except Legendre and peoplewho've been developing re'nements of his method for thelast 200 years, you mean.>2. A partial difference equation leads to a partial differential>equation.And you've never given even a teensy bit of evidence that thesolution to the thing you call a partial differential equationhas anything whatever to do with counting primes.>I'll test your ability to appreciate uniqueness, as I know that on one>level most of you can, but on the other, your minds tell you that if I>have a unique result in this area that mathematicians *should* care on>some level.Here's a result which has never before been stated in any mathematicalpublication: 876462738768738741723847628374817236472349438572764398756298743 5 * 768574087627364823684745652038748762130748274985719388309874975 6 = 673626549788564774135834065285982366668305733679526191582679045 971306781111397144640567072877231983303649544477145721488373158 60.This is a 'rst - never before in recorded history has anyone even _conjectured_ that 876462738768738741723847628374817236472349438572764398756298743 5 * 768574087627364823684745652038748762130748274985719388309874975 6 = 673626549788564774135834065285982366668305733679526191582679045 971306781111397144640567072877231983303649544477145721488373158 60.Mathematicians should be interested.>Those of you who are mathematicians may simply say number theorists>should care.When I go to number theorists, they run away.So I'll be going into more detail about what it means that I alone>presented a way to count prime numbers using a partial difference>equation.It may take some time to explain what a partial difference equation>is!!!Tee-hee. After you do that, 'nd a book, _learn_ what a partialdifferential equation is, and then take the time to explain why thething you continue to call a partial differential equation is notone.>However, now that I understand more, I'm willing to take that time.Check out my blog archives:>It's probably too complicated for most of you, but it might take me a>while to 'gure out a way to simplify further, so in the meantime, you>can at least look at it, as I analyze how human beings think in>greater detail.I have to break the code of how you work, and I have made a lot of>progress over years of effort, and I feel like I am close to 'guring>out all the inner details of human wiring.Time will tell.>James Harris************************David C. Ullrich => I feel better now because for a while I believed that some of you were> *deliberately* lying about my work, but I'm starting to think that> it's far simpler: none of you can yet grasp it.All these dumb people have somehow made it to professor positions andsuch. So clearly they have been right about enough things before. Youhave done only two things in your life and for both you're getting norespect. Statistically that means you have far less -- one might say: no-- evidence for you being more clever than other mathematicians here.Maybe there is an even simpler explanation than the one you justsuggested.V.-- homepage: cs utk edu tilde lastname => I feel better now because for a while I believed that some of you were> *deliberately* lying about my work, but I'm starting to think that> it's far simpler: none of you can yet grasp it. So let's start again. I'll work harder to simplify, and try to 'nd> some pieces of my work which I think your minds should grasp. I'll start with my prime counting research: 1. I use a partial difference equation, which no on else in recorded> history has managed to do, to count prime numbers. 2. A partial difference equation leads to a partial differential> equation. I'll test your ability to appreciate uniqueness, as I know that on one> level most of you can, but on the other, your minds tell you that if I> have a unique result in this area that mathematicians *should* care on> some level. Those of you who are mathematicians may simply say number theorists> should care. When I go to number theorists, they run away. So I'll be going into more detail about what it means that I alone> presented a way to count prime numbers using a partial difference> equation. It may take some time to explain what a partial difference equation> is!!! However, now that I understand more, I'm willing to take that time. Check out my blog archives:> It's probably too complicated for most of you, but it might take me a> while to 'gure out a way to simplify further, so in the meantime, you> can at least look at it, as I analyze how human beings think in> greater detail. I have to break the code of how you work, and I have made a lot of> progress over years of effort, and I feel like I am close to 'guring> out all the inner details of human wiring. Time will tell.> James HarrisJames, you are a total and utter moron who is uneducable. You need to growup and act like a man instead of some child who throws tantrums everytime hedoesn't get his way. Either that or get a life.David Moran => I feel better now because for a while I believed that some of you were> *deliberately* lying about my work, but I'm starting to think that> it's far simpler: none of you can yet grasp it. So let's start again. I'll work harder to simplify, and try to 'nd> some pieces of my work which I think your minds should grasp. I'll start with my prime counting research: 1. I use a partial difference equation, which no on else in recorded> history has managed to do, to count prime numbers.And which you eloquently repudiated with your .95discovery' that the squareroot function is *inherently ambiguous* and, in fact, cannot be a functionat all since it always returns two values, neither one of which can beignored. Because your method uses the square root, and you are unable toeliminate this defect, your method fails by your own standard.> 2. A partial difference equation leads to a partial differential> equation.Your method for deriving a .95partial differential equation' from a partialdifference equation consists simply of replacing a unit difference with anon-unit, 'nite difference which you eventually reduce to zero. Sorry, nocigar on this approach. The resulting equation does *not* solve theoriginal problem, as you clearly understand. You have been challenged toproduce a single meaningful result from this amazing equation but haveproduced exactly ---- nothing!--There are two things you must never attempt to prove: the unprovable -- andthe obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com => I have to break the code of how you work, and I have made a lot of> progress over years of effort, and I feel like I am close to 'guring> out all the inner details of human wiring.Being so inhuman yourself, this task may be beyond your abilities. => I feel better now because for a while I believed that some of you were> *deliberately* lying about my work, but I'm starting to think that> it's far simpler: none of you can yet grasp it.> So let's start again. Like we used to say in high school: Let's not, and say we did. =>>The reason I need to break log(X+Y) up is because the problem is to>>minimize (X+Y-C)^2>>where C is known. So it is something like quadratic optimization, even>worse, X is de'ned to be a^x, Y is de'ned to be a^y, C is de'ned tobe>a^c, where c is known. Hence the true variables are x, y.>>The problem becomes>>minimize (a^x+a^y-a^c)^2>>where a is the common base for all these terms...>>If I take log a, it will make things a lot easier, that's why I want to>break>>log_a_(a^x+a^y-a^c)>>up...>Can you give me some more ideas?>>Huh? Clearly, the function is minimized for any X and Y such>>that X + Y = C. Are x and y constrained?-- >>Stephen J. Herschkorn and y are constrained... Sorry I did not make it clear... x and y are integer and are over> dertermined by more equations than unknown variables... hence there isno> exact solution...and there are a bunch of such variables X and Y's...hence> a bunch of x and y's ... 'nally what I want is to minimize the MSE fora> bunch of such X and Y's> i.e. minimize sum{ (a^x_i+a^y_i-a^c_i)^2 , i from 0 to N} ... What is it that you really want to solve> (objective, constraints, ranges of variables)?> Revealing only small bits at a time will not give> you a satisfying answer. You are well advised to learn Arnold Neumaier>Man, I had throw out the big question(the original one) one month ago... butwhen problem is too big and too dif'cult... I got even no response atall... People are busy, it is my responsiblity to divide and conquare if Iwant it to be solvable at all...Also gathering enough information anyway, it is another language! I have learned so manylanguages(I mean C, matlab, pascal... basic, etc... the =>Can 1/(x+y) be de'ned in terms of 1/x and 1/y ? > Of course. Call these a,b, and c respectively. Then a = 1/(1/b+1/c)=b c / (b+c). Naturally, some division is required, e.g. 1/5 is not in the ring> Z[1/2,1/3].But expression of division function is demanded, itself should not beused again in function de'nition. Likewise, using essentially samereciprocal functions log and exp as elsewhere stated is not leading tonew functional relationship that the original poster wants, so itappears.>For Sin (or any trig function),Exp etc.,there is no such problem.> Really? If cos(x)=0 and cos(y)=0, what is cos(x+y)?cos(x+y)= cos(x) cos(y) - Sqrt [ ( 1-cos(x)^2) (1-cos(y)^2) ] leads to-1 and/or +1 ; ( x and y are odd multiples and x+y is even multiple ofPi/2 ). = I met the following problem i one book,which can be solved either by>measure theory or by Lebesque Dominated Convergence theorem,but the>author says it is too dif'cult to handle without these means .I think>I need help on it.Let fn:[0,1]--->[0,1] are continuous functions and fn--->0 for each x>in [0,1].Then Int(fn(x)dx,0,1)--->0!>First note Int(f_n(x)dx,0,1)=f_n(x_n) for some x_n in [0,1]. I think the claimis that {f_n(x_n)} has a unique limit point, L, and L=0. This does not seemhard to prove. Perhaps I am wrong...rich => I met the following problem i one book,which can be solved either by>measure theory or by Lebesque Dominated Convergence theorem,but the>author says it is too dif'cult to handle without these means .I think>I need help on it.Let fn:[0,1]--->[0,1] are continuous functions and fn--->0 for each x>in [0,1].Then Int(fn(x)dx,0,1)--->0! > First note Int(f_n(x)dx,0,1)=f_n(x_n) for some x_n in [0,1]. I think the> claim> is that {f_n(x_n)} has a unique limit point, L, and L=0. This does not seem> hard to prove. Perhaps I am wrong...Yes. You're wrong.It's a nice idea, but it won't work, because in order to conclude thatf_n(x_n) --> 0, you need the f_n to converge to 0 uniformly. (In fact,this is more or less the de'nition of uniform convergence to 0. Anyway, it's equivalent.)Others have given examples. Just consider a function f_n on [0,1]whose graph consists of the two line segments joining (0,0) to (1/n,1)to (1,0). Now, you say that int f_n(x) dx = f_n(x_n) for some x_n in[0,1]. But why couldn't that x_n be 1/n, where f_n(x_n) = 1? Itisn't, but how do you rule that out in general?--Ron Bruck =>> I met the following problem i one book,which can be solved either by>>measure theory or by Lebesque Dominated Convergence theorem,but the>>author says it is too dif'cult to handle without these means .I think>>I need help on it.Let fn:[0,1]--->[0,1] are continuous functions and fn--->0 for each x>>in [0,1].Then Int(fn(x)dx,0,1)--->0! >> First note Int(f_n(x)dx,0,1)=f_n(x_n) for some x_n in [0,1]. I think the>> claim>> is that {f_n(x_n)} has a unique limit point, L, and L=0. This does not>seem>> hard to prove. Perhaps I am wrong...Yes. You're wrong.It's a nice idea, but it won't work, because in order to conclude that>f_n(x_n) --> 0, you need the f_n to converge to 0 uniformly. (In fact,>this is more or less the de'nition of uniform convergence to 0. >Anyway, it's equivalent.)Others have given examples. Just consider a function f_n on [0,1]>whose graph consists of the two line segments joining (0,0) to (1/n,1)>to (1,0). Now, you say that int f_n(x) dx = f_n(x_n) for some x_n in>[0,1]. But why couldn't that x_n be 1/n, where f_n(x_n) = 1? Because f_n(0)-->0? Again, I don't really know.rich =>>> I met the following problem i one book,which can be solved either by>measure theory or by Lebesque Dominated Convergence theorem,but the>author says it is too dif'cult to handle without these means .I think>I need help on it.>>Let fn:[0,1]--->[0,1] are continuous functions and fn--->0 for each x>in [0,1].Then Int(fn(x)dx,0,1)--->0!