mm-325 === Subject: : Re: complex integral.....?? > um...... i think that root of 1+(z^3)+(z^5)}] exist between -1 ~ -0.5> thus, f(z) is analysis of |z|<1/2> thus.......i think that answer is 0> it is right??> All the roots of a_n*z^n + a_{n-1}z^(n-1) + ... + a_1*z + a0 verify> 1/(1 + B/|a_0|) < |z| < 1 +A/|a_n|If a_n * a_0 =/= 0, of course ...> Where A = max(|a_0|, |a_1|, ..., |a_{n-1}|) and B = max(|a_1|, ...,> |a_{n-1}|, |a_n|)> Then, you are right ...> -- > Best regards,> Ignacio Larrosa Ca.96estro> A Coru.96a (Espa.96a)> ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: : DifferentiationIs differentiation a one way function?More exactlyIs differentiation a one way functional?So if I ask how to integrate complicated expressionam I actually asking for help to break a code? ;-)=== === Subject: : Re: Two questions in propositional logic>[...]>2. Let C be a set of propositions. We say C is a chain if for every p, q >in C either p proves q or q proves p (and not both). If the set of >primitive propositions is allowed to be uncountable, can there be an >uncountable chain?>I doubt it. Say p < q if p |- q but not q |- p; then your chain is>totally ordered by <. First, either there exists an increasing>sequence with at least two upper bounds>This is presumably under the assumption that C is uncountable?>Yes.>or a decreasing sequence with at least two lower bounds>(by the traditional proof that any sequence of reals contains>a monotone subsequence: Say you have elements p_a, indexed by the>countable ordinals. Say a is dominant if p_a > p_b for all b > a.>Either there are uncountably many dominant a or not. If there>are uncountably many dominant a then the p_a with a dominant>give a cofinal decreasing set, while if there are only countably>many dominant a then the dominant a are bounded above>and you get a cofinal increasing subset.) So, changing the>notation and replacing all the wffs by their negations if>needed, you have> p_1 < p_2 < ... < q_1 < q_2.>It seems clear to me that this implies q_1 must be a>tautology (it seems like this is by compactness or>something, but I can't quite prove it this second)>and then q_2 is weaker than a tautology, contradiction.>???>Unless you're using some extra property about the p_i, q_i in saying >this that I'm not seeing >No.>(which is very possible, on account of it being >Early here and I just got up. :), that needn't be true.>For example take the following sequence:>p_3, p_3 v p_4, p_3 v p_4 v p_5 ... , p_1 v p_3 , p1 v p_2 v p_3.>??? Is it true that p_3 v p_4 |- p_1 v p_3 ? I don't think so...>Not that I can see how to prove what I said, and it may well>be false, but this is not a counterexample, unless _I'm_>missing something - I did think about simple examples>like this. In fact assuming that the p_j are propositional>_variables_ then if p_3 v ... v p_n |- w for all n then it>does follow that w is a tautology: If w is not a tautology then>it is falsified by some truth assignment. Since w contains>only finitely many variables, it is falsified by some>truth assignment that sets some irrelevant variable to T,>and that shows that p_3 v ... v p_n |- w is false for some n.>I actually figured that out last night - realizing that _that_>chain cannot be followed by anything but a tautology is>why I conjecture that no increasing sequence can be>followed by anything but a tautology. (Any increasing>sequence has the same form as above (identifying>wffs that are logically equivalent), except that the p_j>are wffs instead of variables. It seems like one should>be able to use more or less the same argument,>sort of, although I haven't worked out _exactly_>how it goes...)Never mind - ths above is not a counterexample, but ofcourse what I conjectured is false. Realized that Iwanted to show that the intersection of any strictlydecreasing sequence of open subsets of the Cantorset had empty interior, realized immediately that thatwas false, hence the examplep_1 & (p_2), p_1 & (p_2 v p_3), ... p_1.(Well, if you had an uncountable chain you could get anincreasing chain isomorphic to omega_1 as above, andI bet it's impossible to have a strictly decreasingomega_1-sequence of open subsets of the Cantorset... but never mind, you say you have a proof.)>I thought at one point that you could have a chain isomorphic to any >countable ordinal, although in retrospect I'm not entirely convinced by >my argument (I'm slightly worried about what happens with some of the >bigger limit ordinals), but this doesn't really matter. Of course not >every chain is isomorphic to an ordinal, as the reverse of a chain is >also a chain.>Anyway, as I said in the other post I think I've got this sorted out now >(although I'll want to write my own proof of it before I'm fully happy >with it).>If you're interested, John's proof went roughly as follows:>(Sorry if there's something below I should have replied>to that I'm appearing not to notice - you've forced me to>stop reading at this point..)>[snipped with eyes shut] === Subject: : Re: torque T = r x F and basic tensors> 2. How is torque a tensor? Following Feynman vol 2 ch 31, a tensor is a> linear map A relating 2 vector quantities, for example p = Ae (where I am> using p for the polatization vector of a dielectric and e for the electric> field vector) and hence if we change the basis of R^3, A must transform as> A' = Q^{-1} A Q where V is the new ordered basis, U is the old and> V = UQ^{-1}> I can see Feynman gets the 3 by 3 anti-symmetric matrix> T_{ij} = x_i F_j - x_j F_i i,j = 1,2,3> How would this matrix object ever be used for torque?> If it has any relevance, I can derive the formula> Q( a x b ) = 1/(detQ) ( Qa x Qb )> where Q is the matrix of the change of basis, but I'm not quite seeing> how that matches the A' = Q^{-1} A Q form.As you say, under the change of orthonormal basis, the vectors a and bgo to a' = Qa and b' = Qb. In component form,a_i = (sum over m) Q_(im) a_m and b_i = (sum over n) Q_(in) b_nI've used different summation indices in order that the substitutions below makes sense.Because both the new and old bases are orthonormal, Q is an orthogonalmatrix, i.e., Q^T = Q^(-1), i.e if M = Q^(-1), M_(ij) = Q_(ji). Thus,det Q = +/- 1. If in addition, right-handed orthonormal bases aretaken to right-handed orthonormal bases, then det Q = 1, and Q is calleda special orthogonal matrix. The set of all special orthogonal matricesforms the group SO(3), where the notation is fairly self-explanatory -S for special (det +1), O for orthogonal (inverse = transpose), 3 for3 by 3.Define matrix A by A_(ij) = a_i b_j - a_j b_i. Under a transformation,A'_ij = a'_i b'_j - a'_j b'_i = (sum over m,n) [Q_(im) a_m Q_(jn) b_n - Q_(jn) a_n Q_(im) b_m] = (sum over m,n) [Q_(im) a_m b_n Q_(jn) - Q_(im) a_n b_m Q_(jn)] = (sum over m,n) [Q_(im) (a_m b_n - a_n b_m) Q_jn] = (sum over m,n) [Q_(im) A_(mn) Q^(-1)_nj]A = Q A Q^(-1)This is a little different than A = Q A Q^(-1) because, the way Feynmandefines things, a'_i = (sum over n) Q^(-1)_(in) a_n. It's a goodexercise to show this.George === Subject: : Re: Key Core Error Argument> ...> Actually it is, as Dik Winter is trying to find a way that 7, 7 and 22> become 1, 1 and 22 based on a *varying* x.> That is, he's trying to make the change in *constants* dependent on a> variable.> Eh?> I want readers to understand that his behavior is crank, while I guess> many of you may sympathize with his strong desire for me to be wrong,> remember, it's not about people as the math didn't just decide to> change.> What you should be sympathetic to, is the truth.> You claim that having P(x) = g1(x).g2(x).g3(x) with g1(0) = g2(0) = 7> and g3(0) = 22; the *only* way to divide P(x) by 49 is by dividing g1(x)> and g2(x) by 7 and g3(x) by 1. Because now g1(0)/7 = 1, g2(0)/7 = 1 and> g3(0)/1 = 22.> I claim there are other ways to do that. Have w1(x), w2(x), w3(x), such> that w1(0) = w2(0) = 7, w3(0) = 1, w1(x).w2(x).w3(x) = 49. Because now> g1(0)/w1(0) = 1, g2(0)/w2(0) = 1 and g3(0)/w3(0) = 22.> Where do I change constants?> The basic algebra is that if you have 7 and divide it by *something*> and get 1, then you divided by 7. And no tricks will change that fact, and it's simply crank behavior to> try and act like there's some complicated way you can divide 7 to get> 1, without actually just dividing it by 7.> Since nobody is claiming to be doing anything else, your> diatribe seems a bit pointless.> At x=0: You divide by 7, Dik divides by 7> At x<>0: You divide by 7, Dik divides by something other than 7> The two factorizations are thus different. But the constant> terms depend only on what happens at x=0. Thus the constant> terms are the same.