mm-209 === Subject: Re: De facto censorship, counting primes http://www.giganews.com/info/dmca.html>[...]What Jesse said:>A sociologist of science with whom I am cordial has taken up the topic>of who does what to whom and why, in NG's such as sci.math, sci.logic,>and sci.physics. That's interesting.> This is an area that has not been explored in the>detail that it deserves to be, perhaps because even sociologists are>not entirely comfortable with what these NGs show about those, many of>them professors, who inßict as much harm as possible on>seekers-after-knowledge such as yourself in these groups.But suggesting that Harris is a seeker-after-knowledge is hilarious.He has repeatedly said he's not interested in learning any math.He's the only person I've ever seen state on sci.math that if whathe just said was wrong we shouldn't bother saying so because hedidn't want to know. Seeker after knowledge? Right.Does your conjecture about why this interesting topic has notbeen studied come from your sociologist friend, or is it justyour own book, it is W. Rudin, Principles of>Mathematical Analysis, chapter 2 ( Basic Topology), problem 18:>[A set E is perfect if E is closed and every point of E is a limit>point of E] Is there a nonempty perfect set in R which contains no>rational number?> ... [ contructs open set containing rationals and with measure < 1,> takex X as its complement ] ...>> Try showing (1) X is not countable>> (2) the isolated points of X constitute at most a>> countable set>> (3) X {isolated points of X} is closed I don't think this will work. Try considering points of X such that> every neighborhood contains an uncountable infinity of points of X> instead. By Cantor-Bendixson, (2) must be true. (3) seems clear since > X, being closed, contains all its accumulation points, and these > are also exactly the accumulation points of X{isolated points}.> Is there a problem with (1) or C-B too much to take as known?The theorem is not listed in the index to the book, === Re: DO MATHEMATICIANS READ WITH HALF A LIGHTBULB?>Prove that 1 + (a + a^2 + a^3 . . .) = 1 / (1 - a)>>So I did:>>(1 - a) [1 + (a + a^2 + a^3 . . .)] = (1 - a) (1 / (1 - a)) >{multiplying both sides by (1 - a)}And what if a = 1?>(1 - a) + [(1 - a) (a + a^2 + a^3 . . .)] = 1 {interim result}>>(1 - a) + [(a + a^2 + a^3 . . .) - (a^2 + a^3 + a^4 . . .)] = 1>{interim result}Ah, a bit of shufßing the terms of a conditionally convergent series,that's always a neat trick in any proof.>(1 - a) + a = 1 {interim result}>>1 = 1 {final proof of equality}Very nice, for hundreds of years we have erroneously assumed thegeometric series converges only for |x|<1, but you have === large prime numbers Let all known primes be > {2,3,7,13,43,139,3263443},> the largest prime being> 3263443. If you change Richard's proof, you can make it wrong. Richard did not say,> ``take all _known_ primes'', or ``take all primes found by process X.'' He> said, ``take _all_ primes.''Euler permits all primes to be {2,3,7,13,43,139,3263443}.Richard's wording does not explicitly forbid that.My definition of all and of is known is such that if there's a finite number of primes, the premise to be disproved, then all primes = all known primes, i.e. knowledge simply consists of listing them in the set of primes.I deliberately try to avoid saying all primes, as when I say things likelet {2,3,7} be all the primes, some doofus pipes up but you're missing 5.> Your list above is missing a few primes less than 3263443. If you multiply> _all_ primes up to 3263443 and add 1, the result will not be divisible by any> prime less than or equal to 3263443. that there are a finite number of primes. Not a finite number of _known_> in principle, take the product of _all_ primes and add 1, and label that N. > Then there has to be a prime dividing N. Yes.> That prime is necessarily larger than> P.No.> I agree that if you take something less than _all_ primes and apply the above> process, that N could be divisible by something less than P. So what? Also, we all understand that this is not Euclid's proof. Again, so what?At the moment it doesn't prove what it was claimed to prove, and thereforeisn't a proof at all. It's a proof of C->F >> No prime less than or equal to P>> divides N. i.e. you want to add the assumption> that you know all primes up to P. I don't see this as adding an assumption.My form is take the set of all primes, your/Richard's version is take all primes and exclude the possibility that there can be any prime less than the largest element. If you _don't_ see that as adding an assumption, then there's nothing more I can say. I've tried to explain it _repeatedly_. It's an extra assumption. > If there's a largest prime, there> are finitley many primes, and in principle they can be listed. There is no> addtional assumption after that of a largest prime.Nope. > I did not use the word ``blathering.''Nor do I say you do. You're not the only responedent in this thread.Phil-- Unpatched IE vulnerability: window.open search injectionDescription: cross-domain scripting, cookie/data/identity theft, command executionReference: http://safecenter.net/liudieyu/WsFakeSrc/ WsFakeSrc-Content.HTMExploit: http://safecenter.net/liudieyu/WsFakeSrc/WsFakeSrc-MyPage.htm= === ==Subject: Re: Question on generation of large prime numbers If Richard has simply added let all primes <=P be known to his premise> I wouldn't have jumped on it that way. Let all primes <=P be known. :-)If you assume that then proof by contradiction leads to the possibilty that that assumption is false.I.e. the final conclusion is that either there's a new prime >P or there's actually a new prime

Ockham has some wise words for moments like this. I was about to say don't needlessly multiply spellings, but my dictionary > agrees with you that Ockham is an acceptable variant of Occam. I find > this deliciously ironic.It's a town name after all, and the town's name was and is Ockham. http://uk2.multimap.com/map/browse.cgi?X=505000&Y=155000&width =500&height=300&client=public&gride=&gridn=&srec=0&coordsys=gb &addr1=&addr2=&addr3=&pc=&scale=100000&advanced=&multimap.x= 338&multimap.y=92Some francophiles prefered a more italic rendering, but others, such as the IEP (which I think contains the single most detailed biography of him that I've seen anywhere), list only Ockham.Phil-- Unpatched IE vulnerability: Basic Authentication URL spoofingDescription: Spoofing the URL displayed in the Address barReference: === Subject: Problem with a seriesPlease, can someone help me with this (difficult) exercise ?Let u_k be a positive real sequence, such that the series sum( 1/u_k,k=1..infinity) converges.Let T_n = u_1 + ... + u_n. Prove that the series sum (n/T_n,n=1..infinity) converges and that sum (n/T_n, n=1..infinity) <= 2 *sum( 1/u_k, k=1..infinity).Hint : Use the === Factorial ending in 8000000In sci.math, Dale ShoultsIf the last seven digits on n! are 8000000, compute the value of n. > Hint: Since the number ends with exactly six 0's, it must contain 5^6 as a > factor, but not 5^7.> Given that hint, it's trivial; it has to be at least 25!, andless than 30!.27! = 10888869450418352160768000000-- #191, ewill3@earthlink.netIt's === sci.math, Olivio This is a (elementary?) geometry problem and I'm looking for a simple> solution. In a unit circle a chord is drawn.The distance of the center> from the chord is x (0<=x<=1).What is the length of the chord?> Is it proportional to sqrt(1-x^2)?Half of the chord, the radius to the end of the chord half,and the line from the circle center to the bisection pointof the chord results in a right triangle. Therefore:hypotenuse = 1side adjacent = xside opposite = sqrt(1 - x^2)Bear in mind that this is only half of the chord, but theother half is symmetrical; the answer is 2 * sqrt(1 - x^2).Or one can do it analytically. Place the chordperpendicular to the X axis and to the right of the origin;therefore the circle point above the X axis is (x,y). Whatis y? Well, x^2 + y^2 = 1 as we're hypothesizing a unitcircle; the length of the chord is then 2y = 2* sqrt(1-x^2). Olivio> -- #191, ewill3@earthlink.netIt's === limits - Need Help!In sci.math, Roy<7a108ddd.0311161336.5e01fb3e@ limits.> I mustn't use L'hospital rule:> a) lim(x*(2^(1/x))-x) where x increases to infinite.> b) lim(cosh(x)-1)/(x^2) where x approaches 0.(a) = lim{x->oo} x*(2^(1/x) - 1) = lim{x->oo}x*(exp(ln2/x) - 1) = lim{y->0} (exp(y*ln 2) - 1) / y (y = 1/x)It's now clearly a derivative. Note that we do *not* needto worry about the chain rule here; all we're doing isswitching variables within a limit. In fact, usingz = ln2/x is instructive; one gets (a) = lim{z->0} (exp(z) - 1) * ln(2) / zwhich gives the same answer anyway.Or one can write (a) = lim{y->0} (2^y - 1)/y = lim{y->0} {exp(y*ln2) - 1)/y(b) might be doable by setting x^2 = y (y = sqrt(x)); note that one *has* to use the chain rule in this case. A little confusing perhaps but remember that a derivative is lim{d->0} (f(x+d) - f(x))/d; (the traditional notation is delta x, but ASCII is a pain at times :-) ); the value lim{d->0} (f(x+d)-f(x))/d^2 is something else entirely. However, since cosh'(0) = sinh(0) = 0, it works in this case.Note that both are a special case of L^Hopital's rule(note spelling):lim{x->c}f(x)/x = lim{x->c}f'(x)/1 = f'(c)since the derivative of x is the constant 1; be careful howyou explain your answer.-- #191, ewill3@earthlink.netIt's still legal to go === Help!In sci.math, Robin athena.ex.ac.uk>:>> Roy escribi.97 en el I mustn't use L'hospital rule:> a) lim(x*(2^(1/x))-x) where x increases to infinite.> b) lim(cosh(x)-1)/(x^2) where x approaches 0.>> Can you use series developments?>> No, I can't. What a shame. They make such problems much easier :-(> All this one needs is a simple variable change.-- #191, ewill3@earthlink.netIt's still legal to go === categories?>certainly the special case where [for any object x in c, f(x) has a>unique object] can be considered, but i'm still confused about whether>what you tried to describe is really the same thing. if [for any>object x in c, f(x) has a unique object], then what we're dealing with>is something like a functor from c to (some version of) the category>of monoids, but i didn't see anything in your description that really>sounded like that. but it's not implausible to me that you might be>trying to describe the same thing in different language, because>despite what you say i honestly find your notation and terminology>confusing, and it makes me wonder whether you might have made what i>call a level slip somewhere- getting concepts on the level of>objects mixed up with concepts on the level of morphisms, or something>like that.I don't think I've made a level slip. To be sure I'm writing thewhole lot from scratch: maybe my wording will be more fortunate... (orprecise!)Let's start with monoids, say A,B. Aut(A) is a monoid itself, so theremay well be something in Hom(B,Aut(A)), and indeed -incidentally-there always is!So, chosen f in Hom(B,Aut(A)), we can define the semidirect product asusual:(*) (a2,b2),(a1,b1)|->(a2 f(b2) a1,b2 b1).Now, as a matter of a fact a Category turns out to be a sort of bigmonoid with possibly more than one identity and a product not definedfor all elements, right? (intendedly loosely speaking!)Now instead of Aut(A) we have the monoid of functors A->A. But amonoid IS a category, with just one object. So we take into accountHom(B,Hom(A,A)): again it is not empty and if we choose f in it we candefine a composition *exactly* as in (*) provided that allcompositions in it are well defined, i.e. the domain of a2 is theimage through f(b2) of the codomain of a1 and the domain of b2 is thecodomain of b1.Please do not make me write down extensively (in an ASCII environment)the (trivial) proof that both associativity and identity propertieshold!TIA,Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore Aursand on === between C and M(2,R)I've written at least twice about this subject in the past, withoutreceiving any feedback. I'd be glad to read any kind of comment!>>Exponentials, and logarithms of invertible elements, exist in any Banach >>algebra. Look up holomorphic functional calculus.>>I should qualify that. The holomorphic functional calculus exists in>complex Banach algebras, while the usual quaternions are an algebra>over the reals. Of course there's no trouble the OP, I would like to expand to someextent on the relationships between C (the complex *field*) and thethe *algebra* M(2,R) of 2x2 real matrices.Of course nothing of what I'm saying has a sound mathematical meaning,but IMO my observations yield a very natural point of view in somerespects, e.g. when dealing with some particular problems. (HoweverI'll give as a brief account as possible!)The point is that M(2,R) can be thought of (being isomorphic to) a2-dimensional complex algebra with a basis given by {1,chi}satisfying chi^2=1, ichi + chi i=0.On the other hand (the algebra isomorphic to) M(2,R) is a4-dimentional real algebra with a basis given by {1,i,chi,ichi}: inparticular these elements are not (the images of) the standard basisvectors of M(2,R).Note that in this sense the elements of M(2,R) are (numbers thatconstitute) another hypercomplex extension of C.Now, you can work abstractly with this extension of the ring ofcomplexes, and it is not important how you do intepret them. But ifyou want to have a direct expression of z+wchi (z,w in C) in terms ofsquare matrices, then a *possible choice* of i and chi for thetranslation is: i=[0 -1] chi=[1 0] [1 0], [0 -1].It's worth to notice that it is not a mere chance that the secondmatrix acts on a column vector as complex conjugation.Now, it is straightforward to realize that for A=z+wchi det(A)=|z|^2-|w|^2, A^{-1}=det(A)^{-1} (z*-wchi) if det(A)neq 0.(the latter identity works also if you abstractly *define* det(A) asabove). Interestingly the operatorial norm of A is ||A||=|z|+|w|.Now, as an example, let's find the solutions of the equation x^2=-1.Let x=a+bchi, a,b in C: the following two equations must besatisfied: a^2+|b|^2=-1, 2Re(a)b=0.If (i) b=0 then a^2=-1 => a=+i or a=-i; if (ii) bneq 0, then a=ik, kin R, |b|^2=k^2-1 => |k|>1. By allowing |k|>=1 one can express all thesolutions including those found at point (i) as x=ik+sqrt(k^2-1)e^{itheta}chi,period!If do the similar calculation for x^2=1, then you find that thesolutions found at the corresponding point (i) cannot be incorporatedin a general expression and one has x=1 or x=-1 or x=ih+sqrt(h^2+1)e^{iphi}chi, h in R.Another interesting exercise is to look for numbers/matrices I,X thatsatisfy the same identities as i,chi. (Since 1,-1 commute with everyelement of the algebra, then X must be chosen of the latter form!)Hope this was a TEASER!!Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore === <>sSHfTy;{Dhe&:+?b`9fUj5A~$gIYlYT0/$-asR-K~3S3[]q.R3YSmpR|$- GiZp>UN2a}!Fmw+%h}YL`!h_XXr5Q>_nGsY2_also:Pfeffer: The Riemann Approach to Integration (CUP)-- === calculating limits - Need Help!> I'm having difficulties solving these two limits.> I mustn't use L'hospital rule:> a) lim(x*(2^(1/x))-x) where x increases to infinite.> b) whereF:(0,infty)-->R , F(x):= x*(A^{1/x}-1) , with A>0 , A=/=1 lim_{z-->0}(1+z)^{1/z}= e , where ,,e is Napier's constant.Let z:= A^{1/x}-1 . Then x-->infty iff z--->0 .Also x=ln(A)/ln(1+z) . Therefore F(x)= ln(A)/(ln(1+z)^(1/z))and L=lim_{z-->0} supposing known the theory of Riemann integral, and inparticular case when x take only integer values. More precisely considerthe sequence with general term X_n:= n(A^{1/n}-1) . Let us assume thatA>1. On interval [1,A] take the division (D_n) (D_n) 1=x_0(n)infty iff norm ||D_n||:=max_{k=0,1,...,n}(x_{k+1}(n)-x_k(n))= =A*(A^(1/n)-1)-->0 .Take the continuous function f:[1,A]-->R , f(x)=1/x and considerthe integral sumS_n(f)=S((D_n), f )= SUM_{k=0 to k=n}(x_(k+1)-x_k(n))f(x_k(n)).We have X_n= S_n(f) . In this mannerlim_{n-->infty}X_n = lim_{||D_n||-->0}S_n(f)= =INTEGRAL_{t=1 to t=A}dt/t= ln(A).In your case === the partial difference integration> Why don't you just post the program? Ok. Here's a straight-forward Java implementation. Nothing fancy. > It just does the job. ___JSH It seems to work, at least for a few input values: # primes < 100 = 25> # primes < 1000 = 169That should be 168. That is, there are 168 primes up to and including 1000.James HarrisMy math discoveries, found for === on generation of large prime numbers> Summary of Euclid's proof: 1) Suppose that there is a largest prime; call it P> 2) Calculate N = (product of all primes from 2 to P, inclusive) + 1Are you trying to say that the set of all primes has no primes missing? That is vacuously true, and holds for {2,3,7}. If {2,3,7} is the setof all primes, which we are permitted to assume according to Euclid, then {2,3,7} is the set of all primes, bar none.Or are you trying to say and we assume that we know all primes <=P.> 3) N has one or more prime factors, all of which must be > P> 4) #3 contradicts #1, so #1 is wrong, so there is no largest primeOr the assumption in #2 was wrong.I never thought that Euclid's proof would be hard for people to grasp. I think I can only recommend that people look at Kummer's proof instead, as that's even simpler.Phil-- Unpatched IE vulnerability: DNSError folder disclosureDescription: Gaining access to local security zonesReference: === Subject: Re: Advanced techniques, non-polynomial factorization so that all the functions in (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) equal 0, when x=0.> We have been here many times. The problems is that a_1(x), a_2(x) andb_3(x) are not polynomials. Therefore, we do not know thatthe way in which the 49 distributes itself among thethree factors on the LHS is independent of x. Thus we cannotconclude that 7 divides (5 a_1(x)+ 7) for all x. Thus wecannot conclude that 7 divides a_1(x) for all x. -William === well-known in science and mathematics, but whilefinding roots of polynomials is typically the aim of the averageresearcher, polynomials themselves can be used as powerful tools foranalyzing the roots of *other* polynomials.The concepts are advanced, but can be approached by first consideringa basic example.The basic factorization to start is(c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = 49(x^3 + 5x^2 + 3x + 1)with the c's algebraic integers, notice that only two of the c's have7 as a factor.It might help to go the *other* way, and start with (d_1 x + 1)(d_2 x + 1)( d_3 x + 1) = x^3 + 5x^2 + 3x + 1and now multiply by 49.In the first example you're looking at a product and realizing thatfrom the distributive property a(b+c) = ab + ac, you know there's*one* way it could be produced, which is to multiply something likethe second example by 49.The distributive property is key here. Understanding it thoroughly,is of prime importance.Now notice that you can abstract from here as you're looking at*functions* of x, as introducingf_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, you have(f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1).Notice that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1as long as you're in a ring where 7 is not a factor of 1.Which is consistent with what was found before, as only two of thefunctions have the property that 7 is a factor.Now I'll move on to a more complicated example.Let(5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)where the a's are roots ofa^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)so they are functions of x, and since one of the roots equals 3 atx=0, I haveb_3(x) = a_3(x) - 3, so that all the functions in(5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)equal 0, when x=0.Those of you who find it hard to use the distributive property withthe *product* can imagine the factorization from *before* 49 beingmultiplied.It's harder to show here as the polynomial which defines the functionin that factorization is not displayable in general.So I started at the end, with 49 already multiplied because then I cangivea^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x).That slight change, starting at the end, means that you have tounderstand the distributive property fully and *trust* it.Now notice that I have the result that only two of the roots of thecubica^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)can have factors in common with 7, so the 49 splits between those two.What's so startling is that the result is for a *family* ofpolynomials as it applies for any algebraic integer x.James HarrisMy math discoveries, found for === efficiently do polynomial division? (I am looking for techniques similar to Karatsuba used formultiplication === maths not psych)> You say that the force on a charge due to an electric field acts>> instantaneously. Correct?>>Why do you ask what I am saying, when what I am>saying is quoted right above?>>I am saying:> as it enters a static electric field.>> So there is also an opposite force acting on the electrodes.> even if the electrodes are light years apart.>> IS THAT WHAT YOU ARE SAYING?I am saying: as it enters a static electric field.We have two electrodes - say 1 km apart.(Or a light year apart - if you insist)The potential difference is 1 million volts.(Or a zillion volts - if the distance is a light year)There is a small hole in the negative electrode.We inject an electron through this hole.When will a force act on the electron?I am still saying: as it enters a static electric field.But what are YOU saying?Not untill the electrode 1 km away feels the opposing force? IS THAT WHAT YOU ARE SAYING?Come on, make your point.What is the action time of the force on the electron?Why do you think the distance to the other electrode is relevant?How does the distance to the other electrode affectthis action time?Please don't say something like we don't know.Because we DO know.Do YOU know?The rest is === measures of errorSuppose that g(x) is proposed as an approximation of f(x) on [a,b]. Whatare the most popular ways of measuring how well g approximates f over thatinterval?Here are several measures. In each case, the smaller the measure is, thebetter the approximation is considered to be.AInf. The maximum of |absolute error| over the interval, where absolute error = g(x) - f(x).RInf. The maximum of |relative error| over the interval, where relative error = (absolute error)/f(x).A2. The root-mean-square of |absolute error| over the interval.R2. The root-mean-square of |relative error| over the interval.A1. The average of |absolute error| over the interval.R1. The average of |relative error| over the interval.All of these measures may be thought of as power means (also called Hoeldermeans). They have the form* ( Integral( |error|^p ) / (b-a) ) ^ (1/p)where the integral is taken with respect to x from a to b, and error iseither absolute or relative. Obviously, in the cases of A1 and R1, p = 1,and in the cases of A2 and R2, p = 2. The value of p is not so obvious,however, in the cases of AInf and RInf. But in the limit as p increaseswithout bound, the power mean gives simply the maximum, as needed in AInfand RInf. As such, for those cases, we may say that p = +oo.Here are some questions of mine.Are there any important measures of error in form * which use values of pother than 1, 2, and +oo?Are there any important measures of error which are not in form * ?Clearly, using p = 2 yields a measure which is intermediate between thosewith p = 1 and p = +oo. In that sense, p =2 represents a nice compromise.But is there anything really special about p = 2 (say, as opposed to p = 4or p = 3/2) ? (Of course, I grant that the integral is typically far easierto evaluate analytically when p = 2 than when p = 4 or 3/2. But I'mwondering if p = 2 is special for a more fundamental reason === of large prime numbersCorrect me if I'm wrong, but isn't the proof much simpler if you saysomething like:1) Assume P is the largest prime.2) Calculate P!+1 (I.E. the product of all numbers from 1 to P, plus one)3) Dividing that number by any number <= P will give a remainder of 14) Therefore, P!+1 is either prime or a multiple of a prime > PSaSW, Willem (at === record.>> I went to Indian Institute of Technology (IIT), Powai, Mumbai to> explain mechanism of my Action Device and to seek technical help. I> met Dr. Amitay Issac of Aerospace Engineering Department and I tried> to explain very basic component/idea of this action device. I have> given in my homepage what exactly I tried to convince him.>> http://www.geocities.com/actiondevice>> But he insisted that point B will shift its position along Y axis!.> I had to return in few minutes.>> Now I tried to convince again to Dr. G Arvind Rao of Aerospace> Engineering Department by email, but he also said that point B will> shift its position along Y axis !. Hmmm... did you consider that they could be right, and you could be wrong?Laura, where from you suddenly dropped in this mess? You just don'tknow, what is going on. I thought about this thousands of times inlast 13 months. I had posted idea of whole device in many newsgroup.This is just one of the basic component or idea behind this invention.At least this problem was not arised. And now suddenly this problempropped up.> Indian Institute of Technology is most prestigious college in India.> This institute gives people for Aviation Industry around the world.> And I just wonder, why so highly educated people fail to understand> such simple problem. Maybe, just maybe, they do understand it.Have you done elementary Geometry Laura? Take a look at my homepage.http://www.geocities.com/actiondevice> In fact, this is not problem at all. But what a tragedy, I am facing> such ridiculous problems.>> I can end my all problems anytime, but I am following the rules of> this battle, waiting game. Build a working model and submit it to them for examination.> Doesn't matter how much force it produces, as long as it proves that your> idea works.No, not yet. You just don't know what is going on around me. Thingsare under absolute control. You will never believe it.> I am just watching how the minds of highly educated people around the> world are controlled by that Supreme Force named God. Let me get this straight.... *God* doesn't want this device discovered? Why> not? And if not, what's stopping him from destroying you to make sure you> stay quiet?He does want this device to be discovered. This is exactly why Hecontrolled absolutely everything in my personal life. He navigatedthings in last 17 years in such a way that my thought process movesonly in one direction. He trained me to gain absolute power ofimagination.This device is very simple. But there is no victory withoutsufferings. And He has discovered His own ways to trap me.Things are being controlled very cleverly. Don't believe me? People in this NG will not answer clearly the question I have posed.Will point B move along Y axis in XY plane? It needs just yes/no.But they will remain silent(or they will be humorous). They willignore me. Because they are controlled.Laura, Watch Out Apocalypse In === issue>If mathematicians hadn't decided to break faith with you and the rest>of the world, probably there'd be a book, some popular work,>explaining the story.>>But how can you get that story if mathematicians are playing their>academic games?>>Bottom line: What I have works.>>So what if I sell my story and get rich. Psst, James, there is a very small market for stories about mathematics.Better find some way to work in spies and the CIA, and pretty girl agents,and such like. And sex. Sex always sells, even when the sex scenes areseparated by boring mathematical explanations. People just skip those.-- Wolf Kirchmeir, Blind River ON CanadaNature does not deal in rewards or punishments, but only in consequences.(Robert === LIGHTBULB?> The Ôproof' you did must be wrong somewhere as the equation doesn't> work with any value outside the range <-1,1>. I believe the error is> that the operations shown can only be applied to absolute convergent> series (Ôabsoluut convergente reeksen' in dutch). Since the series is> not convergent at all for a being outside <-1,1>, the whole proof is> nonsense. I'm sure someone else can explain this better...I posted virtually the same arguments at the www.johnpatrick.commessage board and recieved a similar response from The Truth.________________________________________________________ _____________>>Essential in your derivation is the step [(a + a^2 + a^3 . . .) -(a^2 + a^3 + a^4 . . .)] = a.But this equivalence only holds if the series a + a^2 + a^3 . . .converges, and it only converges for certain a, not for any a.< Suppose that g(x) is proposed as an approximation of f(x) on [a,b]. What> are the most popular ways of measuring how well g approximates f over that> interval?I guess it depends on the field of application, and also in theparticular application.In engineering, I believe the one used most frequently is themean-square error, since it is related to energy.> Here are some questions of mine. Are there any important measures of error in form * which use values of p> other than 1, 2, and +oo?I'm not familiar with any that are commonly used (I'm an electricalengineer -- maybe in other fields there might be)> Are there any important measures of error which are not in form * ?Not that I'm familiar with.> or p = 3/2) ? (Of course, I grant that the integral is typically far easier> to evaluate analytically when p = 2 than when p = 4 or 3/2. But I'm> wondering if p = 2 is special for a more fundamental reason than that.)Calculating the integral is usually irrelevant. What you want itfind conditions that guarantee that the error is minimized, notfinding out what the error is.For instance, when you solve an overdetermined set of linearequations, you are calculating the optimal solution; you arenot calculating the error (though you know that it is minimum,and you could calculate it after === Re: probability 2......> thank...you....very much....>> i think......if we only use 0-x-y-1>> in this case, probability is 1/8>> but, if we use 0-x-y-1, 0-y-x-1>> in this case, probability is 1/4>> which of case is right??>> === (sorry, maths not psych)>>And you have not provided any theory of E&M that allows any such>thing as a reverse field. Nor why there should be any kind of>speed limit involved. Nor why it should follow any such thing>as the kinetic energy formula observed in accelerators. Nor have>you provided a relation between energy and mass if you don't>accept relativity.>Socks>> radiation from an acceleraed charge!>> fields associated with a moving charge!>> The ÔBack EMF' concept.>> I would be most amazed if a moving charge DID NOT alter the field around> itself, wouldn't you?Quite.are accelerated.You KNOW the following, Henry.In an accelerator going at full efficiency, we KNOW thatbecause it looses this energy as synchrotron radiation in the bendsof the circuit.(Very obvious and easily measurable.)So we - and YOU - know that the RF-cavities never ceasesis only few mm/s below the speed of light.So why do you keep pretending that the E-field is notspeed approaches c, when you KNOW that isn't true?Another case of selective memory loss?What you admit knowing in one posting,you have forgotten in the next, === polynomial division?> (I am looking for techniques similar to Karatsuba used for> multiplication ...)long === don't you try and get a life instead of repeatedly swamping thisnewsgroup with endless variations of the same material? Is this what youmeant when you said you were going to turn up the volume? If so, why notpost in all caps? Maybe that will prove the contents are correct.--There are two things you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of popularity over === generation of large prime numbers> Correct me if I'm wrong, but isn't the proof much simpler if you say> something like: 1) Assume P is the largest prime.> 2) Calculate P!+1 (I.E. the product of all numbers from 1 to P, plus one)> 3) Dividing that number by any number <= P will give a remainder of 1> 4) Therefore, P!+1 is either prime or a multiple of a prime > PThat's fine. (Pedantically you could change the conclusion to either prime or a multiple of primes >P, but what you've said is correct, and sufficient for the proof.)Phil-- Unpatched IE vulnerability: Basic Authentication URL spoofingDescription: Spoofing the URL displayed in the Address barReference: === Subject: Re: Roman carvingWhy do these old posts keep showing up here throughsome address at mathforum.org? Is there some sort ofglitch at the Math Forum causing this to === Integrity> Maybe so, but in the mathematics community, truth takes precedence over> ego, which makes the field very different from business, politics, and> many other human endeavors. Well, I think that the major difference is that in mathematics it is muchmore difficult to get away with bull, because of the nature of thediscipline. Hand waving can get you very in business, politics,philosophy, etc.; not so in mathematics. Now my opinion is that ego is just as strong in the mathematics communityas elsewhere, as can be seen in lots === MATHEMATICIANS READ WITH HALF A LIGHTBULB?> I'm listening ... so tell me in English why I'm wrong.Do you know what it means for a series to converge?Are you talking about a series when === Question on generation of large prime numbers[...]|> The proof could be expressed|> in terms of a finite number of known primes, as Phil seems to have assumed|> it was, but that's not the way Richard expressed it -- he spoke explicitly|> of the product of *all* primes.||If you view it in terms of sets, subsets of Z, my view is perfectly standard |one, it's what Ribenboim calls Euclid's proof, and as such is applicable to |the premise that Richard was assuming.The proof you like is standard, but I don't believe your stance toward hisway of formulating it is standard.|Richard was the one who was|introducing|more assumptions. All I need is a finite set of primes. A maximum prime P,|and |the fact that we're in Z, gets me that. Euclid's proof just follows |immediately. Let all primes be all primes in that set, and the connection|is made.Not being as elegant as possible doesn't invalidate a proof. If he wants toprove that there are infinitely many primes, he is free to assume that thereare only finitely many of them.Starting with that assumption is unnecessary, but it's apparently a commonway to present the proof. For instance, Robert Wisner's _A Panorama ofNumbers_: But Euclid proved that such a thing [all numbers past a certain point being sieved out] does not happen. Here's how he went about it. He said (to himself, and later to the rest of mankind) that if the list of primes has an end, then we could stare at the complete list as 2 3 5 7 11 13 ... P where P is the largest prime, whatever its name. [...] This is essentially the proof of Euclid....The word essentially is his escape. :-) I also have a Ivan Niven bookwhich starts by assuming there is a largest prime, but he doesn't attributethe proof to Euclid.