>> First note Int(f_n(x)dx,0,1)=f_n(x_n) for some x_n in [0,1]. I think the> claim> is that {f_n(x_n)} has a unique limit point, L, and L=0. This does not>>seem> hard to prove. Perhaps I am wrong...Yes. You're wrong.It's a nice idea, but it won't work, because in order to conclude that>>f_n(x_n) --> 0, you need the f_n to converge to 0 uniformly. (In fact,>>this is more or less the de'nition of uniform convergence to 0. >>Anyway, it's equivalent.)Others have given examples. Just consider a function f_n on [0,1]>>whose graph consists of the two line segments joining (0,0) to (1/n,1)>>to (1,0). Now, you say that int f_n(x) dx = f_n(x_n) for some x_n in>>[0,1]. But why couldn't that x_n be 1/n, where f_n(x_n) = 1? Because f_n(0)-->0? Again, I don't really know.Well, I'm missing the point to Ron's example above, because thosef_n do not tend to 0. But regardless, in case you're still not sure,his Yes. You're wrong and his explanation why were exactlyright.(Or who knows, maybe you _can_ come up with a simple proofthat f_n(x_n) -> 0. That _is_ true, after all, and it happensthat people 'nd simple proofs of things that previously seemedhard. But does not seem hard to prove seemsto indicate you don't know exactly how to prove it, and itseems hard to prove to the rest of us, precisely because wedo not know that f_n -> 0 uniformly. Do you have an actualplan in mind for proving it?)>rich************************David C. Ullrich =I have recently encountered the problem of creating the standard notationfor transversal intersection (unicode U02ADB) which looks like a cap with amid through it. Is there any token for this symbol in TeX? If not, howwould I glue together a cap and a mid so as to reproduce it.Tyler SmithUIUC => I have recently encountered the problem of creating the standard notation> for transversal intersection (unicode U02ADB) which looks like a cap with a> mid through it. Is there any token for this symbol in TeX? If not, how> would I glue together a cap and a mid so as to reproduce it.> Tyler Smith> =tgsmith scribbled the following:> I have recently encountered the problem of creating the standard notation> for transversal intersection (unicode U02ADB) which looks like a cap with a> mid through it. Is there any token for this symbol in TeX? If not, how> would I glue together a cap and a mid so as to reproduce it.There does not seem to be to 'nd some way to glue cap and mid together. SorryI couldn't be of more help.-- /-- Joona Palaste (palaste@cc.helsinki.') ------------- Finland ---------- http://www.helsinki.'/~palaste --------------------- rules! --------/How can we possibly use sex to get what we want? Sex IS what we want. - Dr. Frasier Crane =Before Genzten proved Con(PA), Wilhelm Ackermann proved it. Is a sketchof his proof readily available?Was Gentzen's proof better in some sense?-- G.C. =I'm reading Browder, Mathematical Analysis (Springer). In page 242,Lemma 10.47, where he is trying to prove the change of variableformula for the lebesgue integral, we 'nd the following assertion:Let phi : Usubset R^n -> R^n be C^1, Ksubset U compact. If T_y (x) =phi(y)+Dphi(y)(x-y) is the af'ne map approximating phi near y, for yin K, we have, given e>0, |phi(x)-T_y(x)|< e |x-y| if 0 < |x-y| Let phi : Usubset R^n -> R^n be C^1, Ksubset U compact. If T_y (x) => phi(y)+Dphi(y)(x-y) is the af'ne map approximating phi near y, for y> in K, we have, given e>0, |phi(x)-T_y(x)|< e |x-y| if 0 < |x-y| d(e,y). And he asserts: since K is compact and Dphi is continuous, we> can choose d=d(e) to be independent of yin K.that K_r = {x in R^n : d(x, K) <= r} is a compact subset of U. If x and y are in K and |x-y| < r, then the line segment joining x and y lies in K_r. Now apply the mean value theorem on this line segment and use the fact that Df is continuous, hence uniformly continuous, on K_r. => http://graphics.stanford.edu/~afra/goodies/markov.pdf> This seems very interesting! => Is there any list of known undecidable problems, like those for> NP-Complete problems (Garey-Johnson, etc)?> The only places I know of with lists are:> 1) For formal languages/automata, there's a table in Hopcroft & Ullman.This is a bit outdated, though, as some of th eproblems that were openthen have been solved (such as equivalence of deterministic contextfree grammars). Torben =First of all thank you all, I am beginning to understand this issuebetter.> 4. It doesn't halt and we can't prove that it doesn't.But if we can't prove that it doesn't halt, how do we know that itdoesn't halt ?It seems that knowing that it doesn't halt must be a series of logicalsteps that shows you that it doesn't halt. This reasoning is theproof that it doesn't halt. So how it is that it doesn't halt andyou can't prove if it doesn't ?Can you give me an example of something speci'c that doesn't halt,but you can't prove it ? Remeber that I had asked my questionregarding a speci'c machine on a speci'c input, and my argument withmy teacher was about him saying (and you have also agreed), that youdon't know if the machine halts,but you have an algorithm (the bi-valent one).Alex. => First of all thank you all, I am beginning to understand this issue> better.>> 4. It doesn't halt and we can't prove that it doesn't. But if we can't prove that it doesn't halt, how do we know that it> doesn't halt ?We don't, but it is a fact. Two simple cases : 1) a TM (with no input)programed to halt if it 'nd a odd number not sum of two primes. We are allwilling to bet it doesn't halt, but we cannot prove it.2) a TM generating all proofs in ZFC (by Godel numbering in ascendingorder) till it 'nds a proof of 0=1. Again, we are all willing to bet itdoesn't halt, and G.9adel did show that we will never be able to prove it.> It seems that knowing that it doesn't halt must be a series of logical> steps that shows you that it doesn't halt. This reasoning is the> proof that it doesn't halt. So how it is that it doesn't halt and> you can't prove if it doesn't ?See above Can you give me an example of something speci'c that doesn't halt,> but you can't prove it ? Remeber that I had asked my question> regarding a speci'c machine on a speci'c input, and my argument with> my teacher was about him saying (and you have also agreed), that you> don't know if the machine halts,> but you have an algorithm (the bi-valent one). Alex. => Dave Rusin scribbled the following:>>The math course I'm taking right now recommends buying the TI-83 ($100),>>but for $40 more I could get the TI-89 ($140). I'm assuming that I will>>be using the calculator for future higher-level math courses.> Not necessarily a good assumption. Typical high-school and some university> math classes in the US involve lots of symbolic computation, graphing,> and number crunching, but only up through roughly the end of calculus;> if you're already near that stage, you may never use a calculator again> in your higher-level math courses. If you can 'nd a calculator which,> can, for example, prove the set of rational numbers to be countable,> do let me know; it could be very useful.> Don't hold your breath. Last I checked, none of the calculators on the> market supported the concept of in'nite sets at all, countable or> uncountable.The proof will be 'nite even if the sets aren't.-- G.C. =Some new interesting formula's have been discovered that produceexponents from A^2 through A^7The info is athttp://www.sollog.com/Discoveries-exponentequivalency.shtml_ __________The Exponent Equivalency Factoring FormulasBy Sollog are the Exponent Equivalency Factor Formulasfor exponents N^2 through N^7.When N is = or greater than 1N^2 = 1 + (N-1) * (N+1)Example: 2^2 = 1 + (1) * (3)4 = 1 + 3N^3 = N + (N-1)*(N)*(N+1)Example: 2^3 = 2 + (1)*(2)*(3)8 = 2 + 6N^4 = N^2 + (N^2)*(N-1)*(N+1)Example: 2^4 = 4 + (4)*(1)*(3)16 = 4 + 12N^5 = N + (N^2+1)*(N-1)*(N)*(N+1)Example: 2^5 = 2 + (5)*(1)*(2)*(3)32 = 2 + 30N^6 = N^2 + (N^3+N)*(N-1)*(N)*(N+1)Example: 2^6 = 4 + (10)*(1)*(2)*(3)64 = 4 + 60N^7 = N^3 + (N^4+N^2)*(N-1)*(N)*(N+1)Example: 2^7 = 8 + (20)*(1)*(2)*(3)128 = 8 + 120http://www.sollog.com/exponentfactors.pdfShalom,Sollog =I know there are test to prove linearity for data sets withrepetitionsfor example: conc g/ml | response day 1 | response day 2 0.5 | 0.6989 | 0.6888 0.9 | 0.8888 | 0.8799 1.5 | 1.2510 | 1.2493for this type of sets, I'm able to prove linearity (the example is nota real data set, I don't think it's lineair).But what if I have only one day, and no repetitions?I can prove r2, but how do I prove linearity?Bart =>Could anyone give a (Hamel) basis of the reals over the rationals?> No. If the Axiom of Determinacy is consistent, and I> believe even otherwise, there are models where there> are no Hamel bases for the reals over the rationals.> If the reals can be well-ordered, a Hamel basis can> be the set of all reals which are not 'nite linear> combinations of previous reals (in the well-ordering).>How can these be classi'ed?> This question is unclear.Isn't it true that the statement every vector space has a basis isequivalent to the axiom of choice? =>Could anyone give a (Hamel) basis of the reals over the rationals?> > No. If the Axiom of Determinacy is consistent, and I> believe even otherwise, there are models where there> are no Hamel bases for the reals over the rationals.> If the reals can be well-ordered, a Hamel basis can> be the set of all reals which are not 'nite linear> combinations of previous reals (in the well-ordering).>How can these be classi'ed?> This question is unclear.Could some Borel set be a Hamel basis for the reals over Q ?David Bernier =>Could some Borel set be a Hamel basis for the reals over Q ?Re: where's Tex? + a problem of topology [Hamel basis]Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 =>>Could anyone give a (Hamel) basis of the reals over the rationals?>>> No. If the Axiom of Determinacy is consistent, and I>> believe even otherwise, there are models where there>> are no Hamel bases for the reals over the rationals.>>> If the reals can be well-ordered, a Hamel basis can>> be the set of all reals which are not 'nite linear>> combinations of previous reals (in the well-ordering).>>>How can these be classi'ed?>>> This question is unclear.Could some Borel set be a Hamel basis for the reals over Q ?Surely not (where of course the word surely indicatesI don't know for sure...) Surely one can prove this bysomething analogous to the standard proof that thereexists a non-Lebesgue-measurable set.>David Bernier************************David C. Ullrich =>> Could anyone give a (Hamel) basis of the reals over the rationals?>Such a basis is necessarily uncountable. Since the set if real numbers>that can be de'ned by 'nite symbol strings in any given language is>necessarily countable, any Hamel basis must contain numbers that are>almost all indescribable in any given language.But a set can be describable even if almost all of its members are not.After all, we can describe some uncountable sets (e.g. intervals).So this does not say that a Hamel basis is indescribable.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CBQqf10308; => We also have, for all rings of integers of number> 'elds, that they are a UFD if and only if they are a PID.>>What are the essential ideas to prove it ?The key result is unique factorization of ideals into prime ideals,>and that the ideals form a group under ideal multiplication, via the>equivalence relation. If you have that, then this is easy:PID->UFD holds in general, so there is no problem.To prove that UFD->PID, let I be an arbitrary nonzero ideal. Since the>ideal classes form a group, let J be a nonzero ideal such that IJ =>(a) is principal. Note that IJ is contained in I, so that I divides>the ideal (a). Since we are assuming that the ring is a UFD, we can>factor a into a product of irreducibles, > a = p_1^{a_1}*...*p_r^{a_r}. It is easy to verify that each p_i is also a prime element, and that>(p_i) is a prime ideal, so that(a) = (p_1)^{a_1} *...*(p_r)^{a_r}is a factorization of (a) into primes. Since I divides (a), by unique>factorizatio of ideals we must have that the prime factorization of I>is of the formI = (p_1)^{b_1} *...* (p_r)^{b_r}with 0<= b_i <= a_i, for i=1,...,r; therefore,>I=(p_1^{b_1}*...*p_r^{b_r}) is principal. QED >It's not denial. I'm just very selective about> what I accept as reality.> --- Calvin (Calvin and Hobbes)> =Arturo approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CBQrb10354; =>If you want us to comment on a proof, then >you will have to show us a proof. I saw a lot >of words, a few terms that might have been >intended as mathematics I have written the proof in this way just for the people who dont want understand it, Are you one of them?