> -William Hughes> That's the kind of odd illogic which shows a crank, and I'll explain> quickly why your approach is specious.> So I have the factors (a_1(x) + 7) and people like yourself and Dik> Winter want desperately to believe that some variable factor of 49> divides through so that the factor itself has a varying factor of 7.> I've pointed out that the constant term of its corollary factor is 1.> So let's pick x=9 and see what happens, as then you have> a_1(9) + 7> This is nonsense. The constant term of (a1(x) +7)/w(x) is > (a1(0) +7)/w(0). The value (a_1(9) + 7)/w(9) has no bearing.> Since the factor is a_1(x) + 7, if x=9, it is a_1(9) + 7 as I said.> At x = 9 Dik divides by w1(9) > The constant term doesn't change with x, because it is constant.> The constant term does not depend on what Dik divides> by at x=9.> Do you understand that in general the factor is a_1(x) + 7?> Now then, if you understand that, do you understand that at x=9, it is> a_1(9)+7?> Maybe if that's gotten out of the way you can see the rest.> The factor is (a_1(x) +7) > The value of the factor is (a_1(x) +7)> The constant term of (a_1(x) +7) is (a_1(0) + 7) = 7> At x=0> The value of (a_1(x) +7) is (a_1(0) + 7)> The constant term of (a_1(x) +7) is (a_1(0) + 7) = 7> At x=1> The value of (a_1(x) +7) is (a_1(1) + 7)> The constant term of (a_1(x) +7) is (a_1(0) + 7) = 7> At x=2> The value of (a_1(x) +7) is (a_1(2) + 7)> The constant term of (a_1(x) +7) is (a_1(0) + 7) = 7> At x=9> The value of (a_1(x) +7) is (a_1(9) + 7)> The constant term of (a_1(x) +7) is (a_1(0) + 7) = 7> Note that the value of (a_1(x) + 7) depends on x> Note that the constant term of (a_1(x) +7)> does not depend on x> Note that the constant term of (a_1(x) + 7) does not> depend on the value (a_1(9) + 7)> Now let's add the factor w(x)> The factor is (a_1(x)/w(x) + 7/w(x)) > The value of the factor is (a_1(x)/w(x) +7/w(x))> The constant term of (a_1(x)/w(x) +7/w(x)) is > (a_1(0)/w(0) + 7/w(0)) = 7/w(0)You keep writing a ratio because apparently you think a ratio is morepowerful or mysterious, capable of doing something that it can't.Now then, if you admit that a_1(x)/w(x) is an algebraic function, itcan be replaced by f(x), and if you admit that 7/w(x) is an algebraicfunction, it can be replaced by g(x), so then you have f(x) + g(x).But the constant term of f(x) + g(x) is 1, so let h(x) + 1 = f(x) +g(x), to isolate constant terms as before.It's that result h(x) + 1 that you're terrified of, which is why youkeep writing ratios as if algebra is mysticism!You are a crank William Hughes who has seized upon ratios as a way topromote your illogical position.The important factorization is(5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22where you you have a clear separation between constant terms andvarying terms, which is why 1(1)(22) equals the constant term of300125 x^3 - 18375 x^2 - 360 x + 22.What you're trying to do is *hide* the simple reality by usingratios!!!It's *crank* behavior and anti-math, as you're trying to refute alogical, mathematical position by attempting to fool people using adodge.James Harris === Subject: : Re: Usenet Posting Guide?> With all respect, dear man, you strike me as that type more enamored> with process rather than function. Most of us out here have far better> things to do than screw around for weeks configuring newsreaders and> the like, even if we were so inclined, which most of us clearly are> not. I suppose we could memorize the phone book, too, but would that> help us communicate our ideas any better?> Some of us prefer to view the forest rather than count the trees down> there. Better view, too.>Well, I find the technical aspects of how USENET functions much more>interesting than most of the discussions that take place on it. By the>same token, I am much more interested in the hardware and operating system>software of the systems I administer than in any of the applications>for which they are used.>Once I hoped that the growing popularity of personal computers meant>that nearly everyone would learn to think like programmers. It never>occurred to me that, sadly, the opposite would happen: That computers>would be designed to be used by people who *can't* program them.But that's always been the case. We made a lot of money doingwork that our customers didn't want or have to do. What wedidn't do (when we weren't ing idiots) was hide all thewarts under pretty pictures./BAHSubtract a hundred and four for e-mail. === Subject: : parallelizability of manifoldsX-ID: XMyd06ZbYe1qqeJyfD-rbIp58cBrznLrQ5aSRnbmhWwKSYEevXbOcHIn my self-studies of differential topology, I am not sure if I understoodthe concept of parallelizability correctly.A n-dim. manifold M is said to be parallelizable iff its tangent bundle TMis diffeomorphic to (M x R^n). Is this equivalent to the following: twotangent vectors (at maybe different points on the manifold) are identicalin one chart if and only if they are identical in all charts (with theappropriate range).The following argument makes me think that this is false, but I do not quiteunderstand why.I have read that S^1, S^3 and S^7 are parallelizable. Because there are nospecific charts mentioned, I assume that this is the case for every atlasequivalent to the atlas consisting of the two stereographic projectioncharts. But this is not true; even for S^1, it is easy to show that, givenany non-constant differentiable curve on S^1, tangent vectors in one chartmay be parallel while they are not in the other one.So how to imagine parallelizability? And how can it be proven? Are thetori T^n:=(S^1)^n parallelizable?TIA,Tobias-- reverse my forename for mail! === Subject: : Re: Usenet Posting Guide?> Once I hoped that the growing popularity of personal computers meant> that nearly everyone would learn to think like programmers. It never> occurred to me that, sadly, the opposite would happen: That computers> would be designed to be used by people who *can't* program them.>That ought to be positive, it is called job security. I would shudder>when everybody at our institute thought like a programmer. Yep. Even in our neck of the woods, where we were gettingpaid to be programmers, we had strict rules about what not todo.> ..You would>have umpteen versions of all software around, and hope you could in some>way couple the lot.There would always be two versions that did exactly opposite things.The way it used to be was that enough information was shippedwith each computer so that mere users had access to more detailedknowledge about how the system worked. They could get as sophisticatedas they wanted to. Thus a system could service those who reallyweren't interested in how the system did their bidding and providethe hard/software to those who got interested in how a systemgot things done./BAHSubtract a hundred and four for e-mail. === Subject: : Re: Greek Alphebet Ioannis [NonBreakingSpace].8b.96.87.8c .97.99.95 .93fi.92.9b.93.87> Thomas Bushnell, BSG [NonBreakingSpace].8b.96.87.8c .97.99.95 .93fi.92.9b.93.87> Nowadays there is only one accent, but in ancient Greek and in its> modern> equivalents, there were two accents: The grave (tilde) and thesharp> (regular accent). What you saw is probably an omega with a grave.> No, ancient Greek had three accents: acute, grave, and circumflex.> The circumflex is sometimes written as a tilde or a macron.> Well, I really don't know their names in English, but upon more careful> inspection, you are probably right. I meant to write circumflex and> acute,which is still missing the grave.> Acute = ' (oxeia)> Circumflex = ~ (perispomeni)> Grave = ? (bareia)> I don't recall what the grave looks like, as in modern Greek it doesn't> exist, at all.It faces the opposite way of the oxeia.To mention that other simbols not in use today are:-psili (pneyma).-dasia (pneyma).-ypogegrameni (under the eta and the omega).-- E' mai possibile, oh porco di un cane, che le avventure in codesto reamedebban risolversi tutte con grandi puttane!F.d.A === Subject: : Factorization disputeIt turns out that I can isolate the current dispute easily enough byfocusing on the factorizations:Consider(5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22where even by inspection you can see that the constant terms areseparated out, so that you have 1(1)(22) = 22, the constant term ofthe polynomial.I'll add that at x=0, a_1(0) = a_2(0) = b_3(0) = 0.Notice, (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 again you see that the constant terms match as now you have7(7)(22) = 1078, which is again the constant term of the polynomial.If 22 does not have 7 as a factor, the former factorization is the*only* allowed way for 49 to divide through.