|This is why I was blathering about what all meant. The meaning is plain; all primes means all of them: {n : n is prime}.|> But Richard's conclusion that P has been shown not to be the largest|> prime number is also wrong; what has been shown is that the assumption|> of a finite number of primes is wrong.||Yup, which is why I said Nope..But his first step was to say that if P is the largest prime, then thereare finitely many primes. That's a correct deduction. If the conclusion ofthat step then leads to a contradiction, it is valid to conclude that thepremise of it was false too.|> The only reason to assume that|> at least one of the additional primes discovered must be larger than|> P is that we *thought* P was the largest prime, before we discovered them.||Euclid permits me to assume {2,3,7,13,43,139,3263443} is the finite set |of primes, with maximum prime P=3263443. In what way does |{2,3,7,13,43,139,3263443}, P=3263443 violate the premise Let P be the |largest prime?3263443 is not the largest prime. Obviously, there's no satisfying theassumption that P is the largest prime. Nor is it possible to satisfythe requirement that these are all the primes while leaving gaps!|In which case, surely Richard, as he originally stated his|argument should permit it too. However, _none_ of the primes that Euclid's |construction discovers is larger than P. If the premise is satisfied, but |the conclusion isn't, then there's a syllogistic error somewhere.But the premise is NOT satisfied. It is typically trickier to evaluate thesoundness of a proof by contradiction, since the assumption *isn't* evercorrect.Compare the proof with what is in Hardy and Wright, which they also callEuclid's proof: Let 2,3,5,...,p be the aggregate of primes up to p, and let q = 2.3.5. ... .p + 1. Then q is not divisible by any of the numbers 2,3,5,...,p. It is therefore either prime, or divisible by a prime between p and q. In either case, there is a prime greater than p, which proves the theorem.It is valid to reason in this way if the set of primes in question is allof the primes up to a given prime p. This is mainly what Richard does.It appears to me, then, that a lot of your objection hinges on the readernot being ready to regard as equivalent the fact that a set of naturalnumbers is finite, that it has a largest element P, and that it consists of(all) those elements which are <= P. I think anybody who is unable torecognize these as equivalent is not really ready to be reading Euclid'sproof. And these equivalences continue to hold even when we're consideringa counterfactual condition such as P being the largest prime.his version of it, but since his P is introduced as a hypothetical largestprime, references to it only make sense so long as we are still under theassumption that there is such a P. So in particular, the statement thatthere is a prime greater than P is still among the consequences of P beingthe largest prime. That's valid. True, the same reasoning can be used toshow that there is a prime > P without assuming that P is the largest primenumber.His last step, asserting that his original assumption is false, is correcttoo. It refers to P again, but simply denies the original assumption.|If Richard has simply added let all primes <=P be known to his premise|I wouldn't have jumped on it that way.I don't think this would help. Talking about what primes are known issubjective.Without the subjective reference, what's the alleged extra assumption?That all the primes <= P are actually among the set of all primes? Orvice-versa? It's true that he refers to a list, but I don't see anybetter antecedent for that than his reference to the primes.|However, if he had, then it would |_not_ have been Euclid's proof (and he claimed what followed was Euclid's |proof so I would have jumped on that instead),I think you would have been better off saying that instead. Of course, anerror of a similar nature is in Hardy and Wright. They don't start with acounterfactual assumption, but they do ask us to consider all the primesup to p, and as I understand it Euclid doesn't.I would guess that in fact the typical mathematician would consider all ofthese distinctions to be minor stylistic variations.|and wouldn't (on its own) |have been a proof of the infinitude of primes (as you've added another |assumption, so the proof by contradiction only disproves the _conjunction_|of the assumptions, not either one individually).Are you saying the hypothetical reader is supposed not to recognize thatthe one implies the other?|Ockham has some wise words for moments like this.I'm sure Ockham would have appreciated an elegant argument, but I doubt hewould say that an argument becomes invalid if it contains redundancies.[...]|This is why I asked Richard to define all primes. He gave no definition.|Euclid's gave one, which permits {2, 3, 7}. Which is why I don't view my|insistance that we all know what all primes means to be blathering.Euclid did not define all primes, let alone define it to permit gapswhere there happen to be additional primes.|Richard's insistance that all primes up to P be known and there be no primes|greater than P is cannot work directly, as is, to prove that there's a prime |greater than P, as the contradiction doesn't tell you whether there be no |primes greater than P is false or all primes up to P be known is known.||Adding extra assumptions is almost always a bad thing to do when performing|a proof by contradiction.I admit that for someone lacking the mathematical competence to recognizethat a set of natural numbers being finite, its having a largest element,and its consisting of precisely those elements <= its maximum element areequivalent, this presentation would not be enough of a proof.I'm just having a hard time seeing any of these critiques as a meaningfulobjections to the proof. We should not be trying to create the impressionthat in mathematics, it's normal to engage in hair-splitting like that.You already have Richard chalking up a new, phony reason to think he's notquite cut out to be a mathematician (not that that necessarily makes anydifference). That strikes me as silly.Keith === research so far in using optic ßow for aerial robot navigation hasbeen mainly experimental (seehttp://www.pages.drexel.edu/~weg22/research.html if interested). However, I want to get more involved in the theoretical aspects of itby creating an optic ßow sensor numerical model. When the same inputis fed into my model and a sensor (www.centeye.com for example), theoutput should be the same. I have never developed any numericalmodels before and I was just wondering if anyone out there could offersome advice or suggestions on where to begin? Also, I wouldappreciate any information that can be shared about models that arealready in === you did must be wrong somewhere as the equation doesn't> work with any value outside the range <-1,1>. I believe the error is> that the operations shown can only be applied to absolute convergent> series (Ôabsoluut convergente reeksen' in dutch). Since the series is> not convergent at all for a being outside <-1,1>, the whole proof is> nonsense.>> I'm sure someone else can explain this better...>> I posted virtually the same arguments at the www.johnpatrick.com> message board and recieved a similar response from The Truth.> ______________________________________________________________ _______>>Essential in your derivation is the step [(a + a^2 + a^3 . . .) -> (a^2 + a^3 + a^4 . . .)] = a.> But this equivalence only holds if the series a + a^2 + a^3 . . .> converges, and it only converges for certain a, not for any a.<<>> There is only one element of infinity that is not common to both> infinite series and that is a to the first power, and thus the> difference between the two infinite series is simply a to the first> power.>> If you will take the time to explain in English why the a I've> isolated is not the certain a that you require, I'll listen, but> you've otherwise said nothing.> --------------------------------------------------------------- --------->> I'm listening ... so tell me in English why I'm wrong.The so-called associative property of the addition a + ( b + c ) = ( a + b ) + callows us to write both sides of the equality as a + b + c.This property is *not* valid for so-[badly]-calledinfinite sums. You cannot write something like a1 + ( a2 + a3 + ... ) = (a1 + a2) + a3 + ...In fact, the thing a1 + a2 + a3 + ....is not even a sum to begin with!It is a so-called limit of a series of partial sums: s1 = a1 s2 = a1 + a2 ... sn = a1 + a2 + ... + anIf this series has a limit for n -> infinity, then one isallowed to use the abbreviation limit(sn; n -> infinity) = a1 + a2 + a3 + ...The property of having a limit in the previous sentenceis something that can === of>S^n x S^m. Can anyone give me some ideas? I am thinking of using the MV>sequence (exact):>>... -->H_n(A and B)-->H_n(A) + H_n(B)--> H_n(A union B)-->H_(n-1) (A and>B) --> ...I'm not sure that's the best approach but you could make it work.What will you choose for A and B? You might try decomposing S^m = C union D and then take A = S^n x C, B = S^n x D. You'llneed to do the calculation for all m this way so that you canuse induction. Don't forget to use the naturality of the MV sequenceso that you can identify the maps between the groups.By the way, Mr V just died about a year ago -- at the time he was theoldest living person in Austria (110.8 === algebra book?> Looking for a good lie algebra book at the introductory level.Humphreys, James E. Introduction to Lie algebras and representation theory. (2.ed)Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978.should still serve as a good introduction. It requires not muuchmore === good lie algebra book?> Looking for a good lie algebra book at the introductory level.Varadarajan's Lie groups, Lie algebras, and their representations andSerre's Lie Algebras and Lie === important measures of error in form * which use values of p> other than 1, 2, and +oo?There are some harmonic means that, I think, involve expressions with p=-1.$.02 -Ron === factorization> Polynomials are well-known in science and mathematics, but while> finding roots of polynomials is typically the aim of the average> researcher, polynomials themselves can be used as powerful tools for> analyzing the roots of *other* polynomials. The concepts are advanced, but can be approached by first considering> a basic example. The basic factorization to start is (c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = > 49(x^3 + 5x^2 + 3x + 1) with the c's algebraic integers, notice that only two of the c's have> 7 as a factor. It might help to go the *other* way, and start with (d_1 x + 1)(d_2 x + 1)( d_3 x + 1) = > x^3 + 5x^2 + 3x + 1 and now multiply by 49. In the first example you're looking at a product and realizing that> from the distributive property a(b+c) = ab + ac, you know there's> *one* way it could be produced, which is to multiply something like> the second example by 49. The distributive property is key here. Understanding it thoroughly,> is of prime importance. Now notice that you can abstract from here as you're looking at> *functions* of x, as introducing f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, you have (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1). Notice that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1 as long as you're in a ring where 7 is not a factor of 1. Which is consistent with what was found before, as only two of the> functions have the property that 7 is a factor.> This part so far is OK.> Now I'll move on to a more complicated example. Let (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where the a's are roots of[***] a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so they are functions of x, and since one of the roots equals 3 at> x=0, I have b_3(x) = a_3(x) - 3, so that all the functions in (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) equal 0, when x=0. Those of you who find it hard to use the distributive property with> the *product* can imagine the factorization from *before* 49 being> multiplied.> As in the first example you gave above, the 49 can bedistributed among the three factors in several differentways. There is no justification for your implied claim below that two constant terms 7 should be error.> It's harder to show here as the polynomial which defines the function> in that factorization is not displayable in general. So I started at the end, with 49 already multiplied because then I can> give a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). That slight change, starting at the end, means that you have to> understand the distributive property fully and *trust* it. Now notice that I have the result that only two of the roots of the> cubic a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) can have factors in common with 7, so the 49 splits between those two.> When you were discussing the simpler polynomial above, yousaid that the coefficients c_1, c_2, c_3 were algebraic integers,so presumably you are talking about the same thing here. Thatis, you are saying the a's are algebraic integers and two of themare divisible by 7 in the ring of algebraic integers. Thus assume a_1/7 is an algebraic integer; equivalently a_1 = 7*b_1,where b1 is an algebraic integer. But since you are saying thata_1 is a root of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) we must have that b_1 is a root of 7^3*b^3 + 3*(-1 + 49*x)*7^2*b^2 - 7^2*(2401*x^3 - 147*x^2 + 3*x) = 0.Now divide through by 7^2: 7*b^3 + 3*(-1 + 49*x)*b^2 - (2401*x^3 - 147*x^2 + 3*x) = 0.Finally, let x = 1: 7*b^3 + 144*b^2 - 2257 = 0.This polynomial is primitive, irreducible, and non-monic. Thereforenone of its roots can be algebraic integers. Therefore a_1/7 isnot an algebraic integer. Therefore a_1 is not divisible by 7. You continue to think that your proof that a_1 *is* divisible by7 is valid. If it were, it would imply a mathematical contradiction.Mathematics would be inconsistent. There is no sense in claimingthat the algebraic integers are incomplete. They are a perfectly well-defined subset of the real numbers. Something cannot both be inthat subset and not in that subset. Please explain your own conclusions on this. > What's so startling is that the result is for a *family* of> polynomials as it applies for any algebraic integer x.> For example, x = 1, which is what I used above. Your proof implies a contradiction, so it is either incorrect or math is inconsistent. I have identified exactlyabove where you have made an unjustified assumption. Your argument here is wrong. Nora B. James Harris My math discoveries, found for profit> === TRADING THE MARKETS BY UNIFYING CYCLES AND === sin(1)/1 + sin(2)/2 + sin(3)/3 + sin(4)/4 + ... converge?How does one prove convergence (or divergence)?. If it converges what is a good way to estimate its value?Jim Buddenhagen------------To reply copy jbuddenh@REMOVEtexas.net === censorship, counting primes> Some of you were probably surprised to learn that I did indeed find a> way to count prime numbers by integrating a partial difference> equation. Some of you probably STILL doubt that no one else in> recorded history has managed such a feat because you need to believe> in mathematicians. Numerous others have created prime counting functions. Take a look athttp://mathworld.wolfram.com/PrimeCountingFunction.html> But my point is that mathematicians have gone rogue and act against> the needs of society by de facto censorship of information that they> don't think makes them look good, like the information about my> partial difference equation. What do I mean by de facto censorship? Well, besides the active activity, like webpages labeling me a crank,> there's the passive act of refusing to acknowledge the discovery> itself. After all, it's very compact, as here are the instructions, yet again: dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))], S(x,1) = 0. And p(x, y) = ßoor(x) - S(x, y) - 1, and you get S as the sum of dS> from dS(x,2) to dS(x,y).I did a little checking, and this is just a rehash of Legendre'sFormula.http://mathworld.wolfram.com/ LegendresFormula.html> That's it. That's the knowledge which mathematicians have purview> over, in terms of the expectation from society that important> information of a mathematical nature will be acknowledged by> mathematicians. Note that it's a *discrete* function, so for you programmers that> means you need to use int's or long's or some discrete variable type.> Also, if you wish to implement it, please sum from dS(x,2) up to and> INCLUDING dS(x,y). Now if you're a programmer or have been taught as a programmer, did> you ever get an assignment to count prime numbers? Now then, think about kids currently in school who I doubt will see> the method I've just shown you, unless maybe they're out on Usenet> reading my posts, because the mathematical establishment thinks it can> ignore my results. Have I contacted mathematicians? Yes. I've contacted mathematicians all over the world.My 6 year old niece can contact mathematicians all over the world. The real question is how many of them replied.> But you see, what benefit do they see to their society by allowing> that someone NOT a mathematician found such a result? Worse, I have other math knowledge. But mathematicians can let the> world be convinced I'm just a crank, most of them passively just> sitting by, and keeping quiet about my results--after all, that's> quite effective, eh? Then they have de facto censorship because people BELIEVE they> wouldn't do such a thing if my work were important!!!> So you have a standstill with me pushing my research, and a few> mathematicians actively fighting its acceptance on Usenet, while most> just do their best to ignore it. For instance, I contacted Georgia Tech and talked to a Professor Ernie> Croot, giving him more information about my prime counting research> than I've posted here. He replied back *once*, and seemed friendly> enough. I answered him and awaited further replies. After some weeks> I James, No, I haven't gotten around to looking at it. I'll let you know> when I do. Best, ERnie Professor Croot: Just checking to see if you still have any interest in my find of a way to > count prime numbers by integrating a partial difference equation, as I > haven't heard from you since my last reply. If you've lost interest can you refer me back to the professor who sent me > to you because I'd like to check ------------------------- Intellectual laziness is about deciding> ahead of time what you wish to believe,> and daring God to be different.> http://lostincomment.blogspot.com/> Will I ever hear back from Professor Croot? Well, consider the> evidence: I've given something new, a partial difference equation integration> for counting prime numbers, a first in recorded human history.Wrong there. Difference equations are not integrated.> Professor Croot has had some time to consider my work, but now begs> off, claiming not to have looked at it.If he did look at it, he would probably just pile it on top of all theother letters from amatures claiming to have a breakthrough whenactually the result is either wrong or previously known.> It turns out that he's a first year professor and I was referred to> him by another professor at Georgia Tech who *asked* him to look over> my work. I daresay that Professor Croot lied in his email. That's a professor at Georgia Tech. So I'm an *independent* researcher, which means that mathematicians at> universities have a lot of power when it comes to acceptance of my> work, but may see little point in helping me. That leaves me Usenet, where there are mathematicians, like David> Ullrich, a tenured math professor at Oklahoma State University, to> TELL people that my work is useless or wrong.In this case, it is useless. There are faster prime counting functionsout there.> Yet, I found a partial difference equation that you can integrate to> count prime numbers which NO ONE ELSE in recorded human history has> managed.Again, partial difference equation are NOT integrated.> I have other mathematical research, but as long as mathematicians> stick to their guns, who gets to hear it? Sure I can talk about it on Usenet, and watch as posters malign my> work, lie and generally act like asses, knowing that others will just> sit, and wait, waiting for mathematicians in the mainstream to let> them know that it's important. To a large extent I now censor my *own* work in talking about it, as I> focus on things that are hard for people to lie about, and hope for> the best.Go for the ultimate self-censorship: stop talking, writing andposting.> Right now, locked inside of me is information that could be lost to> humanity because I'm the genius maligned, trapped by a system that> lets mathematicians get away with hurting the society that feeds and> clothes them, by de facto censorship.> I know things, important things, that you may never know about> numbers, and mathematics. Mathematicians are no longer part of decent society, but are now rogue> having taken their own path into darkness. Don't believe me?I don't believe you.> Check my instructions for integrating that partial difference> equation. Check for yourself. > James Harris My math discoveries, found for profit> === on prime counting issueIn sci.physics, Christian Bau> was a somewhat tongue-in-cheek contest I sponsored 2 months back>> that produced a few bizarre results and some interesting>> algorithms. (However, Christian Bau has a better one anyway,>> although he didn't submit that particular one for my contest.>> Perhaps it was because my contest was unworthy thereof. :-) ) No, it was not finished at that time, and I have to find some spare time > to improve it anyway. What I am quite interested in at the moment is > that there seems to be a substantial improvement possible if you want to > calculate pi (N) for many different values of N, for example N = k * 10^14 for 1 <= k <= 10000. My implementation should take about O (N^(2/3)) to find pi (N). However, > it might be possible to find pi (x) for n different values x <= N in > about O (N^(2/3)) * sqrt (n) instead of O (N^(2/3)) * n.I suppose it might depend in part on the value of max(N_i),where N_i are the numbers fed into pi(N). I really don'tknow, and haven't researched the issue.Good luck. :-)-- #191, ewill3@earthlink.netIt's still === sci.math, Hauke Reddmann:> There are also multiple solutions -- although one of them>> should jump right out and bite the OP (were the value to>> have any teeth, that is -- and this particular numeral>> obviously hasn't hatched any yet :-) ). Just saw that my handy plotting software (MathGV) has> sec(x) built-in. Looks like an EPR spectrum on crack :-)Or an EPR spectrum of someone *on* crack... :-)-- #191, ewill3@earthlink.netIt's still legal to go === positive real sequence, such that the series sum( 1/u_k,> k=1..infinity) converges.> Let T_n = u_1 + ... + u_n. Prove that the series sum (n/T_n,> n=1..infinity) converges and that sum (n/T_n, n=1..infinity) <= 2 *> sum( 1/u_k, k=1..infinity).> advance.D.8esol.8e je t'avais oubli.8e !! (Solution du Monier, 3.2.23):By Cauchy-Schwarz inequality applied to the vectors (sqrt(u_1), ...,sqrt(u_n)) and (1/sqrt(u_1),...,n/sqrt(u_n)),(1+2+...+n)^2 <= (u_1+...+u_n)*(1/u_1+2^2/(u_2)^2+...+n^2/(u_n)^2).It follows that: (2n+1)/(u_1+...+u_n) <= 4*(2n+1)/(n^2*(n+1)^2)*sum(k^2/u_k,k=1..n); summing for N>0,sum((2n+1)/(u_1+...+u_n), n=1..N) <= 4*sum((2n+1)/(n^2*(n+1)^2),n=1..N)*sum(k^2/u_k, k=1..n) = 4*sum(k^2/u_k*sum((2n+1)/(n^2*(n+1)^2),n=k..N), k=1..N) <= 4*sum(k^2/u_k*1/k^2, k=1..N)because:sum((2n+1)/(n^2*(n+1)^2), n=k..N) = sum(1/n^2-1/(n+1)^2, n=k..N) = 1/k^2 -1/(N+1)^2 <= 1/k^2,whence:sum((2n+1)/(u_1+..+u_n), n=1..N) <= 4*sum(1/u_k, === %L;%tM$D+%zkQ$zp8f/vAx*mr6T79jgxh,SC!$,8.r%HBe}KZ)iMb$tB.Z,30 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21:> To every polytope, we associate a graph in the following way: take its> vertices as nodes. The nodes are joined by an edge if and only if the> corresponding vertices are adjacent. How can we decide, given any graph, whether it is the graph of some> 0/1-polytope or not? > If there is no exact criterion known, is there a good sufficient one?I have no idea, but an obvious necessary condition is that G is (log_2 |V(G)|)-connected.-- David Eppstein http://www.ics.uci.edu/~eppstein/Univ. of California, Irvine, School of Information & Computer === ?> Does the series sin(1)/1 + sin(2)/2 + sin(3)/3 + sin(4)/4 + ... converge?> How does one prove convergence (or divergence)?. If it converges what> is a good way to estimate === difficulty proving that ||v||_p <= ||v||_2 for p>=2, here v> is a vector in R^n. Prove it when ||v||_p = 1 and then remember === About naturalsLet S(n) the decimal sum of n, this isS(17)=1+7=8,S(98)=9+8=17I have the following question: There are a, b and c, natural numbers, suchthatS(a+b)<5S(a+c)<5S(b+c)<5 andS(a+b+c)>50 ?I think that there are not.(Exuse-me I write english very bad)Pepe === integration> Why don't you just post the program?>> Ok. Here's a straight-forward Java implementation. Nothing fancy. >> It just does the job. ___JSH>> It seems to work, at least for a few input values:>> # primes < 100 = 25>> # primes < 1000 = 169 That should be 168. That is, there are 168 primes === Re: Relativity is based on assumption.>assumption, and is intuitive as well. Making the assumption that the time it>takes for a signal to reach an object is the same as the time it take for>the signal to return, when in the meantime [sic] you've moved away or>toward the object [...]Motion is relative. You were standing still. It was the object thatwas === large prime numbers> Starting with that assumption is unnecessary, but it's apparently a common> way to present the proof. For instance, Robert Wisner's _A Panorama of> Numbers_: But Euclid proved that such a thing [all numbers past a certain point> being sieved out] does not happen. Here's how he went about it. He said> (to himself, and later to the rest of mankind) that if the list of> primes has an end, then we could stare at the complete list as 2> 3> 5> 7> 11> 13> ...> P where P is the largest prime, whatever its name.> [...]> This is essentially the proof of Euclid.... The word essentially is his escape. :-) I also have a Ivan Niven book> which starts by assuming there is a largest prime, but he doesn't attribute> the proof to Euclid.Forget the largest prime thing. That's what's confusing everyone.Magnitude of the primes is _utterly_ irrelevant to Euclid's proof.In black and white - and you may quote me on this -EUCLID DID NOT MAKE REFERENCE TO THERE BEING A LARGEST PRIME IN HIS PROOFOF THE INFINITUDE OF PRIMES. (Of course, Euclid had no concept of infinite, as such, so he didn't word it that way.)The only time he refers to magnitude is that of the set size, not the numeric size of the elements. i.e the set with 4 elements is larger in magnitude than the set with 3 elements.> |This is why I was blathering about what all meant. The meaning is plain; all primes means all of them: {n : n is prime}.I think the meaning is plain. However, I, like Euclid, think that {2,3,7}is a perfectly valid example for what could be posited as a finite set of all primes. i.e. in taht case {2,3,7} _is_ all primes. (And that leads to a contradiction, and therefore it isn't.) There seems to be as muc a fundamental misunderstanding of how proof by contradiction works as well as a misunderstanding of Euclid's proof.If you have a problem with {2,3,7} being posited as all primes, then youhave a problem with the mechanics of proof by contradiction.> |> But Richard's conclusion that P has been shown not to be the largest> |> prime number is also wrong; what has been shown is that the assumption> |> of a finite number of primes is wrong.> |> |Yup, which is why I said Nope.. But his first step was to say that if P is the largest prime, then there> are finitely many primes. That's a correct deduction. If the conclusion of> that step then leads to a contradiction, it is valid to conclude that the> premise of it was false too.Or one of the other premises. Richard introduced other assumptions, not realising that he'd done so.> |> The only reason to assume that> |> at least one of the additional primes discovered must be larger than> |> P is that we *thought* P was the largest prime, before we discovered them.> |> |Euclid permits me to assume {2,3,7,13,43,139,3263443} is the finite set > |of primes, with maximum prime P=3263443. In what way does > |{2,3,7,13,43,139,3263443}, P=3263443 violate the premise Let P be the > |largest prime? 3263443 is not the largest prime. I'm sorry, I thought we were playing a game of proof by contradiction.As far as I cen tell immediate gainsaying is not a valid move in that game.If the set of all primes is {2,3,7,13,43,139,3263443} then 3263443 _is_ the largest prime. That's a simple unassailable mathematical fact.Euclid's proof does not say: Let PP be the set of all primes, but PP musn't be {2,3,7,13,43,139,3263443}.does it? It says (when reworded in more modern language): Let PP be the set of all primes.Why do you have a problem with {2,3,7,13,43,139,3263443}?Euclid didn't. Kummer didn't. I don't.> Obviously, there's no satisfying the> assumption that P is the largest prime. Nor is it possible to satisfy> the requirement that these are all the primes while leaving gaps! |In which case, surely Richard, as he originally stated his> |argument should permit it too. However, _none_ of the primes that Euclid's > |construction discovers is larger than P. If the premise is satisfied, but > |the conclusion isn't, then there's a syllogistic error somewhere. But the premise is NOT satisfied. It is typically trickier to evaluate the> soundness of a proof by contradiction, since the assumption *isn't* ever> correct. Compare the proof with what is in Hardy and Wright, which they also call> Euclid's proof:>> Let 2,3,5,...,p be the aggregate of primes up to p, and let q = 2.3.5. ... .p + 1. Then q is not divisible by any of the numbers 2,3,5,...,p. It is> therefore either prime, or divisible by a prime between p and q.> In either case, there is a prime greater than p, which proves> the theorem.That's quite a way from Euclid's formulation.It presupposed that you can generate all primes up to P.Euclid's proof doesn't.I don't care that one can generate all primes up to P trivially, and can prove it can be done pretty trivially, it's just _unnecessary_ as part of a proof of the infiniteness of the set of primes.Euclid's proof was that the set of primes is larger than any finite set. At no point did it make reference to the magnitude of any of theelements in the set. (except that primes aren't units, of course.)> It is valid to reason in this way if the set of primes in question is all> of the primes up to a given prime p. This is mainly what Richard does. It appears to me, then, that a lot of your objection hinges on the reader> not being ready to regard as equivalent the fact that a set of natural> numbers is finite, that it has a largest element P, and that it consists of> (all) those elements which are <= P. I think anybody who is unable to> recognize these as equivalent is not really ready to be reading Euclid's> proof. And these equivalences continue to hold even when we're considering> a counterfactual condition such as P being the largest prime.No. My objection is to people presupposing they know what all means, when they've not considered what assumptions they've made in order to come up with that meaning.> his version of it, but since his P is introduced as a hypothetical largest> prime, references to it only make sense so long as we are still under the> assumption that there is such a P. So in particular, the statement that> there is a prime greater than P is still among the consequences of P being> the largest prime. That's valid. True, the same reasoning can be used to> show that there is a prime > P without assuming that P is the largest prime> number.>> His last step, asserting that his original assumption is false, is correct> too. It refers to P again, but simply denies the original assumption.But he had extra unstated assumptions. That's what I've been jumping on.Repeatedly. Proof by contradiction denies one of the assumptions, but doesn't tell you which one is denied. As I have said repeatedly.> |If Richard has simply added let all primes <=P be known to his premise> |I wouldn't have jumped on it that way. I don't think this would help. Talking about what primes are known is> subjective.Not really, would assigned make you happy? The set of all primes is the set of all known primes in this proof. I've said that repeatedly. The set-theoretic notation for what my sentences expressed would beno different if I included or excluded the word known. I was simply trying to avoid the naked word all as people immediately misinterpret that based on their knowledge about the primes. > Without the subjective reference, what's the alleged extra assumption?> That all the primes <= P are actually among the set of all primes? This is why I jump on people's wording - the above is vacuously true as worded (and uses the naked term all twice, which immedately biases the inexpert reader as to what it might refer to). But yes, that is the assumption. It is possible to deny that clause and still prove the infiniteness of the primes using Euclid's proof.> Or> vice-versa? The proof relied on both directions, but you only assume one, you derive the other quite easily.> It's true that he refers to a list, but I don't see any> better antecedent for that than his reference to the primes. |However, if he had, then it would > |_not_ have been Euclid's proof (and he claimed what followed was Euclid's > |proof so I would have jumped on that instead), I think you would have been better off saying that instead. if he had... . He hadn't, so I didn't.