>but were not de'ned (for example measurable >by linear units). That belongs to the analogy. Anyway, if you think that the air can be measured with a metre(for instance), then you have a problem. >There are quite a lot of non-sequiturs (for >example, you de'ne countable set as meaning >that there exist a one to one correspondence >between the set and N (which is correct) and >then assert that that criterion doesnt work >for the real numbers since it is said in Point 2 >that no such correspondence. That criterion works >quite nicely, thank you: it says that the set of >real numbers is NOT countable. What did you think >a criterion was for?)Obviously you have not read the proof. For a better understanding my advice is:1) Read and understand the proof of Point 2. There the conclusion is than the naturals fail counting R due to its nature (i.e, it is not because there be more reals than naturals).2) Go to the analogy and grasp the equivalent concept, i.e., the linear units failmeasuring A (the air).3) Finally, go back to the proof, and read it carefully. If there is something that you don't understand, seek the equivalent concept in the analogy.>While Point 2, that there is no one to one >correspondence between the set of real numbers >and the set of positive integers is true, your >purported proof, that (to summarize) the set >of real numbers contains number with an in'nite >number of nozero digits while no integer does, >therefore, there can be no such correspondence, >is completely invalid.>The set of all rational numbers also includes >numbers with an in'nite number of non-zero >digits (1/3 for example) but there IS such a >correspondence- the set of rational numbers IS >countable.This must be refuted in other way, because the proof with Q has no signi'cance in this case. When Cantor counts the rationals he is counting the process (n/n) with different arguments (values of n and m). He doesn't count the results of this process i.e. 0.33333 in your example. That only means that naturals succeed in counting processes (or the natural arguments used) but fails counting the results of these processes (due to the in'nity of the digits). If you can prove that each real can be generated by a process using naturals as arguments (as n/m, sqrt(n), cbrt(n), 4-rt(n), e, pi, etcetera,) then the reals would be countable (with naturals) by means of its respective processes, but uncountable by its results, as currently are and always will be using just naturals.>Finally, I can't see what this has to do with Cantor's >proof. You don't actually say anything about >Cantor's proof.Sorry, I should have said the conclusion of Cantors proof and others proofs>By the way, you end with >>Conclusion:>>The trans'nite construction N <->R is not possible >>if we do not admit naturals with an in'nite number >>of nonzero digits. If some day we admit that, then >>the reals would be countable.>Yes, and if we de'ne countable to mean simply >is in'nite then the reals would be countable- and >if we de'ned blue to mean is in'nite then the >reals would be blue- but we wouldn't have learned >anything important in either case.Out of context, and meaningless in this case>It's not a matter of admitting naturals with an >in'nite number of nonzero digits. The naturals, as >de'ned in our basic arithmetic, do not have such. >If you want to try to change elementary school >arithmetic just to make the reals countable (and so >make mathematics trivial and unable to do most of >the things we rely on it for) I don't think you >will be very sucessful.This is the only point I agree with you. It would be an enormous mistake to change the nature of the naturals, just to count the reals. However, it is not necessary. R would be countable if we de'ne and use another kind of naturals (I have already done it, and I think it works), although Im not interested at all in that solution because it is completely arti'cial, and the only conclusion I have got at the moment from it is that R and N have a different nature, something that we already know. In other words it has a little sense to create a new kind of numbers (trans'nite naturals) just to approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CBQrP10361; => What (if at all) the existence of higher ordinal in'nities (such as>> strongly inaccessible cardinals and higher) imply in terms of CH or>> GCH ?The general rule so far is that adding large cardinal axioms does not>decide CH one way or the other.Of course, some as-yet-unconsidered large approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CBR3810419; =>Going from a solid to a liquid is .95melting'; from a liquid to a solid is .95freezing'; from a liquid to a gas is .95evaporating'; from a gas to a liquid is .95condensing'; from a solid to a gas is .95subliming'.> What does one call the transition from a gas to a solid? In chemistry the word approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CE8Kr21708; =The proof (plus analogy) in three acts has been written in this way just for those who do not want to understand the proof. As it is not in a correct and formal mathematical language, I understand that you don't want to refute it.However, the proof of Point 2 ful'ls all the requirements of a mathematical proof. Therefore, as I suppose that most of you are professional mathematicians, instead of talking and talking, the only thing you have to do is to refute this proof. This is called the mathematical method.As the conclusion of this proof clearly states that the natural numbers fail in counting the reals because they cannot undertake the in'nity of R, it is obvious that the naturals cannot be used to count the reals, and therefore the Cantor's criterion (that you call de'nition) cant be used with R. In short, refute the proof of Point 2 and every thing will be as it was before.PD: If someone else wants to use the criterion that Q is countable, please show me a proof using the decimal expansion of n/m. I'll try to 'nd out why naturals succeed in counting rationals in this way, and however they fail doing the same with the reals.Nicolas de la Foz => As the conclusion of this proof clearly states that the natural > numbers fail in counting the reals because they cannot undertake the > in'nity of R, it is obvious that the naturals cannot be used to > count the reals, and therefore the Cantor's criterion (that you call > de'nition) cant be used with R. De'nition. A triangle is a three-sided polygon.Let's call the above Euclid's criterion.Question: Is a square a triangle? Let's apply your reasoning to thequestion. A square has four sides, and therefore the sides of a trianglefail in counting the sides of a square. It is obvious that thetriangle-sides cannot be used to count the square-sides, and thereforethe Euclid's criterion (that you call de'nition) can't be used withsquares.Perhaps it is undecidable whether a square is a triangle or not.Dave Seaman-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. I need some help with my history essay.> Does anyone know what Archimedes corollary is that was included in his > Measurement of a Circle? Or have any ideas how i could 'nd it, as i'm having > dif'cultity 'nding it in the books i have got. You'd need to try a library, probably a university library, for The Works of Archimedes translated by Sir Thomas Heath. Measurement of a Circle is on pp.91-98. However, there's no corollary in it, so I can't see what it is that you've been asked to do. Ken support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CE8Ma21766; any function that relates log(x+y) with log(x) and log(y)...In an optimization problem, I basically need to seek the minization of the>function of the form log(x+y)... but in order for further mathematical>manipulation, I need to break log(x+y) up into log(x) and log(y),>right?>-Walala> No, there approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CKkUl21736; nico80@jazzfree.com (Nicolas>I would like to know if this is a mathematical proof of the Cantors >>goof, or conversely it is just another mathematical goof about the >>Cantors proof.That's a lie. It's clear from your behavior in other threads that>you have no interest in having your mathematical errors corrected.>Please, be a little more speci'c and analyze the things in a whole. I only have been in a thread, and there I just met people who do not want to understand the proof (you among them). I said you don't want because the proof is so easy that I can believe that you don't understand it.Anyway, if you are mathematician, or you consider yourself that, the only thing you have to do is to refute the proof of Point 2. If you do that, you'll be right.Nicolas de la (from approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CL67523587; => Could a proof of the Goldbach Conjecture be worth of a Fields prize? >>> Don't you know the regulations for your own prize?>Fields Medals are restricted to those under age 40.In case someone proved it in the 'rst forty years support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CKkU621728; =>it's called Donald Duck in Mathemagic Land we watched it in math...its approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CL67V23591; =I have built an argument for a proof of Goldbach Conjecture. Itconsists of 8 pages and I would be happy to hear your comments.Since I am almost convinced of my argument, any mistake only youcould 'nd will be extremely helpful for my pursuit of mathematics.The paper could be found inhttp://www.geocities.com/erdosfan/solution.htmlerdos support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0D0PxR06638; =>What is this theorem called?>>If f: R->R is a continuous function, and f(a)<0 and f(b)>0, then there>>exists a c between a and b so that f(c)=0.Usually that theorem (but also sometimes a trivially more>general theorem) is called the Intermediate Value Theorem>in (American) English-language textbooks.>And in Hebrew ones too, if I remember correctly :)>Lee support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0CKkU621722; => My latest brainchild, as minor as most of the others, is this>> mini-theorem. Let s_0 and s_1 be nonnegative integers, not both zero.>> De'ne inductively s_{n+2} = | s_{n+1} - s_n |>> for n >= 0. Show that the sequence (s_n) eventually looks like>> 0, x, x, 0, x, x, 0, x, x, ...>> and x = gcd(s_0,s_1).>>> For example, if s_0 = 69 and s_1 = 39, the sequence runs>> 69, 39, 30, 9, 21, 12, 9, 3, 6, 3, 3, 0, 3, 3, 0, ...I think this is fairly easy to verify,>as the sequence is bounded and so must become periodic.>In fact s_n >= 0 for all n, ands_n <= max(s_{n-1},s_{n-2}).It follows thatmax(s_{n+1},s_n} <= max(s_{n-1},s_{n-2})with strict inequality unless some s_n = 0.>Hence some s_n = 0, and your conclusion follows.>-- >Timothy Murphy >tel: +353-86-2336090, +353-1-2842366 s_{n+2} = |s_{n+1} - s_{n}| = ||s_{n} - s_{n-1}| - s_{n}|and once s_{n} > 0 but s_{n-1} = 0 so it will be s_{n+2} = s_{n-1}=0then once s_{n-1} = s_{n-2} - s_{n-3} = 0 so s_{n-2} = s_{n-3} = xand we have sequence: s_{n-1}; s_{n-2}; s{n-3} 0 ; x ; xnow moving up s_{n} = |s_{n-1} - s_{n-2}| = |0 - x| = x s_{n+1} = | s_{n} - s_{n-1}| = |x - 0| = x s_{n+2} = | s_{n-1) - s_{n-2}| = |x - x| = 0 and so on sequence up consists from 0;x;x;0;x;x;... but moving down we'll collect some bigger valuesand it looks as some product of sequence r_{k} and x ,where r_{k} = 1;1;2;1;3;4;7;3;10;13;23;36;59;108... and some order like r_{1} = s_{n-2} / x r_{2} = s_{n-3} / x now if I am not mistaken, somebody can help for to'nd some more elegant development of r_{k} and as well to proof x = support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0D245j14040; =Prove that ((a % m) + (b % m))%m = (a + b)%mwhere a,b,m are support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DDHwo26616; =>After a bit of experimenting numerically, I've noticed that the percentage>of numbers in an interval [1,k] that is divisible by an n-th power larger>then 1 tends to a constant as k -> oo.The sequences below list the number of mentioned numbers for>k=10^1,10^2,10^3,10^4,10^5.n=2: 3,39,392,3917,39206>n=3: 1,15,167,1681,16810>n=4: 0,7,75,760,7605>n=5: 0,3,35,355,3560What are the theoretical values of the corresponding percentages?>[d(2) = +/- 39%, d(3) = +/- 17%, d(4) = +/- 7.6%]d(n) = 1 - 1/zeta(n)http://jove.prohosting.com/~skripty/page_807. support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DDHxn26646; =Timothy Murphy asks:>Slightly OT,>>but does anyone publish maths in Hebrew nowadays?