(For more detail, like what the a's are, seehttp://mathforprofit.blogspot.com/where more is explained.)I have a result that shows a problem with a definition thatmathematicians have used for over a hundred years, and rather thanface the result which follows from rather basic algebra mathematiciansare being pussies and running like scared cowards from the result.Some posters, not professional mathematicians from what I've gathered,are at least trying to stand and fight, but because what I have is amath proof, their claims are necessarily irrational.I need you to stand up for the truth, here and now.Let's chase these mathematician cowards down, and make them face themusic.After all physics had its challenges and physicists faced them, whilemathematicians seem to believe that they can just ignore problems.Let's take 'em out.James Harrishttp://mathforprofit.blogspot.com/ === Subject: : Re: Two questions in propositional logic>>I've just finished being supervised on an example sheet for our 'Logic, >>Computation and Set Theory' course, and there were two questions which >>neither I, my partner, nor my superviser could actually answer. I was >>wondering if someone could give me some hints so that I could try and >>sort these out. (Preferably not more than hints, as some people haven't >>been supervised on this yet. Don't want to spoil it for them. :)>We're working in propositional logic, with our basic symbols being >>implies, falsehood and p_1, ... . The axioms are>p -> (q -> p)>>[ p -> (q -> r) ] -> [ (p -> q) -> (p -> r) ]>>p -> p>(Where p = ( p -> F ) ).>Fairly standard, but people use some slightly different axioms so I just >>thought I should say which ones I'm using.> You also need to specify what the inference rules are...> Sorry, of course. The only rule of inference is modus ponens.>2. Let C be a set of propositions. We say C is a chain if for every p, q >>in C either p proves q or q proves p (and not both). If the set of >>primitive propositions is allowed to be uncountable, can there be an >>uncountable chain?> I doubt it. Say p < q if p |- q but not q |- p; then your chain is> totally ordered by <. First, either there exists an increasing> sequence with at least two upper bounds> This is presumably under the assumption that C is uncountable?> or a decreasing sequence with at least two lower bounds> (by the traditional proof that any sequence of reals contains> a monotone subsequence: Say you have elements p_a, indexed by the> countable ordinals. Say a is dominant if p_a > p_b for all b > a.> Either there are uncountably many dominant a or not. If there> are uncountably many dominant a then the p_a with a dominant> give a cofinal decreasing set, while if there are only countably> many dominant a then the dominant a are bounded above> and you get a cofinal increasing subset.) So, changing the> notation and replacing all the wffs by their negations if> needed, you have> p_1 < p_2 < ... < q_1 < q_2.> It seems clear to me that this implies q_1 must be a> tautology (it seems like this is by compactness or> something, but I can't quite prove it this second)> and then q_2 is weaker than a tautology, contradiction.> ???> Unless you're using some extra property about the p_i, q_i in saying > this that I'm not seeing (which is very possible, on account of it being > Early here and I just got up. :), that needn't be true.> For example take the following sequence:> p_3, p_3 v p_4, p_3 v p_4 v p_5 ... , p_1 v p_3 , p1 v p_2 v p_3.Pointed out a few minutes ago that this is not a counterexample.Of course what I said is wrong - a counterexample isp_1 & (p_2), p_1 & (p_2 v p_3), ... p_1.(Realized I wanted to show that the intersection of a strictlydecreasing sequence of open subsets of the Cantor set hadempty interior. Realized that that was false. Now, I betthat there's no strictly decreasing omega_1-sequence ofsubsets of the Cantor set, but never mind, the proof youalready have is simpler.)Sorry if I already said this - I just had a PC go up in smoke,not sure where I was when that happened...> I thought at one point that you could have a chain isomorphic to any > countable ordinal, although in retrospect I'm not entirely convinced by > my argument (I'm slightly worried about what happens with some of the > bigger limit ordinals), but this doesn't really matter. Of course not > every chain is isomorphic to an ordinal, as the reverse of a chain is > also a chain.> Anyway, as I said in the other post I think I've got this sorted out now > (although I'll want to write my own proof of it before I'm fully happy > with it).> If you're interested, John's proof went roughly as follows:> Define an equivalence relation on C by p ~ q if there is a bijection f > from the set of primitive propositions to itself, such that q = p with > all the p_i in p replaced with f(p_i).> He then showed that there can be at most one element of each equivalence > class in C, and that there are countably many equivalence classes > greater than p (because you can take some countable subset of the whole > set of primitive propositions and represent every formula in C as a > formula using only this countable subset and the primitives that appear > in p).> He then went on to show there were only countably many elements less > than p in a similar manner. I pointed out that this was rather easier, > as you could just reverse the chain by negation and use the previous > result.> I think I can make that proof slightly shorter by considering the set of > propositions in the chain provable in <= n lines and considering how > 'new' primitives are introduced. Then, up to equivalence, there should > be only countably (finitely?) many of these. Then a countable union of > countable sets is countable. Not sure if that's going to work, but it > looks plausible.> David> (E-mail address spam-blocked in the obvious way) === Subject: : Re: parallelizability of manifolds>In my self-studies of differential topology, I am not sure if I understood>the concept of parallelizability correctly.>A n-dim. manifold M is said to be parallelizable iff its tangent bundle TM>is diffeomorphic to (M x R^n). Is this equivalent to the following: two>tangent vectors (at maybe different points on the manifold) are identical>in one chart if and only if they are identical in all charts (with the>appropriate range).It's hard for me to make sense of this last sentence, but inthe best rendering I can give it, the answer is, no, that'snot what parallelizable means.>So how to imagine parallelizability? And how can it be proven? Surely the simplest way to understand the concept of parallelizabilityis this: the n-manifold M is parallelizable if and only if there existn vectorfields V_i on M with the property that, at each point x of M, the n vectors V_i(x) are a basis of the tangent space of M at x.No imagination required.>Are the>tori T^n:=(S^1)^n parallelizable?Yes, indeed. (And, more generally, the product of parallelizablemanifolds is parallelizable.)Lee Rudolph === Subject: : Re: Two questions in propositional logic>[...]>2. Let C be a set of propositions. We say C is a chain if for every p, q >>in C either p proves q or q proves p (and not both). If the set of >>primitive propositions is allowed to be uncountable, can there be an >>uncountable chain?> [...]> If you're interested, John's proof went roughly as follows:> Define an equivalence relation on C by p ~ q if there is a bijection f > from the set of primitive propositions to itself, such that q = p with > all the p_i in p replaced with f(p_i).> He then showed that there can be at most one element of each equivalence > class in C, Because if p ~ q and p |- q then a permutation of the variablesin the proof shows that q |- p.>and that there are countably many equivalence classes > greater than p (because you can take some countable subset of the whole > set of primitive propositions and represent every formula in C as a > formula using only this countable subset and the primitives that appear > in p).??? Isn't it clear that there are only countably many equivalenceclasses altogether, and so we're done? I mean any wff involvingonly n variables is ~ - equivalent to one involving only p_1,...p_n, and up to logical equivalence there are only finitely manyof those.Oh. That's not right with the ~ you defined. But why not insteadsay p ~ q if there exists a permutation of the variables whichmakes q into a wff logically equivalent to p, instead of requiringthat it literally become p? > He then went on to show there were only countably many elements less > than p in a similar manner. I pointed out that this was rather easier, > as you could just reverse the chain by negation and use the previous > result.