> Of course, an> error of a similar nature is in Hardy and Wright. They don't start with a> counterfactual assumption, but they do ask us to consider all the primes> up to p, and as I understand it Euclid doesn't.Yup, Euclid asks us to consider an arbitrary finite list.Euclid, like Kummer (whose proof hasn't been distorted over time)_doesn't_ even require _2_ to be in the list of primes.This comes a shock to many people, but it's God's honest truth. Euclid's proof doesn't presume _any_ particular number is prime, not even 2.When people see Let p1 I would guess that in fact the typical mathematician would consider all of> these distinctions to be minor stylistic variations. |and wouldn't (on its own) > |have been a proof of the infinitude of primes (as you've added another > |assumption, so the proof by contradiction only disproves the _conjunction_> |of the assumptions, not either one individually). Are you saying the hypothetical reader is supposed not to recognize that> the one implies the other? |Ockham has some wise words for moments like this. I'm sure Ockham would have appreciated an elegant argument, but I doubt he> would say that an argument becomes invalid if it contains redundancies.But, as I said before, if they do introduce new assumptions, which this did, then it throws a spanner into the works when it comes to proof by contradiction.With the assumption, you prove A |= B, without the assumption you prove |= B.In order to get |= B from the first you need |= A. I.e. it's _not_ a proof of B until you add a proof of A.If you bring Euclid up to date with terminology, then you can't get much more elegant than Euclid's proof, IMHO. Kummer's is the closest to that, but is worded in terms of his ideal numbers (and therefore applies to more general rings, not just the integers).> [...]> |This is why I asked Richard to define all primes. He gave no definition.> |Euclid's gave one, which permits {2, 3, 7}. Which is why I don't view my> |insistance that we all know what all primes means to be blathering. Euclid did not define all primes, let alone define it to permit gaps> where there happen to be additional primes.I have no idea what he said in Greek, but the usual translation is assigned primes. He made no reference to gaps, at all.Euclid's construction permits _any_ list of primes. It makes _no_ statement about the magnitude either of any of the primes in the finite set or of thenewly proved to exist prime(s). e.g. {3} yeilds {2}. {5} yeilds {2,3}. {3,5} yeilds {2}. (Yup, Euclid's proof doesn't even require a plurality of primes; anyone who told you it did was deceiving you.)> |Richard's insistance that all primes up to P be known and there be no primes> |greater than P is cannot work directly, as is, to prove that there's a prime > |greater than P, as the contradiction doesn't tell you whether there be no > |primes greater than P is false or all primes up to P be known is known.> |> |Adding extra assumptions is almost always a bad thing to do when performing> |a proof by contradiction. I admit that for someone lacking the mathematical competence to recognize> that a set of natural numbers being finite, its having a largest element,> and its consisting of precisely those elements <= its maximum element are> equivalent, this presentation would not be enough of a proof.I'm glad I don't have the mathematical competence to think that a set of natural numbers a) being finite b) having a largest element c) consisting of precisely those elements <= its maximum elementare equivalent.{2,3,7} satisfies (a), and (b), but does not satisfy (c).So - are you mathematically competent enough to think (a), (b), and (c)are equivalent? > I'm just having a hard time seeing any of these critiques as a meaningful> objections to the proof.What is the proof? The original leaves us with A |= B. That's _not_ a proof of B. Sure, we know that A is provable, but as it stands without aproof of A, we don't have a proof of B. (e.g. all things that are dependent on RH aren't proven yet.) > We should not be trying to create the impression> that in mathematics, it's normal to engage in hair-splitting like that.> You already have Richard chalking up a new, phony reason to think he's not> quite cut out to be a mathematician (not that that necessarily makes any> difference). That strikes me as silly.It's unnecessary to introduce things into a proof that are not necesary for the proof. That strikes me as silly. I saw it, I said it. If it takes hair-splitting to separate an unconditional proof from a conditional proof reliant on an unproved assumption, then split hairs I will.Phil-- Unpatched IE vulnerability: Click hijackingDescription: Pointing IE mouse events at non-IE/system windowsReference: http://safecenter.net/liudieyu/HijackClick/ HijackClick-Content.HTMExploit: http://safecenter.net/liudieyu/HijackClick/HijackClick2- === infinity ?Julien Santini a .8ecrit dans le message de> Does the series sin(1)/1 + sin(2)/2 + sin(3)/3 + sin(4)/4 + ...converge?> How does one prove convergence (or divergence)?. If it converges what> is a good way to estimate its value?> Abel's rule>>OK what about === counting issueIf mathematicians hadn't decided to break faith with you and the rest>of the world, probably there'd be a book, some popular work,>explaining the story.>>But how can you get that story if mathematicians are playing their>academic games?>>Bottom line: What I have works.>>So what if I sell my story and get rich. Psst, James, there is a very small market for stories about mathematics.> Better find some way to work in spies and the CIA, and pretty girl agents,> and such like. And sex. Sex always sells, even when the sex scenes are> separated by boring mathematical explanations. People just skip those.A very small market in today's world can be worth millions of dollarsUS.The bottom line is that what I have works, people expectmathematicians to report on discoveries, but they are not doing theirjobs.It's easy to check using Google. Go search on partial differenceequation which can verify for you that they are real. Then search onprime counting or counting primes to see if ANYONE besides me hasever used a partial difference equation to count prime numbers.For those wondering what they might do to help, I think that maybesending an email to some news organization might have an impact. Forinstance, you can email TIME magazine at letters@time.com, and whoknows what might happen?James HarrisMy math discoveries, found for === Relativity is based on assumption.>What experimental evidence?Transverse Doppler effect; Relativistic corrections to the spectrumof the Hydrogen atom; all experimental evidence for QED, which conßatesto evidence for Relativity; E = mc^2 directly observed in Hiroshimapermanently settling the issue; relativistic momentum and energyhalf life of 15 minutes (neutrons) to travel light years for thousands ofyears across the cosmos to reach Earth; Michelson-Morley experiment;the constitutive relations D = epsilon_0 E, B = mu_0 H.search.yahoo.com/search?p=Evidence For Relativity>Moving clocks running slow? They don't.Directly observed to do so, in fact.>The GPS clocks run fast.... precisely as predicted by Relativity, and in the very amount predictedto do === x/(tanx) [0,pi/2]?> Ray Steiner> I came up with another way of showing that x/tan x has no elementary> antiderivative.> We need only one result from Wiener's 1997 paper:> arcsin(x)/x does not have an elementary antiderivative.>> Let I = int(x/tan x dx)= int (x cot x dx)> Use integration by parts to get> I = x ln(sin x) - int( ln(sin x) dx)> Let I2= int( ln(sin x) dx)>> Let u= sin x, x = arcsin(u), dx = 1/sqrt(1-u^2) du> Then> I2= int ( ln(u)/sqrt(1-u^2) du)>> Finally, use parts again to get> I2= ln(u) arcsin(u) - int(arcsin(u)/u du).> So, by Wiener's result, the original integral is not elementary.>> More results:> By exactly the same method one can show that> I3 = int (x tan x dx) is not elementary.> Now, let's substitue u= tan x, x = arctan u, dx= 1/ (u^2 + 1) du in I3.> Then it reduces to> I4 = int( u*arctan(u)/(1+u^2) du).> so the second integral of my previous post is non-elementary.> Finally, consider> I5= int ( (arctan(x))^2 dx).> By parts, one can reduce it to integrating I4, so I5 is also> non-elementary.> way of doing it.Again We need the result from Wiener's 1997 paper:arcsin(x)/x does not have an elementary antiderivative.In int(arcsin(x)/x dx) substitute x= sin u, dx= cos u du; arcsin x= uThen it becomesint( u cos u/sin u du) which is int(u/tan u).If the latter were elementary thenint(arcsin(x)/x dx) would be elementary, which is not the case.So the result follows easily.BTW, if we substitute x = cos u in the same integral, we alsofind that int(x tan x dx) is === factorizations> Polynomials are well-known in science and mathematics, but while> finding roots of polynomials is typically the aim of the average> researcher, polynomials themselves can be used as powerful tools for> analyzing the roots of *other* polynomials. The concepts are advanced, but can be approached by first considering> a basic example. The basic factorization to start is (c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = > 49(x^3 + 5x^2 + 3x + 1) with the c's algebraic integers, notice that only two of the c's have> 7 as a factor. It might help to go the *other* way, and start with (d_1 x + 1)(d_2 x + 1)( d_3 x + 1) = > x^3 + 5x^2 + 3x + 1 and now multiply by 49. In the first example you're looking at a product and realizing that> from the distributive property a(b+c) = ab + ac, you know there's> *one* way it could be produced, which is to multiply something like> the second example by 49. The distributive property is key here. Understanding it thoroughly,> is of prime importance. Now notice that you can abstract from here as you're looking at> *functions* of x, as introducing f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, you have (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1). Notice that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1 as long as you're in a ring where 7 is not a factor of 1.> This is the only way to divide both side by 49 if the f_i arelinear functions of x. If you use more complicated functionsall bets are off.> Which is consistent with what was found before, as only two of the> functions have the property that 7 is a factor. Now I'll move on to a more complicated example. Let (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)> More complicated functions. All bets are off. === difference integrationI haven't programmed this algorithm yet. But here are a few questionsfor the newsgroup.1) Aside from the complaints about terminology (e.g. incorrectly usingthe term integration to describe a discrete summation), does thisformula work?2) If it does indeed work for certain input values, does it fail forothers?3) Is this formula just a restatement of something we already knowfrom number === ?James Buddenhagen says...>>Does the series sin(1)/1 + sin(2)/2 + sin(3)/3 + sin(4)/4 + ... converge?>How does one prove convergence (or divergence)?. If it converges what >is a good way to estimate its value?I'm not sure whether it converges, but if it does, I know what it convergesto 8^)Note that the infinite series S = sin(1)/1 + sin(2)/2 + ...is the imaginary part of the series (using the relation exp(ix) = cos(x) + isin(x)). E = exp(i)/1 + exp(2i)/2 + ...This is a special case of the series x/1 + x^2/2 + ...(To see this, let x = exp(i))If this converges, it converges to the function log(1/(1-x)), so E = log(1/(1-exp(i)))To take the imaginary part, we need to rewrite1/(1-exp(i)) in the for A exp(iB) where A andB are real. To get it in this form, note 1/(1-exp(i)) = exp(-i/2)/(exp(-i/2) - exp(i/2)) = i exp(-i/2)/(2 sin(1/2)) = exp(i pi/2) exp(-i/2)/(2 sin(1/2)) = exp(i (pi/2 - 1/2))/(2 sin(1/2))Taking the log gives E = i (pi/2 - 1/2) - log(2 sin(1/2))Taking the imaginary part gives: S = (pi/2 - === naturalsI think you are right but I can't demonstrate thatPepe Bosch ha scritto nel messaggio> Let S(n) the decimal sum of n, this is>> S(17)=1+7=8,> S(98)=9+8=17>> I have the following question: There are a, b and c, natural numbers, such> that>> S(a+b)<5> S(a+c)<5> S(b+c)<5 and> S(a+b+c)>50 ?>> I think that there are not.>> (Exuse-me I write english very === some people?>> Am I too dumb for math?>Define dumb. Math is a tool, and for some a pleasure>in its own right Ahem. It's not polite to talk about matherbation in public.... except among === math so difficult for some people? Just as the > answer to a mathematical question is either right or right, I'd like a few of those questions!Some of us learned three-valued logic in high school. There are threeways to solve a problem: the right way, the wrong way, and the way youwere told to do it (which may have nothing to do with right or wrong).David === LIGHTBULB?Look, math is basically a formalization of common sense and logic. So,if something produces nonsense, that means along the way you have madean erroneous step. Otherwise logic is illogical, so you haveundermined the foundations of logic, a feat similar to what BertrandRussel and later Kurt Godel were able to do.Now, as to your question...You want to prove that 1 + a + a^2 + a^3 . . . = 1 / (1 - a) FOR ALLa.First of all notice that this statement does not make sense when a =1, because the right side is undefined, hence you are trying to saysomething using things without definitions -- i.e. potential nonsense.So you may say, fine, I'll DEFINE 1/0 for ya. It's infinity!Then you have to invent your own infinity arithmetic. For you cannothave infinity follow the same rules that we all agree regular numbersfollow. For example, 1 / 0 = infinity, 2 / 0 = infinity, so bydefinition of division 0 * infinity has all the values of the rainbow.So infinity * 0 is not unique. Incidentally, 0 / 0 is therefore notunique either, since the answer is the number which multiplied by 0gives 0 but this is true for all numbers. So you can replace thewith and and have yourself a nice little world with your infinityand zero arithmetic.No one says you can't do that. It just has to be consistent andlogical. You can invent your own math. If it's interesting and no onehas done it before, you can even publish it. :-)So, notice:You want to prove that 1 + a + a^2 + a^3 . . . = 1 / (1 - a) FOR ALLa.This means you have to define what the above statement MEANs for alla. We've just treated the case a = 1. Now let's see the case a > 1.What does the . . . mean in your express? if it means add infinitelymany numbers together I say there is still a lot of things undefined.What do you mean by add infinitely many numbers together? Instead, Iwant to use the definition that is used by all mathematicians, and iswell defined:SERIES [i=0...infinity] a^i = lim [n->infinity] SUM[i=0...n] a^iand I want to prove that = 1/1-a)The standard proof goes like this:for any partial sum,a * SUM[i=0...n] a^i = SUM[i=1...n+1] a^i = a^(n+1) - 1 + SUM[i=0...n]a^iSo we solve for SUM[i=0...n] a^i and we getSUM[i=0...n] a^i = (a * SUM[i=0...n] a^i) + (1 - a^n+1)(1 - a) * SUM[i=0...n] a^i) = (1 - a^n+1)SUM[i=0...n] a^i = (1 - a^n+1) / (1-a) // IF a != 1Taking the limit as n goes to inity NOW givesSUM[i=0...infinity] a^i = 1 / (1-a) // IF |a|<1SUM[i=0...infinity] a^i = a - INFINITY / (1 - a) // IF |a|>1This is why limits were invented, also. You don't want to deal withinfinities straight out. You will get confused about what the heck isgoing on. TO make things precise, you usually want to deal with FINITErepresentations of the same thing.For example, the SERIES (a_n) is defined as the limit of the partialsums s_n = SUM [i=1...n] (a_i). If this limit is infinity, we stopcaring about its value.If two series both diverge to infinity but we want to see whether oneapproximates the other very well, we take the limit of the RATIOS ofthe partial sums, and that's how we reach our conclusion.All this rests on the concepts of FOR EVERY and THERE EXISTS. Insteadof talking about infinity and FOR EVERY, you can use De Morgan's Laws,which are simply logical to our minds, and prove the negativestatement about THERE EXISTS. People who have difficulty understandinginfinite constructions (I also do, sometimes), should realize thatthey can replace FOR EVERY x, P(x) with NOT [THERE EXISTS x, NOTP(x)]. This is useful for example for Cantor's DiagonalizationArgument.Some things are just impossible to define if you want them to fitexisting standards. For example, it's impossible to define an orderrelation between complex numbers without violating the axioms oforder. In the same way, you can make up your own arithmetic with zeroand infinity, just make clear what, if anything, you are violating innormal arithmetic (as I did above with dividing by zero). Once youhave a consistent system, and you're sure it works, it might give youuseful results, but you always have to be careful about translatingthe results into regular === efficiently do polynomial division? > (I am looking for techniques similar to Karatsuba used for> multiplication ...) === Nothing, as usual, and was prolix about it.>> Society national meeting in Denver next year? Uncle Al knows a fellow> who was in your audience last year. He said you had everybody rolling> on the ßoor clutching their tummies - some from laughter, others> puking, and the remainder collecting lint for tinder for igniting your> faggots.>> --> Uncle Al> http://www.mazepath.com/uncleal/qz.pdf> http://www.mazepath.com/uncleal/eotvos.htm> (Do something === a .8ecrit dans le message de>> Does the series sin(1)/1 + sin(2)/2 + sin(3)/3 + sin(4)/4 + ...> converge?>> How does one prove convergence (or divergence)?. If it converges what>> is a good way to estimate its value?>> Abel's rule OK what about tan(n)/n ?The sequence tan(n)/n does not converge to 0; therefore, the seriestan(1)/1 + tan(2)/2 + tan(3)/3 + ... diverges.Best === with four identical digits>> More generally, for any odd b, if (b-1) x^2 + d == 0 mod b^n but not mod>> b^(n+1), where gcd(d,b) = 1, then gcd(x,b) = 1 and >> (b-1) (x + y b^n)^2 - d == (b-1) x^2 - d + 2 (b-1) b^n x y mod b^(n+1)>> which is 0 for the appropriate value of y mod b. So if d is a quadratic >> residue mod b with gcd(d,b)=1 there are squares ending in arbitrarily >> many d's in base b.>What about higher powers (instead of squares)?Sure, why not? A similar proof will work for k'th powers if gcd(b,k) = 1.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === response to my workThis is where you are totally off. possible prime factors =/= primenumbers . If you were silly enough to attempt to test primality of a largenumber by trial division, then you would only have to check that the numberwasn't divisable by all the primes less than the square root of the number.So the possible prime factors might be every every prime less than half of anumber, the ones that you have to check to establish primality are thosethat are less than the square root. Obviosly, a if n < m, m cannot be afactor of n.The fact that your kids understand this proof at such a young age seems tobe an attempted slight on my knowledge or intelligence, since I am mucholder and you imply that I do not understand it. It must be ratherembarrassing then to have insulted me based on a complete lack ofunderstanding of the thread. I'll refrain from insulting you, since itwould be pointless and immature to advise you to have your kids double checkyour usenet posts.Justin Van Winkle> This doesn't really make sense to me. If you can generate all the> possible> prime factors, then you know how many there are. If you can't generate> them, then certainly you can't use the fact that there are more factorsto> somehow generate more factors. Seriously, if you know that there mustbe> at> least 10^20 more primes to try, that still leaves you to find all these> primes. If you know that there aren't any more primes to try, then you> know> how many you've tried. (I'm not an expert on this topic, so I may beway> off.)>> There is a simple proof that there can be no largest prime. Even my kids> === how to detach cycles/transpositions however I've already managed to make up something fortranspositions disjoining (c++):Transposition: a class, with some methods and fields ( int* for permuation)void Perm::detachTranspositions(){ if (isIdent()) { cout<0) { if (tmp.p[pos-1] > tmp.p[pos]) { swap(tmp.p[pos-1],tmp.p[pos]); X1[--index] = pos-1; X2[index] = pos; } } } while(x != tmp.getPos(x)); } for(x=inv-1; x>=0; x--) cout<<[< Let S(n) the decimal sum of n, this is S(17)=1+7=8,> S(98)=9+8=17 I have the following question: There are a, b and c, natural numbers, such> that S(a+b)<5this is true then :a < 50 and b < 50> S(a+c)<5this also :a < 50 and c < 50> S(b+c)<5 andand this, of course :b < 50 and c < 50> S(a+b+c)>50 ?Uhm..if this statement were true, this would be true as well : a+b+c>599999 (as this is smallest number n for which S(n) > 50)And..that can't be true? I'm not sure if this is right since it's awfullysimple..perhaps your definiton of === measures of error> Suppose that g(x) is proposed as an approximation of f(x) on [a,b].> What are the most popular ways of measuring how well g approximates f> over that interval?As pointed out by another responder, the answer depends on the application.For example, in time-dependent analysis, where x represents time, one mightwant to compare two time series. The comparison is made more complicated ifthe two signals look similar to the eye, but differ more in phase than inamplitude. For example, compare two sine waves of equal === what they might do to help, I think that maybe> sending an email to some news organization might have an impact. For> instance, you can email TIME magazine at letters@time.com, and can happen with education gone bad, a person with delusions ofgrandeur, fame and fortune. This individual believes he is one of thegreatest number theorists and analytical researchers of ALL TIME!Can you perhaps run a story on NPD (you have a clinical definition of NPD:***Diagnostic criteria for 301.81 Narcissistic Personality Disorder (cautionarystatement)A pervasive pattern of grandiosity (in fantasy or behavior), need foradmiration, and lack of empathy, beginning by early adulthood and present ina variety of contexts, as indicated by five (or more) of the following:(1) has a grandiose sense of self-importance (e.g., exaggerates achievementsand talents, expects to be recognized as superior without commensurateachievements)(2) is preoccupied with fantasies of unlimited success, power, brilliance,beauty, or ideal love(3) believes that he or she is special and unique and can only beunderstood by, or should associate with, other special or high-status people(or institutions)(4) requires excessive admiration(5) has a sense of entitlement, i.e., unreasonable expectations ofespecially favorable treatment or automatic compliance with his or herexpectations(6) is interpersonally exploitative, i.e., takes advantage of others toachieve his or her own ends(7) lacks empathy: is unwilling to recognize or identify with the feelingsand needs of others(8) is often envious of others or believes that others are envious of him orher(9) shows arrogant, haughty behaviors or attitudesReprinted with permission from the Diagnostic and Statistical Manual ofMental Disorders, fourth Edition. === Stirling Numbers Approximation?Does anyone know an approximation for Stirling Numbers of the Secondkind, S(n,k), for very large values of n?S(n,1) = S(n,n) = 1 are easy, as are S(n,2) and S(n,n-1). It's theintermediate values of k that are difficult. I can't use thealternating series because of === reference to understand topological properties of L^p spaces === n, this is>>S(17)=1+7=8,>>S(98)=9+8=17>>I have the following question: There are a, b and c, natural numbers, such>>that>>S(a+b)<5 > this is true then :> a < 50 and b < 50Surely not? What about a = b = 100? a+b = 200 and s(200) = 2 + 0 + 0 < 5. >>S(a+b+c)>50 ? > Uhm..if this statement were true, this would be true as well : a+b+c>599999 (as this is smallest number n for which S(n) > 50) And..that can't be true? I'm not sure if this is right since it's awfully> simple..perhaps your definiton of decimal sum is different..> Doesn't work. You can get arbitrarily big values of a + b + c with S(a+b), etc < 5. I can't see whether or not you can make S(a+b+c) bigger than 50 or not, but will think about it. (Probably won't have too much success - number theory isn't my === at the University of Montana.> Let S(n) the decimal sum of n, this is>> S(17)=1+7=8,>> S(98)=9+8=17>> I have the following question: There are a, b and c, natural numbers, such>> that>> S(a+b)<5>>this is true then :>a < 50 and b < 50No. Say a=1010, b= 200. Then a+b = 1210, and S(a+b) = 4 < 5.Write x = a_0 + 10a_1 + 10^2a_2 + ... + 10^n a_n y = b_0 + 10b_1 + 10^2b_2 + ... + 10^n b_n with 0<= a_i,b_i < 10, and at least one of a_n,b_n nonzero. So S(x) = a_0 + a_1 + ... + a_n S(y) = b_0 + b_1 + ... + b_nDefine d_0,....,d_n recursively as follows:d_0 = 0 if a_0+b_0 < 10d_0 = 1 if a_b+b_0 > 9.d_{i+1} = 0 if a_{i+1} + b_{i+1} + d_i < 10d_{i+1} = 1 if a_{i+1} + b_{i+1} + d_i > 9(The d_i are the carries)Then (x+y) + (a_0+b_0 - 10d_0) + 10(a_1+b_1+d_0-10d_1) + ... +10^n(a_n+b_n+d_{n-1}-10d_n) + 10^{n+1}d_n. Then S(x+y) = a_0+b_0 + a_1 + b_1 + ... + a_n+b_n + +(d_0+...+d_n)- 10(d_0+...+d_n) = S(x) + S(y) - 9(d_0+...+d_n).Now, you assume that S(a+b) <= 4 S(b+c) <= 4 S(a+c) <= 4Say we take 2(a+b+c) = (a+b) + (b+c) + (a+c).Now, let's consider the carries involved: Each of a+b, a+c, b+c has atmost 4 nonzero digits, and each nonzero digit is at most 4. How manycarries can there be? For there to be carries when we add a+b, a+c,and b+c together, there must be corresponding entries, at least one ofwhich is a 4, and the other 2 are either 4's or 3's. But that meansthat there is at most 1 carry. That is:S(2(a+b+c)) = S(a+b) + S(a+c) + S(b+c) orS(2(a+b+c)) = S(a+b) + S(a+c) + S(b+c) - 9.Now we want to relate S(2(a+b+c)) to S(a+b+c)Again, let x = a_0 + 10a_1 + 10^2a_2 + ... + 10^n a_nwith 0<= a_i <10, a_n>0So S(x) = a_0 +...+ a_n.Define e_0,....,e_n by lettinge_i = 0 if 0<=a_i <5e_i = 1 if 4A very small market in today's world can be worth millions of dollars>US.IMO, you need to brush up on your arithmetic, too.-- Wolf Kirchmeir, Blind River ON CanadaNature does not deal in rewards or punishments, but only === Emergent GravityJack, you ask:... Tony what do you mean by D5 describing Gravity?What that would mean to me is that start from the D5 groupand end up with Einstein's local field equationGuv = (superstring tension)^-1Tuvcan you actually do that?Similarly, start from D4 and get the U(1)xSU(2)xSU(3) principal fiberbundle of theelectroweak-strong gauge forces with the associated vector bundle ofthe lepto-quarksources. Can you do that? ....Yes, to both questions (although I don't use the conventionalsuperstring structure that you mention).I am trying to see if there is a precise mathematical connection between what you are doing andwhat I am doing in http://qedcorp.com/APS/EmergentGravity.pdfIn my theory Einstein's gravity field plus exotic vacuum w = -1 dark energy/matter fields are all ODLRO c-number collective emergent low energy effective MACRO-QUANTUM fields in which Diff(4) is an emergent symmetry from the spontaneous breaking of the U(1) EM symmetry at the unstable false vacuum micro-quantum level of the lepto-quark sources/electroweak-strong gauge forces level.More specifically, I get an emergent c-number LOCAL giant MACRO-QUANTUM VACUUM COHERENT WAVEVacuum Coherent Field = (Higgs Amplitude Field)e^i(Goldstone Phase Field)World Crystal Lattice Distortion Field = Lp*^2(Goldstone Phase Field),uEinstein's guv field is the ßat Minkowski metric + the Strain Tensor of the Distortion FieldTherefore, Einstein's Gravity Field emerges as modulation of the Goldstone Phase Field consistent with Andrei Sakharov's metric elasticity, which is the complementary view of P.W. Anderson's generalized phase rigidity in his More is different paradigm. That is, the basic emergent gravity coupling is precisely Ed Witten'salpha' = 1/(string tension) = 8piG*/c^4where I allow G* to be a scale-dependent variable that is 10^40 G(Newton) at the 1 fermi scale.Unified Exotic Vacuum Dark Energy/Matter Field is from the modulation of the Higgs Amplitude Field which also provides the rest mass of the lepto-quarks asmc^2 ~ e^2/zpf^1/2where /zpf ~ -1/(1 fermi)^2 at the 1 fermi scale in the vibrating dark matter quantized vortex string core where theHiggs Field is zero.Where with h = c = 1 convention/zpf = (alpha')^-1[(alpha')^3/2|Higgs|^2 -1]alpha' is a scale-dependent variable not fixed at 10^-66 cm^2.BTW I have come to the tentative conclusion that the alleged Holographic Universe formulaLp* = Lp^2/3L^1/3found in the LNL e-prints has serious problems of interpretation.My basic theory does not essentially depend on that additional assumption.While I see no basic problem in you getting the micro-quantum U(1)xSU(2)xSU(3) from your group theory, I donot understand how you get Einstein's Gravity. The Ed Witten argument that one gets a spin 2 quantum is not good enough for me since in the More is different emergence the consensus quantum gravity idea is not correct at all and that is what non-renormalizability of Einstein's GR in the low-energy sector is telling us.There may be some linear spin 2 perturbative random gravitons as micro-quantum noise or normal ßuid in the emergent curved spacetime superßuid background from the modulation of the c-number Goldstone phase.So I am asking where is the MACRO-QUANTUM EMERGENCE in your group theory approach to deriving Einstein's Gravity from a more fundamental level?Tony:Those things (the forces of Gravity and the Standard Model) allcome from the 28-dim adjoint rep of the D4 subalgebra of D5.More particularly:As to Gravity (and Higgs, and special conformal generators):D4 has a 15-dimensional D3 subalgebra that is the conformalalgebra SU(2,2) = Spin(2,4). It has:1 dilation generator (corresponds to Higgs)OK4 special conformal generatorsWhat do they locally gauge to?My hunch is /zpf,u10 anti-deSitter generators.The 4 Pu of T4 locally gauge to Einstein's guv.The 6 Muv of O(1,3) locally gauge to a Torsion Field.By a modified conformal MacDowell-Mansouri mechanism,Is this where ODLRO MACRO-QUANTUM EMERGENCE is buried?you get the Einstein-Hilbert Lagrangian.All this is at conventional textbook level, for example,section 14.6 of Unification and Supersymmetry, 2nd edition,by Rabindra Mohapatra, Springer-Verlag 1992.If you want a prominent establishment name dropped, Frank Wilczekmentions the MacDowell-Mansouri mechanism inhttp://xxx.lanl.gov/abs/hep-th/9801184where he notes that the mechanism was also independentlyformulated by Chamseddine and West.The mechanism was invented to make it possible to get gravityfrom the anti-deSitter part of Lie superalgebras used insupergravity theories.As to the Standard Model,look at the 28 generators of the D4 Lie algebra.Use 15 of them as above, and 1 moreto complete the SU(2,2) to U(2,2) = SU(2,2)xU(1).Then you have 12 generators left.The form the 12-dim Standard Model SU(3) x SU(2) x U(1).Here is how you can see that structure geometrically,and unambiguously:If you look at things (here I was inspired by Saul-Paul) interms of the Weyl reßection group of the root vector space,and take away the 12 root vectors of the U(2,2) and the 4 Cartanalgebra elements of the 16-dim rank 4 U(2,2), that leaves youwith 28-12-4 = 12 root vectors.Since the root vectors of D4 form a 24-cell, which can be seenas a 12-vertex cuboctahedron plus two 6-vertex octahedra,you see that the remaining 12 root vectors form a pair of octahedra.Line the two octahedra up so that they share a common axis,and project the two octahedra into a space perpendicular to that axis,so that the 4 axis vertices fall on a line that forms theroot vector diagram (including Cartan origin vertices) ofthe 4-dim Lie algebra U(2) = SU(2) x U(1).The remaining 8 can be seen as the vertices of a cube: tb----xb | | | zb----yb | | | | yr-|--zr | | | xr----trNow look at the cube along its tb-tr diagonal axis,and project all 8 vertices onto a plane perpendicularto the tb-tr axis, giving the diagram yb xb zb tb tr zr xr yrwith two central points surrounded by two interpenetrating triangles,which is the root vector diagram of SU(3).Therefore,the 16 of the 28 D4 generators give us Gravity, Higgs, and special conformal,andthe remaining 12 give us the Standard Model SU(3) x SU(2) x U(1).Consideration of the relevant geometries and combinatorics givelevel calculations, so such higher order things as neutrino masses(they are in my model tree-level massless) remain to be fully calculated.Details are in my papers, including a paper athttp://www.innerx.net/personal/tsmith/TQ3mHFII1vNFadd97. pdfthat contains material barred from the Cornell arXiv due === Mathematical Stupidity Constant in all assessed Intelligent must be also based on> stupidity.I'll concede that yours is.--There are two things you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of === Re: Axiom of Foundation (absymally stupid question)>Since we're discussing the axiom of foundation (in the textbooks I've seen, >it's called the axiom of regularity), does anyone know what the intuitive >justification for this axiom is? I mean, all the other axioms seem pretty >natural to me, even the infamous axiom of choice. But where in world did >they come up with the axiom of regularity?It was to avoid having a set x which is its only element,and more complicated versions of this. It is easy to seethat both it and its negation are consistent.