>>Are school maths textbooks normally in Hebrew?I've seen a calculus book in Hebrew. I own a book by Azriel Levy, in Hebrew,>consisting of notes for a course in mathematical logic. Such notes are>available from the campus bookstore, Academon. I've tried for a long time>to get information about the math books they have in Hebrew. I'm told there>are Hebrew books on differential geometry, for example. According to Academon,>when a friend of mine called them on the phone to ask for a catalogue, all>the information about their books is available online through their website.>When I tried several months ago to look at it, I found it was under>construction. My own browser can't navigate the site, since I never 'gured>out how to install the necessary Hebrew fonts, but the library computers haveI got the Hebrew word for nilpotent from a Ph.D. thesis written in Hebrew>at Hebrew University in Jerusalem.Allan Adler>ara@zurich.ai.mit.edu>I'll start with the basics of the situation : Hebrew was fairly recently revived as a spoken everyday language. The key man in this context is Eliezer Ben-Yehuda who observed that in the condition it was at the time, it was totally irrelevant to many aspects of modern life, including science. It was virtually impossible back then to teach science in Hebrew, because every other word was Loazit (of foreign origin), and scientists of the time who lived here at the early stages of the 20th century, refused to use it, if I remember correctly.Eliezer used the basic roots and structure of the Hebrew language, and quite brilliantly and inspiringly built a whole new dictionary of Hebrew words for a wide range of modern concepts. He was successful in getting the public to use most of them, and was inherited by the Hebrew Language Academy which produces words on demand, or if they see a need for it. Some of it works, some of it doesn't. For example, the word for telephone didn't catch, so in everyday spoken Hebrew, a telephone is a telephone though written in Hebrew letters, and to call someone is letalphen, based on the derived foreign four letter root (Hebrew roots have three letters, some say origin is two letters, but four letter roots are now also used).So that's the rule with many scienti'c concepts. Either the Hebrew Academy provided a Hebrew term, by request or not, or usually the term has a Hebrew parallel (such as continuous - ratzif, or measure - mida) that caught (this is actually problematic because a few concepts get to have more than one parallel that caught, and I recently heard people saying we need a good way to standardize the practice so a scientist would know what to do and what not to do in such a situation) , or it simply remains as it is and phonetically translated into Hebrew letters - such as nilpotent coming from nilpotency.This said one can imagine how math classes in an Israeli University looks like : The language is Hebrew, which is one of the three of'cial languages of the state of Israel (the others are English and Arabic for historical reasons), and the dominant everyday language. (an interesting anecdote - I remember hearing about a case in the place I teach in which Russian repatriates asked to teach the class in Russian, and was asked for my opinion on this. As Hebrew is very strong in Israel, I thought they should). Advanced math books, however, are usually in English, because our Math elite are usually busy doing research rather than translating or writing textbooks for students, and usually recommend books they see 't regardless of their language. However, all basic University math is available with quite a few Original Hebrew texts on the subject, whether it's Algebra, Calculus, Set Theory, or whatever. Also, some of the usual basic training books are translated like Scaum's book!Hope this helps,The Levantine.>*************************************************** *************************>* *>* Intelligence Lab. My actions and comments do not re¤ect *>* in any way on MIT. Moreover, I am nowhere near the Boston *>* metropolitan area. *>* *>************************************************************ support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DDHwG26621; =I am going crazy trying to 'nd a formula/equation for the equivalent resistance asross diagonally opposite ends of a resistive mesh. Can someone help.Xo--+--Rh--+--Rh--+--Rh--(n columns)--Rh--+--Rh--+ | | | | | | | Rv Rv Rv | | Rv Rv | | | | | | | +--Rh--+--Rh--+--Rh--(n columns)--Rh--+--Rh--+ | | | | | | | Rv Rv Rv | | Rv Rv | | | | | | | . . . .(m rows) .--Rh--+--Rh--+ | | | | | | | Rv Rv Rv | | Rv Rv | | | | | | | +--Rh--+--Rh--+--Rh--(n columns)--Rh--+--Rh--+ | | | | | | | Rv Rv Rv | | Rv Rv | | | | | | | +--Rh--+--Rh--+--Rh--(n columns)--Rh--+--Rh--+--o Y Imagine a rectangular mesh with resistors at every small segment. The horizontal segment resistances are Rh and vertical ones are Rv. Need to get a formula to calculate the effective resistance between X and Y. support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DDI1126694; =I'll start with the basics of the situation : Hebrew was fairly recently revived as a spoken everyday language. The key man in this context is Eliezer Ben-Yehuda who observed that in the condition it was at the time, it was totally irrelevant to many aspects of modern life, including science. It was virtually impossible back then to teach science in Hebrew, because every other word was Loazit (of foreign origin), and scientists of the time who lived here at the early stages of the 20th century, refused to use it, if I remember correctly.Eliezer used the basic roots and structure of the Hebrew language, and quite brilliantly and inspiringly built a whole new dictionary of Hebrew words for a wide range of modern concepts. He was successful in getting the public to use most of them, and was inherited by the Hebrew Language Academy which produces words on demand, or if they see a need for it. Some of it works, some of it doesn't. For example, the word for telephone didn't catch, so in everyday spoken Hebrew, a telephone is a telephone though written in Hebrew letters, and to call someone is letalphen, based on the derived foreign four letter root (Hebrew roots have three letters, some say origin is two letters, but four letter roots are now also used).So that's the rule with many scienti'c concepts. Either the Hebrew Academy provided a Hebrew term, by request or not, or usually the term has a Hebrew parallel (such as continuous - ratzif, or measure - mida) that caught (this is actually problematic because a few concepts get to have more than one parallel that caught, and I recently heard people saying we need a good way to standardize the practice so a scientist would know what to do and what not to do in such a situation) , or it simply remains as it is and phonetically translated into Hebrew letters - such as nilpotent coming from nilpotency.This said one can imagine how math classes in an Israeli University looks like : The language is Hebrew, which is one of the three of'cial languages of the state of Israel (the others are English and Arabic for historical reasons), and the dominant everyday language. (an interesting anecdote - I remember hearing about a case in the place I teach in which Russian repatriates asked to teach the class in Russian, and was asked for my opinion on this. As Hebrew is very strong in Israel, I thought they should). Advanced math books, however, are usually in English, because our Math elite are usually busy doing research rather than translating or writing textbooks for students, and usually recommend books they see 't regardless of their language. However, all basic University math is available with quite a few Original Hebrew texts on the subject, whether it's Algebra, Calculus, Set Theory, or whatever. Also, some of the usual basic training books are translated like Scaum's book!Hope this helps,The Levantine.Timothy Murphy asks:>Slightly OT,>>but does anyone publish maths in Hebrew nowadays?>>Are school maths textbooks normally in Hebrew?I've seen a calculus book in Hebrew. I own a book by Azriel Levy, in Hebrew,>consisting of notes for a course in mathematical logic. Such notes are>available from the campus bookstore, Academon. I've tried for a long time>to get information about the math books they have in Hebrew. I'm told there>are Hebrew books on differential geometry, for example. According to Academon,>when a friend of mine called them on the phone to ask for a catalogue, all>the information about their books is available online through their website.>When I tried several months ago to look at it, I found it was under>construction. My own browser can't navigate the site, since I never 'gured>out how to install the necessary Hebrew fonts, but the library computers haveI got the Hebrew word for nilpotent from a Ph.D. thesis written in Hebrew>at Hebrew University in Jerusalem.Allan Adler>ara@zurich.ai.mit.edu*********************************** *****************************************>* *>* Intelligence Lab. My actions and comments do not re¤ect *>* in any way on MIT. Moreover, I am nowhere near the Boston *>* metropolitan area. *>* *>************************************************************ support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DF9QH02993; =>> I've noticed lately that the de'ntion of science I picked up from>> Karl Popper (that it is a critical activity including criticism by>> experimentation where possible) must be drastically different than the>> de'ntion of science assumed by some of more visible posters here.>>> How about a little poll on this.>>> In one or two paragraphs in your own words, describe what science>> means to you, and if possible from whom your de'ntion is in¤uenced>> by.>Science is the branch of philosophy dealing with attempting to>determine truth via experimental means,...with the ultimate goal of achieving some kind of (imprecise)>certainty,...but 'nally discovering that the Whole Mess is based upon UNCERTAINTY>in the end...>Science: the branch of art/religion/mathematics(/politics) dealing>with that seen *outside* what is known to our own inner-eyes, even if>our inner-eyes are what actually have been seeing it all, and perhaps>with a tad bit of hallucination yet is in mathematics then it is sciencelord Kelvinhjs.just because you don't understand what they mean, does not mean approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DL6SQ30642; =>no right angle in the triangle?>I'm boning up on my geometry and having to use trig.. I never took trig.>Go 'gure, I'm having trouble!>If any one can help with this confusion I would be very approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DL6SL30646; =>they based on? was Re: Geodetic Measurements of the Great Pyramid aka>says...>[many snips]>says...>>I asked you how you could tell that a ratio was part of the design>rather than coincidental. You did not answer. I ask if all ratios were>important, you responded other than no. What I really want to know,>because the solution would be so useful, is how you can tell if the>ratio is part of the design.>A design implies a plan. A plan implies intent.>>Really? A beehive has a rather nice design that appears not to be the>result of plan or intent.No, a beehive is an organic arrangement. There is a difference.>>You might also have cited a crystaline structure. Mathematical>>regularity which is not the result of a plan may result in a>>pattern which has a similar shape or form to one which has >>been designed, or indeed a design may pattern itself after >>such a form, but a pattern may be unintentional and in that>>instance would not be considered a design.>If the presence of a ratio is an intentional part of a design, >>it might have as its purpose a desire to organise, or control, >>or modulate the arrangement of the work in some harmonious >>arithmetic or geometric progression.>>This is a completely non-responsive reply.Which part of the response don't you understand?Try again. It was not that your response was so intellectual and>clever that we don't understand it. The problem is that your response>does not answer the question. The question you have been asked by at>least two of us is, *how* can you tell if the ratio is part of the>design. Your response above and elsewhere does not answer that>question.>Or in other words, you still haven't answered the question. >We don't want to know what the purpose of an intentionally >incorporated ratio might be;>we want to know how to tell when the ratio's presence is intentional. If the purpose of incorporating a ratio in a design is >>to organise, or control, or modulate the arrangement of the work>>you would want to look for repetitions of the elements of>>the proportional system transformed in a systematic way.