> I think I can make that proof slightly shorter by considering the set of > propositions in the chain provable in <= n lines and considering how > 'new' primitives are introduced. Then, up to equivalence, there should > be only countably (finitely?) many of these. Then a countable union of > countable sets is countable. Not sure if that's going to work, but it > looks plausible.> David> (E-mail address spam-blocked in the obvious way) === Subject: : Re: parallelizability of manifoldsX-ID: VC5Ub8Zp8edFmqeyNGF3bYu8oDlFkT-AXYjDmLN7Fh8y97WADCICE1> Surely the simplest way to understand the concept of parallelizability> is this: the n-manifold M is parallelizable if and only if there exist> n vectorfields V_i on M with the property that, at each point x of M,> the n vectors V_i(x) are a basis of the tangent space of M at x.> No imagination required.Ah, that's interesting! So why are there no such vector fields for S^2?> Yes, indeed. (And, more generally, the product of parallelizable> manifolds is parallelizable.)Ok, this is clear because the tangent bundle of a product is the product ofthe tangent bundles.-- reverse my forename for mail! === Subject: : Re: Two questions in propositional logic > Anyway, as I said in the other post I think I've got this sorted out now > (although I'll want to write my own proof of it before I'm fully happy > with it).> If you're interested, John's proof went roughly as follows:> Define an equivalence relation on C by p ~ q if there is a bijection f > from the set of primitive propositions to itself, such that q = p with > all the p_i in p replaced with f(p_i).> He then showed that there can be at most one element of each equivalence > class in C, and that there are countably many equivalence classes > greater than p (because you can take some countable subset of the whole > set of primitive propositions and represent every formula in C as a > formula using only this countable subset and the primitives that appear > in p).> He then went on to show there were only countably many elements less > than p in a similar manner. I pointed out that this was rather easier, > as you could just reverse the chain by negation and use the previous > result.I found this the most natural approach. However (and IBL will probablykill me for saying anything at all about this sheet in public) youshould also be aware there is a one word answer to this question. Thatis, there is one particular word (familiar to everyone) whichinstantly makes this question a triviality. I'll leave you to guesswhat it is.By the way you didn't ask about the last question (if the set ofprimitive propositions is allowed to be uncountable is it true given aset S of propositions that you can find an independent set ofpropositions equivalent to S); does that mean you've solved it? Istill don't have it, one year after taking the course. No hintsplease!Michael === Subject: : Decimal ExpansionIf m Surely the simplest way to understand the concept of parallelizability> is this: the n-manifold M is parallelizable if and only if there exist> n vectorfields V_i on M with the property that, at each point x of M,> the n vectors V_i(x) are a basis of the tangent space of M at x.> No imagination required.>Ah, that's interesting! So why are there no such vector fields for S^2?There's not even the *beginning* of such a sequence of vectorfieldsfor S^2: no matter what the vectorfield V_1 on S^2, there is always some point x of S^2 at which V_1(x) does not belong to any basis of the tangent space of S^2 at x. That's just a (seemingly morecomplicated, but ultimately worthwhile) way of saying that every(continuous!) vectorfield on S^2 has at least one zero. And that, in turn, follows (after building the appropriate machinery relating differential topology to algebraic topology) from the fact that the Euler characteristic of S^2 is non-zero.In general, you could (and people do) ask, given an n-manifold M,what is the smallest k such that there is some sequence of k vectorfields V_i on M such that, at every point x of M, thevectors V_i(x) span the tangent space of M at x. Alternatively(somewhat dually) you could (and people do) ask, given a genericsequence of k vectorfields V_i on M, what is the nature of the subset of M at every point of which the vectors V_i(x) are linearly dependent. If you pursue such questions, you eventuallyfind yourself studying characteristic classes (in homology as wellas in cohomology); the Euler class is just the tip of that particulariceberg.Lee Rudolph If you saw(c_1 x + 7)(c_2 x + 7)( c_3 x + 2) = 49(x^3 + 5x^2 + 3x + 2)with the c's algebraic integers, I think few of you would have aproblem realizing that only two of the c's have 7 as a factor.But, of course, you're looking at *functions* of x, as you have f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, so I could also write it as(f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 2) = 49(x^3 + 5 x^2 + 3x + 2).Notice that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 2) = x^3 + 5 x^2 + 3x + 2as long as you're in a ring where 7 is not a factor of 22.I want to emphasize that point as notice there's only *one* way todivide through by 49 if 7 is not a factor of 22.Usually you can *see* the other factors of 7, but I want you toabstract, and generalize.Please pay careful attention to that example.You may see people who reply claiming that the word polynomial hassome significance, as if it's a mystical thing which refutes basiclogic, so if something isn't polynomial it no longer behaveslogically.Now then, in my advanced factorization work, I just use functions of xthat are a lot more complicated than f_1(x) = c_1 x, and unfortunatelythere are people who can use an unfamiliar leap in complexity toconfuse others.Some of you have learned various advanced math topics, now imagine ifin your classrooms there were some hecklers who continually holleredout at your teacher, or otherwise disrupted the class?What if when there were difficult concepts those hecklers would try toconfuse everyone as they sought to discredit the mathematics?If you find that hard to imagine, imagine me in your class with youquestioning the professor and calling him names.How much would you have learned?I need those of you interested in mathematics to focus on the basics,so that you can understand the advanced.James Harrishttp://mathforprofit.blogspot.com/ === Subject: : Re: Factorization dispute> It turns out that I can isolate the current dispute easily enough by> focusing on the factorizations:> Consider [snipped]Crank Information http://www.crank.net/harris.html http://www.crank.net/usenet.html http://www.google.com/search?q=harris+site%3Awww.crank.net http://www.google.com/search?q=%22james+harris%22+site% 3Ausers.pandora.be === Subject: : Re: Difficult social problem> It looks like I'm swinging at tissue paper with a sledgehammer when it> comes to getting acceptance of my work, as while I can get initial> contact with mathematicians they tend to run as soon as I give them> enough information to realize the implications of my work and that I> am correct.even if your work were correct (and it is not), it would be as awe inspiringas my morning piss after waking up.It would hardly be worth a mention anywhere, but hey, it has only taken you7 + years to form a wrong page of nonsense (I dare not refer to it asproof).You are so full of yourself and don't even consider the possibility that youcould be wrong.Get a reality check. === Subject: : Re: parallelizability of manifoldsX-ID: VxsqABZeoe36TSsokXoidqV3Hzw8uWVzs05YvH5gPgsv52b7d+tZrV> There's not even the *beginning* of such a sequence of vectorfields> for S^2: no matter what the vectorfield V_1 on S^2, there is always> some point x of S^2 at which V_1(x) does not belong to any basis of> the tangent space of S^2 at x. That's just a (seemingly more> complicated, but ultimately worthwhile) way of saying that every> (continuous!) vectorfield on S^2 has at least one zero. And that,> in turn, follows (after building the appropriate machinery relating> differential topology to algebraic topology) from the fact that the> Euler characteristic of S^2 is non-zero.Indeed, algebraic topology seems to be an extremely powerful tool: S^2 issimply connected, so every continuous image of it also is; in contrast,R^2{(0,0)} is not. Realizing that this gives the proof for S^2 was apretty cool heureka-moment! Maybe it is possible to use higher homotopygroups for the n-spheres with n>2, but one has to be careful for n=7 :-)> In general, you could (and people do) ask, given an n-manifold M,> what is the smallest k such that there is some sequence of k> vectorfields V_i on M such that, at every point x of M, the> vectors V_i(x) span the tangent space of M at x. Alternatively> (somewhat dually) you could (and people do) ask, given a generic> sequence of k vectorfields V_i on M, what is the nature of the> subset of M at every point of which the vectors V_i(x) are> linearly dependent. If you pursue such questions, you eventually> find yourself studying characteristic classes (in homology as well> as in cohomology); the Euler class is just the tip of that particular> iceberg.Sounds very interesting, but I'm afraid that it is a lot ahead of me now...