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: === between > basis transformation and a similarity transformation.A similarity transform of A is any transformation of theform inv(S)*A*S with nonsingular S.According to my Horn and Johnson, every invertiblematrix is a change-of-basis matrix, and every change-of-basis matrix is invertible. Thus, every similaritytransform is a change of === About naturals>Let S(n) the decimal sum of n, this is>>S(17)=1+7=8,>S(98)=9+8=17>>I have the following question: There are a, b and c, natural numbers,>such that>>S(a+b)<5>> this is true then :>> a < 50 and b < 50 Surely not? What about a = b = 100? a+b = 200 and s(200) = 2 + 0 + 0 < 5.Right.. of course!..I already thought there was somethign wrong.There's really nothing you can get for S(a) or S(b) out of S(a+b)<5.. So thesame counts for the relationship between S(a+b) and S(a+b+c)Sounds like a very complex problem then, unless you perhaps put a reasoningbehind Ôcarries' when summating a+b with c and a with b+c (2*(a+b+c) =(a+b)+c + a+(b+c)), since the carries working in the sum can't have anincreasing effect on the decimal sum?.. So for the decimal sum to increasewhen adding two numbers, you'd need to have values that fill each other upand don't cause carries..which eerily revolves around the boundary of 5.Well I'll stay out of === Wilzcek's Emergent GravityBy a modified conformal MacDowell-Mansouri mechanism,I asked:Is this where ODLRO MACRO-QUANTUM EMERGENCE is buried?you get the Einstein-Hilbert Lagrangian.All this is at conventional textbook level, for example,section 14.6 of Unification and Supersymmetry, 2nd edition,by Rabindra Mohapatra, Springer-Verlag 1992.If you want a prominent establishment name dropped, Frank Wilczekmentions the MacDowell-Mansouri mechanism paper seems very relevant in accord with what I amdoing in a simpler way independently. Wilczek is one of the besttheoretical physicists around for sure. I heard him speak twice nowthis year.where he notes that the mechanism was also independentlyformulated by Chamseddine and West.The mechanism was invented to make it possible to get gravityfrom the anti-deSitter part of Lie superalgebras used insupergravity === integration> I haven't programmed this algorithm yet. But here are a few questions> for the newsgroup.>> 1) Aside from the complaints about terminology (e.g. incorrectly using> the term integration to describe a discrete summation), does this> formula work?James has coded the algorithm for the original case involving a discretesummation and it does produce correct results in that case.> 2) If it does indeed work for certain input values, does it fail for> others?He has claimed that replacing the Ô1's in the original equation with'delta y' converts the difference equation to a partial differentialequation -- but this formulation is *not* a partial differential equationand it does *not* work as a method of counting primes.> 3) Is this formula just a restatement of something we already know> from number theory?Yes.--There are two things you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of popularity over === generation of large prime numbers> You were trying to conclude that, for any prime P, N is a larger> prime.No, I wasn't. I was saying that it might be a larger prime, but that it might be a composite of primes at least one of which is larger than P.-- Richard Heathfield : binary@eton.powernet.co.ukUsenet is a strange place. - Dennis M Ritchie, 29 July 1999.C FAQ: http://www.eskimo.com/~scs/C-faq/top.htmlK&R answers, C books, === Permutation: how to detach cycles/transpositions[something not in english!]You know, whenever one posts source code in a NG that is notcomp.lang.thatlanguage or a subhierarchy of it, it would be a VeryNice Thing(TM) to specify in which language it is written!Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore Aursand on === some people?>I have taught math for many years, both in a classroom setting and one>on one, and more often than not, the main problem lies in students not>understanding the basic ideas behind the formulas, rather than just>memorizing the formulas. For many students, there is the mistaken>belief that knowing the formulas is enough.This is not surprising, considering that this is essentiallyhow everything is taught in the elementary and high schools,and also in the courses through calculus.As just about all textbooks take that approach, and the greatbulk of teachers believe in teaching that way, it is hard tosee what can be done. We are NOT going to be able to teachthe teachers; this failed for the new math, and it was verydefinitely tried.It is not even learning the basic ideas behind the formulas;this might get through to a percentage of the teachers. Itis knowing the properties and the concepts of the mathematicalobjects used, and these are not going to be learned by themistaken methods of the educationists.We can teach variables in the general sense (do NOT limit them to numbers) as linguistic entities with beginning reading. We can teach sound mathematical logic to at leasthalf of those in elementary school; it has been done. But we cannot succeed if learning mathematics is measuredby the multiple choice computational stuff on the exams nowbeing mandated. Nobody learns what multiplication means bymemorizing the tables, and one can understand what it meanswithout having any facility in computing answers.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: === <>sSHfTy;{Dhe&:+?b`9fUj5A~$gIYlYT0/$-asR-K~3S3[]q.R3YSmpR|$- GiZp>UN2a}!Fmw+%h}YL`!h_XXr5Q>_nGsY2_> Does the series sin(1)/1 + sin(2)/2 + sin(3)/3 + sin(4)/4 + ... converge?> How does one prove convergence (or divergence)?. If it converges what > is a good way to estimate its value?> Recognize sin(x)/1 + sin(2x)/2 + sin(3x)/3 + sin(4x)/4 + ...as the Fourier series of a certain function, then evaluate it at x=1.Answer: (pi-1)/2 = 1.0708, approx.-- G. A. Edgar === talk factorizations> Polynomials are well-known in science and mathematics, but while> finding roots of polynomials is typically the aim of the average> researcher, polynomials themselves can be used as powerful tools for> analyzing the roots of *other* polynomials. The concepts are advanced, but can be approached by first considering> a basic example. The basic factorization to start is (c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = > 49(x^3 + 5x^2 + 3x + 1) with the c's algebraic integers, notice that only two of the c's have> 7 as a factor. It might help to go the *other* way, and start with (d_1 x + 1)(d_2 x + 1)( d_3 x + 1) = > x^3 + 5x^2 + 3x + 1 and now multiply by 49. In the first example you're looking at a product and realizing that> from the distributive property a(b+c) = ab + ac, you know there's> *one* way it could be produced, which is to multiply something like> the second example by 49. The distributive property is key here. Understanding it thoroughly,> is of prime importance.> I like this example. Of course if you want to multiply (d_1 x + 1)(d_2 x + 1)( d_3 x + 1)by 49 so that you have 7 as the constant term in two ofthe factors, you are right, there is basically one way to do it (modulo permutations of d1, d2, and d3). However you can distributefactors of 49 through the three linear factors above in infinitely many *other* ways, and the resulting polynomialis still the same. For example, (7^{4/3}*d1*x + 7^{4/3})*(7^{1/2}*d2*x + 7^{1/2})*(7^{1/6}*d3*x + 7^{1/6}).If you do this, you still get 49(x^3 + 5x^2 + 3x + 1)just as before. The thing is, there is no particular reason you needto get 7 as the constant term for two of the factors. In theexample you give, when x = 0, the constant term P(0) is 49,and of course 49 = 7^2 = 7^{12/6} = 7^{4/3} * 7^{1/2} * 7^{1/6},right? So the constant terms multiply together as they should. And this is only *one* way to use the distributive law to distribute the factors of 49 across the three factors. You caneven define the factorization in three parts as a function of x if you want.> Now notice that you can abstract from here as you're looking at> *functions* of x, as introducing f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, you have (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1). Notice that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1 as long as you're in a ring where 7 is not a factor of 1. Which is consistent with what was found before, as only two of the> functions have the property that 7 is a factor. Now I'll move on to a more complicated example. Let (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so they are functions of x, and since one of the roots equals 3 at> x=0, I have b_3(x) = a_3(x) - 3, so that all the functions in (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) equal 0, when x=0. Those of you who find it hard to use the distributive property with> the *product* can imagine the factorization from *before* 49 being> multiplied. It's harder to show here as the polynomial which defines the function> in that factorization is not displayable in general. So I started at the end, with 49 already multiplied because then I can> give a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x).> Yes - note that now the a's are indeed functions of x. When youconsider 5*a1(x) + 7,and you want to factor a factor of 7 out of it, there are many ways that could be done, depending on the value of a1(x). When x = 0, a1(x) = 0. You know that a1(1), for example, is not 0. You argue, for no visiblereason, that a1(1) must be divisible by 7 because the constant termof the product is 7*7*22, and you think that if you divide 7*7 outof the whole expression, you cannot divide any piece of 7 outof the constant term 22 because 7 and 22 are relatively prime in the algebraic integers. You are indeed right that 7 and 22 are relatively prime. However, you don't *need* to have 22 itself divisible by any part of 7. The only thing you need to have divisible by some part of 7 is 5*b3 + 22.Now: you know that 22 is coprime to 7 and you know that 5 is coprimeto 7. Suppose b3 is also coprime to 7. Can I conclude from thatthat 5*b3 + 22 is also coprime to 7? Isn't it possible under the hypotheses juststated that 5*b3 + 22 is NOT coprime to 7 ? (Hint: say b3 = 4). In fact, there is a decomposition of 7*7 of the form r*s*t suchthat: 1. 7*7 = r*s*t 2. 7/r and 7/s are algebraic integers 3. a_1/r and a2_/s are algebraic integers 4. (b_3*x + 22)/t is an algebraic integer. You may well shriek, WHAT ABOUT X ? SEE THAT X IN THERE?IT'S A VARIABLE! WHAT'S IT DIVISIBLE BY? Here's the key thing. The numbers r, s, and t, just like a_1,a_2, and b_3, are *** functions of x ***. That's exactly how it works.Divisibility of the constant terms is not important. What is important is that when you multiply everything out after youhave distributed the parts of 49 among the three factors, theproduct of the constant terms is still 22. Here's what you get: (7/r) * (7/2) *(22/t) = (7*7/(r*s*t))*22 = (49/(r*s*t))*22 = 22,just as it should. ***IT IS NOT IMPORTANT THAT 22/t IS NOT ANALGEBRAIC INTEGER***. WHAT IS IMPORTANT THAT (5*b_3 + 22)/t IS AN ALGEBRAIC INTEGER. The other key idea here is the functions-of-x idea. How 49splits into three parts depends on the value of x. You may say,How can r, s, and t know about x? They know about x becausethey are intimately connected to a_1, a_2, and b_3, and clearlythese coefficients are functions of x because a_1 and a_2 anda_3 = b_3 + 3 are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 See those x's in there? There is NO REASON to think that because, when x = 0,r, s, and t are respectively 7, 7, and 1, that the same is truefor all values of x. That is *not* a requirement which one candeduce from the fact that the constant terms are 7, 7, and 22.> That slight change, starting at the end, means that you have to> understand the distributive property fully and *trust* it.> No problem with the distributive law. The problem is, you seemto be able to imagine using it in only one way. However 7*7 can be split up as a product of three algebraic integers in infinitely manyways, and thus distributed among the three factors of your polynomial ininfinitely many ways. Not all of those ways give algebraic integercoefficients. As we have shown MANY TIMES, a_1 and a_2 cannot bedivisible, in the algebraic integers, by 7. Assuming they are leadsto a contradiction when x <> 0. Bottom line: for x <> 0, 49 does not split up the way you think itdoes. There is *another way* to split it up which does not resultin the contraction just mentioned. You appear to be making the mistakeof thinking that 22/t has to be an algebraic integer. > Now notice that I have the result that only two of the roots of the> cubic a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) can have factors in common with 7, so the 49 splits between those two.> No - you have seen the proof *many* times. No root of the polynomial above can be divisible by 7, and no root can be relatively prime to 7. If you assume so you get a contradiction.It's inescapable! Nora B.> What's so startling is that the result is for a *family* of> polynomials as it applies for any algebraic integer x. > James Harris My math discoveries, found === to solve the system of differential equation?>Consider a system of DE:> 1/c dx_i/dt + x_i = sum_{jk} a^i_{jk}x_j*x_k, where a^i_{jk} are> non-negative real numbers and a^i_{jk}=a^i_{kj} (i.e. the matrix a^i> is symmetric).> Given the initial values of x_i, can the system be analytically> solved:> a) in general case;> b) under the condition sum_i a^i_{jk}=1; > c) under the condition b) and sum_i x_i=1?Probably not in general. Maybe for the case of a 2 x 2 system, althoughthe solution might be horrendously complicated. In case (b), ifX = sum_i x_i, you have 1/c dX/dt + X = X^2 which can be solved (andX=1 is a solution). So this effectively reduces the dimension of the system by 1.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of === Russian claims of torsion weaponsCommentary 3The hyperspace H consists of fibers f(x) that areeither copies of or representations of the symmetrygroup G.Jack, this is not quite correct. They are homogenous spaces onwhich the group operate transitively. Example, for the group SU(2),you can take as the fibre a copy of SU(2) itself (3-dimensional), oryou can take sphere S^2, on which SU(2) operate (2-dimensional).Notice that S^2 is not a representation of SU(2). It is a quotientSU(2)/SO(2).Early Kaluza-Klein theories were operating with group Manifolds.Souriau, and later Witten, suggested more realistic theories wherefibers could be of lesser dimensions. Thie rigorous mathematics andexamples of this latter approach have been developed in themonograph:Riemannian Geometry, Fibre Bundles, Kaluza-Klein Theories and AllThat... (World Scientific Lecture Notes in Physics, Vol 16)by Robert Coquereaux, for the local gauge forces:1. A transformation g of the symmetry group G acts on the ordered pair X = (x, fo) in hyperspace H with output gX.Question: Can gx = x' =/=x i.e. can one move the base point in this operation or must G always be the identity in the base space? That is, we always need, in addition to G a connection and a path in order to change location in the horizontal base space and the vertical fiber space that is beyond space-time. G certainly moves fo up and down the vertical fiber for every element g =/= identity. Does it also move x -> x' = gx =/= x horizontally along the base manifold without a connection field and a path specified? Clearly the answer must be NO. See below.The modern understanding of gauge invariance, as a symmetry under transformations ofquantum-mechanical wave functions, was reached by Weyl himself and also by London veryshortly after the new quantum mechanics was first proposed. In this understanding ofabelian gauge invariance, and in its nonabelian generalization [2], the space-time aspect islost. The gauge transformations act only on internal variables. This formulation has hadgreat practical success. Still, it is not entirely satisfactory to have two closely related, yetdefinitely distinct, fundamental principles, and several physicists have proposed ways tounite them.One line of thought, beginning with Kaluza [3] and Klein [4], seeks to submerge gaugesymmetry into general covariance. Its leading idea is that gauge symmetry arises as a reßec-tion in the four familiar macroscopic space-time dimensions of general covariance in a largernumber of dimensions, several of which are postulated to be small, presumably for dynam-ical reasons.Here we should take the opportunity to emphasize a point that is somewhatconfused by the historically standard usages, but which it is vital to have clear for whatfollows. When physicists refer to general covariance, they usually mean the form-invarianceof physical laws under coordinate transformations following the usual laws of tensor calculus,including the transformation of a given, preferred metric tensor. Without a metric tensor,one cannot form an action principle in the normal way, nor in particular formulate the ac-cepted fundamental laws of physics, viz. general relativity and the a purely mathematical point of view one might consider doing without the metric tensor;in that case general covariance becomes essentially the same concept as topological invari-ance. The existence of a metric tensor reduces the genuine symmetry to a much smaller one,in which space-times are required not merely to be topologically the same, but congruent(isometric), in order to be considered equivalent. In the Kaluza-Klein construction, for thisreason, the gauge symmetries arise only from isometries of the compactified dimensions.Another line of thought proceeds in the opposite direction, seeking to realize generalcovariance [CapitalEth] in the metric sense [CapitalEth] as a gauge symmetry. arXiv:hep-th/9801184 v4 23 Apr 1998IASSNS-HEP-97/142Riemann-Einstein Structure from Volume alerting me to this relevant paper by Wilczek.BTW Wilczek shows that Gennady Shipov's torsion theory is closely related toRoger Penrose's spinors in curved spacetime with the anti-symmetricspin connection as the locally induced compensating torsion field.It all comes from locally gauging the O(3,1) subgroup of the Conformal Groupas I said previously based on Utiyama's and Kibble's papers from the mid-1960's.Whether or not Akimov's claims from Moscow that torsion waves from O(1,3) ofsufficient intensity to have psychotronic weapons bio-toxic effects can easily be generated when,in contrast, gravity waves from T4 are so hard to find is another issue not considered here.The gravity wave T4 coupling parameter is essentially Ed Witten's alpha' = (superstring tension)^-1.What is the corresponding O(1,3) spin connection coupling parameter? Akimov's claims hangon the answer to that question. Is it easier to make propagating torsion dislocation topological string defectsthan to make propagating curvature disclination topological string defects in the MACRO-QUANTUMVacuum Coherence Field's Goldstone Phase? That's what Akimov's claims come down to in terms ofmy new theoretical paradigm for the emergence of Einstein's Gravity and the Unified Exotic Vacuum Field ofw = -1 Dark Energy/Matter.2. The action of the symmetry group G on the total hyperspace H induces an equivalence relation ~ .That is, if X' = gX, g < G, then X' ~ X.3. ~ partitions hyperspace H into disjoint non-overlapping equivalence classes called G-orbitsG(X) = {gX, for all g < G}Remember that in this principal bundle fo is also a g < G.All G-orbits have identical structure and are diffeomorphic to G.4. This disjoint partition of hyperspace H gives the quotient space H/G that is the base space M with points x.Every point x of the base space M is really an equivalence class or G-orbit of a continuous infinity of points of a larger dimensional Hermetic or occult hidden hyperspace implicate inside it. Worlds within worlds. Wheels within wheels. Shades of Bohm's Implicate Order?5. The Projection Map P is simply P:G-orbit -> x.This means that each individual G-Orbit is really associated with a single vertical fiber at a single horizontal base space event. The G-orbit is the vertical fiber beyond, in the usual physics applications, a localized spacetime event x, although we can have delocalized base spaces of twistors whose intersections are points. We can also perhaps have base spaces of finite strings both open and closed and even base spaces of higher dimensional brane worlds?Commentary 2Given coordinate patch C(x) in the base space M in a neighborhood of point x and fiber f(x)form the local Cartesian product C(x)f(x) with ordered pair X = (x,fo).Take the union C(x)f(x)/C(x')f(x')/... of all such local products.There are redundant ordered pairs X because the coordinate patches C(x) and C(x') as sets overlapwith non-vanishing intersection C(x)/C(x')=/= Empty Set.Identify the redundant multiple images of the same actual point of the base space M usingthe symmetry group G as an equivalence relation. That is, two ordered pairs X and X' areidentified or equivalent if x = x' < C(x)/C(x') and if fo' = gfo where g < G to form disjointequivalence classes {f(x)} that are the distinct points of the fiber in hyperspace H.This is all local at a fixed base point x like in an internal gauge force symmetry.g is also called a transition function.The hyperspace H is the factor space of the union C(x)f(x)/C(x')f(x')/ ... mod G.The projection map P:(x,{fo}) -> xWhen M is the curved space-time of Einstein's gravity theory in addition to the G equivalencein the extra space dimensions of the fiber, x'(E) = Diff(4)x(E) at fixed event Eto make disjoint equivalence classes {x(E)} mod Diff4(E).One can imagine a hybrid where the fiber is a discrete space of strings of c-bits.One can also imagine a fiber of strings of qubits.1 qubit is a parallel infinity of c-bits.i.e.|qubit> = |1 c-bit><1c-bit|qubit> + |0 c-bit><0 c-bit|qubit>Where there is a continuous infinity of different c-bit basesor orthonormal frames each corresponding, for example,the the angular orientation of an inhomogeneous fieldmagnet in a Stern-Gerlach filter for spin qubitsin the DARPA spintronics project or like the billion billionSingle Electron Transistors inside the human brain at thesub-microtubular protein dimer hydrophobic cage level formingthe hardware interface with external world whose software is our stream of inner consciousness.Each possible orientation is a primitive parallel quantum universe.The quantum computer computes in all possibleorientations simultaneously like a continuousinfinity of classical Turing machines in adistributed network working on the same problem - or so the folklore goes.to be continued.Commentary 1The fiber bundle as an idea has 4 parts.1. A structure symmetry group G.2. The total hyperspace H or, in some applications Wheeler's BIT.3. The projection map P.4. The base space M or, in some applications. Wheeler's IT.The hyperspace H consists of fibers f(x) that areeither copies of or representations of the symmetrygroup G.The projection map P collapses a fiber f(x) in the hyperspace H toa point x in the base space M.All of these objects are continuum differential manifoldsdepending on the continuum of real numbers which itsassociated issues of Cantor's infinity of infinities ofCabalistic Aleph's in an ascending Jacob's Ladder.This is not a discrete combinatoric mathematics althoughsuch a skeletal structure is associated with it as inHerman Weyl's Theory of Groups and Quantum Mechanicsand as in Saul-Paul Sirag's presentation of V.I. Arnold'sA-D-E mathematics of everything.The base space is covered by an atlas of local coordinate patcheswith all important overlap transition functions sewing thepatches together like a quilt.M is space-time in local micro-quantum field theory of pointThe extra-dimensions of hyperspace formthe Calabi-Yau space of vibrations of thesuperstring beyond space-time.The connection on the total hyperspace H is the potentialof a local gauge force.Examples of connections is the 4 potential Au(x) inMaxwell's electromagnetism with G as U(1).There are similar connections for the Yang-Mills weak forcewith G = SU(2) and the strong force with G = SU(3).Classical general relativity, as distinct from local micro-quantumfield theory, has the torsion-free symmetric three-index non-tensorLevi-Civita connection with G as the Diff(4) group.The latter comes from locally gauging the 4 parameter translation subgroup(generated by the 4-momentum Pu of globally ßat special relativity )of the 15 parameter conformal group of Roger Penrose's massless twistors.Bottom -> Up: Given base space M and symmetry group G construct thehyperspace H as a quilt patchwork.Top -> Down: Given hyperspace H and symmetry group G construct thebase space M as the non-overlapping partition of hyperspace into G-orbitscalled the quotient space of H mod G in the principal bundle.Micro-quantum source renormalizable local fields of spin 1/2 lepto-quarks are associated vector bundles.Micro-quantum force renormalizable local fields of spin 1 gauge force bosons (electro-weak and strong) arefrom the principal bundle.There is no renormalizable quantum gravity in this precise sense.This is because classical Einstein gravity is a More is different (P.W. Anderson)emergent collective effect as in Andrei Sakharov's metric elasticity of aninstability in the globally ßat false vacuum of the interacting lepto-quark source/electroweak-strong force.Einstein's gravity + unified exotic vacuum dark energy/matter with Andrei Linde's chaotic inßationary cosmology are the result of the continual phase transitions from globally ßat false high entropy micro-quantum vacua to locally curved macro-quantum low entropy metastable === professorX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS>I am not a mathematician by trade,That's obvious. More important, you lack the ability to listen.>but I was talking to a maths>professor and he absolutely refused to acknowledge the concept of a>cross product for two vectors that are not 3 dimensional.No doubt he also refused to acknowledge the concept of a pair ofintegers i and j such that i+j>j+i. Im sure that there are all sortsof impossible concepts that he refused to acknowledge.>For me,>Let A be vector in N space>Let B be vector in N space>A x B = CYou haven't defined anything. What is C? Let's be concrete: ifA=(1,0,0,0) in R^4 with the L2 metric and B=(0,1,0,0), what is C=AxB? >Now, I can prove that C has the propertyHow, when you haven't defined it. Worse, one of the properties of thecross product in R^3 is that AxB is orthogonal to A and B, so ifC=AxB, C.A=C.B=0. Thus your>C.A / |C||A| = C.B / |C||B|Doesn't say much.>So, one could define the cross product between two n-vectors as an>n-vector with the propertySure: just define AxB=0 for every A and B. It would not, however, havethe properties of the cross product in R^3. If you don't want to dothat, then you would have to define *WHICH* vector with thoseproperties, and there you run into difficulties.>But, am I wrong?Of course you're wrong. Google for Exterior or Grassmann.>Or is the professor wrong? Not on this he isn't. at 10:29 PM, j.schoenfeld@programmer.net (John Schoenfeld) said:>What is there was no professor? Then you work it out a step at a time, without any handwaving, and yousee where you went wrong. Or you don't, and you remain deluded.>Don't you look stupid, believerSomebody does, but it isn't him. You look stupid for presenting as adefinition something that in fact did not define anything, and youlook stupid for rejecting good advice instead of thinking thingsthrough.But tell me, tonto, why are you paying tuition if you believe that theprofessor doesn't understand things as elementary as that?-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === groupsX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 08:02 PM, Arthur said:>I'm reading about topological groups and I am having trouble with the> definition of such a group. What exactly are the open sets?That's like asking what the open sets are in a topological space. Partof specifying a topological group is specifying a topology; the opensets of a topological group are the open sets of its topology.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === torsion weapons> Commentary 3 > The hyperspace H consists of fibers f(x) that are> either copies of or representations of === Re: probability 2......>> thank...you....very much....>> i think......if we only use 0-x-y-1>> in this case, probability is 1/8>> but, if we use 0-x-y-1, 0-y-x-1>> in this case, probability is 1/4>> which of case is right??>> advice....please....sir~> 1/4>>Problem may be restated - if we have two independent uniformly distributedvariables on [0,1] variables x,y representing distances from one end ofwire, where we cut the wire - what is probabilities that pieces may formtriangle? If we consider only 0-x-y-1 then we have additional constraintthat y>x and space of all possible events are not represented by square butupper left triangle. If we allow y I haven't programmed this algorithm yet. But here are a few questions> for the newsgroup. 1) Aside from the complaints about terminology (e.g. incorrectly using> the term integration to describe a discrete summation), does this> formula work? 2) If it does indeed work for certain input values, does it fail for> others?Correctly implemented, it works fine, albeit slowly.Plenty of people have successfully tested one versionor another of James' recursion. 3) Is this formula just a restatement of something we already know> from number theory?Legendre's inclusion-exclusion formula, ca. 1790 I === HALF A LIGHTBULB?> I'm trying to get the deadwood off my reading shelf, and I just> finished BITCH, by Elizabeth Wurtzel, which turned out to be worth> reading for her powerful, insightful, and soul-baring epilogue, but> too much of the rest of the book was just too much, over and over,> about O.J. Simpson. Worse, Ms. Wurtzel is a cinema-holic who> dismisses the woes of real-life victims who don't play their parts> well, as if we should all have just the right life scripts for our> tragedies and have the necessary acting abilities to earn her respect> and sympathy, or at least a tepid encore? And, one of the next space-hoggers on my shelf is INFINITY AND THE> MIND: THE SCIENCE AND PHILOSOPHY OF THE INFINITE, by Rudy Rucker. Rudy ticked me off yesterday -- I had hoped to get a good start on> finishing the rest of the book, but then he stopped writing his barely> comprehensible Math-eze and added some stinking number questions,> including the following: Prove that 1 + (a + a^2 + a^3 . . .) = 1 / (1 - a)> for all a?Looks like you're better off sticking with E. === infinity ?>> OK what about tan(n)/n ?>The sequence tan(n)/n does not converge to 0; therefore, the series>tan(1)/1 + tan(2)/2 + tan(3)/3 + ... diverges.Correct. And for a proof that tan(n)/n does not converge to 0, you canA limit problem with explanation.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of === Selecting the correct graphI would like to know some important points when selecting a certaingraph for the data I have i.e. Using a Pie graph for displaying thelargest % of Annual sales, === Frank Wilczek's Emergent GravityOn closer look Wilczek has a different idea than mine. His idea is interesting.I am claiming to derive Einstein's gravity with the cosmological term as a large scalelimit of the exotic vacuum field from an instability at least in the QED sector ofthe micro-quantum vacuum. Wilczek in contrast never does any micro-quantumtheory that I can see in his paper? He already starts with classical fields andmakes no attempt to derive gravity as emergent from the micro-quantum standardmodel.By a modified conformal MacDowell-Mansouri mechanism,I asked:Is this where ODLRO MACRO-QUANTUM EMERGENCE is buried?you get the Einstein-Hilbert Lagrangian.All this is at conventional textbook level, for example,section 14.6 of Unification and Supersymmetry, 2nd edition,by Rabindra Mohapatra, Springer-Verlag 1992.If you want a prominent establishment name dropped, Frank Wilczekmentions the MacDowell-Mansouri mechanism paper seems very relevant in accord with what I amdoing in a simpler way independently. Wilczek is one of the besttheoretical physicists around for sure. I heard him speak twice nowthis year.where he notes that the mechanism was also independentlyformulated by Chamseddine and West.The mechanism was invented to make it possible to get gravityfrom the anti-deSitter part of Lie superalgebras used insupergravity === anyone know an approximation for Stirling Numbers of the Second>kind, S(n,k), for very large values of n?Graham, Knuth and Patashnik Concrete Mathematics refers to David and Barton, Combinatorial Chance, Hafner 1962, chap. 16, for asymptoticsof Stirling numbers.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of === I NEED HELP BADLY (sorry, maths not psych)Expires: 28 days>> You say that the force on a charge due to an electric field acts> instantaneously. Correct?>>Why do you ask what I am saying, when what I am>>saying is quoted right above?>>I am saying:>> as it enters a static electric field.>> So there is also an opposite force acting on the electrodes.>> even if the electrodes are light years apart.>> IS THAT WHAT YOU ARE SAYING?>>I am saying:> as it enters a static electric field.>>We have two electrodes - say 1 km apart.>(Or a light year apart - if you insist)>The potential difference is 1 million volts.>(Or a zillion volts - if the distance is a light year)>There is a small hole in the negative electrode.>We inject an electron through this hole.>When will a force act on the electron?>I am still saying:> as it enters a static electric field.>But what are YOU saying?>Not untill the electrode 1 km away feels the opposing force?>> IS THAT WHAT YOU ARE SAYING?NO PAUL. I am saying that your claim infers that the effect of an injectedelectron will be felt INSTANTLY at the far electrode. Of course, since theelectron existed BEFORE it was injected, the effect would have already beenthere even though the near electrode was in the way.. The only way thisexperiment can even be hypothesized is by either Ôannihilating' a very largenumber of electrons or by monitoring the force on the far electrode withmovement of the electron mass towards it.If the effects are instantaneous as you claim, you will have achievedinstantaneous communication.>>Come on, make your point.>What is the action time of the force on the electron?>Why do you think the distance to the other electrode is relevant?>How does the distance to the other electrode affect>this action time?>>Please don't say something like we don't know.>Because we DO know.>Do YOU know?OK I will agree with you. Assume it is instant. Therefore I can send messagesto the far electrode instantaneously.>>The rest is a repetition of the again!>>Paul>Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)Expires: 28 days>> as it enters a static electric field.>>Let us consider a charged sphere somewhere in the universe. It exerts a force>>on every other charge. If we can arrange for it to lose that charge somehow,>>you are claiming that all those forces disappear INSTANTLY.>>Parse the bloody sentence in quotes, Henry. It doesn't say that. Isn't>English your first language?