>>The whole would be comprised of proportional modules.>Perhaps you could explain your usage of the term proportional. All>this sounds like is that you would expect the item would have ratios.>Once again, this says Chings examples>>are a useful analog. Start with a square. Put a circle inside it.>>Is the circle a space or hole in the form of the square, or is>>the square the background for the form of the circle? When we>>measure a design do we measure the form?, measure the space? >>or come to realise that both together have a relationship?This has all kinds of just fascinating esthetics philosophical>insight. However, the question at hand is entirely different. We are>asking how you can tell a particular ratio was in the design and was>the the framing, masonry, plywood, >>drywall; the doors and windows and their hardware is designed >>to be an element of a modular system.Now you are making an assertive statement about design. How do we know>that every element is so designed? I can understand this a statement>of principle that an architect should use, but in this context it>sounds like you are claiming this is practice? Is that your claim and,>if so, support proportion. If you>>want to divide the circumference of a circle into degrees, does>>it help if you can take one standard unit measure for the diameter >>and another for a degree on the circumference???? Degrees is a unit of measure.>A circle with a circumference of 90 feet has 1 palm to a degree>>and a diameter of 19 Biblical cubits within the range of error >>in establishing the Egyptian and modern values for the foot.Huh??? What does this mean?>A circle with a circumference of 3 furlongs or 40 perches>>where 160 perches = 1 acre has one Greek cubit of 22 inches >>to a degree and is 120 Royal cubits in diameter within>>the range of error of establishing those values.An acre is a measure of area, not length. What is a perch? And even>so, what does this mean?>(The Biblical cubit is assumed to be about 1.5 foot but 18.09>>would probably not be out of the range of possibility where>>our inch is 304.8 mm, the Roman inch ours is developed from>>was 296 mm and the Egyptian foot was 300 mm whereas the Greek>>measures were in the ratio of 25/24 the Roman)Wow, an inch is 304.8 mm. Must be an important number. I wonder what>other things are in a 1 to 304.8 ratio?>>Oh, and unit fractions are *still* ratios.The are a subset of ratios. The concepts and uses of ratios and >>unit fractions are not exactly identical. Would you call a square >>a rhombus?Yes I would. What is your point? Both factions and ratios show a>proportionate relationship. >>3) Generally standard I don't know what that means. But I doubt it>means that they would change the units in mid construction.>The best reference to standards would be the Platonic Dialogs>>which explains very well the Egyptian conmcept of Ma'at.>>That, sir, is a fake citation. What dialogue? Where? And how is>this concept involved in changing units in mid-construction?The best discussion of standards in the Platonic Dialectic>>is probably given in the introduction to the 1932 Random House>>edition written by Raphael Doumas. All of the dialogs are concerned>>with a discussion of what is right and proper (standards) which>>closely parallels the Egyptian concept of Ma'at. Which is very different than your 'rst claim, that the Fibonacci sequence?Corbusiers Modulor was a module based on the Golden Section or phi.>So? What in the world does that have to do with Egypt or the Great>Pyramid?Matt Silberstein>-------------------------------Rossignol's curious, albeit simply titled book, .95The Origins of a>World War', spoke in terms of .95secret treaties', drawn up between the>Ambassadors from Plutonia and Desdinova the foreign minister. These>treaties founded a secret science from the stars. Astronomy. The>career of evil.> I dont know how you came with up with that quote in wcich appears on a Blue Oyster Cult album entitled Secret Treaties, however, i further dont understand what that has to do with any of this support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DK8NM26147; =>The reason I need to break log(X+Y) up is because the problem is tominimize (X+Y-C)^2where C is known. So it is something like quadratic optimization, even>>worse, X is de'ned to be a^x, Y is de'ned to be a^y, C is de'ned to be>>a^c, where c is known. Hence the true variables are x, y.The problem becomesminimize (a^x+a^y-a^c)^2where a is the common base for all these terms...If I take log a, it will make things a lot easier, that's why I want to>>breaklog_a_(a^x+a^y-a^c)up...>>Can you give me some more ideas?>> Huh? Clearly, the function is minimized for any X and Y such>> that X + Y = C. Are x and y constrained? -- >> Stephen J. Herschkorn and y are constrained...Sorry I did not make it clear... x and y are integer and are over>dertermined by more equations than unknown variables... hence there is no>exact solution...and there are a bunch of such variables X and Y's... hence>a bunch of x and y's ... 'nally what I want is to minimize the MSE for a>bunch of such X and Y's>i.e.minimize sum{ (a^x_i+a^y_i-a^c_i)^2 walala,call this to be a^z = Z =(a^x_i+a^y_i-a^c_i)^2 =(X+Y-C)^2now you have 3 variables X,Y, and Z WHERE x and y are integers [AS YOU SAY]but not z necessarily (C is known as you say ?)so minimize this instead![I, CAN SAY THAT X,Y (BUT NOT C as negative number)and Z are points on the logarithmiccurve of base a and x,y, and z lie on the Archimedes Curve as de'ned by : http://www.stefanides.gr/why_logarithm.htm Under these conditions there is no provision of minimization.Any views welcome.]Panagiotis Stefanideshttp://www.stefanides.gr, i from support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0DHs7U15837; =X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oftX-Terminate: SPA(GIS) =In Foz) said:>It is not the same to be INTRINSICALY uncountable, that to be >uncountable because we havent the suitable means to do it.What do you mean by suitable means? If, using the rules o'nference, I can generate the statment from the axioms then it isproven. It doesn't matter what technique my proof uses, as long aseach step is valid.>Please, read carefully my latest version of the proof,You didn't post a proof, and others have already explained what is>analogyAnalogies are not Mathematics.>An useless job.You're going in circles. The issue is not whether a proof is useful;the issue is whether it is correct.>Why do you always take out the things out of context? Why do you always ask questions that assume things contrary to fact.>there are undecidable statements.Correct, but irrelevant.>Concluding whether the reals are countable or not it probably is >one of them.No, because proofs are already known.>Sorry but, is it not the same?Because the way you worded it implies that such a tool might exist butbe undiscovered. If, as we already know, N<->R is not possible, thensuch a tool cannot exist.>Statement: sentence or af'rmation that can be true, false or>undecidable (within the current mathematics).What statement does the properties of N and R are incompatible referto?-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In at 09:13 PM, gds@best.cut.here.com said:>Of course not. Your reply to Diego implied that (engineering) math>applications are in some way analogous to covering cracks with>artwork. Correct. Mathematics is a discipline in its own right, not just a boxof tools for other disciplines.>Here's another example. If someone wants to impart a love and>understanding of music to someone else, they might:>* have them learn an instrument and how to read music>* have them learn the history of music>* take them to concerts>* listen to recordings with themNone of those involves showing them that music has practicalapplications, so none of them is analogous.>What basis do you have of determining that she wouldn't 'nd>applications fun right now?What basis do you have for assuming that she would, instead of beingguided by her inclinations? What I do know is that children of thatage are not concerned with the far future; if she's interested in anapplication, it will almost certainly be an application that ismeaningful to a 7 year old, not something that an adult would consideruseful. Again, I say that the best approach is to teach what ischallenging, fun and interesting to her; not what would be interestingto someone much older.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to =>What basis do you have for assuming that she would, instead of being>guided by her inclinations?I went back to the original post, and this is all we've been toldabout her inclinations:Her arithmetic skills are now good enough that I could begin to teachher high school Algebra, but I don't know to what end. No matter whatI do or don't do, she will eventually know high school Algebra.So we do not know whether or not she would be interested inapplications.>What I do know is that children of that age are not concerned with>the far future; if she's interested in an application, it will almost>certainly be an application that is meaningful to a 7 year old, not>something that an adult would consider useful. Again, I say that the>best approach is to teach what is challenging, fun and interesting to>her; not what would be interesting to someone much older.Perhaps you and I have different de'nitions of applications. To usea recent topic, I was thinking along the lines of doing certain typesof geometric constructions, which is something I was able to put topractical use in some drafting classes I took in high school.--gregbogds at best dot com => I am soliciting some advice and opinions about how I should interact with my> granddaughter.> First some history. I am a 72 year old Mathematician. I got my degree in> Recursive Function theory in 1955. I was a bit of a prodigy in that by the time> I graduated high school I know nearly all the Mathematics I would need for a> Ph.D (I had done little research; I did not yet have my thesis).> My daughter is now nearly 40 years old. When she was growing up, I taught her> elementary Mathematics, in part, as a form a high quality father-daughter> interaction. The effect of this was that when she was in the 9th grade she took> AP Calculus and aced the test.> I have often wondered whether it would have made much of a difference in her> life if she learned Calculus, three years later, the usual time for bright kids.> She never learned much more Mathematics and eventually went to Caltech, where> she got a degree in Molecular Biology.> Her daughter is now seven years old. She is also intellectually precocious, I> am wondering how I should interact with my granddaughter .> > Her arithmetic skills are now good enough that I could begin to teach her> beginning high school Algebra, but I don't know to what end. No matter what I> do or don't do, she will eventually know high school Algebra.> Perhaps the best legacy I could leave her, in the few years I may have left, is> an an understanding and love for the beauty of Mathematics. But I don't know> how to impart that.> Three other things I've thought of teaching her are:> Beginning Number Theory (divisibility, primes, modulo arithmetic ...)> Combinatorics (how many subsets does a 'nite set have, how many> permutations, ...).> Very elementary group theory, with an eye toward being able to produce a> rigorous proof.> I would very much appreciate some opinions. Also any suggestions about text> books, outlines of material, etc. would be most welcome.I wouldn't teach her too much formal theory, or methods. I think mostimportant especially at that age would be to learn the basicfundamental questions in mathematics(When are two things equal? Whatcomes next in this sequence? How many prime numbers are there? What isthe area of rectangle?, etc.), and pattern recognition of sequencesand properties of shapes. If she learns the problems either she won'tcare about them, or she will get an opportunity to try solving them.Also, it might useful to teach her a little set theory and somethingabout the concept of proof, so she could then have the possibily ofreading mathematical texts if so-inclined.If you have good grasp of Algebra, Calculus is quite easy to learnfrom any decent text assuming you haven't been taught too much garbageat school. It only took me about an hour to learn Calculus (I.E.integration, differentiation, what they mean, etc.) Schools generallyteach mathematics in such a way so that the simplest concepts becomethe most complicated hard to understand things.Gershon BialerX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In (brian tenneson)said:>in other words, this is a strict extension of zfc.No, just an alternative to ZFC. For it to be an extension, you'd haveto keep the existing axioms.The real question, of course, is whether you can get interesting oruseful results with your approach. Good luck.