-- reverse my forename for mail! === Subject: : Re: Usenet Posting Guide?> Once I hoped that the growing popularity of personal computers meant> that nearly everyone would learn to think like programmers. It never> occurred to me that, sadly, the opposite would happen: That computers> would be designed to be used by people who *can't* program them.> That ought to be positive, it is called job security. I would shudder> when everybody at our institute thought like a programmer. You would> have umpteen versions of all software around, and hope you could in some> way couple the lot.Well, I wasn't talking *just* about programming computers. Primarily Imeant that I'd hoped that everyone would get so far into the programmingmindset that they'd approach everyday life as a sort of programmingexercise. It's much easier to deal with people who approach everythingin a series of logical steps and who don't introduce extraneous issues.-- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: : Re: {Group Theory} Confusing group theory conundrum> For instance, an algebraist says: We have a group G. Then it must> have the identity element G_e. But by saying this, she has> implicitly used such a function.> I don't see why she has implicitly used such a function.Well, let's look at an analog. We have a number N. Then it musthave the square N_s. Admittedly, this is rather peculiar terminologyand notation, but I did that purposefully to show how we can thusthrow a veil over the fact that we are, in this case, implicitly usingthe function f(x)=x^2. When we conjure up the identity of an abstractgroup, the situation is very similar, with the exception that there isno easy formula for the group's identity like there is for anumber's square.I admit I am probably much less experienced than you. Please know Ihave much respect for your advanced mathematical knowledge and, if youstill believe I am mistaken, I look forward to being corrected and seton the path toward increased understanding of these clandestinematters.Sniz Pilbor === Subject: : Re: Key Core Error Argument>So let me give you the example that I've used elsewhere which is>a_1(x) + 7, which has a constant term that is 7. Do you understand>what it means for it to be constant?But by your definition, the constant term of b(x) = a_1(x)+7 would bethe value of b(0) = a_1(0) + 7.Who says a_1(0) has to be 7? What if a_1(x) = sqrt(x^2+1) +0.5(x^3-5)?>Here's a test.>If I have x=11, then I necessarily have a_1(11) + 7, right?>Notice the constant term is STILL there, do you understand?Yes, but if a_1(0) is divisible by 7, e.g. a_1(x) = sqrt(x+49), whosays that a_1(11) is still divisible by 7?>Now then, if I now divide P(11) by 49, what should the constant term>be?I dunno. What's the constant term of sqrt(x+49)/7 + 7/7?What's the value at x=0? - RandyI'm trying to find an e-version of Church's On the Concept of a Randomhelp?gimma === Subject: : Re: Uncle Al is Sadistic .>>My kids (all 5 of them) have nightly and weekly chores. The older kids>>(15 & 16) each make dinner at least once a week. Occasionally my wife>>or I help the kids with some of the housework but they are expected to>>keep it clean.>>Side note: Last night a parent took one of the kids on my soccer team>>home early. The guy said that his son had to finish his homework.>>Right. The SOB needed a drink and couldn't manage to wait another 15>>minutes.>You and your kids have/had chores. These kids have jobs.>http://www.theatlantic.com/issues/96feb/pakistan/ pakistan.htm>not that they get paid.>One of the societal problems in the USA is that kids are not>allowed to be gainfully employed.> Well, of course if you want your six year old to support you, you are outluck. OTOH, kids older that 14 can work, and older than 12 in agriculturaljobs in the US. The rules, of course, vary from country to country.That you don't see them, is your problem.josh halpernLast I looked there were lots of kids working in lots of jobs.>/BAH> === Subject: : Re: Ex(~x=x), counterpart theory, and contingent identity> In another post I suggested that John Correy's ideas about non-reflexive> identity might have a rational formulation with respect to something> called counterpart theory.> (I think categoreal would be the right term here--in my humble> opinion, the apparent self-contradictions associated with John's> intuitions arise from vagueness and ambiguity associated with standard> presuppositions rather than irrational error on his part. For the> record, Langholm's investigation of determinability and> indeterminability in first-order contexts includes incoherent formulas.> Exclusion negation is not informationally well-behaved.)> The link,> http://www.sussex.ac.uk//Users/muralir/kct_final.pdf> <>((x=x) & ~(x=x))> is discussed as well as what is actually done in counterpart theory to> exclude such an incoherent result.> :-)> mitch> http://www.sussex.ac.uk//Users/muralir/kct_final.pdf> <>((x=x) & ~(x=x))> is discussed as well as what is actually done in counterpart theory to> exclude such an incoherent result.> :-)> mitch> The tie-in with contingent identity is as asserted by (1).> (1) AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))> That is, identical(x,y) are necessarily identical if, ond only if,> N(x=x) and N(y=y). Thus, granted that the Morning Star and the> Evening Star are each necessarily self-identical, if the Morning Star> and the Evening Star are identical, they are necessarily identical.> Hence the Morning Star and the Evening Star are necessarily> identical. The same is not true, however, for Benjamin Franklin> and the inventor of bifocals. For although Benjamin Franklin is> necessarily self-identical, the inventor of bifocals is not> necessarily self-identical. This is why Benjamin Franklin> and the inventor of bifocals, although identical are not> necessarily identical.> --John> I am not sure that my remarks are welcome here, I have had enough of the> remarks from (G. Frege)(II)> : stupid asshole, ing bitch, and all of that childish rhetoric, I cannot> reply to him at all.> I assume that you guys are more mature than He.> As to John's claim that: (John Correy = John Correy) is necessarily true, I> disagree.> It is not sufficient to say, x=y <-> AF(Fx <-> Fy).> Rather, it is sufficient to say, x=y <->. E!x & E!y &AF( Fx <-> Fy).> After all, (ix: x=John Correy) is (John Correy).> E!(John Correy) is just as doubtful as E!(The poster who claims that> ~Ax(x=x)).> We cannot assume that E!(John Correy) any more than we can assume> E!(Vulcan)!> (x=y) -> [](x=y) and E!x -> [](E!x), are consequences of Leibnitz's Law.> There are no contingent existences nor contingent identities.> Exisence and Identity (and Membership) are analytic properties.> Can we assume that: []Exists(George W. Bush)?> I don't think so, do you?> Surely, the existence of, George W. Bush, is contingent.> There cannot be an assumption that all names of purported physical entities> refer!> Santa dosen't work, Vulcan doesn't work, Pegasus dosen't work, etc.> For although Benjamin Franklin is necessarily self-identical, the inventor> of bifocals is not> necessarily self-identical.> Benjamin Franklin = Benjamin Franklin, is not necessarily true.> (the inventor of bifocals)=(the inventor of bifocals), is also not> necessarily true, imo.> Necessary identity and existence, applies to logical/mathematical objects,> not to emprical objects, imho.> Witt> (1) states necessary and sufficient conditions for the necessity of> (material) identity:> (1) AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))> If identicals x and y are necessarily self-identical, then--and> only then--is their identity a necessary one. Beyond this,> (1) states nothing more: from (1) it neither follows that> John Correy is necessarily self-identical nor that John> Correy is contingently self-identical--or indeed that John> Correy is self-identical at all. (1) does not say which of> the foregoing is the case.