>> - Randy, grabbing his popcorn and going back to watch the>entertainment>Moron. Stop kissing Andsernon's arse.Henri Wilson. See the Stupidity of === Basic relational theoryMy professor and my TA skipped their office hours and Im totally lost.Can someone give me a hand? They also rarely respond to emails. Justlooking for guidance since Im confused.Consider relation schema R(A,B,C) and the set of functionaldependenciesF={B->A,A->C}1. All non-trivial relations. Is this correct? Im just guessing.B->A, A->C,B->C,AB->BC,AC->C, BC->CA,AB->C2. Find a non-empty instsance of R(give a number of rows) thatsatisfies every Functional Dependency in F.Is this correct?A B C2 1 33 2 44 3 53. Find an instance in R that satisfies every FD in F accept A->BHow do you get A->B???? I dont see it. 4. Possible to find an instance an instance that satisfies every FD inF, but does not satisfy the FD AB->C. I have no idea. Im totally lost.Can === calculating limits - Need Help!> I'm having difficulties solving these two limits.>> I mustn't use L'hospital rule:>> a) lim(x*(2^(1/x))-x) where x increases to infinite.>>We want lim_{x->oo} [2^(1/x)) - 1]/(1/x). As x -> oo, 1/x -> 0. So this is >nothing but the derivative of 2^x at x = 0.> b) lim(cosh(x)-1)/(x^2) where x approaches 0.>>Use Taylor series as others have suggested, or if you're feeling >adventurous, show that for any C^2 function f in a neighborhood of 0 with >f'(0) = 0, (f(x) - f(0))/x^2 -> f''(0)/2 as x -> 0. For b), think cosh(x) = sqrt(1 + sinh(x)^2). Rationalize the numerator by multiplying both numeratorand denominator by 1 + sqrt(1 + sinh(x)^2) = 1 + cosh(x).Or substitute x = 2*y and use double-angle formulae.sinh(x) is better-behaved in a limit problem sincesinh(x) approaches zero as x -> 0.-- After California's recall election, wildfires Schwartz-en-ed the Bush-lands on its geographic right (when we wanted the forests to be Green). pmontgom@cwi.nl Home: San Rafael, California === (sorry, maths not psych)Expires: 28 days>And you have not provided any theory of E&M that allows any such>>thing as a reverse field. Nor why there should be any kind of>>speed limit involved. Nor why it should follow any such thing>>as the kinetic energy formula observed in accelerators. Nor have>>you provided a relation between energy and mass if you don't>>accept relativity.>>Socks>> radiation from an acceleraed charge!>> fields associated with a moving charge!>> The ÔBack EMF' concept.>> I would be most amazed if a moving charge DID NOT alter the field around>> itself, wouldn't you?>>Quite.>are accelerated.>>You KNOW the following, Henry.>In an accelerator going at full efficiency, we KNOW that>because it looses this energy as synchrotron radiation in the bends>of the circuit.(Very obvious and easily measurable.)>So we - and YOU - know that the RF-cavities never ceases>is only few mm/s below the speed of light.>>So why do you keep pretending that the E-field is not>speed approaches c, when you KNOW that isn't true?I DID NOT SAY THAT.The question is how much energy?You are making no attempt to answer that one. In typical fashion, you pretendthe relevant question does not exist.>>Another case of selective memory loss?>What you admit knowing in one posting,>you have forgotten in the next, eh?>>Paul>Henri Wilson. See the Stupidity of === transfinite series> Take the infinite series expansion for e and put it into the infinite> series expansion of e to the power of x and multiply the terms and you> end up with a series of aleph 1 terms>> This is nonsense. Even when you multiply everything out, there are>> still only countably many terms.>> Yes, compare countability of the rational numbers.> that sum to a finite result e to> the power of e. If you repete this process for e to the power of e to> the power of e do you end up with aleph 2 terms?>> For series with an arbitrary number of terms, look up the term summable>> family. But a necessary condition for convergence is that at most>> countably many terms are nonzero, this follows from the archimedean axiom>> of the real numbers.>If aleph 1 is the set of all subsets of aleph 0 No, aleph1 is first uncountable ordinal. You must mean bet1, which is usually written c. Whether or not aleph1=c is called the Continuum Hypothesis, and is undecidable in ZFC.> Anyway, your argument is incorrect for c as well. then take the counting>numbers the first subset is the empty set next come the sets with only>one member that is 1,2,3 etc next is the sets with two members these>can be arranged in a two dimensional array 1 2,1 3,1 4 etc 2 3,2 4,2>5 etc 3 4,3 5,3 6 etc etc all the way up to those with an infinite>number of terms which can be arranged in an array with an infinite>number of dimensions. Now if you want each one of these sets has a>complement however I think when you get to the sets with an infinite>number of terms then each subset is duplicated. which subset did I>miss? You missed infinite subsets. Your counting scheme simply won't work for the infinite number of dimensions case. You did not describe the counting scheme for that case.As for e to the power of e the first term is 1 next comes 1 + x>+ (x^2)/2 + then comes 1/2 + x/2 + (x^2)/4 + , x/2 + (x^2)/2 + (x^3)/4>+ , (x^2)/4 + (x^3)/4 + (x^4)/8 + etc arranged in a two dimensional>array all the way up to those with an infinite number of terms that>can be arranged an array with an infinite number of dimensions. you>have a one to one correspondence or at least a one to two>correspondence. CronOK lets review e^e=1+e+(e^2)/2+(e^3)/6+...= 1+ 1+1+1/2+1/6+1/24+...+ 1/2+1/2+1/4+1/12+1/48+ 1/2+1/2+1/4+1/12+1/48+ sorry about the messthis is the best I can do 1/4+1/4+1/8+1/24+1/48+ 1/6+1/6+1/12+1/36+1/144+ 1/6+1/6+1/12+1/36+1/144 1/12+1/12+1/24+1/72+ 1/6+1/6+1/12+1/36+1/144+ 1/6+1/6+1/12+1/36+1/144 1/12+1/12+1/24+1/72+ 1/12+1/12+1/24+1/72+1/288+ 1/12+1/12+1/24+1/72+1/2881/24+1/24+1/48+1/144+ this series has 1+aleph 0+(aleph 0)^2+(aleph0)^3+...epsilon 0 terms now I know that aleph 0 =epsilon 0 but thisseries contains more than epsilon 0 terms. the value of an infiniteseries depends on how you add up the terms this series is even worsehowever I want to leave it how it is to see if it can define curvesthat a === conjecturemean of two consecutive primes.it's easy to see how primes behave like spectral co-linear equations.Q: Resonance === measures of error> Here are some questions of mine. Are there any important measures of error in form * which use values of p> other than 1, 2, and +oo?I wouldn't dare try to answer this - I haven't learned enough yet (as ifdoing so is ever possible), but everything I do learn usually teaches methat I should have paid more attention to subjects I had previouslythought were unimportant. ;-) > Are there any important measures of error which are not in form * ?The error norms in higher order Sobolev spaces aren't quite in that form;but at least all the integral order spaces have norms in the form(Integral( Sum_i (|error_i|^p) ) ^ (1/p))which isn't much more general.> Clearly, using p = 2 yields a measure which is intermediate between> those with p = 1 and p = +oo. In that sense, p =2 represents a nice> compromise. But is there anything really special about p = 2 (say, as> opposed to p = 4 or p = 3/2) ?With any p, the absolute error norms are metrics on appropriately definedfunction spaces, so everything you can prove about normed vector spacesapplies to them.It's only for p = 2 that the metric comes from an inner product, however,and you can prove a lot more about Hilbert spaces. It's easier to findupper bounds for finite element method errors === transfinite series > Shmuel (Seymour J.) Metz, SysProg and === Popular measures of error> Suppose that g(x) is proposed as an approximation of f(x) on [a,b]. What> are the most popular ways of measuring how well g approximates f over that> interval? Here are several measures. In each case, the smaller the measure is, the> better the approximation is considered to be. AInf. The maximum of |absolute error| over the interval,> where absolute error = g(x) - f(x).> RInf. The maximum of |relative error| over the interval,> where relative error = (absolute error)/f(x). A2. The root-mean-square of |absolute error| over the interval.> R2. The root-mean-square of |relative error| over the interval. A1. The average of |absolute error| over the interval.> R1. The average of |relative error| over the interval. All of these measures may be thought of as power means (also called Hoelder> means). They have the form * ( Integral( |error|^p ) / (b-a) ) ^ (1/p) where the integral is taken with respect to x from a to b, and error is> either absolute or relative. Obviously, in the cases of A1 and R1, p = 1,> and in the cases of A2 and R2, p = 2. The value of p is not so obvious,> however, in the cases of AInf and RInf. But in the limit as p increases> without bound, the power mean gives simply the maximum, as needed in AInf> and RInf. As such, for those cases, we may say that p = +oo. > Here are some questions of mine. Are there any important measures of error in form * which use values of p> other than 1, 2, and +oo? Are there any important measures of error which are not in form * ?How about the geometric mean, A0 exp ( integral ( log |absolute error| ) / (b - a) ).-- Gerry Myerson === Permutation: how to detach cycles/transpositionsX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>>[something not in english!]>You know, whenever one posts source code in a NG that is not>comp.lang.thatlanguage or a subhierarchy of it, it would be a Very>Nice Thing(TM) to specify in which language it is written!It's Python. You can tell because it looks like pseudocodebut it's not. You know, nobody ever complains about not recognizing C.Things are gonna be different after the revolution...>Michele>-- >> Comments should say _why_ something is being done.>Oh? My comments always say what _really_ should have happened. :)>- Tore Aursand on === === Re: sum( sin(n)/n , n=1..infinity) < infinity ?X-DMCA-Notifications: http://www.giganews.com/info/dmca.html> Julien Santini a .8ecrit dans le message de> Does the series sin(1)/1 + sin(2)/2 + sin(3)/3 + sin(4)/4 + ...>> converge?> How does one prove convergence (or divergence)?. If it converges what> is a good way to estimate its value?> Abel's rule>> OK what about tan(n)/n ?>>The sequence tan(n)/n does not converge to 0; I imagine this is true, since Israel says he's proved it.But it's not all that obvious (it's not clear to me whetheryou were meaning to say it was obvious or not...)>therefore, the series>tan(1)/1 + tan(2)/2 + tan(3)/3 + ... diverges.>>Best === replies to this message constitute permission for an emailed EC F3 04 26 4E BF 1A 92X-Tom-Swiftie: These are very nice apples, Tom said tartlyFor what it's worth, there *is* an extension of the cross product forvectors in higher dimensions. The problem is that the cross productis a cheat. The generic operation is a tensor outer product, and it so happensthat the outer product of two 3D vectors has three components, so youcan map it into a vector easily. But this is a cheat of sorts, and itleads to confusions in many areas (especially when people start usingit in studying === reliably count primes may be> substantial, perhaps many millions of $/yr. Such a method would> enable someone to decide the probability of whether he he has tried> all possible prime factors of a number for code breaking. Such a> > method combined with others could be worth a great deal of money.>> This doesn't really make sense to me. If you can generate all the possible> prime factors, then you know how many there are. If you can't generate> them, then certainly you can't use the fact that there are more factors to> somehow generate more factors. Seriously, if you know that there must be at> least 10^20 more primes to try, that still leaves you to find all these> primes. If you know that there aren't any more primes to try, then you know> how many you've tried. (I'm not an expert on this topic, so I may be way> off.)>> If you know that there are 10^20 primes to try it will take you 10^11> seconds to try them all when you could try each prime in a nanosecond.> That is something like 3000 years. That would even get you moderate> 40 digit numbers out of reach.>> There are very fast methods to> count how many primes are less than a given number. There are VERY fast> methods to determine the primality of even extremely large numbers, with> some small chance of error.>> There are even VERY fast methods to determine the primality of even> extremely large numbers without a chance of error.Just to clarify, as I understand it... knowing a number is composite does nottell you the factors, to be commercially useful you have to find the factors, otherwisethe Ôdiscovery' is mostly academic.The main techinique is finding witnesses to compositeness. Over half of all numbersless than any composite are Ôwitnesses' to that composite, so to determine a number hasfactors just requires more and more random attempts until a witness is found, unfortanetlythis only establishes a confidence of primality, you seem to be suggesting a === assumption.> 1. Are you saying that the speed of light is source dependent? Yes. 2. What is your definition or explanation of Ôsource dependency'?>> Peter Riedt> Just throw a ball forward from your car window as you move.> It should be obvious, and no different from throwing a photon from a star.> The velocity of light in interstellar space is c, with respect to the star.> If the star moves, then it still c with respect to the star. To an observer> the velocity is c+v, where v is the velocity of the star.And for waves ... === you may have noticed frenetic activity from posters trying to> convince you that there's nothing sinister about mathematicians doing> their best to downply my find [...]And again you find it perfectly acceptable to hurl insults at millions-- while reserving to play the indignant sensitive little ßower whensomeone hands you a tiny fraction of your insults back.> of a way to count prime numbers by> integrating a partial difference equation, but what's the bottom line?You have yet to present any kind of way to count prime numbers thatactually has anything to do with integration at all.Hint: a sum is not an integral.> Does what I found work or not?> It has been conclusively proven that it doesn't.I presented the implementation of the exact literal lines you postedhere and you yourself could not find anything wrong with it.If you had a quarrel with the implementation, you could even simplyhave posted your own little fortran or basic or c-routine. No bigdeal. But of course you can't.It does not work. That is all there is to it.I have given you thebenefit of the doubt long enough to implementexactly what you posted here to see for myself whether you're on tosomething or not. That's called Ôscience': I go and examine theevidence myself.And I have seen with my own eyes that you don't have anything herethat counts primes. And further *lies* of yours to the contrary willnot sway someone who's actually examined the evidence himself.> It does. End of story, so mathematicians should acknowledge it. Ah: you say so and thus it is so. ÔTis a simple world you live in.So why does this go for you but not for everybody else on the planet?Because there's a lot of people out there that say you stuff doesn'twork. And contrary to you they have evidence for their claim.> But they're fighting to totally ignore it. Translation: Sinister> attempt by academic types to hide something really important.Ask yourself: how does this line distinguish you from everyrun-of-the-mill dime-a-dozen psychotic crackpots with a new theory ofeverything to sell, without a shred of evidence to present and withdemonstrated lack of grasp of what they're talking about?> Otherwise, why go to so much effort to fight me, when a simple way to> shut me up on the issue is just record it somewhere?Nobody is going to any particular effort fighting you.Nobody is going to record anything anywhere because there's nothing torecord here.> These posters trying to convince you otherwise are just insulting your> basic intelligence.Just to clarify for to odd reader out there: I am not trying toconvince you of anything at all. (Contrary to Mr. Harris.) Go and seefor yourself, as I === OK lets review e^e=1+e+(e^2)/2+(e^3)/6+...= 1+ 1+1+1/2+1/6+1/24+...> + 1/2+1/2+1/4+1/12+1/48+ > 1/2+1/2+1/4+1/12+1/48+ sorry about the mess> this is the best I can do > 1/4+1/4+1/8+1/24+1/48+ > 1/6+1/6+1/12+1/36+1/144+ 1/6+1/6+1/12+1/36+1/144 > 1/12+1/12+1/24+1/72+ 1/6+1/6+1/12+1/36+1/144+ > 1/6+1/6+1/12+1/36+1/144 1/12+1/12+1/24+1/72+ > 1/12+1/12+1/24+1/72+1/288+ 1/12+1/12+1/24+1/72+1/288> 1/24+1/24+1/48+1/144+ this series has 1+aleph 0+(aleph 0)^2+(aleph> 0)^3+...epsilon 0 terms now I know that aleph 0 =epsilon 0 but this> series contains more than epsilon 0 terms. the value of an infinite> series depends on how you add up the terms this series is even worse> however I want to leave it how it is to see if it can define curves> that a simple series can'tYou are mixing ordinals with cardinals. Aleph_0 is a cardinal, butepsilon_0 is an ordinal. The least transfinite ordinal is called omega(or sometimes omega_0). Both omega and epsilon_0 are countable ordinals,which is to say that their cardinality is aleph_0.If A is any countably infinite set, then AxA is likewise countable. Byinduction, we can conclude that A^k is countable for each natural numberk > 0. The union of { A^k : k > 0 } is likewise countable. Thecardinality of each of these sets is aleph_0.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.>S(17)=1+7=8,>S(98)=9+8=17>>I have the following question: There are a, b and c, natural numbers, such>that>>S(a+b)<5>> this is true then :>> a < 50 and b < 50>>Surely not? What about a = b = 100? a+b = 200 and s(200) = 2 + 0 + 0 < 5.>>S(a+b+c)>50 ?>> Uhm..if this statement were true, this would be true as well : >> a+b+c>599999 (as this is smallest number n for which S(n) > 50)>> And..that can't be true? I'm not sure if this is right since it's awfully>> simple..perhaps your definiton of decimal sum is different..>>Doesn't work. You can get arbitrarily big values of a + b + c with >S(a+b), etc < 5. I can't see whether or not you can make S(a+b+c) bigger >than 50 or not, but will think about it. (Probably won't have too much >success - number in the obvious way)> You can get at least 24:a = b = c = 5050505.a+b = a+c = b+c = 10101010a+b+c = 15151515S(a+b+c) = 24 An upper bound is 60.S(a + b + c) = S(10a + 10b + 10c) <= S(5a + 5b) + S(5a + 5c) + S(5b + 5c) <= 5S(a + b) + 5S(a + c) + 5S(b + c) <= 15*4 = 60. -- After California's recall election, wildfires Schwartz-en-ed the Bush-lands on its geographic right (when we wanted the forests to be Green). pmontgom@cwi.nl Home: San Rafael, California Microsoft === talking pure archeology?>>>> Forbidden archeology by Cremo, Thompson!>> Like the name says: Forbidden>> > I urge you not to read it!>> Have you been to a museum lately? They have these things call> Dinosaurs> that lived millions of years ago. Don't you think if the bible isthe> word> of the lord that the good lord would have maybe warned us about> gigantic> carnivorous creatures roaming the earth! Afterall, he did mentionthe> measly little snake.>>> Lurch>>> The argument from my Southern Baptist Preacher relatives in> Kentucky ( I have several) say that the world was created complete> with the bones and other found elements of the historical recordcreated> intact to give a old construct when it was created 6000 years ago....>> And I cannot dispute it... Because any old stuff sticking out of the> ground> was made by the hand of god sticking out of the ground and any testonly> confirms gods ability to set a good stage for mans little drama toplay> out....>> We are but god theatrical group and nothing more than entertainmentfor> his and his kids enjoyment in the evening, them throwing in an> occasional> ELE to start the next act.......>> That's why it's important to note that Genesis contradicts Genesis. If> you're looking for facts, the Bible is bull.>> Mark Folsom>> I don't agree it's totally bull.. While I don't in anyway> think there is anything like god as constructed in its words it> still has some historical value on both a sociological> standpoint and in the study of physiology with a bit of> geography to boot.You mean like the geographical facts you can believe if you find that theyagree with a reliable source--those facts?> Some is good data but must be considered> as a man made group of many writings put together> by committee ( mainly a king) for a purpose.Please cite some good data in the bible and tell us what it's good for.Tell us which facts you would believe without independent verification.> But within> its text are references that can be verified and are worth> noting. While the B'Levers will point to the occasional reference> of a verifiable event, person or subject as proof that the> god construct is valid, they are mistaken. As with any bit> of prose, parts may be valid and parts may be complete> fabrications and parts may be incorrect interpretations of> valid parts.>Still waiting for the valid === Theory & Fiber BundlesCommentary 4Synopsis of where we are at so far in the emergent evolution of our understanding of how the mathematics of fiber bundles with a natural idea of hyperspace and Super Cosmos (Linde's chaotic inßation) is interpreted as the physics of classical relativity, local quantum field theory with the objective of using it also in the macro-quantum theory of emergent Einstein gravity with exotic vacuum dark energy/matter for metric engineering and possibly also in micro-quantum delocalized string theory.We have taken a top -> down approach for the principal bundle. Start with a large higher dimensional hyperspace H. Do not assume any metric in it to begin with. Assume a CONTINUOUS Lie symmetry group G equivalence relation ~ that partitions H into disjoint G-orbits that are equivalence classes of points X of H where X' ~ X mod G. Each distinct point of the base space M is a projection from a single G-orbit where M = H mod G or H/G. The G-orbit is an internal hidden structure of the base space event M that can include extra compactified boson dimensions and also the fermi dimensions of supersymmetry. How Planck's h and Heisenberg's uncertainty fit in is not apparent yet; The construction so far seems classical. h seems to demand fractals that are continuous but not differentiable like the classical manifolds are.Hyperspace H is locally a product of a the beyond space-time fiber and a small neighborhood of the base space.Around each point x of base space M there is a coordinate patch C(x) and a fiber f(x) and a special diffeomorphism Trivial (x) that maps H at x into the product C(x)f(x). If the hyperspace is globally not oriented like a one-sided Mobius strip or a Klein Bottle then Trivial(x) locally unwraps the global twists. A transition function isTrivial(x)Trivial(x')^-1 in the overlap of the local coordinate patch neighborhoods around x and x' with different G-orbits (I think?)6. There is a purely vertical inverse bottom -> up emergent projection P^-1 from base space C(x) to fiber f(x).P^-1 is a rule for associating each point fo in the fiber f(x) with a group element g < G of the principal bundle for the gauge forces i.e. electroweak + strong NOT gravity yet.7. P^-1 does not establish a horizontal connection for identifying points on different fibers f(x) and f'(x') in different regions of the base space with the same continuous symmetry group element g in the global group G.The global Cartesian product space is like a broad staircase with vertical handrails. In contrast the fiber space is like a set of identical escalators moving up and down independently. S. Y. Auyang How is Quantum Field Theory Possible?p. 217, Oxford, 1995.8. The local gauge force potential interaction dynamics allows parallel transport of fiber information along continuous paths in the base space of control parameters, which in special applications can be the space-time manifold, but generally it can be other kinds of spaces.9. The ALL-IMPORTANT section: A section is an inverse projection C(x) -> P^-1[C(x)] mapping a neighborhood of base space back into a region of hyperspace H. The section creates a local coordinate patch in the hyperspace from the local coordinate patch in the base space by arbitrarily CALIBRATING a single point in the vertical fiber fo(x) above each x in C(x) as the identity e of G. If a single section works globally for the whole hyperspace then the bundle is trivial like a two-sided orientable cylinder not like a one-sided non-orientable Mobius strip that resembles a spinor needing a 4pi rotation to return to its original normal vector.The idea of connection is implicit in the idea of the section.10. The special section called the principal connection maps the tangent spaces of the base space to the tangent spaces of the hyperspace. Let Tx(M) be a tangent space of M at point x. Let TX(H) be the tangent space of hyperspace at hyper-point X. Then the principal connection isP^-1[Tx(M)] = TX(H)X = (x,fo)TX(H) = TX(H)horizontal + TX(H)verticalTX(H)horizontal =Tx(M).Note that Einstein's smooth c-number gravity is essentially from the tangent bundle {M, Tx(M)} with an additionalmetric or alternatively a tetrad spanning both x < M and Tx(M) that embodies Einstein's Equivalence Principle (EEP). The symmetry group G acts like the identity in Tx(M) and should not be confused with Diff(4) in Einstein's gravity theory. The principal connection splits any path in hyperspace into a horizontal path in base space and a vertical path in the extended fiber region of hyperspace. Presumably we can extend this from paths to world sheets for strings rather than points?11. Given some principal connection |~ and a worldline in base space M. The worldline can be horizontally lifted into the extra space dimensions of the Calabi-Yau spaces (anticipating the string generalization yet to come) such that all tangent vectors of the hyper world line are horizontal. This is PARALLEL TRANSPORT IN HYPERSPACE as distinct from parallel transport of world tensors along worldlines in Einstein's gravity theory in the special tangent bundle [M, Tx(M)].12. The horizontal lift of a M world line into hyperspace is UNIQUE and this allows us to associate different points fo(x) and fo'(x') in different NON-OVERLAPPING regions of hyperspace with disjoint patches C(x) & C'(x') with the same g < G relative to that specific M worldline connecting the two points.13. EEP (Einstein's Equivalence Principle) of GR is an approximate statement that: i. far from a space-time singularity and ii. at a scale larger thanLp^2 = hG(Newton)/c^3One can freely ßoat/fall feeling no weight (i.e. no g-force) along a slower-than-light time-like geodesic in a non-rotating LIF (Local Inertial Frame) with comfortably small stretch-squeeze torture rack local curvature tidal force inhomogeneities in the g-force.13. Thus GR is a specialized kind of fiber bundle not the same as the fiber bundles in local quantum field theory. Indeed, I claim that the former is emergent from a false vacuum instability in the latter.Commentary 3The hyperspace H consists of fibers f(x) that areeither copies of or representations of the symmetrygroup G.Jack, this is not quite correct. They are homogenous spaces onwhich the group operate transitively. Example, for the group SU(2),you can take as the fibre a copy of SU(2) itself (3-dimensional), oryou can take sphere S^2, on which SU(2) operate (2-dimensional).Notice that S^2 is not a representation of SU(2). It is a quotientSU(2)/SO(2).Early Kaluza-Klein theories were operating with group Manifolds.Souriau, and later Witten, suggested more realistic theories wherefibers could be of lesser dimensions. Thie rigorous mathematics andexamples of this latter approach have been developed in themonograph:Riemannian Geometry, Fibre Bundles, Kaluza-Klein Theories and AllThat... (World Scientific Lecture Notes in Physics, Vol 16)by Robert Coquereaux, for the local gauge forces:1. A transformation g of the symmetry group G acts on the ordered pair X = (x, fo) in hyperspace H with output gX.Question: Can gx = x' =/=x i.e. can one move the base point in this operation or must G always be the identity in the base space? That is, we always need, in addition to G a connection and a path in order to change location in the horizontal base space and the vertical fiber space that is beyond space-time. G certainly moves fo up and down the vertical fiber for every element g =/= identity. Does it also move x -> x' = gx =/= x horizontally along the base manifold without a connection field and a path specified? Clearly the answer must be NO. See below.The modern understanding of gauge invariance, as a symmetry under transformations ofquantum-mechanical wave functions, was reached by Weyl himself and also by London veryshortly after the new quantum mechanics was first proposed. In this understanding ofabelian gauge invariance, and in its nonabelian generalization [2], the space-time aspect islost. The gauge transformations act only on internal variables. This formulation has hadgreat practical success. Still, it is not entirely satisfactory to have two closely related, yetdefinitely distinct, fundamental principles, and several physicists have proposed ways tounite them.One line of thought, beginning with Kaluza [3] and Klein [4], seeks to submerge gaugesymmetry into general covariance. Its leading idea is that gauge symmetry arises as a reßec-tion in the four familiar macroscopic space-time dimensions of general covariance in a largernumber of dimensions, several of which are postulated to be small, presumably for dynam-ical reasons.Here we should take the opportunity to emphasize a point that is somewhatconfused by the historically standard usages, but which it is vital to have clear for whatfollows. When physicists refer to general covariance, they usually mean the form-invarianceof physical laws under coordinate transformations following the usual laws of tensor calculus,including the transformation of a given, preferred metric tensor. Without a metric tensor,one cannot form an action principle in the normal way, nor in particular formulate the ac-cepted fundamental laws of physics, viz. general relativity and the a purely mathematical point of view one might consider doing without the metric tensor;in that case general covariance becomes essentially the same concept as topological invari-ance. The existence of a metric tensor reduces the genuine symmetry to a much smaller one,in which space-times are required not merely to be topologically the same, but congruent(isometric), in order to be considered equivalent. In the Kaluza-Klein construction, for thisreason, the gauge symmetries arise only from isometries of the compactified dimensions.Another line of thought proceeds in the opposite direction, seeking to realize generalcovariance [CapitalEth] in the metric sense [CapitalEth] as a gauge symmetry. arXiv:hep-th/9801184 v4 23 Apr 1998IASSNS-HEP-97/142Riemann-Einstein Structure from Volume alerting me to this relevant paper by Wilczek.BTW Wilczek shows that Gennady Shipov's torsion theory is closely related toRoger Penrose's spinors in curved spacetime with the anti-symmetricspin connection as the locally induced compensating torsion field.It all comes from locally gauging the O(3,1) subgroup of the Conformal Groupas I said previously based on Utiyama's and Kibble's papers from the mid-1960's.Whether or not Akimov's claims from Moscow that torsion waves from O(1,3) ofsufficient intensity to have psychotronic weapons bio-toxic effects can easily be generated when,in contrast, gravity waves from T4 are so hard to find is another issue not considered here.The gravity wave T4 coupling parameter is essentially Ed Witten's alpha' = (superstring tension)^-1.What is the corresponding O(1,3) spin connection coupling parameter? Akimov's claims hangon the answer to that question. Is it easier to make propagating torsion dislocation topological string defectsthan to make propagating curvature disclination topological string defects in the MACRO-QUANTUMVacuum Coherence Field's Goldstone Phase? That's what Akimov's claims come down to in terms ofmy new theoretical paradigm for the emergence of Einstein's Gravity and the Unified Exotic Vacuum Field ofw = -1 Dark Energy/Matter.2. The action of the symmetry group G on the total hyperspace H induces an equivalence relation ~ .That is, if X' = gX, g < G, then X' ~ X.3. ~ partitions hyperspace H into disjoint non-overlapping equivalence classes called G-orbitsG(X) = {gX, for all g < G}Remember that in this principal bundle fo is also a g < G.All G-orbits have identical structure and are diffeomorphic to G.4. This disjoint partition of hyperspace H gives the quotient space H/G that is the base space M with points x.Every point x of the base space M is really an equivalence class or G-orbit of a continuous infinity of points of a larger dimensional Hermetic or occult hidden hyperspace implicate inside it. Worlds within worlds. Wheels within wheels. Shades of Bohm's Implicate Order?5. The Projection Map P is simply P:G-orbit -> x.This means that each individual G-Orbit is really associated with a single vertical fiber at a single horizontal base space event. The G-orbit is the vertical fiber beyond, in the usual physics applications, a localized spacetime event x, although we can have delocalized base spaces of twistors whose intersections are points. We can also perhaps have base spaces of finite strings both open and closed and even base spaces of higher dimensional brane worlds?Commentary 2Given coordinate patch C(x) in the base space M in a neighborhood of point x and fiber f(x)form the local Cartesian product C(x)f(x) with ordered pair X = (x,fo).Take the union C(x)f(x)/C(x')f(x')/... of all such local products.There are redundant ordered pairs X because the coordinate patches C(x) and C(x') as sets overlapwith non-vanishing intersection C(x)/C(x')=/= Empty Set.Identify the redundant multiple images of the same actual point of the base space M usingthe symmetry group G as an equivalence relation. That is, two ordered pairs X and X' areidentified or equivalent if x = x' < C(x)/C(x') and if fo' = gfo where g < G to form disjointequivalence classes {f(x)} that are the distinct points of the fiber in hyperspace H.This is all local at a fixed base point x like in an internal gauge force symmetry.g is also called a transition function.The hyperspace H is the factor space of the union C(x)f(x)/C(x')f(x')/ ... mod G.The projection map P:(x,{fo}) -> xWhen M is the curved space-time of Einstein's gravity theory in addition to the G equivalencein the extra space dimensions of the fiber, x'(E) = Diff(4)x(E) at fixed event Eto make disjoint equivalence classes {x(E)} mod Diff4(E).One can imagine a hybrid where the fiber is a discrete space of strings of c-bits.One can also imagine a fiber of strings of qubits.1 qubit is a parallel infinity of c-bits.i.e.