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to <3ffa32dd$8$fuzhry+tra$mr2ice@news.patriot.net> <8765fp8z1v.fsf@becket.becket.net> <3ffc9d5e$9$fuzhry+tra$mr2ice@news.patriot.net> <87lloj9qj9.fsf@becket.becket.net> <87hdz79ppk.fsf@becket.becket.net> <4000aafc$21$fuzhry+tra$mr2ice@news.patriot.net> <87llofyvmu.fsf@becket.becket.net> <4001da76$11$fuzhry+tra$mr2ice@news.patriot.net> <878ykdx0pr.fsf@becket.becket.net>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In tb+usenet@becket.net (Thomas Bushnell, BSG) said:>What connection between irrational>numbers and in'nity do you think the Greeks might have noticed?The method of exhaustion comes to mind.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to <3ffa32dd$8$fuzhry+tra$mr2ice@news.patriot.net> <8765fp8z1v.fsf@becket.becket.net> <3ffc9d5e$9$fuzhry+tra$mr2ice@news.patriot.net> <87lloj9qj9.fsf@becket.becket.net> <4000a9c3$20$fuzhry+tra$mr2ice@news.patriot.net> <87hdz3yvik.fsf@becket.becket.net> <4001dc35$12$fuzhry+tra$mr2ice@news.patriot.net> <874qv1x0m8.fsf@becket.becket.net>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In tb+usenet@becket.net (Thomas Bushnell, BSG) said:>When you say that anything can be limited, you are saying (mirabile>dictu) that the concept of something entirely unlimited doesn't make>sense.No, just that it must be carefully de'ned to make sense.>Huh? You attributed to me something that I did not write; I returned thefavor.>You seem to say X has no meaning when all>you should say is I didn't understand when you said X.No. I write X has no meaning when X has no meaning. If you reallymeant Y, please clarify.-- Shmuel (Seymour J.) Metz, <3ffa32dd$8$fuzhry+tra$mr2ice@news.patriot.net> <8765fp8z1v.fsf@becket.becket.net> <3ffc9d5e$9$fuzhry+tra$mr2ice@news.patriot.net> <87lloj9qj9.fsf@becket.becket.net> <4000a9c3$20$fuzhry+tra$mr2ice@news.patriot.net> <87hdz3yvik.fsf@becket.becket.net> <4001dc35$12$fuzhry+tra$mr2ice@news.patriot.net> <87y8sdvlqq.fsf@becket.becket.net>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In tb+usenet@becket.net (Thomas Bushnell, BSG) said:>So the concept of in'nite physical extension has no meaning? Again, you are challenging things that I didn't write instead ofthings that I did.>(Your insistence on the details of Hebrew spelling, on another>thread,Yes, it is odd that I would comment on the details of Hebrew spelling'nally convinced me that you are just arguing for the sake ofarguing, and I will adjust my 'lters accordingly.*PLONK*-- Shmuel (Seymour J.) Metz, SysProg <40009798$9$fuzhry+tra$mr2ice@news.patriot.net> <4001e0b6$15$fuzhry+tra$mr2ice@news.patriot.net> X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In question I was asking, I was not using that conventional>notation. Sheesh, is David Ullrich the only one to understand me?I understood you perfectly. I asked whether you had mistakenly givenone symbol in place of another. I then proceeded to answer yourquestion in a contingent fashion. You didn't understand my response,identical by de'nition.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In PM, Russell Easterly said:>For example, let's say I want to represent the>motion of the center of gravity of a rabbit.>I want to put together a set of all the points>the rabbit passes through and I want to>order this set in the same order the rabbit>passed through the points.Simple enough, at least locally.>Standard set theory says this is impossible.No it doesn't, unless the path crosses itself.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to => In PM, Russell Easterly said:For example, let's say I want to represent the>motion of the center of gravity of a rabbit.>I want to put together a set of all the points>the rabbit passes through and I want to>order this set in the same order the rabbit>passed through the points. Simple enough, at least locally.It is impossible if the rabbitmoves any measurable, non-zero distance.>Standard set theory says this is impossible. No it doesn't, unless the path crosses itself.>Assume the rabbit starts at point 0.000...and goes to point 1.000...Which real number represents the point the rabbitgoes to from 0.000... ?Russell- Zeno was right. Motion is impossible. <3ffc9a68$7$fuzhry+tra$mr2ice@news.patriot.net> <4000a20b$18$fuzhry+tra$mr2ice@news.patriot.net> <4001e786$20$fuzhry+tra$mr2ice@news.patriot.net> X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In 10:48 PM, Dan Christensen said:>In much the same way, you can develop number theory in my system.Adding axioms is not the same as using the existing axioms.>Why should they necessarily be treated differently>by set theory?I don't know. They're not treated differently in ZF, but they are inyour set theory.>Call them de'nitions if you like.Axioms are not de'nitions. De'nitions simply allow you to abbreviatestatments. Axioms change what you can prove.>Have you tried to use my system?No. Please tell me what I need to do to prove results about Euclideanand[1] Hyperbolic Geometry. In, e.g., ZF I can de'ne as manystructures as I want any not worry about con¤icting axioms, because Idon't add any axioms. No, I'm sure that I could shoehorn that in toyour system, but is there a way to do it that is natural and notclumsy?[1] Not just one, both.-- Shmuel (Seymour J.) Metz, SysProg and <40014E86.4060308@comcast.net> <4001A856.6040809@comcast.net> tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In 09:52 PM, William Elliot said:>Oh sure, the topology generated by the open balls. The real problem>is to show that the topology has a base of open balls. For that you>need show the intersection of two balls is a union of balls. Thus>every open set is a union of open balls.Triangle inequality.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot 11:55:20 +0100, Paul B. Andersen skrev i melding>> The above are merely statements of fact. The clocks are built>> to run at 10.22999999543MHz prior to launch so that the signal>> received once they are in orbit is measured as 10.23MHz on the>> ground receivers. so that the signal received once they are in orbit is measured>> as 10.23MHz on the ground receivers. Really??? >;^) The satellite is in a state of continuous motion always.>> Why do you have forgotten about Doppler effect?It doesn't work quite like that.>The P-code (military) pseudo random pattern is transmitted>with 10.23Mbps, but this bit rate isn't normally measured>from the ground.>The important point is that the 10.23MHz signal is>the frequency standard of the satellite clocks.>That is, the satellite clock counts 10 230 000 cycles for>each second it advances. The clock time is transmitted.>(Coded as a bit pattern modulated on two carriers.)>We know the frequency must be right since the satellite>clocks stay in synch with the GPS time, not because>the frequency is measured from the ground.Paul> It is all explained by the fact that lioght speed is source dependent.So source dependent speed of light will make clocks in GPS orbitfasten by 38 us a day compared to ground clocks?Not even funny.Just dumb.Paul It is all explained by the fact that lioght speed is source dependent.Not according to the Sagnac experiment.That is a wrong interpretation. The sagnac effect DOES comply with sourcedependency.It refutes relativity.George>Henri Wilson. Einsteinian relativity is the arse end of physics.www.users.bigpond.com/hewn/index.htm Dishman It is all explained by the fact that lioght speed is source dependent.Not according to the Sagnac experiment. That is a wrong interpretation. The sagnac effect DOES comply with source> dependency.Go ahead and show how you get a 'rst order result from asource dependent model. Last time we talked, you were unableto do this.> It refutes relativity.Sorry, SR predicts the result precisely.George Dishman If the orbiting clocks appear to run slow it is because they DO run>slow.They run fast, not slow. Yes I know. I said if. It doesn't matter for the purpose of the argument, anyway.I assumed it was a typo but sometimes what looks like a typo>can be deliberate so I try to just note them in passing just>in case.It is of no signi'cance whether or not it was a typo. The point has been made.The GO counts every tick emitted by the OC per orbit. And for your information, ignorant one, according to GR, clocks in orbit nearthe surface SHOULD run slow, not fast.Are you trying to win the .95moron of the week' prize?>> The ground observer counts every .95tick' emitted per orbit.Well a ground observer can only see any satellite for a fraction>>of its orbit but if the Earth didn't get in the way, you would>>be right.> The GPS clock preset is required simply to compensate for an increase>in>>clock> rates due to being in free fall.The change of coordinate rate is correlated to the difference in>>gravitational potential, not force, but with that clari'cation>>you are correct again. Is it just a matter of de'nition then? No I don't think so. The clocks>are>> relieved of their own mechanical strain due to gravitational force.>> They run slightly faster when in free fall.I thought this looked more deliberate but it could also have>been just a slip. Let me put it in Newtonian terms to keep it>simple, gravitational force varies as r^-2 while potential>varies as r^-1. The change of rate varies as r^-1, not r^-2.That is because potential and force are quantities with different dimensions,idiot.> The fact that the change is of the>> same order as the so called GR correction is purely coincidental.> .....unless the .95tick-fairies' have been active again!If a satellite signal could be received throughout its orbit the>>ground observer could count every tick emitted, so there are no>>tick fairies. It doesn't have to be seen over the whole orbit. The orbiting clock (OC)>knows>> how many ticks it emits per orbit. The GO also knows, simply by reading>the>> OC's time.Sure, but I was agreeing with you anyway.Funny way to agree!> The orbit itself provides a common clock.Sorry, it doesn't, the duration of the orbit is not the same>measured on the ground as in orbit.Oh! really!You obviously have the IQ of a relativist.One ORBIT can be used to de'ne the basic TIME unit. The orbit IS the duration.If it is measured to be different, then the clocks used to measure it arewrong.>Nothing you have said con¤icts in any way with what was stated,>>so what were you objecting to? The claim that relativity is incorporated into the GPS system. It isn't.I assume that was a typo and you meant The claim that a>correction for relativistic effects ..., you don't>incorporate relativity. That is untrue as you can see from>the quote above:To compensate for this large secular effect, the clock is given>a fractional rate offset prior to launch of -4.465 [Times]10^-10 from>its nominal frequency of exactly 10.23 MHz, so that on average>it appears to run at the same rate as a clock on the ground.Regardless of your views, the clocks are built with a correction>term applied that is not included in ground-based clocks.Naturally, they are corrected for the free fall effect before they arelaunched. They are software 'ne tuned when in orbit. The .95GR correction' amounts to an error of only 7cms per orbit anyway.That is regularly corrected out empirically.George>Henri Wilson. Einsteinian relativity is the arse end of physics.www.users.bigpond.com/hewn/index.htm Dishman If the orbiting clocks appear to run slow it is because they DO runslow.They run fast, not slow. Yes I know. I said if. It doesn't matter for the purpose of the argument, anyway.I assumed it was a typo but sometimes what looks like a typo>can be deliberate so I try to just note them in passing just>in case. It is of no signi'cance whether or not it was a typo. The point has beenmade.> The GO counts every tick emitted by the OC per orbit.It was never disputed. Why are trying to make an argument outof it?