> We can sit around and argue about whether you or I are necessarily> self-identical. However, although logicians do argue about such> matters--Who else would bother?--it is not as logicians that they> argue but as metaphysicians, or as pataphysicians, or as what have> you?Good point. That is one of the reasons that I have always known that there is a philosophical element involved withdiscussions of identity.> So, when I claim that (e.g.) Benjamin Franklin is necessarily> self-identical while the inventor of bifocals is not, my main> warrant for this claim is that if Benjamin Franklin is necessarily> self-identical but the inventor of bifocals is not, then Benjamin> Franklin and the inventor of bifocals are not necessarily identical> (although they are identical). In other words, I take the necessary> self-identity of the former and the contingent self-identity of the> latter to constitute, together with (1), an *explanation* for the> contingent identity of Benjamin Franklin and the inventor of bifocals.> To this you might object that these would be also be contingently> identical if both Benjamin Franklin and the inventor of bifocals were> contingently self-identical. To which I would respond that identities> involving what linguistically oriented analytical philosophers refer> to as rigid designators, are identities whose terms are necessarily> self-identical; whereas identities involving what such philosophers> refer to as non-rigid designators, are identities whose terms are> contingently self-identical. Therefore, granted that I take rigid and> non-rigid designation as the linguistic marks of necessary and> contingent self-identity--putting the cart back behind the horse,> rather than approaching the matter bass ackwards as it is> fashionable to do these days--and granted that I take> Benjamin Franklin and John Correy to be 'rigid' designators> and 'the inventor of bifocals' to be 'non-rigid', I conclude> that the contingent identity of Benjamin Franklin and the inventor> of bifocals has as its basis the contingent self-identity of the> inventor of bifocals, while Benjamin Franklin is necessarily> self-identical.Personally, I still have some doubts about self-identical and not self-identical. But I am now fairly certain that ifone accepts Frege's argument for the definition of number as attaching to an object, one should logically accept thisdistinction.> As to whether physical or mathematical objects are contingently> self-identical or necessarily so, some sort of metaphysical argument> (rather than a logical one) warranting one or the other of these> conclusions would have to be made. I suspect that mathematical> properties are both essential in, and necessary to, mathematical> objects--but this is an intuition and nothing more.> It won't surprise me if Paul Holbach or G. Frege bring in talk> about 'scope', which I think is only peripherally relevant to> discussions of necessary and contingent identity.Well, when you claim that you can define scope so that self-identity is always implicit, you must deal with a Kantianpossibility--...The logical determination of a concept by reasonis based upon a disjunctive syllogism, in which themajor premiss contains a logical division (the divisionof the sphere of a universal concept), the minorpremiss limiting this sphere to a certain part, andthe conclusion determining the concept by meansof this part. --Immanuel Kant Critique of Pure Reason A577/605...--namely, the non-self-identical.Of course, the concept is still a fiction in Kantian epistemology. However, it is also the reason for the apparentcomplexity of his ideas--he does not trivially assert self-identity as self-evident.:-)mitch === Subject: : Re: Usenet Posting Guide?> Some of us prefer to view the forest rather than count the trees down> there The point was this, back in the days when you had to think before youposted, those who, to borrow your analogy, visited the forest actuallyappreciated it, and contributed to the ecology of the forest, rather thannow, where too many people use the forest as a place to hold their drug andalcohol parties, while thrashing the forest.-- Entities are not to be multiplied without necessity.--Bishop William William of Occam === Subject: : Re: Factorization dispute[all newsgroups except sci.math snipped] I have a result that shows a problem with a definition that> mathematicians have used for over a hundred years, and rather than> face the result which follows from rather basic algebra mathematicians> are being pussies and running like scared cowards from the result.No you don't. And nobody's running. You can't refute the errors othershave pointed out and have resorted to ad hominem attacks and cut and pastingof your entire argument assuming that if you repeat it often enough, thenpeople will fail to respond to one of you posts and you can then assume youmust finally be right.> Some posters, not professional mathematicians from what I've gathered,> are at least trying to stand and fight, but because what I have is a> math proof, their claims are necessarily irrational.Nonsense. Maybe if your proof was flawless you could claim something likethat, but you have a serious error in step 6 of your argument. (think aboutthis - *everybody* who has responded to you has pointed this out). Youassume that because you can divide the terms of your polynomial by 49 thatyou can show that two of the factors must be divisible by 7, but that's justnot true. You haven't come close to proving that.And in your new and improved short form argument you start with:> (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) => 300125 x^3 - 18375 x^2 - 360 x + 22You start off with your conclusion and work backward, proving what? Nothingat all. You *assume* that the variable terms of the first two factors aredivisible by 7. Nice trick.> I need you to stand up for the truth, here and now.Making appeals to the gallery again... I'm in the gallery, and I'll standwith the mathematicions on this one. You're argument is flawed.> Let's chase these mathematician cowards down, and make them face the> music.Hilarious. Make my music King Crimson's 21st Century Schizoid Man.> After all physics had its challenges and physicists faced them, while> mathematicians seem to believe that they can just ignore problems.HIlarious.> Let's take 'em out.To do that you will need to correct your errors (if you can...). I willecho the advice another poster gave you yesterday. Get a text on AbstractAlgebra and work through the theorems. Ask for help here when you getstuck. In a few months you will be a better amateur mathematician. You'vealready given 6-7 years to this effort. Taking a few months off foreducation will give you the potential for not wasting the next 7 years.--Stan Gula === Subject: : Re: Greek Alphebet by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876Jr16266;Back in the 1970's when I took a course in astrophysics, the professor told us that this was called a Cambridge pi. He didn't know why it was called that.---Steve Maye> G. A. Edgar [CapitalEth][EDoubleDot][Micro] .b3 .b9[EDoubleDot].b9> Here is a web page on Greek letters.> <http://www.math.ohio-state.edu/~edgar/GreekChartLarge.jpg nowadays still used in astronomy for perihelion.>Hmmm, so that explains _why_ some students in my elementary school and high>school were writing this symbol instead of pi on their theses.>Apparently it was used interchangeably with the regular pi. I have no idea>why.> -- > G. A. Edgar>http:// www.math.ohio-state.edu/~edgar/-->Ioannis Galidakis>http:// users.forthnet.gr/ath/jgal/------------------------------- ----------->Eventually, _everything_ is understandable === Subject: : Re: ELLIPTICAL INTEGRAL PROBLEM by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876Ju16262;I don't have the handbook,so could u please send me copy of the section 17 if have itthanxSamer>
 Hi All I have a problem in solving the following integrals> which
falls in to the form of elliptical integrals.> K = LImits
0-PI/2 Integral of 1/sqrt(1-k^2sin^2Theta) w.r.t Theta.> E =
Limits 0-PI/2 Integral of sqrt(1-k^2sin^2Theta) w.r.t Theta. Please help in finding the values of K and E> Sairam>
-------------------==== Posted via Deja News
====-----------------------> http://www.dejanews.com/
Search, Read, Post to Usenet>Sairam,>Look in Abramowitz and
Stegun, _Handbook of Mathematical Functions_ , >section
17.>John>