|qubit> = |1 c-bit><1c-bit|qubit> + |0 c-bit><0 c-bit|qubit>Where there is a continuous infinity of different c-bit basesor orthonormal frames each corresponding, for example,the the angular orientation of an inhomogeneous fieldmagnet in a Stern-Gerlach filter for spin qubitsin the DARPA spintronics project or like the billion billionSingle Electron Transistors inside the human brain at thesub-microtubular protein dimer hydrophobic cage level formingthe hardware interface with external world whose software is our stream of inner consciousness.Each possible orientation is a primitive parallel quantum universe.The quantum computer computes in all possibleorientations simultaneously like a continuousinfinity of classical Turing machines in adistributed network working on the same problem - or so the folklore goes.to be continued.Commentary 1The fiber bundle as an idea has 4 parts.1. A structure symmetry group G.2. The total hyperspace H or, in some applications Wheeler's BIT.3. The projection map P.4. The base space M or, in some applications. Wheeler's IT.The hyperspace H consists of fibers f(x) that areeither copies of or representations of the symmetrygroup G.The projection map P collapses a fiber f(x) in the hyperspace H toa point x in the base space M.All of these objects are continuum differential manifoldsdepending on the continuum of real numbers which itsassociated issues of Cantor's infinity of infinities ofCabalistic Aleph's in an ascending Jacob's Ladder.This is not a discrete combinatoric mathematics althoughsuch a skeletal structure is associated with it as inHerman Weyl's Theory of Groups and Quantum Mechanicsand as in Saul-Paul Sirag's presentation of V.I. Arnold'sA-D-E mathematics of everything.The base space is covered by an atlas of local coordinate patcheswith all important overlap transition functions sewing thepatches together like a quilt.M is space-time in local micro-quantum field theory of pointThe extra-dimensions of hyperspace formthe Calabi-Yau space of vibrations of thesuperstring beyond space-time.The connection on the total hyperspace H is the potentialof a local gauge force.Examples of connections is the 4 potential Au(x) inMaxwell's electromagnetism with G as U(1).There are similar connections for the Yang-Mills weak forcewith G = SU(2) and the strong force with G = SU(3).Classical general relativity, as distinct from local micro-quantumfield theory, has the torsion-free symmetric three-index non-tensorLevi-Civita connection with G as the Diff(4) group.The latter comes from locally gauging the 4 parameter translation subgroup (generated by the 4-momentum Pu of globally ßat special relativity ) of the 15 parameter conformal group of Roger Penrose's massless twistors.Bottom -> Up: Given base space M and symmetry group G construct thehyperspace H as a quilt patchwork.Top -> Down: Given hyperspace H and symmetry group G construct thebase space M as the non-overlapping partition of hyperspace into G-orbitscalled the quotient space of H mod G in the principal bundle.Micro-quantum source renormalizable local fields of spin 1/2 lepto-quarks are associated vector bundles.Micro-quantum force renormalizable local fields of spin 1 gauge force bosons (electro-weak and strong) are from the principal bundle.There is no renormalizable quantum gravity in this precise sense.This is because classical Einstein gravity is a More is different (P.W. Anderson) emergent collective effect as in Andrei Sakharov's metric elasticity of an instability in the globally ßat false vacuum of the interacting lepto-quark source/electroweak-strong force.Einstein's gravity + unified exotic vacuum dark energy/matter with Andrei Linde's chaotic inßationary cosmology are the result of the continual phase transitions from globally ßat false high entropy micro-quantum vacua to locally curved macro-quantum low entropy metastable === Myth and Reality> Equivalently, M*N is the same as M*N mod (M + N - 1).>> Sorry, this should be multiplication of M digits with N digits, base b,> is equivalent to multiplication modulo b^(M + N - 1), i.e. M+N-1 digits.>>Oh well, so FFT or not, looks like multiplying M by N by any method means>>MN multiplications! Well, when these multiplications are hardwired (as in>>human memory for single digits) the computational issues (On*n) becomes>>really irrelevant, for they all are done in no time at. Like, the video>>extraction for radar data processing is done by NAND gates - its all done in>>real time!>> You are in error. The number of multiplications required for>> multiplying two numbers with the FFT method is O(n*log(n) where n is>> the larger of the two numbers; it is not m*n.>>Fine, just multiply 12345 by 67809 using FFT with less than 25>multiplications. Do it here.Seemingly you do not understand what the O() notation signifies. Whenone says that the FFT method is O(n*log(n)) one is saying that thereis some constant C such that for n sufficiently large,(# of required multiplies) is less than C*n*log(n)This does not mean that the cost of the FFT method is less than n*nfor all n, just that it is for n sufficiently large. Thus, yourproposed test is irrelevant to the point under discussion. That said, the simple two point formula runs as follows: 12*67 = 804 (4 one digit multiplies)345*809 = 279105 (9 one digit multiplies)(12+345)*(67+809) =357*876 = 312732 (9 one digit multiplies)Term 0 = 279105Term 1 = 312732 - 279105 - 804 = 32823 Term 2 = 80412345*67809 = 804000000 + 32823000 + 279105 = 8371021054 + 9 + 9 = 22 multiplies < 25Be that as it may, the cross product method for multiplication isquite obvious and is regularly rediscovered. I discovered it myselfas a child and even then was under no illusion that I had doneanything remarkable.Richard Harter, cri@tiac.nethttp://home.tiac.net/~cri, http://www.varinoma.comWe have people from every planet on the earth in this State.-- California Governor Gray === people?>I've come across various students who viewed that they were missing the>mathematics gene (or programming gene, or whatever the particular>subject happened to be). In those cases it was uniformly the case that>their difficulty was emotional/attitudinal, rather than cognitive. As you>mention, one unhelpful attitude is perfectionism, especially in hard-edged>subjects where some answers are clearly objectively *wrong* and thus the>student has no wiggle room to avoid the conclusion that they made an>error.Have you ever heard of THE MATH GENE by Keith J. Devlin? He arguesthat what mathematicians generally think of as mathematics arises fromthe ability (of humans) to acquire language.>Several of the missing gene students had the unhelpful attitude that>they expected maths to be easy, since they had found their schooling easy>so far.No doubt this is a contributing factor. I have observed that somestudents feel incredible pressure to excel academically (and felt thatway myself as a student, at times), which can add to the stress onefeels when one has hit a mental roadblock.>ISTM that cognitive issues do kick in when dealing with high levels of>abstraction where there are no readily-accessible concrete models. For>example, my brain hit the wall trying to visualise non-Hausdorff spaces,>and my painful memory of the rest of that topology course is of generally>mindless memorizing and proof cranking.I had that problem when learning computability theory as anundergraduate. I didn't find any books to help me develop myintuition until several years later, when I discovered David === Criteria for Complete Metrizability?Could anyone point me towards some theorems which give criteria fortopological completeness, without refering to metrics?Rex === weapons>Commentary 3>>The hyperspace H consists of fibers f(x) that are>>either copies of or representations of the symmetry>>group G.> Well, that convinces me... It should. They are bible fibers. Wouldn't want to blaspheme.-- I love the smell === Re: Question on generation of large prime numbers > Summary of Euclid's proof: > 1) Suppose that there is a largest prime; call it P > 2) Calculate N = (product of all primes from 2 to P, inclusive) + 1 > Are you trying to say that the set of all primes has no primes missing? Why should there primes be missing from all primes? > That is vacuously true, and holds for {2,3,7}. If {2,3,7} is the set > of all primes, which we are permitted to assume according to Euclid, > then {2,3,7} is the set of all primes, bar none.But is there a way in which you can say that {2, 3, 7} are all primes<= 7? My opinion (but see more about this below) is that when somebodytalks about all primes <= P, the meaning is that all numbers less thanP are tested for primeness, as the proposition is only that there isa largest prime, not as in Euclid that the set of primes is finite, itamounts to the same thing, but that is something different.So, when you assume that the set of primes is {2, 3, 7}, indeed theste of all primes <= 7 are those three, vacuously as you say. When youdo *not* assume such the set of all primes <= 7 is {2, 3, 5, 7}.Archimedes Plutonium had a proof that worked (approximately) as follows.1. Definition: 1 is not prime.2. Definition: a number > 1 is prime if it is not divisible by a prime smaller than itself.3. Assume the set of all primes is finite, say {p1, ..., pn}.4. Form the number P = p1*p2*...*pn + 1.5. P is not divisible by any prime smaller than itself.6. So by the definition of prime P must be prime.7. But it is not in the list of all primes: contradiction.It is always stated by many people that this proof is logicallyincorrect. However, there is *no* logically incorrect step involved.(The counterexample is one of the well-known one, but that is not acounterexample to the logic involved, only to the conclusion. Butwe all know that when you start with a false premissa the set ofprimes is finite, you can arrive with logically correct means toa false conclusion...)There is a well-known person in this newsgroup who does the same.He starts with a false premissa (in a ring (a + b)/c = a/c + b/c),although he does not state it in so many words. Using that he doesindeed arrive at a false conclusion, with logically consistent steps.But while Archimedes concludes that the premissa is false, thewell-known poster does not conclude such, he only concludes thatmathematics is in error. > Or are you trying to say and we assume that we know all primes <=P.You are thinking about Euclid's proof, which this is not.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, === generation of large prime numbers > |If Richard has simply added let all primes <=P be known to his premise > |I wouldn't have jumped on it that way. > I don't think this would help. Talking about what primes are known is > subjective. > Not really, would assigned make you happy? > The set of all primes is the set of all known primes in this proof. > I've said that repeatedly. I do not understand why you would not have jumped in Richard's proof whenhe said let all primes <=P be known. This does not help a bit. Ifthe set of primes is {2, 3, 7} than all primes <= 7 are known and theyare 2, 3 and 7. > The set-theoretic notation for what my sentences expressed would be > no different if I included or excluded the word known. I was > simply trying to avoid the naked word all as people immediately > misinterpret that based on their knowledge about the primes. Yup, so what? If somebody talks about all primes <= P I would thinkhe would assume that all numbers <= P have been tested for primenessand would have taken only those that are proven to be prime. Noteagain, this is *not* Euclid's proof, but a well-known and muchoccuring variation. And is *just as valid*.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; is W. Rudin, Principles of>>Mathematical Analysis, chapter 2 ( Basic Topology), problem 18:>>[A set E is perfect if E is closed and every point of E is a limit>>point of E] Is there a nonempty perfect set in R which contains no>>rational number?>> ... [ contructs open set containing rationals and with measure < 1,>> takex X as its complement ] ...>> Try showing (1) X is not countable> (2) the isolated points of X constitute at most a> countable set> (3) X {isolated points of X} is closed>> I don't think this will work. Try considering points of X such that>> every neighborhood contains an uncountable infinity of points of X>> instead.>> By Cantor-Bendixson, (2) must be true. (3) seems clear since >> X, being closed, contains all its accumulation points, and these >> are also exactly the accumulation points of X{isolated points}.>> Is there a problem with (1) or C-B too much to take as known? The theorem is not listed in the index to the book, so I don't think> it can be assumed in the problem.OK. To see (2): p in X is isolated iff it has a neighborhood Np such that Np{p} contains no point of X, and in particular, no (other) isolated point. So the isolated points are in 1-1 correspondence with a set of pairwise disjoint intervals (say for each p the maximal Np), and so form a countable set. === probably a well-known subject for number theorists,but I've never read anything about it. The question is how closely canwe approximate pi by rationals. More specifically:For integer n>0, let f(n) be the largest integer m such m/n < pi.Let d(n) = pi - f(n)/n.Then d(n) measures how accurately we can approximate pi by a rationalwith denominator n.How small can d(n) be? Clearly, d(n) < 1/n. But can we make d(n) muchsmaller than that? Q1: Can we find arbitrarily large values of n such that d(n) < 1/n^2? Q2: Can we find arbitrarily large values of n such that d(n) < 1/n^3? Q3: In general, for each p>1, can we find arbitrarily large values of n such that d(n) < === random processI have a question.For a randomly moving object in two-dimensional plane, the object has tomove from point X to point Y. During the movement, there are two randomprocesses posing on the object. For example, one process is the irregulargeograph and the other process is the varying weather. The two processes maybe correlated. Plz give some suggestions onwhere can I find the related reference and which === talk factorizationsLet me try again: > Polynomials are well-known in science and mathematics, but while > finding roots of polynomials is typically the aim of the average > researcher, polynomials themselves can be used as powerful tools for > analyzing the roots of *other* polynomials. > The concepts are advanced, but can be approached by first considering > a basic example. > The basic factorization to start is > (c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = > 49(x^3 + 5x^2 + 3x + 1) > with the c's algebraic integers, notice that only two of the c's have > 7 as a factor. > It might help to go the *other* way, and start with > (d_1 x + 1)(d_2 x + 1)( d_3 x + 1) = > x^3 + 5x^2 + 3x + 1 > and now multiply by 49. > In the first example you're looking at a product and realizing that > from the distributive property a(b+c) = ab + ac, you know there's > *one* way it could be produced, which is to multiply something like > the second example by 49.But you must realise that there are other ways to produce it, the wayyou do it is the only way possible, *only* if you require polynomials.Define: w3(x) = min(gcd(c3 x + 1), 7), 7) {is 1, 7, or some factor of 7} w2(x) = 7/w3(x) {also 1, 7, or some factor of 7} g1(x) = (c1 x + 7)/7 g2(x) = (c2 x + 7)/w2(x) g3(x) = (c3 x + 1)/w3(x)we have that f1 to f3 are functions from algebraic integers to algebraicintegers and g1(x).g2(x).g3(x) = x^3 + 5x^2 + 3x + 1.So this is an alternative way it can be produced. > The distributive property is key here. Understanding it thoroughly, > is of prime importance.Nope, it is not the key. Going from the second equation to thefirst uses the distributive property. Going the other way cannot use it, unless you assert that in a ring (a + b)/c = a/c + b/c,which is not necessarily true (it is only true if all three termsare elements of the ring). > Now notice that you can abstract from here as you're looking at > *functions* of x, as introducing > > f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, g > you have > (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1). > Notice that dividing both sides by 49 gives > (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1 > as long as you're in a ring where 7 is not a factor of 1.Two fallacies.1. Dividing this way is *also* valid in a ring where 7 is a factor of 1.2. There are other ways to do the divisions, see above. > Which is consistent with what was found before, as only two of the > functions have the property that 7 is a factor.But that is blatantly false. They have 7 as a *polynomial factor*. Butwhen you are working with arbitrary functions there are other*functional factors*. > Now I'll move on to a more complicated example. > Let > (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > where the a's are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > so they are functions of x, and since one of the roots equals 3 at > x=0, I have > b_3(x) = a_3(x) - 3, > so that all the functions in > (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > equal 0, when x=0. > Those of you who find it hard to use the distributive property with > the *product* can imagine the factorization from *before* 49 being > multiplied. > It's harder to show here as the polynomial which defines the function > in that factorization is not displayable in general. > So I started at the end, with 49 already multiplied because then I can > give > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > > That slight change, starting at the end, means that you have to > understand the distributive property fully and *trust* it.Note again: the distributive does *not* require (a + b)/c = a/c + b/cin s specific ring. So you *can not* divide off factors of 7 by thedistributive property. > Now notice that I have the result that only two of the roots of the > cubic > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > can have factors in common with 7, so the 49 splits between those two.You have not. For most x *all* of the a's have a factor in common with7. It is only when (2401 x^3 - 147 x^2 + 3x) = 0 that exactly two ofthe a's are divisible by 7 (because they are 0). > What's so startling is that the result is for a *family* of > polynomials as it applies for any algebraic integer x.Eh? Oh well, does not matter.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; === for some people?> I dont get it.Im a perfect 0 at math.>> Some people have no problems at all with it.>> Am I too dumb for math?>> Actually most of us are. For example I was good at math all my life. > Hotshot in grade school, high school, college. Got to grad school and > for the first time in my life found out I was no better than average. > Lots of math hotshots there, most better than me. After a couple of > years I realized I didn't have the brains and/or the study habits to > compete. I was too dumb for math. I hang out on this newsgroup, take > shots at JSH from time to time, occasionally answer a question. But > most of the stuff here is way over my head.I've always been amazed by some people's study habits; people who wereable to take several difficult university classes at the same time anddo well. One guy who was on my high school's math team triple majoredin biomechanical engineering, biology, and math. (His name is RichardTello. Does he post in this newsgroup? I don't read this newsgroupregularly; I just drop in occasionally looking for topics I might findinteresting.)> Just about everyone has this problem. The only question is, at what > point do you hit your limit? For some it's elementary school. For > others it happens in grad school. I would certainly not be surprised to > find Ph.D. mathematicians who suddenly realise that most of the other > postdocs in their field are smarter than them.Interesting you should say this. How people deal with these wallsprobably has a lot to do with the kind of success they have in math(or for that matter, in life). I hit two walls: one in my freshmanyear of high school, the other in my undergraduate freshman year. Inhigh school, I was placed in an advanced math class (unified math) dueto scoring well on the entrance exam. The entrance exam had algebraproblems on it that I was able to solve, but I didn't have as muchknowledge of algebra as other students in my class. I didn't knowthis until well into the class, when I was struggling while mostothers were not. I almost had to leave the class and go into regularalgebra. But I worked hard to catch up (at the expense of some otherclasses, unfortunately), and eventually became one of the topstudents.The second wall was in the first computer science course people takeat MIT (http://sicp.ai.mit.edu, if you're interested). I hadprogrammed in FORTRAN previously, and had even taken a one monthintroduction to LISP before the class started, so I thought I was inpretty good shape. But I had a lot of trouble with the class. (Itdidn't help that I missed a couple of lectures because I was sick.)There was a lot of material covered which I didn't pick up rightaway. Also, learning how to use the computer systems didn't comeeasily to me. (I sometimes wonder what was the bigger hurdle.) But Ididn't recover from this setback quite as quickly, and it was a coupleof years before I really felt comfortable with the kind of programmingthat was taught in the class.Since there are many people who felt (and feel) the way I did, Iwonder if there is a way to ease people's transition into higher mathor some other discipline. It can be demoralizing to discover that youdon't have the preparation you need when you're trying to pass aclass, and/or other classes are suffering because you need to takeextra time to deal with the difficult === subject for number theorists,> but I've never read anything about it. The question is how closely can> we approximate pi by rationals. More specifically:>> For integer n>0, let f(n) be the largest integer m such m/n < pi.> Let d(n) = pi - f(n)/n.>> Then d(n) measures how accurately we can approximate pi by a rational> with denominator n.>> How small can d(n) be? Clearly, d(n) < 1/n. But can we make d(n) much> smaller than that?>> Q1: Can we find arbitrarily large values of n such that d(n) < 1/n^2?>> Q2: Can we find arbitrarily large values of n such that d(n) < 1/n^3?>> Q3: In general, for each p>1, can we find arbitrarily large values of> n such that d(n) < 1/n^p?>> --> Daryl McCullough> Ithaca, NY>See:http://forums.wolfram.com/mathgroup/archive/2000/May/ msg00188.htmlhttp://forums.wolfram.com/mathgroup/archive/1998/ May/msg00272.htmlhttp://www.math.iastate.edu/hentzel/class .301.03/Oct.15http://www.isi.edu/~johnh/ABOUT/FEATURES/ === GF(q)>What is the order of the orthogonal group of order n over the> finite field GF(q) , i.e O(n,GF(q)) ?I am not completely sure what you mean by O(n,GF(q)) - does this refer tothe simple composition factor of the group of orthogonal matrices? Also,when n is even, there are two tyeps of groups, the plus and minus types.Let me quote verbatim from the ATLAS - that should contain the informationyou need!When n = 2m+1 is odd, the groups have orders|GO_n(q)| = dN, |SO_n(q)| = |PGO_n(q)| = PSO_n(q)| = N,|Omega_n(q)| = |POmega_n(q)| = |O_n(q)| = N/d,where N = q^{m^2}(q^{2m} - 1)(q^{2m-2} - 1) ... (q^2 - 1)and d = (2,q-1).When n = 2m is even, the groups have orders|GO_n^e(q)| = 2N, |SO_n^e(q)| = |PGO_n^e(q)| = 2N/e, |PSO_n^e(q)| = 2N/e^2,|Omega_n^e(q)| = N/e, |POmega_n^e(q)| = |O_n^e(q)| = N/d,where N = q^{m(m-1)}(q^n - e)(q^{2m-2} - e)(q^{2m-4} - 1) ... (q^2 - 1)and d = (4,q^m - e), e = (2,q^m - e), and I have used e as anabbreviation for varepsilon, === (sorry, maths not psych)> as it enters a static electric field.>>Parse the bloody sentence in quotes, Henry. It doesn't say that. Isn't>>English your first language?>> - Randy, grabbing his popcorn and going back to watch the>>entertainment>>Moron. Stop kissing Andsernon's arse.You're irritating me with your inability to read English.I finally reached threshold.Field already existed.Different from: Distant source of field changes.Situation 1: Field already present. Sources not changing. Field notchanging.Situation 2: Field changing. Sources changing. Different.Static. Dynamic. Close. Far.Moron. === workNntp-Posting-Host: apps.cwi.nl... > There are even VERY fast methods to determine the primality of even > extremely large numbers without a chance of error. > Just to clarify, as I understand it... knowing a number is composite does not > tell you the factors, to be commercially useful you have to find the > factors, otherwise the Ôdiscovery' is mostly academic.Essentially true. Finding factors is an art of itself, and trial divisionis not one of the best solutions. So from counting primes going tofactoring is a *long* way.But commercially useful is a misnomer. There is no money in it. Justforget it. Also finding factors and finding methods to find factors ismostly academic. I think we raked in a bit of money by factoring theRSA challenge. I do not think the check has been cashed. Selling themethod is, eh, not really fruitful. Except for RSA, there are *no*commercial firms interested in factoring.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn === Bible 1, Darwin 0! And we are talking pure archeology?X-DMCA-Notifications: http://www.giganews.com/info/dmca.html>>>> Forbidden archeology by Cremo, Thompson!>> Like the name says: Forbidden>> I urge you not to read it!>> Have you been to a museum lately? They have these things call> Dinosaurs> that lived millions of years ago. Don't you think if the bible is> the> word> of the lord that the good lord would have maybe warned us about> gigantic> carnivorous creatures roaming the earth! Afterall, he did mention> the> measly little snake.>>> Lurch> >>> The argument from my Southern Baptist Preacher relatives in> Kentucky ( I have several) say that the world was created complete> with the bones and other found elements of the historical record> created> intact to give a old construct when it was created 6000 yearsago....>> And I cannot dispute it... Because any old stuff sticking out ofthe> > ground> was made by the hand of god sticking out of the ground and any test> only> confirms gods ability to set a good stage for mans little drama to> play> out....>> We are but god theatrical group and nothing more than entertainment> for> his and his kids enjoyment in the evening, them throwing in an> occasional> ELE to start the next act.......>> That's why it's important to note that Genesis contradicts Genesis.If> you're looking for facts, the Bible is bull.>> Mark Folsom>> I don't agree it's totally bull.. While I don't in anyway> think there is anything like god as constructed in its words it> still has some historical value on both a sociological> standpoint and in the study of physiology with a bit of> geography to boot.>> You mean like the geographical facts you can believe if you find that they> agree with a reliable source--those facts?Yea .. those are them... Any mention of locationthat is not at present known may or may not bevalid until verified. Just like any other piece ofevidence... If you note the known history of howthe biblical accounts were assembled, and considerthey are all after the fact, from old verbal traditionsand were no doubt embellished and reconstructedfrom those camp fire stories. So yes.. It has value as apossible source of fragmented geographic data thatmay have been lost in the intervening time. Not thatyou give it any more value than other data just becausehumans stuck the god did it tag on it....>> Some is good data but must be considered> as a man made group of many writings put together> by committee ( mainly a king) for a purpose.>> Please cite some good data in the bible and tell us what it's good for.> Tell us which facts you would believe without independent verification.Good data:1)Political conditions of specific period as a fragmented reference2)Food availability of specific period as a fragmented reference3)Clothing types of specific period as a fragmented reference4)Methodologies of travel of specific period as a fragmented reference5)Currency systems of specific period as a fragmented reference6)Religious Anomalies of specific period as a fragmented reference7)Sociological impact of religious dogma of specific period as a fragmented reference8)Geographic data of specific period as a fragmented reference9)Engineering and Construction forms of specific period as a fragmented referenceAnd a bunch more.....Its a very old book... whether you are a B'leaver or not itsa combined writings of many old writers that give someusable data even without knowing that what they were doing.We can get descriptive of towns and buildings , people andplaces that only exist as bits and pieces of stone.>> But within> its text are references that can be verified and are worth> noting. While the B'Levers will point to the occasional reference> of a verifiable event, person or subject as proof that the> god construct is valid, they are mistaken. As with any bit> of prose, parts may be valid and parts may be complete> fabrications and parts may be incorrect interpretations of> valid parts.> Still waiting for the valid parts.Then you have not to read it with the eye of ascientist. You read parts and see the irrational...then you ignore the details that do have merit..I can point to out takes of how people dressed, whowas ruling the area, the basic descriptions of buildingsand town layouts, etc. but they are there and I don't want tosearch for such a useless detail...As for the study of physiology I maintain its a goodarea of study of physiology to research how so manycan be convinced in myth and magic. And I can tellyou they are absolutely convinced, That's a subject worthyof grants.....>> Mark Folsom>Paul R. Mays---------------------------------------------------------- -------------------Some where within the Quantum StateHttp://Paul.Mays.Com/story.htmlhttp://paul.mays.com/ mayday.htmlhttp://paul.mays.com/rainy.htmlWhy do our bodies wear out? Why can't we just goon and on and on, accumulating a potentially infinitenumber of Frequent Flyer mileage points? Theseare the kinds of questions that philosophers havebeen asking ever since they realized that being aphilosopher did not involve any heavy lifting. Andyet the answer is really very simple. Our bodies are mechanical devices, they break down. Some devices,such as battery-operated toys costing $39.95, breakdown almost instantly upon exposure to the Earth'satmosphere. Other devices, such as stereo systems owned by your next-door neighbor's 13-year-old son who likes to listen to bands with names like ÔNerveDamage,' at a volume capable of disintegrating limestone,will continue to function perfectly for many years,even if you hit them with an ax. But the fundamentallaw of physics is that sooner or later every mechanismceases to function for one reason or another, and it === Re: DO MATHEMATICIANS READ WITH HALF A LIGHTBULB?> I'm listening ... so tell me in English why I'm wrong. Do you know what it means for a series to converge?> Are you talking about a series when you write a + a^2 + a^3 + ... ?> If not, know what'convergence' means in the above infinite series. I do know thatinfinity has the property of both containing each particular numberthat it comprizes as well as having no end of such numbers, and that'sit.If 2 + 3 + 4 + . . . ad infinitum is subtracted from 1 + 2 + 3 + . . . ad infinitum, then it seems obvious that you will be subtracting the equallyinfinite tails of two infinite series at point 2 on the number line,and thus leave the non-infinite 1 that makes the only differencebetween the two series, i.e., the non-infinite difference of 1 in eachseries' starting points on the number line.Please explain how Ôconvergence' refutes that logic.Very the author?I am interested in his methods, please, help === Re: SUM-PART-PART-PRODUCT> My 9 year old son keeps coming home with these papers called sum-part-part > product. The sum is filled in as is the product and he must fill in the part > boxes. I can't, for the life of me, figure out a mathmatical way to > calculate the parts....there must be a way. Can anyone help? -------------------------------> SUM | 20 | 40 | 16 | 30 | 14 | 60 |> -------------------------------> PRT | | | | | | |> -------------------------------> PRT | | | | | | |> 900|I think you'll find that 255 is supposed to be a 225.-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for === seen this done anywhere> else? Yep. In a 1936 book advocating more Ôadventurous' math education> for school children. (The Dutchman Kruijtbosch, iirc.) I don't know> what was his source, though. He sounds as if he considered it old and> common knowledge.So you don't know his source. Hmm. Well, looks like in 1910 oneTractenberg and in 1936 one Kruijtbosch knew these methods. Yet,evidently they were not popularised, ever, in both East and West. Certainly the method I have described does not seem to be commonknowledge today. Nor was it common knowledge 44 years ago. But thissupposedly fraudulent book on Vedic Mathematics was first published in1965. Maybe the author stole from the European sources, or maybe theEuropean sources got it from Indian/Vedic sources with or withoutacknowledgment, or maybe they developed them independently. Or, as Itend to think, the author is simply narrating the traditionalarithmetical methods of India. After all, the book was published inIndia, and sold in reputed bookshops there. And no credible Indiansource I know of says that they were derived from elsewhere. Not eventhe TIFR person who debunks them. But then, who knows?>Also, have you seen one-line division and square-root> extraction? Kruijtbosch also has an alternative method for square roots,> but it is not as nice as his multiplication. How are the Vedic methods?