> And for your information, ignorant one, according to GR, clocks in orbitnear> the surface SHOULD run slow, not fast. Are you trying to win the .95moron of the week' prize?I'll let you keep it, the GPS satellites are not .95near thesurface'. The paragraph you responded to said this:>The net effect of time dilation and gravitational redshift is that>the satellite clock appears to run fast by approximately 38 [Micro]s per>day when compared to a similar clock at rest on the geoid, including>the effects of the velocity of rotation and the gravitational>potential at the Earth's surface. ....and you replied:> Load of crap. If the orbiting clocks appear to run slow it is because they DO run slow.They don't appear to run slow, they appear to run fast.>... Let me put it in Newtonian terms to keep it>simple, gravitational force varies as r^-2 while potential>varies as r^-1. The change of rate varies as r^-1, not r^-2. That is because potential and force are quantities with differentdimensions,> idiot.varies as indicates the dependency, not the complete equation.You need to learn some basics. The change of coordinate ratematches the the dependency of potential on altitude, it does notmatch the dependency of the force.> .....unless the .95tick-fairies' have been active again!If a satellite signal could be received throughout its orbit the>>ground observer could count every tick emitted, so there are no>>tick fairies. It doesn't have to be seen over the whole orbit. The orbiting clock(OC) knows>> how many ticks it emits per orbit. The GO also knows, simply by readingthe>> OC's time.Sure, but I was agreeing with you anyway. Funny way to agree!Your words suggested you don't believe in tick fairies,and I said there are no tick fairies, it seems obviousto me I was agreeing.>> The orbit itself provides a common clock.Sorry, it doesn't, the duration of the orbit is not the same>measured on the ground as in orbit. Oh! really!> You obviously have the IQ of a relativist. One ORBIT can be used to de'ne the basic TIME unit. The orbit IS theduration.> If it is measured to be different, then the clocks used to measure it are> wrong.If you try to use the orbit as the clock, it is impossibleto measure the duration of the orbit, your suggestion isself-referential so useless. You cannot measure any clockby timing it against itself.>>Nothing you have said con¤icts in any way with what was stated,>>so what were you objecting to? The claim that relativity is incorporated into the GPS system. Itisn't....>Regardless of your views, the clocks are built with a correction>term applied that is not included in ground-based clocks. Naturally, they are corrected for the free fall effect before they are> launched. They are software 'ne tuned when in orbit.Exactly so the relativistic correction _is_ built in.> The .95GR correction' amounts to an error of only 7cms per orbit anyway.> That is regularly corrected out empirically.Bollocks, if the 38us per day applied to all the craft,there would be no positional error but there would be atime error of, guess what, 38us per day. GPS is used tosynchronise atomic clocks to a fraction of a nanosecond.The system would be completely useless as a time sourcein a few hours if the design didn't include the correctionpredicted by GR.George =One of the reasons I made the post Quick Math Guide to core errorissues was to have a handy reference. Having found myself ¤ounderinga bit, I found myself looking back at it, which is when I realized awhile back that I'd wondered far astray, and I posted that I'd made amistake with my addition of y to an example from Rick Decker.I then sat back, thought for a while, and realized that possibly myproblem had to do with dif'cult concepts that so push the human brainthat most are not up to handling the mathematics as I'd been givingit.So I'm starting again, and looking for better ways to to explain.now I'm beginning to realize that I need to understand human beingsbetter as abstractions. I need to 'gure out how their brains work.That puts me back into study mode. I have to understand your mentalwiring to an even greater degree than I do now.I have to 'gure you out.James Harris =>One of the reasons I made the post Quick Math Guide to core error>issues was to have a handy reference. Huh - I didn't see the Quick Math Guide.>[...]I've adjusted those expressions several ways to try and make it easier>to understand.So far those adjustment apparently haven't been very successful, and>now I'm beginning to realize that I need to understand human beings>better as abstractions. I need to 'gure out how their brains work.That puts me back into study mode. I have to understand your mental>wiring to an even greater degree than I do now.I have to 'gure you out.Um, actually studying some math might be more productive.Nah, studying is what lesser intellects like the rest of us do.Never mind.>James Harris************************David C. Ullrich = I then sat back, thought for a while, and realized that possibly my> problem had to do with dif'cult concepts that so push the human brain> that most are not up to handling the mathematics that. <3ffd3edf$21$fuzhry+tra$mr2ice@news.patriot.net> <40019929$48$fuzhry+tra$mr2ice@news.patriot.net> <87hdz28d3u.fsf@phiwumbda.org>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In jesse@phiwumbda.org (Jesse F. Hughes) said:>As a result of pretty sure, you advised me thus?seemed to be rejecting the whole idea of modeling one formal system inanother.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to -0500, Shmuel (Seymour J.) Metz>In jesse@phiwumbda.org (Jesse F. Hughes) said:>As a result of pretty sure, you advised me thus?seemed to be rejecting the whole idea of modeling one formal system in>another.Help us put that in the proper context by quoting what he saidthat seemed to you to be rejecting the whole idea of modelingone formal system in another. (I 'nd it very hard to believe thathe said anything like that, but I can't search the thread to 'ndout, because even if I found the quote you're referring to Isuspect I would not recognize it as seeming to say what yousay it seems to say.)************************David C. <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87smim7rqf.fsf@phiwumbda.org> <4001e592$19$fuzhry+tra$mr2ice@news.patriot.net> X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In is this a misrepresentation?-- Shmuel (Seymour J.) Metz, =On said:>How is this a misrepresentation?>How in the world could you possibly construe that as an answer tomy question?************************David C. Ullrich <3ffadc52$20$fuzhry+tra$mr2ice@news.patriot.net> <1g75muz.1d66n4i14gjvdlN%panoptes@iquest.net> <1g75yr4.1si6z861lg1554N%panoptes@iquest.net> <97adneZXNdeRbWCi4p2dnA@comcast.com> <40009fc8$16$fuzhry+tra$mr2ice@news.patriot.net> X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In PM, Russell Easterly said:>I can program a Turing machine to perform the operations I describe.No.>I only make this assumption to satisfy those that claim the TM will>never produce an output.No, and no. That's not the basic objection and it does not satisfy thebasic objection. The basic objection is that an automaton with anin'nite input is not a TM at all.>If we assume a TM can only perform a 'nite number of>operations then my claims are trivially true.Then it's a good thing that we make no such assumption. What we assumeis that the automaton has only a 'nite number of possible states,there are only 'nitely many distinct symbols available for the tape,that the automaton can only examine a 'xed number of symbols at eacheach stage.>If we assume a TM can only perform a 'nite number of>operations then my claims are trivially true.Only in the sense that P^~=>q is trivially true.>The rationals must be uncountable for a system that can only perform>a 'nite number of computations.No. >If a TM can only perform a 'nite number of operations then no TM>can compute PI or any other irrational.However, a TM that can only perform transitions among a 'nite numberof symbols, and that can process only a 'nite number of cells at eachstage, can compute all of the digits of Pi.>Finding the largest member of a set seems applicable to what is>essentiially Cantor's diagonal proof applied to the natural numbers.Mathematics is not about how things seem to you; it is about what isprovable.-- Shmuel (Seymour J.) Metz, SysProg and => In PM, Russell Easterly said:I can program a Turing machine to perform the operations I describe. No.See below.>I only make this assumption to satisfy those that claim the TM will>never produce an output. No, and no. That's not the basic objection and it does not satisfy the> basic objection. The basic objection is that an automaton with an> in'nite input is not a TM at all.I previously gave Turing's de'nition of a computable number.How can we write out a real number if we don't allow in'nitely long tapes?>If we assume a TM can only perform a 'nite number of>operations then my claims are trivially true. Then it's a good thing that we make no such assumption. What we assume> is that the automaton has only a 'nite number of possible states,> there are only 'nitely many distinct symbols available for the tape,> that the automaton can only examine a 'xed number of symbols at each> each stage.If this is your de'nition of a TM then I can write a TM thatperforms the operations I have described.>If we assume a TM can only perform a 'nite number of>operations then my claims are trivially true. Only in the sense that P^~=>q is trivially true.I don't understand what you mean by this.>The rationals must be uncountable for a system that can only perform>a 'nite number of computations. No.Please demonstrate a system that enumerates the rational numbersusing only a 'nite number of computations.>If a TM can only perform a 'nite number of operations then no TM>can compute PI or any other irrational. However, a TM that can only perform transitions among a 'nite number> of symbols, and that can process only a 'nite number of cells at each> stage, can compute all of the digits of Pi.According to your argument, such a machine could never 'nish.Using your argument, a TM can't compute ALL of the digits of PI.Russell- 2 many 2 count <3ffadc52$20$fuzhry+tra$mr2ice@news.patriot.net> <1g75muz.1d66n4i14gjvdlN%panoptes@iquest.net> <3ffd4579$1$fuzhry+tra$mr2ice@news.patriot.net> <1g78quy.1a1bzig1e5uy44N%panoptes@iquest.net> <400098e9$10$fuzhry+tra$mr2ice@news.patriot.net> <1g7epkg.1b14mvqgglvt4N%panoptes@iquest.net>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In 05:43 PM, panoptes@iquest.net (Daniel W. Johnson) said:>While there are several inconsistencies among his claims, I don't>think that was one of them.That depends on what text you read, and how you construe it. He wasusing the same notation for a 'nite sum and for a series, and thentrying to carry over conclusions for one to the other.-- Shmuel (Seymour J.) Metz, => That depends on what text you read, and how you construe it. He was> using the same notation for a 'nite sum and for a series, and then> trying to carry over conclusions for one to the other.You mean the way he was oscillating between the series for e-2 (wheneverhe claimed that it wasn't on the list) and the sum of an initial segmentof that series (whenever he claimed that it was a rational number)?-- Daniel W. Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87smim7rqf.fsf@phiwumbda.org> <4001e592$19$fuzhry+tra$mr2ice@news.patriot.net> <87brp98ys1.fsf@phiwumbda.org>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In jesse@phiwumbda.org (Jesse F. Hughes) said:>It was *not* intended as a condescending question. It came across that way.>You were discussing modeling ZFC in PA and you referred me to an >natural and honest question: contain material that does not relate directly to the title. By implying that I was tying the consistency of the axiom of choice tothe matter at issue.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87smim7rqf.fsf@phiwumbda.org> <4001e592$19$fuzhry+tra$mr2ice@news.patriot.net> <87brp98ys1.fsf@phiwumbda.org> <40044d8d$31$fuzhry+tra$mr2ice@news.patriot.net> => In jesse@phiwumbda.org (Jesse F. Hughes) said:>It was *not* intended as a condescending question. It came across that way.