> Can someone point me to information on the web about fast factorial>> calculation? This needs to be for exact value. I can handle the>part>about>> it being too large to represent, but would like to find a faster>method>than>> just multiplying every number.> Adam,> Try this one http:// www.luschny.de/math/index.htm.> Could tell what kind of application are you working for?>It's labview. NI is sporting a contest to code the fastest factorial>program. I have a few ideas on how to calculate numbers larger than>representable in normal computer formatting, but wanted to see what>algorithms might I use that would be quicker than 2x3x4... Just for funs>tho.> An interesting property is that for n = 2m,> n! = 2^m (m!)^2>I used to think of myself as somewhat of a math wiz but I'm not getting this>equation. Cant seem to make it balance for sample numbers. In fact looking>at {n,m}={4,2}, I dont see how it could work at all. 4! contains a factor>of 3 which I dont see the right side coming up with no matter how I look at>it. Can you clue me in as to what I am missing?As submitted it only works for m = 0 or 1(2m)! = 2^m(m!)^2 = 2m(2m-1)! = 2^m*m(m-1)!*m(m-1)! = (2m-1)! = 2^(m-1)*m(m-1)!*(m-1)!This implies 2m-1 divides m or m-1 true for 0 and 1 only since m>1would make 2m-1 > m > m-1 or 2m > m+1 > m. Integers a > b => a doesnot divide b. === Subject: : Re: Logic Question by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876Lj16323;>How can one use the numbers 1through 9 only in a 3 digit plus three digit problem and come up with a 3 digit answer without using any of the numbers again?>I am stumped...and so are all of my other master's level friends! Makes us wonder how we got through grad school! Please help!Gosh Genie, I know the anwer to that question. You answer mine and I'll answer yours. How about it? Does an MBA count? Or is it just your speechy buds? Grad school sounds good, but sometimes it is just a bunch of crapola-huh? === Subject: : Re: The crazy counterfeit coin problem strikes again! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876Jm16270;What if one doesn't need to find which blocks are Ds? How many less weighings will it take? === Subject: : Re: lengths of segments in a pentagram....... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876IX16250;>i'm trying to find James Choike's modern proof that there are ratios>of lengths of segments in a pentagram that are irrational. i know >the proof was published in the college of mathematics journal in >1980 on pg 312-316 and using the school library is out of the >question because we only carry the journal as far back as 1984, can >anyone point me to a website or maybe another mathematics book that >may lead to this proof. i would like it to describe it to a >class....>thanks.>SLHyou should have a look athttp://www.maa.org/pubs/cmj.html.Hope it helpsM. Doerfner === Subject: : Re: Fulton/Harris Representation Theory good? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876K516295;I would use Humphreys as the primary text (Springer GTM 9), with Fulton / Harris for additional examples and some interesting tangents. === Subject: : Re: Who contributed most to mathematics? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876Jb16283;Hi John,Thank you and others for the replies. I did a little work on this and have it below. I admit the political statement was offbase and should have not have been included in my origional post. I'm pissed at theFrench Bashing. I was just trying to see who in fact borned the most mathematicians. My SWAG list was not intended to be complete but just a start. Ok, I blew it after France at least according to MacTutor link below. However I was close with Italy, Germany and former Russia but totally underestimated England and the U.S. and way over estimated India and Greece. This is a quantative analysis. The next step is qualitativeanalysis which is why I asked you guys the question.My SWAGItaly Germany Russia India Greece England JapanChinaUnited StatesEgyptHungaryAustriaSerbia>There sure have been a lot of important French mathematicians;>Fourier, Galois, d'Alembert...Fermat - but his last theorem was published after his death. So during his life time it did not play a major role in mathematicaldevelopement. The same can be said of Da Vinci and other Greats whosecertain works were suppressed during their lifetime.>but the Greeks started it.Just about every thing the Greeks did was rediscovered by othersas it was needed in their time. Much of Archemedies work was for War efforts as they were needed. Sure they have being first name recognition but played no more role than India and other countries to thought today.>However, I really suspect that the Germans have to take top honors.>And thanks to just *two* guys, even though there were other German>mathematicians.Germans are third in the list below.>Euler and Gauss.I wonder how much of these guy's work was invented or discovered by>John SavardThis was taken from MacTutor on Mathematicians by country. You can visit the site below to get a good biography of some 1468 mathematicians from 350 A.D. to 1997.http://www-history.mcs.st-andrews.ac.uk/history/ index.htmlI delimited it in Excel and copied below. It is interesting that France ranks no. 1. However, if one placed a weighting on each member, a different first place might occur. For example Sir Issac Newton would pull heavily for England. Then again if you add Scotland=44,Ireland=18,Wales=5 Great Britain would take the helm. Similarly, we can combine Russia and the Ukraine to get 118 surpassing Italy and the United States. Then if we look at just Europe, we will have 1163 or 80% of the total. Then we can do itper capita, by country etc. Just joking!Also interesting is that Iraq produced more than China or Japan.This is probably due to the non-availability of written documentation in China and Japan and the pictographic nature of their languages. Perhaps the arabic derived number system we use today played a great part in the evolution of mathematical thought. I suspect there is a lot more that can be learned from this list by simply asking questions as the ones above. China 10? There was no time for math, only for defense.the wall and agricultural subsistence. The math that was done was probably kept secret and destroyed if necessary so the enemy could not use it against them. France,England 415?Sailing ships to conquer the world required mathematical skills soit was an important for the educational system to nurture future navel officers and war planners. War (hot and cold) was probably the single greatest influence on mathematical progress. This is unfortunate. I think the numbers below if done by year would bear that out. Conquering countriesdemand mathematicians and produce them. Some of you were taken aback by not seeing certain countries on mySWAG list. Please do not be offended by the list below. That is allit is - just a list of Mathematicians by place of birth. It is notby religion or genetics. Isreal shows 2 but Poland and Germany hadmany many Jews which borned many mathematicians in those countries.CinoThe straight list of the number mathematicians by place of birth upto 1997Algeria 1Azerbaijan 1Argentina 1Gibraltar 1Haiti 1Indonesia 1Jordan 1Malta 1Mexico 1Moldavia 1Slovenia 1Tajikistan 1Morocco 1Estonia 2Georgia 2Israel 2Lebanon 2Libya 2Lithuania 2Luxembourg 2New Zealand 2Pakistan 2South Africa 2Croatia 3Portugal 3Uzbekistan 3Latvia 4Australia 5Finland 5Romania 5Slovakia 5Syria 5Wales 5Canada 7Belarus 8Japan 9China 10Norway 10Iran 11Spain 11Sweden 11Czech Republic 13Egypt 13Iraq 17Denmark 18Ireland 18Belgium 19Turkey 22Greece 25Netherlands 25Switzerland 27Austria 28Hungary 29Ukraine 37India 39Scotland 44Poland 68Russia 81Italy 101USA 108Germany 162England 204France 211Total 1462 === Subject: : Re: Uncle Al is Sadistic .>>Sure, the I don't fail you you fail you bull. He didn't>>educate anyone either. He didn't go out and get more>>resources for more students.>When you run a race, you don't automatically award first>place to every runner.> There were six hundred places. What is the difference between the 599th> person in that class and the 601st in terms of ability? The wonderprof> simply abdicated any responsiblity for educating the students. Quite> common.Exactly. That is why a rational society only cares about win, place,and show rather than compassionately about a huge pool ofincompetents. -- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The Net! === Subject: : Re: Factorization disputeNothing. http://www.crank.net/harris.html It's not every braying jackass that gets a whole page at crank.net-- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The Net! === Subject: : Re: Uncle Al is Sadistic .