> Never seen them in detail. But i would be very interested.I am sure you can find them from the book. That book is notexpensive, if you could get to the bookshop. Say, less than US$7. Ifyou have a source in Kolkata, you could get it from either SanskritPustak Bhandar or Aurobindo Ashram (I got my copy - left in Kolkata)from one of these shops, most likely the former. I read it for 15-30minutes, concentrating on the multiplication, but I see that theauthor had described one-line division and square root extracationmethods. I have no idea how they were done.> But... it is arithmetic, rather than mathematics.>> Isn't arithmetic a part of mathematics? Hardly, no (strange as that may sound).It does sound strange. And also unbelievable. Especially sinceArithmetic is taught as a part of Mathematics in school.> That may also explain> why the math teaching world is not very fond of tricks like this.>> It is not a trick, it is a sound method for multiplication. Yes. But i've seen a site about Vedic maths showing some other things.We are talking about multiplication here, and nothing else.> They really are tricks, giving a student no Ôbasis' or Ôinsight' at all.Certainly the multiplication method gives us a terrific insight intothe terrific advantages of place value.> Nice tricks, mind you. But tricks.Fine. All of mathematics is art, or a bag of tricks. Allcivilisation comes from trickery with nature, and its subsequentmanipulation. Those who do it best, are the winners. You need tohave great basis and insight to do any trick properly.> It gives> a very thorough insight to the whole process, once properly> understood. It is a welcome addition to arithmetic education.> But when it comes to insight, the distributive law is much better.Explain the distributive law, and show us how it is much better! Withrespect to what?> Children are likely to like this method, and teachers> will find it useful to teach. Yes, i also definitely think so.We agree on one thing. Good.> Mathematics teachers are desperately fighting the general prejudice> that mathematics is Ôjust' arithmetic.>> Well, that is because they are not doing the arithmetic right. Once> that is done right, following proper understanding of what numbers> really are and what they really mean, then things will be better for> all concerned. Here i disagree. Worshipping vedic maths as a potential redemption for> maths education would be fatal.Not for most Indians and all non-bigots, I hope. It is not a questionof worshipping anything; it is to use some wonderful methods to do ourarithmetic better, and thus improve the === it provides a laugh or two.Proof that 0 is the successor of no natural number.0 is defined to be the null set or { }.S(a) is defined to be {a}u a.{ } can be rewritten as { } u .Therefore 0 is the successor of .What is ?It is Nothing.Thus, 0 is the === claims of torsion weapons>>Commentary 3>>The hyperspace H consists of fibers f(x) that are>>either copies of or representations of the symmetry>>group G. Well, that convinces me... It should. They are bible fibers. Wouldn't want to === theory, and contingent identity> 7xttb.77875$HoK.40932@news01.bloor.is.net.cable.rogers.com>... > And as Ive already stated in my previous posting,>> The rule of necessitation can also be unproblematically> applied to the following NFL-theorem:>> Ax(x = x -> E!x)> []Ax(x = x -> E!x)> [](Ax(x = x) -> AxE!x)> []Ax(x = x) -> []AxE!x>> Since in NFL both Ax(x = x) and AxE!x are axioms, everythings fine> here!> Agreed, and, []Ax(x=x -> E!x) <-> Ax[](x=x -> E!x)> and []Ax(x=x) <-> Ax[](x=x).I reject the latter formula, which is an instance of the combinationof the Barcan Formula and the Converse Barcan Formula, becausetreating de dicto and de re necessities as equivalent has certainmetaphysical implications I consider inacceptable. - This is adelicate issue, arguably the most contentious one in modal logic.> ixFx = ixFx -> E!ixFx is not an instance of Ax(x=x -> E!x), in PM.> In Russell's descriptions theory (ixFx) is not a value of the individual> variable> unless E!(ixFx).Yes, but I wasnt referring to Russells PM but to the descriptiontheory of (negative) free logic.For Russell ixFx cannot refer directly to a particular object sincehe thinks it isnt a genuine singular term at all.> I don't see a problem with (1) []Ax(Fx -> E!x) -> ([]AxFx -> []AxE!x)All right.> I hope Ive managed to convince you that the only thing that can be> deduced in a suitable modal version of negative free logic is []AxE!x> but not Ax[]E!x.> I don't agree here either. Given Ax[]Fx <-> []AxFx is valid.> Surely []AxE!x <-> Ax[]E!x, is a theorem.Thats not so sure, for its supposed validity depends on the kind ofproof-theory and semantics you choose.For philosophical reasons Im not prepared to accept a modal theorythat validates both the Barcan Formula and the Converse BarcanFormula. Kripke, for instance, has devised a modification of thestandard system S5 such that the Barcan formulas are no longerderivable and no longer valid.[See Chris Menzels excellent text (especially 3, ÔKripkes System'):http://plato.stanford.edu/entries/actualism/ ]> Under the right interpretation, there is no deductive way from> necessary self-identity to necessary existence!> What exactly is Ôthe right interpretation'?By that Ive certainly meant right--seen from my point of view.I prefer the actualistic, i.e. world-relative interpretation of modallogic.[See Ô11. Quantifiers in Modal Logic':http://plato.stanford.edu/entries/logic-modal/ ]> Although NFL is close to the spirit of Russells description theory,> there is a significant basic difference, for in free logic definite> descriptions are not treated as Ôincomplete symbols' but as genuine> singular terms, i.e. Frege-fashion! > That is very puzzling to me. Can you expand on this point?It is in free logic that a direct comparison of a Frege-like formaltheory of definite descriptions with that of Russells emerges*without* a detour through their different philosophies of language.[...]There are four main traditions in the formal treatment of definitedescriptions. They can be differentiated with respect to their viewson the logical grammar and/or the referential status of unfulfilleddefinite descriptions. Frege held that unfulfilled definitedescriptions are genuine singular terms, but, in the interest oßogical perfection, artificially assigned to them existents asreferents. In ÔPrincipia Mathematica' Russell held that *all* definitedescriptions, though grammatically correct expressions from a logicalpoint of view, did not belong to the category of singular terms; thatis, none of them, including unfulfilled definite descriptions, evenpurported to refer. Hilbert and Bernays held that unfulfilled definitedescriptions were not even grammatically well-formed. Finally, freedefinite description theories treat definite descriptions as genuinesingular terms (like Frege), but do not assign any existent asreferent to an unfulfilled definite description -- even artificially.If the ontological disposition of the free description theorist isRussellian -- as is my inclination -- then unfulfilled definitedescriptions are simply a subclass of the irreferential singularterms.[Lambert, K. (1997). /Free Logics/. Sankt Augustin (Germany):Academia. (pp. 97 & 99)]> If AxE!x is true, as you say, and Vulcan is a value of the variable x, then,> E!(Vulcan) follows. But, you also claim that ~(E!(Vulcan) ??Since there is no such thing as Vulcan, the nonexistent Vulcan isimpossibly a Quinean value of any variable.In free logic the universal quantifier ranges over the set ofexistents!This is rendered explicit by the following being an FL-theorem:AxFx <-> Ax(E!x -> Fx)If E! is substituted for F we get:AxE!x <-> Ax(E!x -> E!x)Everything exists is equivalent to Every_existing_thing exists. ! Read this way, this axiom can hardly be false since its also aninstance of the tautologicalAx(Fx -> Fx) .Vulcan is no existing thing and so its not a true instance of theFL-axiomAxE!x .The corresponding UI reads:AxE!x & E!a -> E!aSince~E!Vulcanthe existence of the nonexistent Vulcan is, of course, not deduciblein free logic.> No object, given or> described,> Ôhas' the property of non-existence.Thats my philosophy.Property-instantiation implies existence, and vice versa.On the usual understanding of the quantifiers, quantifiers range overall and only existing individuals. A universally quantified formula istrue just exactly when every actually existing individual satisfies(makes true) the formula preceded by the universal quantifier, and anexistentially quantified formula is true just in case some existingindividual satisfies (makes true) the formula preceded by theexistential quantifier. We want to preserve this understanding when weintroduce modality; we do not want our quantifiers to be understood asranging over nonexisting things. Quantifiers that occur within thescope of modal signs, that is to say, quantifiers that occur within*de dicto* contexts, receive no special reading. Quantifiers into *dere* modal contexts are understood to be ranging over actually existingindividuals.[Konyndyk, K. (1986). /Introductory Modal Logic/. Notre Dame, IN:University of Notre Dame Press. (p. 91)]> Only every Ôexistent' thing has the property of self identity.Exactly.The axiomAx(x = x) is equivalent toAx(E!x -> x = x)Everything is self-identical--if existent.> ??> But, Ey[](y = ixFx) -> []Ey(y = ixFx), by Ey[]Gy -> []EyGy.In standard S5 the Barcan formulas are deducible as theorems.As I have already emphasized, I reject the alleged equivalence of dere and de dicto modalities, which position unfortunately forces me toreject standard S5.> If one adopts a world-relative semantics (Kripke models), as I do,> then ixFx might well be regarded as a non-rigid designator that> doesnt single out one and the same individual in every possible> world.>> (A) []Ax(E!ixFx) -> [](ixFx = ixFx))I just happened to notice that this is nonsense, since the Ôx' in'E!ixFx' is not a free variable, and so theres nothing to bind forthe universal quantifier!Must read as follows:E!ixFx -> [](ixFx = ixFx)> Ax(Ey(Ax(x=y <-> Fx))) <-> Ey(Ax(x=y <-> Fx)), by Ax(p) <-> p.> []Ax(E!ixFx) <-> []E!(ixFx),> If you mean []Ax(E!x) -> [](ixFx = ixFx), then I disagree.No, I dont mean that.Ax(E!x -> x = x)is an NFL-theorem, and so we get (by means of the NFL-axiom NA2)AxE!x -> Ax(x = x) .This eventually results === MATHEMATICIANS READ WITH HALF A LIGHTBULB?raydpratt grava .88 la saucisse et au marteau:> Please explain how Ôconvergence' refutes that logic.Because, according to you, what is A = 1-1+1-1+1 .... ?Is it 0 because A=(1-1)+(1-1)+...?But this is also 1-(1-1+1-1....) = 1-AA = 1-A, so A = 1/2But A = 1-(1-1)-(1-1)... = 1So, what do you === Reality>>> Equivalently, M*N is the same as M*N mod (M + N - 1).>> Sorry, this should be multiplication of M digits with N digits, base b,>> is equivalent to multiplication modulo b^(M + N - 1), i.e. M+N-1 digits.>>Oh well, so FFT or not, looks like multiplying M by N by any method means>MN multiplications! Well, when these multiplications are hardwired (as in>human memory for single digits) the computational issues (On*n) becomes>really irrelevant, for they all are done in no time at. Like, the video>extraction for radar data processing is done by NAND gates - its all done in>real time!> You are in error. The number of multiplications required for> multiplying two numbers with the FFT method is O(n*log(n) where n is> the larger of the two numbers; it is not m*n. Fine, just multiply 12345 by 67809 using FFT with less than 25> multiplications. Do it here. Do not misinterpret this beahaviour as typical arrogance of brahmins.How does arrogance of brahmins come into this discussion onarithmetic?> His perception is so narrow, he cannot understand others points.But all I am asking is to understand others' points. I want to knowhow you can use FFT to multiply 12345 by 67809 with less than 25multiplications. If really FFT does O(nlogn), then you should do itin 5log5 (that is, 5*1.6 or 8) multiplications. If indeed FFT is abetter method, then it should be possible to do just that! Or else,FFT is not a good method to multiply 12345 by 67809, while it may havemany other uses. As a matter of fact, FFT is totally irrelevant tothis discussion.> So he demands explanations in his own ways. If he wants to learn how> to calculate squres and if you teach him how to find cubes, he may get> confused> and may stop learning matheamatics.Look, if I want to learn how to calculate squares, I expect to betaught how to calculate squares. Like if I want to buy a dog, I don'twant to be sold a horse. > Richard Harter, cri@tiac.net> http://home.tiac.net/~cri, http://www.varinoma.com> We have people from every planet on the earth in this State.> -- California Governor Gray === are a joke! :) :)> Okay, try multiplying 12345 by 67809 in any better manner. Just do it> here. bash-2.05a$ echo Ô12345 67809*p' | dc> 837102105 That wasn't so hard, was it? I found === > So, too, is> the white wine and brie crowd who by and large make up the Sierra Club> constituency. As for the environmentalists, as a group, if anything> they're even worse than liberals in their politics, the vast majority> > of them being Bolsheviks.> Bolsheviks? Are you a fan of Tom Potter? He's the only> other person I know to use that term to describe people> post-1920 or so.> It is interesting to see that this poster thinks > that the Bolsheviks just vanished into nothingness.> The fact of the matter is that the Bolsheviks instigated the > class wars of the 1900's for power and wealth, > and after their class wars were discredited, > and the Native Russians regained controlled > of their government, millions of the Bolsheviks > who had lived high and mighty in Russia, > migrated to Israel and New York, from where > they are instigating the religious wars of the 2000's > to get back into the chips as the loot from their class wars > is almost gone.> I suggest that intelligent, rational, moral folks, > reject the media brainwashing, open their eyes, > look around and see who instigated the class wars, > who is instigating the religious wars, > and who profits from both, > while others, such as Blacks, Rednecks and Latinos, > sacrifice their lives, limbs, liberties and fortunes > to fight folks they would otherwise get along with just fine.> As can be seen by studying history, > and by observing current events, > the Bolsheviks have a long history of > for power and wealth.> Instigating conßict and war is the stock in trade > > of the Bolsheviks, much as fortune telling > is the stock in trade of Gyspies. Tom, I believe you and I are also on the same page here as it reßects> a proper understanding of Bolshevism. These ill-tutored twits who see> themselves as sophisticated and schooled in world politics generally> don't know their gluteous maximus from a hole in the ground. These> clueless suckers love to snicker at things they know absolutely> nothing about. This whole system is crumbling right before their very> eyes, and it's bread and circuses as usual for these political> geniuses while race is set against race, class is set against class,> gender is set against gender and we all get sold down the river for 30> pieces of silver. We're in meltdown and these fools are asleep at the> control panel. I liked the OP's comments about the environmental white wine and brie> crowd, for they are truly the useful idiots in this whole equation who> do the most damage. These people are not idiots.> They are just highly suggestible. I used to hypnotize people, > and soon discovered that most people > are very susceptible to conditioning, > and are thus easy to hypnotize. As most people are highly suggestible, > they fall victims to the propoganda machine > of the people who manipulate them. These people are victims.> Not villians. They, like almost all people, > believe what they are told, > (In a non-threatening way.) > rather than what they see.As can be seen by looking at the headers,I did not make the post above.Obviously, some dishonest personwho cannot address the issues I raise,is taking the dishonest approach topromoting his agenda.As the best measure of a person, or group,are the tactics they use,I suggest that the poster reßect on his actions,use his real name, and address the issues raised,rather than try to obscure someone else's messagesusing dishonest, immoral tricks.In the long run, honesty is the best policy.--Tom Potter === Achievements of Jews> Scratch's philosophy reminded me of the story> about a farmer who lived next to another farm> that was up for sale.> A potential buyer stopped by and asked him how the neighbors were> in the community, and the farmer asked him how they were> in his community. The man replied,> > The people where I live now, are no good,> bastards, etc.> > The farmer replied:> Yep, that's the way they are around here.> Another potential buyer stopped by and asked him how the neighbors were> in the community, and the farmer asked him how they were> in his community. The man replied,> The people where I live now,> are really great. They help out when folks are sick> or have financial problems, they are friendly, etc.> The farmer replied:> Yep, that's the way they are around here.> Who would you like to have for a neighbor? People as two-faced as that, don't deserve to be neighbors. The question therefore arises, as to wheather my minions even deserve to worship me. I will contemplate this. -Satan The farmer wasn't two-faced.> He was letting the devils > filter themselves out, > as he did not want them for neighbors. I agree with you > that you should direct your worshippers > to someone, or something, else, > or destroy them.As can be seen by looking at the headers,I did not make the post above.Obviously, some dishonest personwho cannot address the issues I raise,is taking the dishonest approach topromoting his agenda.As the best measure of a person, or group,are the tactics they use,I suggest that the poster reßect on his actions,use his real name, and address the issues raised,rather than try to obscure someone else's messagesusing dishonest, immoral tricks.In the long run, honesty is the best policy.--Tom Potter === - Need Help!>Use Taylor series as others have suggested, or if you're feeling >adventurous, show that for any C^2 function f in a neighborhood of 0 with >f'(0) = 0, (f(x) - f(0))/x^2 -> f''(0)/2 as x -> 0. For b), think cosh(x) = sqrt(1 + sinh(x)^2). > Rationalize the numerator by multiplying both numerator> and denominator by 1 + sqrt(1 + sinh(x)^2) = 1 + cosh(x).> Or substitute x = 2*y and use double-angle formulae.> sinh(x) is better-behaved in a limit problem since> sinh(x) approaches zero as x -> 0.It's about as easy and far more general to prove: If f''(0) exists, then [f(x) - f(0) - f'(0)x]/x^2 -> f''(0)/2 as x -> 0.proof: fundamental theorem of calculus. (Letting f(x) = cosh(x), the answer to the OP's === y <- y - r/xI am curious, since I am unable to currently plot anything now,if the following iterated plot produces an interesting graph:x(m+1) = x(m) - r(m)/y(m);y(m+1) = y(m) - r(m)/x(m).({r(m)} is some predetermined sequence, preferably causing thesequences x and y to converge, and r(m) never is x(m)*y(m) for any m.)Now, {x(1),y(1)} is the coordinate of the iterated-upon point;and, say, we plot a color at this coordinate which corresponds tolimit {m -> oo} sqrt(x(m)^2 + y(m)^2),as an example, if r is such that the sequences converge.Any {r(k)} lead to any interesting plots??Also....If r is nonzero,x(2+m) = x(1+m) + (x(1+m) - x(m)) x(m) r(1+m)/(x(1+m) r(m));y(2+m) = y(1+m) + (y(1+m) - y(m)) y(m) r(1+m)/(y(1+m) r(m)).So {x(k)} and {y(k)} are each dependent on the first 2 terms of theirsequences only.But x(2) = x(1) -r(1)/y(1) and y(2) = y(1) -r(1)/x(1), so x(1) stillaffects the y sequence and conversely.Now, ißimit {k ->oo} r(k+1)/r(k) = R exists, and R < 1,then both the x sequence and y sequence converge.Furthermore, if R > 1, both sequences diverge.But I do not know how to determine what happens if R = 1 or R does notexist.I conjecture that, if sum{m=1 to oo} r(m) converges (absolutely anyway), then the x and y sequences bothconverge, and the x and y sequences diverge if the r-sum diverges.Can === for where I can measure the speed of a car whenit hits a still object based on car's weight, object's weight, and totaldistance the object was thrown. I realize that there are many other factorssuch surface friction, in this case road, but i'm just looking for anestimate, not to be as exact as === numbers> Summary of Euclid's proof:> 1) Suppose that there is a largest prime; call it P> 2) Calculate N = (product of all primes from 2 to P, inclusive) + 1> Are you trying to say that the set of all primes has no primes missing? Why should there primes be missing from all primes?There shouldn't, and there aren't. That's why I asked the question.Is the art of rhetoric dead? > Or are you trying to say and we assume that we know all primes <=P. You are thinking about Euclid's proof, which this is not.You sem to have mistaken me or someonewho's not been repeatedly telling other that these various techniques aren't Euclid's proof.For reference, the Prime Pages now has an updated section on Euclid's proof, which addresses some of the issues and misunderstandings in this thread. It even includes a link to a translation of Euclid's proposition 20 from Book IX which people can use to verify my claims thatEuclid's proof doesn't talk about this taht or the other.http://primepages.org/ then follow the link to how many..., thence the link to Euclid proved IIRC.Phil-- Unpatched IE vulnerability: dragDrop invocationDescription: Arbitrary local file reading through native Windows dragDrop invocation.Reference: http://msgs.securepoint.com/cgi-bin/get/bugtraq0302/12. htmlExploit: === multiplicationTwo questions: First, consider positive integers k and j with m and n digitsrespectively. Assume moreover that neither k nor j is equal to 0. Can we determine exactly the number of digits in the product kj? It'seither m+n-1 or m+n, but which one it is changes depending on thenature of k and j. I wanted to use this to determine the number ofdigits in, say 2^64. Now, this one might not be so difficult becausemany people know 2^32 off the top of their head so they know it has 10digits. So they would know immediately that 2^64 either has 20 or 19,but still it's not clear (to me anyway) if it's 20 or 19 withoutmultiplying it out. But still, what about 3^64? And what about thenumber of digits of a^b where a and b are any positive integers?Secondly, is there an easy way to determine exactly the first r digitsof the product of an m-digit and n-digit number, without computing thewhole number? It seems like you can estimate the first r digits bymultiplying together some chunk of the beginning of the first numberwith some other chunk of the beginning of the second number, andlooking at the first r digits of the result. But is there someformula in terms of m and n that gives you exactly the size of thechunk you should take so that the result is guaranteed to be exact? Or do you have to multiply the whole numbers === Reality> Okay, try multiplying 12345 by 67809 in any better manner. Just do it> here. bash-2.05a$ echo Ô12345 67809*p' | dc> 837102105 That wasn't so hard, was it? I found it pretty darn trivial.You needed a machine. I was doing it by hand. I'll return the favour - use your vedic methods to calculate: 2 ^ (5653^16384-5653^8192) % (5653^16384-5653^8192+1)What for? :) All I asked was that people multiply 12345 by 67809 inless than 25 multiplication steps, if indeed they could, as claimed.Vedic arithmetic methods are primarily for school children, and theremay be use for them in electronic computation, once their potential iswell understood. Like, one can multiply two million digit numbers, toget two million digit numbers as a result, very fast, and using verylittle coding effort. Also, every place value will have its ownrecursive structure - these days with emphasis on carry it is all atthe highest level only. But all this is work in areas that do notreally interest me. As for the problem you have given, all I ask is,why all % stuff? Surely that is unnecessary?! Like, if you writeN%(N+1) you can only mean N? Never mind! === Ômost probable' value (explained in NumericalRecipes - http://www.library.cornell.edu/nr/bookcpdf/c15-1.pdf).I don't really follow the math but p=2 is optimal for Gaussian typeerrors whereas other functions q(g(x)-f(x)) are === Interactions as Overlapping Fiber BundlesCommentary 5The charged matter field is the associated vector bundle and the EM field by which the charges interact is the principal bundle. Both bundles share the same base space. Remember however that Wheeler and Feynman eliminated the EM field completely in classical physics by introducing the advanced null geodesic direct action between the absorber charge from the future with the emitter charge here-now on an equal footing with the retarded null geodesic direct action between the same two charges to make a closed loop in time. The here-now classical radiation reaction term is an advanced action from the future, which in the micro-quantum case is mixed up with the virtual zero point photons that trigger the real photon emission via spontaneous emission. See Peter Milonni's The Quantum Vacuum for more details where it is also shown that the zero point energy density of random virtual photons is positive, whilst the zero point energy density of random ionized plasma of virtual electron-positron pairs is negative. In both cases w = pressure/energy density = -1. The gravity effect of these zero point ßuctuations, as for all stress-energy density sources both real and virtual, is to first order ~c^2(string tension)^-1 (energy density + 3pressure).The BIT quantum wave of the charged matter field is a hyperspace section of the vector bundle. This section must be a second quantized local field operator and at this point the fiber bundle math seems strangely silent. If one thinks of a single electron, then a lot of physics is left out. One cannot do multi-electron effects using only c-number complex sections of the vector bundle nor is it obvious how to get, for example, the Pauli exclusion principle unless the section of the vector bundle is second quantized operating on a Fock space in the occupation number basis.If the fiber of the vector bundle is a first quantized wave function in the sense of non-relativistic quantum theory then it is a projective ray in which the modulus of the complex number is of no physical significance only the phase matters and even then not the absolute local phase at x, but only the relative phase differences at x =/= x' and this requires a connection field and a path as in optical interferometers. Note that in the MACRO-QUANTUM case the projective ray property vanishes and now the modulus of the complex number is a condensate density for a huge number of bosons in the same single-boson quantum wave packet in ordinary space. The associated fiber bundle needs to be modified accordingly from a simple phase circle fiber to an amplitude-phase cylinder or possibly something more exotic like a Mobius strip or even a multiply surface with handles? We know from the micro-quantum Bohm-Aharonov Effect and the similar MACRO-QUANTUM Josephson Effect how the EM potential Au evolves the phase of the single-electron BIT wave in the former and the giant quantum superconductor wave in the latter in a gauge invariant way that depends only on the quantized magnetic ßux through a closed loop of possible paths even when there is no magnetic field in local contact with the electrons. This is generally what happens in all gauge force quantum interaction dynamics. Given a worldline in the base space-time. Do the unique horizontal lift into the full fiber bundle space of local products of circles with coordinate derivative is the gauge-covariant derivative;;u = ,u + (e/hc)AuIf we were doing EM fields in curved space-time then;;u = ;u + (e/hc)Auwhere ;u is the Diff(4) covariant derivative.Note the cross terms in the second order derivatives.In the case of FRW cosmology on a scalar field describing MACRO-QUANTUM vacuum coherence, the second Diff(4) covariant time derivative (e = 0) gives Linde's all-important friction term for chaotic inßationary cosmology of the continual creation of universes ßoating in hyperspace.The EM field tensor Fuv is a tidal curvature in fiber space.Einstein's gravity GR uses a tangent bundle of local tetrad frames that express the EEP.O(1,3) does 4D rotations of the tetrad frames in Tx(M) but that is not the same aslocally gauging O(1,3) to extend the symmetric Levi-Civita connection from locally gaugingT4 to get an antisymmetric torsion connection field that couples to We also need to locally gauge the special conformal transformations and the dilatation generatorof the Conformal Group.In terms of Kleinert's world crystal lattice the nonlinear special conformal transformationsmust be some kind of plastic defect perhaps since curvature and torsion are disclination anddislocation string topological defect densities in the large scale limit L >> Lp?Commentary 4Synopsis of where we are at so far in the emergent evolution of our understanding of how the mathematics of fiber bundles with a natural idea of hyperspace and Super Cosmos (Linde's chaotic inßation) is interpreted as the physics of classical relativity, local quantum field theory with the objective of using it also in the macro-quantum theory of emergent Einstein gravity with exotic vacuum dark energy/matter for metric engineering and possibly also in micro-quantum delocalized string theory.We have taken a top -> down approach for the principal bundle. Start with a large higher dimensional hyperspace H. Do not assume any metric in it to begin with. Assume a CONTINUOUS Lie symmetry group G equivalence relation ~ that partitions H into disjoint G-orbits that are equivalence classes of points X of H where X' ~ X mod G. Each distinct point of the base space M is a projection from a single G-orbit where M = H mod G or H/G. The G-orbit is an internal hidden structure of the base space event M that can include extra compactified boson dimensions and also the fermi dimensions of supersymmetry. How Planck's h and Heisenberg's uncertainty fit in is not apparent yet; The construction so far seems classical. h seems to demand fractals that are continuous but not differentiable like the classical manifolds are.Hyperspace H is locally a product of a the beyond space-time fiber and a small neighborhood of the base space.Around each point x of base space M there is a coordinate patch C(x) and a fiber f(x) and a special diffeomorphism Trivial (x) that maps H at x into the product C(x)f(x). If the hyperspace is globally not oriented like a one-sided Mobius strip or a Klein Bottle then Trivial(x) locally unwraps the global twists. A transition function isTrivial(x)Trivial(x')^-1 in the overlap of the local coordinate patch neighborhoods around x and x' with different G-orbits (I think?)6. There is a purely vertical inverse bottom -> up emergent projection P^-1 from base space C(x) to fiber f(x).P^-1 is a rule for associating each point fo in the fiber f(x) with a group element g < G of the principal bundle for the gauge forces i.e. electroweak + strong NOT gravity yet.7. P^-1 does not establish a horizontal connection for identifying points on different fibers f(x) and f'(x') in different regions of the base space with the same continuous symmetry group element g in the global group G.The global Cartesian product space is like a broad staircase with vertical handrails. In contrast the fiber space is like a set of identical escalators moving up and down independently. S. Y. Auyang How is Quantum Field Theory Possible?p. 217, Oxford, 1995.8. The local gauge force potential interaction dynamics allows parallel transport of fiber information along continuous paths in the base space of control parameters, which in special applications can be the space-time manifold, but generally it can be other kinds of spaces.9. The ALL-IMPORTANT section: A section is an inverse projection C(x) -> P^-1[C(x)] mapping a neighborhood of base space back into a region of hyperspace H. The section creates a local coordinate patch in the hyperspace from the local coordinate patch in the base space by arbitrarily CALIBRATING a single point in the vertical fiber fo(x) above each x in C(x) as the identity e of G. If a single section works globally for the whole hyperspace then the bundle is trivial like a two-sided orientable cylinder not like a one-sided non-orientable Mobius strip that resembles a spinor needing a 4pi rotation to return to its original normal vector.The idea of connection is implicit in the idea of the section.10. The special section called the principal connection maps the tangent spaces of the base space to the tangent spaces of the hyperspace. Let Tx(M) be a tangent space of M at point x. Let TX(H) be the tangent space of hyperspace at hyper-point X. Then the principal connection isP^-1[Tx(M)] = TX(H)X = (x,fo)TX(H) = TX(H)horizontal + TX(H)verticalTX(H)horizontal =Tx(M).Note that Einstein's smooth c-number gravity is essentially from the tangent bundle {M, Tx(M)} with an additionalmetric or alternatively a tetrad spanning both x < M and Tx(M) that embodies Einstein's Equivalence Principle (EEP). The symmetry group G acts like the identity in Tx(M) and should not be confused with Diff(4) in Einstein's gravity theory. The principal connection splits any path in hyperspace into a horizontal path in base space and a vertical path in the extended fiber region of hyperspace. Presumably we can extend this from paths to world sheets for strings rather than points?11. Given some principal connection |~ and a worldline in base space M. The worldline can be horizontally lifted into the extra space dimensions of the Calabi-Yau spaces (anticipating the string generalization yet to come) such that all tangent vectors of the hyper world line are horizontal. This is PARALLEL TRANSPORT IN HYPERSPACE as distinct from parallel transport of world tensors along worldlines in Einstein's gravity theory in the special tangent bundle [M, Tx(M)].12. The horizontal lift of a M world line into hyperspace is UNIQUE and this allows us to associate different points fo(x) and fo'(x') in different NON-OVERLAPPING regions of hyperspace with disjoint patches C(x) & C'(x') with the same g < G relative to that specific M worldline connecting the two points.13. EEP (Einstein's Equivalence Principle) of GR is an approximate statement that: i. far from a space-time singularity and ii. at a scale larger thanLp^2 = hG(Newton)/c^3One can freely ßoat/fall feeling no weight (i.e. no g-force) along a slower-than-light time-like geodesic in a non-rotating LIF (Local Inertial Frame) with comfortably small stretch-squeeze torture rack local curvature tidal force inhomogeneities in the g-force.13. Thus GR is a specialized kind of fiber bundle not the same as the fiber bundles in local quantum field theory. Indeed, I claim that the former is emergent from a false vacuum instability in the latter.Commentary 3The hyperspace H consists of fibers f(x) that areeither copies of or representations of the symmetrygroup G.Jack, this is not quite correct. They are homogenous spaces onwhich the group operate transitively. Example, for the group SU(2),you can take as the fibre a copy of SU(2) itself (3-dimensional), oryou can take sphere S^2, on which SU(2) operate (2-dimensional).Notice that S^2 is not a representation of SU(2). It is a quotientSU(2)/SO(2).Early Kaluza-Klein theories were operating with group Manifolds.Souriau, and later Witten, suggested more realistic theories wherefibers could be of lesser dimensions. Thie rigorous mathematics andexamples of this latter approach have been developed in themonograph:Riemannian Geometry, Fibre Bundles, Kaluza-Klein Theories and AllThat... (World Scientific Lecture Notes in Physics, Vol 16)by Robert Coquereaux, for the local gauge forces:1. A transformation g of the symmetry group G acts on the ordered pair X = (x, fo) in hyperspace H with output gX.Question: Can gx = x' =/=x i.e. can one move the base point in this operation or must G always be the identity in the base space? That is, we always need, in addition to G a connection and a path in order to change location in the horizontal base space and the vertical fiber space that is beyond space-time. G certainly moves fo up and down the vertical fiber for every element g =/= identity. Does it also move x -> x' = gx =/= x horizontally along the base manifold without a connection field and a path specified? Clearly the answer must be NO. See below.The modern understanding of gauge invariance, as a symmetry under transformations ofquantum-mechanical wave functions, was reached by Weyl himself and also by London veryshortly after the new quantum mechanics was first proposed. In this understanding ofabelian gauge invariance, and in its nonabelian generalization [2], the space-time aspect islost. The gauge transformations act only on internal variables. This formulation has hadgreat practical success. Still, it is not entirely satisfactory to have two closely related, yetdefinitely distinct, fundamental principles, and several physicists have proposed ways tounite them.One line of thought, beginning with Kaluza [3] and Klein [4], seeks to submerge gaugesymmetry into general covariance. Its leading idea is that gauge symmetry arises as a reßec-tion in the four familiar macroscopic space-time dimensions of general covariance in a largernumber of dimensions, several of which are postulated to be small, presumably for dynam-ical reasons.Here we should take the opportunity to emphasize a point that is somewhatconfused by the historically standard usages, but which it is vital to have clear for whatfollows. When physicists refer to general covariance, they usually mean the form-invarianceof physical laws under coordinate transformations following the usual laws of tensor calculus,including the transformation of a given, preferred metric tensor. Without a metric tensor,one cannot form an action principle in the normal way, nor in particular formulate the ac-cepted fundamental laws of physics, viz. general relativity and the a purely mathematical point of view one might consider doing without the metric tensor;in that case general covariance becomes essentially the same concept as topological invari-ance. The existence of a metric tensor reduces the genuine symmetry to a much smaller one,in which space-times are required not merely to be topologically the same, but congruent(isometric), in order to be considered equivalent. In the Kaluza-Klein construction, for thisreason, the gauge symmetries arise only from isometries of the compactified dimensions.Another line of thought proceeds in the opposite direction, seeking to realize generalcovariance [CapitalEth] in the metric sense [CapitalEth] as a gauge symmetry. arXiv:hep-th/9801184 v4 23 Apr 1998IASSNS-HEP-97/142Riemann-Einstein Structure from Volume alerting me to this relevant paper by Wilczek.BTW Wilczek shows that Gennady Shipov's torsion theory is closely related toRoger Penrose's spinors in curved spacetime with the anti-symmetricspin connection as the locally induced compensating torsion field.It all comes from locally gauging the O(3,1) subgroup of the Conformal Groupas I said previously based on Utiyama's and Kibble's papers from the mid-1960's.Whether or not Akimov's claims from Moscow that torsion waves from O(1,3) ofsufficient intensity to have psychotronic weapons bio-toxic effects can easily be generated when,in contrast, gravity waves from T4 are so hard to find is another issue not considered here.The gravity wave T4 coupling parameter is essentially Ed Witten's alpha' = (superstring tension)^-1.What is the corresponding O(1,3) spin connection coupling parameter? Akimov's claims hangon the answer to that question. Is it easier to make propagating torsion dislocation topological string defectsthan to make propagating curvature disclination topological string defects in the MACRO-QUANTUMVacuum Coherence Field's Goldstone Phase? That's what Akimov's claims come down to in terms ofmy new theoretical paradigm for the emergence of Einstein's Gravity and the Unified Exotic Vacuum Field ofw = -1 Dark Energy/Matter.2. The action of the symmetry group G on the total hyperspace H induces an equivalence relation ~ .That is, if X' = gX, g < G, then X' ~ X.3. ~ partitions hyperspace H into disjoint non-overlapping equivalence classes called G-orbitsG(X) = {gX, for all g < G}Remember that in this principal bundle fo is also a g < G.All G-orbits have identical structure and are diffeomorphic to G.4. This disjoint partition of hyperspace H gives the quotient space H/G that is the base space M with points x.Every point x of the base space M is really an equivalence class or G-orbit of a continuous infinity of points of a larger dimensional Hermetic or occult hidden hyperspace implicate inside it. Worlds within worlds. Wheels within wheels. Shades of Bohm's Implicate Order?5. The Projection Map P is simply P:G-orbit -> x.This means that each individual G-Orbit is really associated with a single vertical fiber at a single horizontal base space event. The G-orbit is the vertical fiber beyond, in the usual physics applications, a localized spacetime event x, although we can have delocalized base spaces of twistors whose intersections are points. We can also perhaps have base spaces of finite strings both open and closed and even base spaces of higher dimensional brane worlds?Commentary 2Given coordinate patch C(x) in the base space M in a neighborhood of point x and fiber f(x)form the local Cartesian product C(x)f(x) with ordered pair X = (x,fo).Take the union C(x)f(x)/C(x')f(x')/... of all such local products.There are redundant ordered pairs X because the coordinate patches C(x) and C(x') as sets overlapwith non-vanishing intersection C(x)/C(x')=/= Empty Set.Identify the redundant multiple images of the same actual point of the base space M usingthe symmetry group G as an equivalence relation. That is, two ordered pairs X and X' areidentified or equivalent if x = x' < C(x)/C(x') and if fo' = gfo where g < G to form disjointequivalence classes {f(x)} that are the distinct points of the fiber in hyperspace H.This is all local at a fixed base point x like in an internal gauge force symmetry.g is also called a transition function.The hyperspace H is the factor space of the union C(x)f(x)/C(x')f(x')/ ... mod G.The projection map P:(x,{fo}) -> xWhen M is the curved space-time of Einstein's gravity theory in addition to the G equivalencein the extra space dimensions of the fiber, x'(E) = Diff(4)x(E) at fixed event Eto make disjoint equivalence classes {x(E)} mod Diff4(E).One can imagine a hybrid where the fiber is a discrete space of strings of c-bits.One can also imagine a fiber of strings of qubits.1 qubit is a parallel infinity of c-bits.i.e.|qubit> = |1 c-bit><1c-bit|qubit> + |0 c-bit><0 c-bit|qubit>Where there is a continuous infinity of different c-bit basesor orthonormal frames each corresponding, for example,the the angular orientation of an inhomogeneous fieldmagnet in a Stern-Gerlach filter for spin qubitsin the DARPA spintronics project or like the billion billionSingle Electron Transistors inside the human brain at thesub-microtubular protein dimer hydrophobic cage level formingthe hardware interface with external world whose software is our stream of inner consciousness.Each possible orientation is a primitive parallel quantum universe.The quantum computer computes in all possibleorientations simultaneously like a continuousinfinity of classical Turing machines in adistributed network working on the same problem - or so the folklore goes.to be continued.Commentary 1The fiber bundle as an idea has 4 parts.1. A structure symmetry group G.2. The total hyperspace H or, in some applications Wheeler's BIT.3. The projection map P.4. The base space M or, in some applications. Wheeler's IT.The hyperspace H consists of fibers f(x) that areeither copies of or representations of the symmetrygroup G.The projection map P collapses a fiber f(x) in the hyperspace H toa point x in the base space M.All of these objects are continuum differential manifoldsdepending on the continuum of real numbers which itsassociated issues of Cantor's infinity of infinities ofCabalistic Aleph's in an ascending Jacob's Ladder.This is not a discrete combinatoric mathematics althoughsuch a skeletal structure is associated with it as inHerman Weyl's Theory of Groups and Quantum Mechanicsand as in Saul-Paul Sirag's presentation of V.I. Arnold'sA-D-E mathematics of everything.The base space is covered by an atlas of local coordinate patcheswith all important overlap transition functions sewing thepatches together like a quilt.M is space-time in local micro-quantum field theory of pointThe extra-dimensions of hyperspace formthe Calabi-Yau space of vibrations of thesuperstring beyond space-time.The connection on the total hyperspace H is the potentialof a local gauge force.Examples of connections is the 4 potential Au(x) inMaxwell's electromagnetism with G as U(1).There are similar connections for the Yang-Mills weak forcewith G = SU(2) and the strong force with G = SU(3).Classical general relativity, as distinct from local micro-quantumfield theory, has the torsion-free symmetric three-index non-tensorLevi-Civita connection with G as the Diff(4) group.The latter comes from locally gauging the 4 parameter translation subgroup(generated by the 4-momentum Pu of globally ßat special relativity )of the 15 parameter conformal group of Roger Penrose's massless twistors.Bottom -> Up: Given base space M and symmetry group G construct thehyperspace H as a quilt patchwork.Top -> Down: Given hyperspace H and symmetry group G construct thebase space M as the non-overlapping partition of hyperspace into G-orbitscalled the quotient space of H mod G in the principal bundle.Micro-quantum source renormalizable local fields of spin 1/2 lepto-quarks are associated vector bundles.Micro-quantum force renormalizable local fields of spin 1 gauge force bosons (electro-weak and strong) arefrom the principal bundle.There is no renormalizable quantum gravity in this precise sense.This is because classical Einstein gravity is a More is different (P.W. Anderson)emergent collective effect as in Andrei Sakharov's metric elasticity of aninstability in the globally ßat false vacuum of the interacting lepto-quark source/electroweak-strong force.Einstein's gravity + unified exotic vacuum dark energy/matter with Andrei Linde's chaotic inßationary cosmology are the result of the continual phase transitions from globally ßat false high entropy micro-quantum vacua to locally curved macro-quantum low entropy metastable === hexagon)(This puzzle combines a couple related puzzles I have posted.)We have a regular hexagon whose sides are lables, clockwise from top,A through F.the hexagon's inner walls as if the walls were mirrored, but it alsoaffects the direction of *itself*; for whenever it crosses its ownpath (as already drawn), it passes through the path, but is reßectedas if a mirror has been placed perpendicularly to the previous path atthe point of intersection.if such a particular path actually exists, because the path's finaldirection is heavily dependent upon the accuracy in which the path isdrawn and the accuracy of the angles reßected.But I will give the order of the hexagon's surfaces as visited by thepath (as drawn at the time of each particular crossing) below.(I know this, if my by-hand approximation was not too ßawed.)(I have no idea. You best use exact-rational arithetic to get this, ifit is possible to figure out at all.)E, B , D, 2 crossings, F, E, 1 crossing, F, 3 crossing, D, B, 3crossings, F, 7 crossings, and back to its staring point.(If no crossings are listed between letters, than no crossings occurbetween them.)(If someone solves this, they are going to have to post some link to awebpage with the === Approximating Pi by Rationalsßip> I feel that this is probably a well-known subject for number theorists,> but I've never read anything about it. The question is how closely can> we approximate pi by rationals. More specifically:>> For integer n>0, let f(n) be the largest integer m such m/n < pi.> Let d(n) = pi - f(n)/n.>> Then d(n) measures how accurately we can approximate pi by a rational> with denominator n.>> How small can d(n) be? Clearly, d(n) < 1/n. But can we make d(n) much> smaller than that?>> Q1: Can we find arbitrarily large values of n such that d(n) < 1/n^2?>> Q2: Can we find arbitrarily large values of n such that d(n) < 1/n^3?>> Q3: In general, for each p>1, can we find arbitrarily large values of> n such that d(n) < 1/n^p?>> --> Daryl McCullough> Ithaca, NY>> http://forums.wolfram.com/mathgroup/archive/2000/May/msg00188. html> http://forums.wolfram.com/mathgroup/archive/1998/May/msg00272. html> http://www.math.iastate.edu/hentzel/class.301.03/Oct.15> http://www.isi.edu/~johnh/ABOUT/FEATURES/RATIONAL_PI/One mo' on continued fractions:http://mathworld.wolfram.com/ HurwitzsIrrationalNumberTheorem.htmlplus this one, which puts upper bounds on the quality of a sequence ofrational approximations:http://mathworld.wolfram.com/ LiouvillesApproximationTheorem.htmlBut pi is a special case. Hundreds of series, some of them amazing, displaypi or 1/pi as the limit of a sequence of rationals. There is a formula for1/pi as a series in powers of 1/99, due to Ramanujan. I think it's on theweb somewhere, but I failed to find === to convince again to Dr. G Arvind Rao of Aerospace> Engineering Department by email, but he also said that point B will> shift its position along Y axis !.>> Hmmm... did you consider that they could be right, and you could bewrong?>> Laura, where from you suddenly dropped in this mess? You just don't> know, what is going on. I thought about this thousands of times in> last 13 months. I had posted idea of whole device in many newsgroup.> This is just one of the basic component or idea behind this invention.> At least this problem was not arised. And now suddenly this problem> propped up.>Abhi,this problem that suddenly propped up is the same one that everybody(including me) points out every time you post this ridiculous drivel. Eachtime you ignore it.Just for reference, here is what I posted last time (what, a couple of weeksago?):Abhi,your math/physics is faulty. The device will not ßoat to the ceiling andstay there, unless you nail it up. It will accelerate towards the ßoor atapproximately 9.8 meters per second squared.Your error is in assuming that the forces The restoring forces from the springs (I'llcall each of those F here, but you can call them what you want) must besummed as a vector. You can use many methods to do this, but I'll use yourcartesian coordinate system. I'll call the angles ABD/CBD, BDA/BDC andDAB/DCB alpha, beta and gamma respectively (but again you can call them whatyou want). In that case, the two forces acting at point B from the left andright hand springs can be written as:{-Fsin(alpha),-Fcos(alpha)} and{Fsin(alpha),-Fcos(alpha)} respectivelynet force acting on point B from the springs is therefore{ 0, -2Fcos(alpha) }the thecomponent acting along your y axis does not. There is a net force acting onpoint B towards point D.If you do the same calculation for point D, you'll find a net force actingon that point of equal magnitude but of opposite sign. In other words, thereis no net force acting on the rod BD. No unitary force.Sorry, but it doesn't workKrill>>> Indian Institute of Technology is most prestigious college in India.> This institute gives people for Aviation Industry around the world.> And I just wonder, why so highly educated people fail to understand> such simple problem.>> Maybe, just maybe, they do understand it.>> Have you done elementary Geometry Laura? Take a look at my homepage.>> http://www.geocities.com/actiondevice>yes, Abhi, we've looked at it. And yes, we understand elementary geometry.Do you understand elementary Newtonian mechanics Abhi?>>> In fact, this is not problem at all. But what a tragedy, I am facing> such ridiculous problems.>> I can end my all problems anytime, but I am following the rules of> this battle, waiting game.>> Build a working model and submit it to them for examination.> Doesn't matter how much force it produces, as long as it proves thatyour> idea works.>> No, not yet. You just don't know what is going on around me. Things> are under absolute control. You will never believe it.We will believe you are right if you build a working model. It isn't exactlyhard. In the meantime, since according to the elementary physics we allknow, it doesn't work, could you maybe understand that perhaps it is youthat is wrong, and not the rest of the world?>>> I am just watching how the minds of highly educated people around the> world are controlled by that Supreme Force named God.>> Let me get this straight.... *God* doesn't want this device discovered?Why> not? And if not, what's stopping him from destroying you to make sureyou> stay quiet?>> He does want this device to be discovered. This is exactly why He> controlled absolutely everything in my personal life. He navigated> things in last 17 years in such a way that my thought process moves> only in one direction. He trained me to gain absolute power of> imagination.>> This device is very simple. But there is no victory without> sufferings. And He has discovered His own ways to trap me.>> Things are being controlled very cleverly. Don't believe me?>> People in this NG will not answer clearly the question I have posed.> Will point B move along Y axis in XY plane? It needs just yes/no.> But they will remain silent(or they will be humorous). They will> ignore me. Because they are controlled.No, Abhi, you have been answered. Repeatedly. Read my reply from a couple ofweeks back. Asking whether the point B moves is a little meaningless givenyour defined frame of reference. But there will be a net force acting onpoint B towards point D, exactly the same as the net force acting on point Dtowards point B. So the rod BD is subject to a compressive force.Krill>> Laura, Watch Out Apocalypse In === division?> Are there any techniques that can efficiently do polynomial division?> (I am looking for techniques similar to Karatsuba used for> multiplication ...)If you just want the remainder, and can reuse the same modulus multipletimes, then you can precalculate the reciprocal (mod x^N) of the reversedmodulus and use that to find the remainder by multiplication with thereversed argument. It's quite the trick, and I can't find any references toit, but I can write it who have the book, it is W. Rudin, Principles of>>Mathematical Analysis, chapter 2 ( Basic Topology), problem 18:>>[A set E is perfect if E is closed and every point of E is a limit>>point of E] Is there a nonempty perfect set in R which contains no>>rational number?>> ... [ contructs open set containing rationals and with measure < 1,> takex X as its complement ] ...>>Try showing (1) X is not countable> (2) the isolated points of X constitute at most a> countable set> (3) X {isolated points of X} is closed>>I don't think this will work. Try considering points of X such that>>every neighborhood contains an uncountable infinity of points of X>>instead. > By Cantor-Bendixson, (2) must be true. (3) seems clear since > X, being closed, contains all its accumulation points, and these > are also exactly the accumulation points of X{isolated points}.> Is there a problem with (1) or C-B too much to take as known?While statements (1)-(3) are correct, it seems that you are hintingthat X isolated points is perfect. That isn't true.Suppose X is the closed subset of the reals consisting of 0, {1/n: n>0},and the closed interval [2,3]. Then X isolated === multiplication> Two questions:>> First, consider positive integers k and j with m and n digits> respectively. Assume moreover that neither k nor j is equal to 0.> Can we determine exactly the number of digits in the product kj? It's> either m+n-1 or m+n, but which one it is changes depending on the> nature of k and j. I wanted to use this to determine the number of> digits in, say 2^64. Now, this one might not be so difficult because> many people know 2^32 off the top of their head so they know it has 10> digits. So they would know immediately that 2^64 either has 20 or 19,> but still it's not clear (to me anyway) if it's 20 or 19 without> multiplying it out. But still, what about 3^64? And what about the> number of digits of a^b where a and b are any positive integers?>log base 10 (denoted log10) is a good measure of the number of digits of anynumber in base 10.Let ßoor(x) denote the integer less than or equal to x, then the number ofdigits of a number x is1+ßoor(log10(x))For 2^64, we get === generation of large prime numbers> |If Richard has simply added let all primes <=P be known to his premise> |I wouldn't have jumped on it that way.> I don't think this would help. Talking about what primes are known is> subjective.> Not really, would assigned make you happy? > The set of all primes is the set of all known primes in this proof. > I've said that repeatedly. I do not understand why you would not have jumped in Richard's proof when> he said let all primes <=P be known. This does not help a bit. If> the set of primes is {2, 3, 7} than all primes <= 7 are known and they> are 2, 3 and 7.Dik,My full paragraph, which was either cynically or carelessly snipped, said<<And your final sentence is _exactly_ what I've said in about 2 dozen posts so far. What made you think that I did not share this view?> The set-theoretic notation for what my sentences expressed would be> no different if I included or excluded the word known. I was > simply trying to avoid the naked word all as people immediately > misinterpret that based on their knowledge about the primes. Yup, so what? If somebody talks about all primes <= P I would think> he would assume that all numbers <= P have been tested for primeness> and would have taken only those that are proven to be prime. You said above:<<<> If> the set of primes is {2, 3, 7} than all primes <= 7 are known and they> are 2, 3 and 7.>Note that that contained the phrase all primes <= 7. Therefore from your:<<<> If somebody talks about all primes <= P I would think> he would assume that all numbers <= P have been tested for primeness> and would have taken only those that are proven to be prime.>kicks in.I should therefore be able to deduce that when you said all primes <= 7 are known and they are 2, 3 and 7, that you'd tested all numbers <= 7for primeness and would have taken only those numbers that are proven to be prime.Therefore you should have included 5.i.e. You've just contradicted yourself.One way of viewing the cause of the contradiction is that it is an instance of equivocation. all is taken to have two different meanings in the two different contexts. However, as you can see from the thread I've been trying to highlight this as a problem in the statement and interpretation of the proof all along.It's caught /you/ out, and you are one of the ones who knows best what you're talking about in this field.Another way of viewing the root of the confusion is that it is based on the if clause - the presumption of a fixed finite set of primes which isn't contiguous. This of course clashes with the second phrase which under your latter interpretation expects a contiguous range of primes. There are two contradictory assumptions about what all primes might be.This is why I've been trying to get people to state unambiguouslywhat they mean by all primes in the context where they've used it(or likewise the complete set of primes, and other synonyms).You have two basic options - you permit an arbitrary set (Euler/Kummer), or you generate a set which can have additional properties (Hardy & Wright). In the latter case you ought to firstly at least state that you're using the latter case, and thenjustify that you can perform the generation (which is trivial) and that the claimed properties also hold (which is also trivial).As we've seen, from your very post here, all primes <=7 can mean two different things if the context, which of the two options above you're chosing for your line of attack, hasn't been given.All of the sparse prime sets have satisfied the assumptions that were given in the original statement of the proof proof. However, I chose those sets with the presumption that it _was_ Euclid's proof that was being talked about. As it was:<<It appears that Richard and others were justified in thinking that the proof given (although I still claim the assumptions weren't stated unambiguously) was Euclid's as it seems that Hardy & Wrighthave propagated it as such. However, I've never seen anything apartfrom Elements Book IX proposition 20 and the Ribenboim version, which is pretty faithful.> Note> again, this is *not* Euclid's proof, Exactly. It shares the product+1 feature, but has essentiallydifferent (more) assumptions. > but a well-known and much> occuring variation. And is *just as valid*.If done correctly, yes (i.e. all assumptions are unambiguously stated). Willem spotted a good way to make the proof not presume all primes <=P were sieves out (erm, in), by simply steam-rollering them all into the product with the factorial function rather than the primorial function. This simplifies the proof _greatly_,IMHO.Phil-- Unpatched IE vulnerability: HTTP error handler Local Zone XSSDescription: HTML/Script injection in the Local ZoneReference: http://sec.greymagic.com/adv/gm014-ie/Exploit: === MATHEMATICIANS READ WITH HALF A LIGHTBULB?> When n goes to infinity, the right hand side of this equation only> becomes equal to the right hand side of the equation given above by> Rudy, if a has a value between -1 and 1. So the complete statement> should have been: 1 + (a + a^2 + a^3 . . .) = 1 / (1 - a) with -1 < a < 1If I'm not mistaken, you are stating that any fraction as a substitutefor a will reveal nonsense, and, yes, Rudy said the same.However, I believe that fractions work just as well as any othersubstitution for a but that we simply do not force a logical processof infinite division for the seemingly simple fractions on the rightside because we believe that these fractions are in their simpleststate and cannot equal the plainly different forms on the left side. I nonetheless alluded to such a process for the fraction 1/2, and Ibelieve that a mentally rigorous but creative mathematician willeventually prove me correct, or, more accurately, that he or she iscorrect.Very === an explanation of why it doesn't work. With details. If you believeyou are correct, I have provided everything you need to point out the ßawin my reasoning. If you can, please do so. But you can't complain thatnobody listens to you when you don't listen to what anybody else saysKrill> Abhi,> your math/physics is faulty. The device will not ßoat to the ceilingand> stay there, unless you nail it up. It will accelerate towards the ßoorat> approximately 9.8 meters per second squared.>> Your error is in assuming that the forces don't. The restoring forces from the springs(I'll> call each of those F here, but you can call them what you want) must be> summed as a vector. You can use many methods to do this, but I'll useyour> cartesian coordinate system. I'll call the angles ABD/CBD, BDA/BDC and> DAB/DCB alpha, beta and gamma respectively (but again you can call themwhat> you want). In that case, the two forces acting at point B from the leftand> right hand springs can be written as:>> {-Fsin(alpha),-Fcos(alpha)} and> {Fsin(alpha),-Fcos(alpha)} respectively>> net force acting on point B from the springs is therefore>> { 0, -2Fcos(alpha) }>> the component of forces acting along your y axis does not. There is a net force actingon> point B towards point D.>> If you do the same calculation for point D, you'll find a net forceacting> on that point of equal magnitude but of opposite sign. In other words,there> is no net force acting on the rod BD. No unitary force.>> Sorry, but it doesn't work>> Krill>> I relate myself to Tom Hank in that movie, CAST AWAY. You people are> like that volleyball Wilson. I am talking to people who lack> consciousness. Now I am lost at sea and I am going to lose === Car crash formula > I'm trying to find a formula for where I can measure the speed of acar when> it hits a still object based on car's weight, object's weight, andtotal> distance the object was thrown. I realize that there are many otherfactors> such surface friction, in this case road, but i'm just looking for an> estimate, not to be as exact as posible.>Crash investigation has it's own field. These links will give youinsight and enough information to make your own estimate.http://www.accidentreconstruction.com/education/ equations.asp http://www.accidentreconstruction.com/discussion/disc5/disc_ toc.htmhttp://www.accidentreconstruction.com/testyourskill/ === generation of large prime numbersX-KorrNews: UsedSomebody claiming to be Phil Carmody Why should there primes be missing from all primes?>>There shouldn't, and there aren't. That's why I asked the question.>Is the art of rhetoric dead?Yes, rhetoric is dead. You *were* expecting an answer to reply by e-mailWhen will I learn? The answers to life's problems aren't at the bottom of a bottle. They're on TV! --Homer === DO MATHEMATICIANS READ WITH HALF A LIGHTBULB?> The so-called associative property of the addition> a + ( b + c ) = ( a + b ) + c> allows us to write both sides of the equality as> a + b + c. This property is *not* valid for so-[badly]-called> infinite sums. You cannot write something like> a1 + ( a2 + a3 + ... ) = (a1 + a2) + a3 + ...> In fact, the thing> a1 + a2 + a3 + ....> is not even a sum to begin with!> It is a so-called limit of a series of partial sums:> s1 = a1> s2 = a1 + a2> ...> sn = a1 + a2 + ... + an> If this series has a limit for n -> infinity, then one is> allowed to use the abbreviation> limit(sn; n -> infinity) = a1 + a2 + a3 + ...> The property of having a limit in the previous sentence> is something that can be verified by other means.Okay, now please give an example of where s1 = a1 and s2 = a1 + a2 but(a1 + a2) - a2 DOES NOT = a1.Although the associative property of addition is important in the endof my proof, the most important property is the distributive propertyof multiplication over addition where the common factor of an infiniteseries (a + a^2 + a^3 . . . ad infinitum) gets distributed andmultiplied over 1-a and leaves two equal infinities with one startingat a and the other at a^2, but where the infinite portions of bothinfinities can be leaving the non-infinite point of a on the numberline.However, I'm listening, so please show my WHY the associative processof addition no longer works with an infinite sum series. If yourproof depends on the seemingly illogical form of various outcomes withsome substitutions, then I invite you to explore those forms deeperrather than risk a circular argument that my math is wrong because itgives wrong outcomes.Very === GameX-Abuse: abuse@usq.edu.au> | A quote from:> | The New Fowler's Modern English Usage, Third Edition, Edited by R. W.> | Burchfield, The acknowledged authority on English usage> | [all of that from the front of the dust jacket...]> |> | Under the topic billion:> | It is best now to work on the assumption that the word means Ôa> | thousand millions' in all English-speaking areas...>I suggest that the word million, billion and trillion be abolished,and usage adapted around SI prefixes.Eg: Bill Gates is not worth billions of dollars; he === HALF A LIGHTBULB?> Very nice, for hundreds of years we have erroneously assumed the> geometric series converges only for |x|<1, but you have proven> otherwise.Although I still do not know what converges refers to, the form ofyour objection is argumentum ad populum. Let us remember that formany centuries, the world was most assuredly ßat.Very === Rationals>But pi is a special case. Hundreds of series, some of them amazing, display>pi or 1/pi as the limit of a sequence of rationals. There is a formula for>1/pi as a series in powers of 1/99, due to Ramanujan. I think it's on the>web somewhere, but I failed to find it.I think you're referring to /infinity | ----- | 1 1/2 | (4 n)! (1103 + 26390 n)| ---- = 2/9801 2 | ) -----------------------| Pi | / 4 (4 n) (4 n) | | ----- (n!) 4 99 | n = 0 /See e.g. the postings of David Findlay and Sebastian Hew on the subject Pi from 7 and 8 September 2001.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === generation of large prime numbers>> You were trying to conclude that, for any prime P, N is a larger>> prime.> No, I wasn't. I was saying that it might be a larger prime, but that it > might be a composite of primes at least one of which is larger than P.Sorry, you're right. The *original poster* was (if I understand themcorrectly) trying to make the conclusion noted above.