>You were discussing modeling ZFC in PA and you referred me to an >>natural and honest question: contain material that does not relate directly to the title. And yet, as it turns out, it was an apt question. At present, itdoes not show Con(PA) -> Con(ZF), since (as a few knowledgeable folkhave pointed out) such a proof amounts to ~ Con(ZF).ask what the connection between the stated result and the title of theIn any case, as I've said, there is typically *some* connectionperfectly natural. By implying that I was tying the consistency of the axiom of choice> to the matter at issue.I didn't imply any such thing. You're the one that said the result-- If you have a really big idea, you can get a measure of how big it isREALLY, REALLY, *REALLY*, BIG DISCOVERY!!! --James Harris, on being ignored <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87smim7rqf.fsf@phiwumbda.org> <4001e36d$17$fuzhry+tra$mr2ice@news.patriot.net> <8765fh8y5b.fsf@phiwumbda.org>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In jesse@phiwumbda.org (Jesse F. Hughes) said:>But the fact that you're now doubting (1about where you misrepresented me, since you've just done it again.No, I am *NOT* doubting 1. I listed 1. solely because a large part ofthe debate revolved around it.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87smim7rqf.fsf@phiwumbda.org> <4001e36d$17$fuzhry+tra$mr2ice@news.patriot.net> <8765fh8y5b.fsf@phiwumbda.org> <40044e9e$33$fuzhry+tra$mr2ice@news.patriot.net> => In jesse@phiwumbda.org (Jesse F. Hughes) said:>But the fact that you're now doubting (1 about where you misrepresented me, since you've just done it again.> No, I am *NOT* doubting 1. I listed 1. solely because a large part of> the debate revolved around it.No? If (1) was beyond doubt, there's no reason to list it as aquestion to be settled.Look, you were talking out your ass. No big deal. We all do that onoccasion. On the other hand, you expressed great certainty in yourass-talking, which is a bit more problematic. If you *don't* knowsomething, but you're simply *pretty sure*, then bloody well say so.Worse, you con'dently asserted that, to con'rm your claims, oneit a thorough reading, I would've wasted several hours. At present, you've offered no apology for this behavior. This isasinine.-- Jesse HughesWell, you know as soon as you have a new number I will be happy toadd it to the list. Don't try those childish tit-for-tat games withme. -- Ross Finlayson on Cantor's <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87vfni7sqx.fsf@phiwumbda.org> <4001e487$18$fuzhry+tra$mr2ice@news.patriot.net> <878ykd8ybf.fsf@phiwumbda.org>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In jesse@phiwumbda.org (Jesse F. Hughes) said:>In any case, the big and important difference is that I never told>anybody that what I was saying is such a well-known fact that he canActually I was refering you there for the methods.>I assume you've since things stacked up.-- Shmuel (Seymour J.) Metz, SysProg and <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87vfni7sqx.fsf@phiwumbda.org> <4001e487$18$fuzhry+tra$mr2ice@news.patriot.net> <878ykd8ybf.fsf@phiwumbda.org> <40044e07$32$fuzhry+tra$mr2ice@news.patriot.net> => In jesse@phiwumbda.org (Jesse F. Hughes) said:>In any case, the big and important difference is that I never told>>anybody that what I was saying is such a well-known fact that he can Actually I was refering you there for the methods.model ZF in PA in order to prove the result. Huh.>>I assume you've since things stacked up.Take your time.-- We want a single platform. We're trying to get there using the -- Madison, WI, superintendent Rainwater grasps subtlety in the operating system wars.X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In at 03:09 PM, kramsay@aol.com (KRamsay) said:>You know, it would have been a little more likely that I would've>been reading this if somebody could have renamed the thread something>a little more descriptive than rationals are uncountable. I mean,>just because someone decides to name a thread something like Yer>Bleedin' Doggies or Help or Proof of 2+2=5 doesn't mean it has>to stay that way....Thread drift is a fact of life in Usenet.question: are there important results in Analysis or Topology thatdepend on axioms for large cardinals?-- Shmuel (Seymour J.) Metz, SysProg and JOATnot <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87smim7rqf.fsf@phiwumbda.org> <4001e36d$17$fuzhry+tra$mr2ice@news.patriot.net> X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft =In haven't read the entire thread carefully, but I don't recall seeing>any response from you to the following.I was hoping to track down copies of some of the relevant papers. >I think that it's all true - which statement below seems false>to you:I'd have to check the originals; my 'rst take is that my recollectionwas wrong.that my recollection is mistaken. It's also clear that there's a lotmore interesting material out there than I have time to read.-- Shmuel (Seymour J.) Metz, =On said:>I haven't read the entire thread carefully, but I don't recall seeing>>any response from you to the following.I was hoping to track down copies of some of the relevant papers. Asking what relevant papers? would be condescending. So I'llbe blunt instead:If you need to track down papers to reply to the question ofwhich step in the proof a few posts above is wrong then itfollows that you really don't know anything at all about thisstuff. Really. The statement that ZF |- Con(PA) is incrediblyobvious, whether you can 'nd it in a paper or not - it's utterly trivial. And the fact that Godel did in fact provethat ZF |- Con(ZF) implies that ZF is inconsistent isabout the most well-known result in logic - it's the onetechnical result in logic that people who know nothingabout logic know.(Comment regarding whether ZF |- Con(PA) is reallyall that obvious: Yes, it is. You've seen me express doubtsabout that, but as I stated the _only_ reason I had doubtswas because if it was as obvious as it seemed to me thenI didn't see why people made a big deal, or a deal of anysize, over Gentzen's result. People have con'rmed thatyes, ZF |- Con(PA) _is_ exactly as obvious as I thoughtit was, the explanation for what I was confused about isthat Gentzen proved much more than ZF |- Con(PA).)>>I think that it's all true - which statement below seems false>>to you:I'd have to check the originals; my 'rst take is that my recollection>was wrong.that my recollection is mistaken. It's also clear that there's a lot>more interesting material out there than I have time to read.And that your Goggle for Godel, and attitude in generaltowards those of us who didn't understand the techniquesone could use in showing that Con(PA) implies Con(ZF),were quite condescending. If only we'd read Godel'swork on consistency of AC (you have, right?) we'dunderstand all this...************************David C. Ullrich =:>:> What is the output of the following program if it never reaches:> the end of input?: I am assuming the computer can perform an in'nite number of: operations in 'nite time.:> while (!end_of_input):> {:> input x;:> if (x>max):> max=x;:> }:> print max;: It will print a natural number.: We are assuming that every input is a natural number.: max must also be a natural number.But it never gets to the print statement. The while statement is always true, so it will never reach the print statement.What do you think this program prints? if (false) print False;:> What do you think the output of this program would be?:>:> while (true):> {:> n=n+1;:> }:> print done;:>: It will print an in'nitely long string of numbers followed by: the word done.That is funny. There is only one print statement. Whywould it print an in'nitely long string of numbers? Whywould it ever exit the while loop in the 'rst place?Stephen => So, unless Shmuel can provide some real details that show Goedel's> paper on the consistency of Choice and CH yields also a proof of > Con(PA) => Con(ZF) (which would be a fairly deep corollary!), It is an easy consequence of the incompleteness theorem thatCon(PA) => Con(ZF) is not provable in ZF (if ZF is consistent). <87u135nv7i.fsf@phiwumbda.org> <874qv5nprb.fsf@phiwumbda.org> <87brpc6yhi.fsf@phiwumbda.org> => So, unless Shmuel can provide some real details that show Goedel's>> paper on the consistency of Choice and CH yields also a proof of >> Con(PA) => Con(ZF) (which would be a fairly deep corollary!), It is an easy consequence of the incompleteness theorem that> Con(PA) => Con(ZF) is not provable in ZF (if ZF is consistent).So, it *would* be deep, huh?-- Meaningless movies on the screen behind the band that's blowing Waterboys,throwing shapes My Love is My RockHalf of the music is on tape in the Weary Land =I have a problem solving the following inequality in L^p spaces over the reals with 2<=pI have a problem solving the following inequality in L^p spaces over the reals >with 2<=p|(f+g)/2|^p + |(f-g)/2|^p <= (|f|^p + |g|^p)/2>when p=2 I know I can use the parallelogram law, but I don't know what to do >about the other p's. I tried for example >to use the Minkowsky inequality and binomial expansions.. but I still can't >solve it. I would appreciate any suggestions or >comments.I think this is one of Clarkson's Inequalities. You want to 'rst proveit for real numbers, and then integrate to get it in L^p (where |.|is replaced by ||.||_p). Reduce to the case where f and g are positive, and use the convexity of t -> t^(p/2) to get(1) ((f + g)/2)^p <= ((f^(p/2) + g^(p/2))/2)^2 In the case f > g, use (f - g)^(p/2) <= (f - g) f^(p/2-1)and g^(p/2) <= g f^(p/2-1) to get(2) ((f - g)/2)^p <= ((f^(p/2) - g^(p/2))/2)^2Then add (1) and (2).Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 =If you have a bunch of numbers and want to 'nd the GCD of them all, what's the best way to do it?The method that suggests itself is to keep a running GCD of the 'rst 1, 2, 3, etc. by computing GCD(x1,x2,x3,...,xn) = GCD(x1,GCD(x2,GCD(...,GCD(xn-1,xn)...)))or reverse order.And then for each one you use Euclid's Algorithm or binary GCD as appropriate.But I can't stop thinking that there's a better way. Is it worthwhile to sort the list of numbers?-- remove reply. => If you have a bunch of numbers and want to 'nd the GCD of them all, what's> the best way to do it?> The method that suggests itself is to keep a running GCD of the 'rst 1, 2,> 3, etc. by computing> GCD(x1,x2,x3,...,xn) = GCD(x1,GCD(x2,GCD(...,GCD(xn-1,xn)...)))> or reverse order.> > And then for each one you use Euclid's Algorithm or binary GCD as> appropriate.> But I can't stop thinking that there's a better way. Is it worthwhile to> sort the list of numbers?> want to reply.There is an old Fortran implementation of GCD for n integers:'le 386for greatest common divisor (GCD)title Greatest Common Divisor of $n$ Integers and Multipliersby G. H. Bradleyref Comm. ACM 13,7 (July 1970) 447see: http://www.netlib.org/toms/386Hugo Pfoertner =@abouthugo.de:> There is an old Fortran implementation of GCD for n integers:binary, but I'll test it for speed and see if it matters or not.-- remove them if you => We all know the 3X+1 conjecture is unprooved to this moment,> But I was wondering if we know by proof that f.i. the 5X+1 ( or 7X+1, ...)> analogue problem has divergent sequences.> My guess is that such a proof could be .95simpler' Could also be harder. See John Conway's paper on Unpredictable Iterations.-- =>AX+D / helpfull links?>Message-id: > Hey,>> thx for your response.>> What o you mean with mensanators? I can't 'nd any the Queen?>A. Neither is a subject.That's Ace to you, sirrah.>-- >Dave Seaman>Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.>AX+D / helpfull links?>Message-id: Hey,>thx for your response.>What o you mean with mensanators? I can't 'nd &as_scoring=r&hl=en&ie=ISO-8859-1&oe=ISO-8859-1&btnG=Google+ Search&as_epq=&as_oq=&as_eq=&as_ugroup=&as_usubject=&as_ uauthors=mensanator&as_umsgid=&lr=&as_drrb=q&as_qdr=&as_mind= safe=imagesand don't forgethttp://members.aol.com/mensanatorBut fair warning, although there is lots on 3x+1, 3x+3 and the generalizationof 3x+C,I haven't had much (if anything) to say on 5x+1 or 7x+1.thx in advGottfried Helms schreef we know by proof that f.i. the 5X+1 ( or 7X+1,>...)>> analogue problem has divergent sequences.>> Look at mensanators place come up soon in this thread...:-) ) Gottfried>>--MensanatorAce of Clubs 
> I haven't
seen a post yet with the answer to your question, so I'll
add>> what little bit I know. In the .9550s and .9560s 
there was a
set of Disney>> educational movies. Hemo the Magni'cent --
about biology -- was one.>> Another was the one you seek. It
involved geometry, was set about the>> 6th to 8th grade level,
and talked a lot about the Golden Ratio. There>> was a
billiards (not pool) segment. I think it was called The
Mathemagician, and featured Donald Duck>in .95math 
land' or
something like that. Robert W. Montcalm | > Tangy Fruit Gang |
rmontcal@osf1.gmu.edu> George Mason U. | http://osf1.
gmu.edu/~rmontcal/index.html | I'm
not talking to myself, I'm just the > only one 
that's
listening.> -- Jimmie's Chicken Shack, 
Milk>