> Exactly. That is why a rational society only cares about win, place,> and show rather than compassionately about a huge pool of> incompetents. Not necessarily incompetent, but folks of middling ability, as most of us are. Remembers it is the grunts who provide the ballast to keep the ship stable in the water. It is the general labor force that make it possible for capitalists to be rich (in addtion to the genius of the capitalist).On a desert island an enterprising capitalist will survive quite nicely, but he will not be able to build an empire unless he can have it done by Friday.We all have some part to play.Bob Kolker === Subject: : Re: Ex(~x=x), counterpart theory, and contingent identity> c37480a7.0311072209.7de21310@posting.google.com>...> (1) states necessary and sufficient conditions for the necessity of> (material) identity:> (1) AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))> If identicals x and y are necessarily self-identical, then--and> only then--is their identity a necessary one. Beyond this,> (1) states nothing more: from (1) it neither follows that> John Correy is necessarily self-identical nor that John> Correy is contingently self-identical--or indeed that John> Correy is self-identical at all. (1) does not say which of> the foregoing is the case.> We can sit around and argue about whether you or I are necessarily> self-identical. However, although logicians do argue about such> matters--Who else would bother?--it is not as logicians that they> argue but as metaphysicians, or as pataphysicians, or as what have> you?> So, when I claim that (e.g.) Benjamin Franklin is necessarily> self-identical while the inventor of bifocals is not, my main> warrant for this claim is that if Benjamin Franklin is necessarily> self-identical but the inventor of bifocals is not, then Benjamin> Franklin and the inventor of bifocals are not necessarily identical> (although they are identical). In other words, I take the necessary> self-identity of the former and the contingent self-identity of the> latter to constitute, together with (1), an *explanation* for the> contingent identity of Benjamin Franklin and the inventor of bifocals.> To this you might object that these would be also be contingently> identical if both Benjamin Franklin and the inventor of bifocals were> contingently self-identical. To which I would respond that identities> involving what linguistically oriented analytical philosophers refer> to as rigid designators, are identities whose terms are necessarily> self-identical; whereas identities involving what such philosophers> refer to as non-rigid designators, are identities whose terms are> contingently self-identical. Therefore, granted that I take rigid and> non-rigid designation as the linguistic marks of necessary and> contingent self-identity--putting the cart back behind the horse,> rather than approaching the matter bass ackwards as it is> fashionable to do these days--and granted that I take> Benjamin Franklin and John Correy to be 'rigid' designators> and 'the inventor of bifocals' to be 'non-rigid', I conclude> that the contingent identity of Benjamin Franklin and the inventor> of bifocals has as its basis the contingent self-identity of the> inventor of bifocals, while Benjamin Franklin is necessarily> self-identical.> As to whether physical or mathematical objects are contingently> self-identical or necessarily so, some sort of metaphysical argument> (rather than a logical one) warranting one or the other of these> conclusions would have to be made. I suspect that mathematical> properties are both essential in, and necessary to, mathematical> objects--but this is an intuition and nothing more.> Best regards,> John> PS It won't surprise me if Paul Holbach or G. Frege bring in talk> about 'scope', which I think is only peripherally relevant to> discussions of necessary and contingent identity.Let me bring in a quote:One must distinguish between the claim that identity sentences arecontingent and the claim that the identity relation itself iscontingent. For the relation to be contingent, there need to be thingsbetween which it holds merely contingent. For it to be necessary, ithas to be that if the relation obtains between things, it obtainsbetween those very things of necessity. [...] One can consistently saythat there are contingent identity sentences, though the relationitself is necessary. Thus one could say that The firstPostmaster-General of the US was the inventor of bifocal lenses. iscontingent and is an identity sentence, but that if we consider theobject, x, which is in fact referred to by the firstPostmaster-General of the US and the object, y, which is in factreferred to by the inventor of bifocal lenses, it is necessary thatx is identical to y.[Sainsbury, M. (1995). Philosophical Logic. In A. C. Grayling (Ed.),/Philosophy. A Guide Through the === Subject: / (pp. 61-122). Oxford: OxfordUniversity Press. (p. 93)]PH === Subject: : Re: Uncle Al is Sadistic .Littlemanwearingbigboypants whines:>Sure, the I don't fail you you fail you bull. He didn't>educate anyone either. He didn't go out and get more>resources for more students.>>When you run a race, you don't automatically award first>>place to every runner.>There were six hundred places. What is the difference between the 599th>person in that class and the 601st in terms of ability? The wonderprof>simply abdicated any responsiblity for educating the students. Quite>common.> Exactly. That is why a rational society only cares about win, place,> and show rather than compassionately about a huge pool of> incompetents. In your form of rational society you'd have been recycled along time ago.You're redefining rational society to suit your rather warpedfundie worldview. A successful society can afford to care aboutall its members. A successful society *is* rational. === Subject: : Re: Uncle Al is Sadistic .> 3 Africans I met in Computer science department in the last 4> years were way above average in thier programming skills in> the midst of Chinese and Indian grad students who are the> overwhelming majority in that department.> You will always find bright smart people in just about any > naturally> occuring group of humans. Race is nothing. Culture is everything.> Bob Kolker> That's my point.> What point? Culture Shmulture....that is simplistic politically > correct> heurist bull of the first kind. >No, it is not.>> If it were that simple > It is THAT simple where I grew up. >Children in the poor neighborhood with their parents making ends meet>>and the children having to do the house work like an adult and start>>working while they themslevs are children (rather than play or study,>>do not get a chance to test their brain since there is no books (not>>referring to school text books and notes) lying around the house.> That's a lot of bull. House work (or farm work) is so boring>> one has nothing else to do but think. It also gives one an incentive>> to go to school and study real hard so that one can get a job that>> doesn't involve either. > Afriend of mine from Wisconsan said justt hat.> He had to get up and helped his father milk the cows in the farm and>it was boring for him. He became a programmer. It was possible for him>because he is the this US of A.> A country which was built by people. Nobody gave them anything;> they did the work themselves. Since you want us to believe that> you are from an underprivileged country that contains no books,> no schools, no business opportunities, I suggest that you begin> to work to build an infrastructure. However, since you're complaining> about having to do housework from dawn to dusk, I conclude that> there exist people who are able to hire other people to clean.> That implies that they make the money they pay those housecleaners> in some other industry than housecleaning. Thus, that country> has opportunities available for thousands.Can you read well? When did I ever say I had to work from dawn to dusk? First of all, Iwasn't talking about myself. Secondly, here is how my day started.Get up, brush my teeth and wash my face. Have breafast which wasalready on the the table. Get ready for school. Grab my lunch box andgo wait for school bus which arrives around 8:20Am.School starts at 9:15 and releaseed at 3:15. School bus brought me home. Change my clothes. Have some snack andwait for my Math tutor (not everyday; in that case, I did my Mathhomework, the only homework I was keen to do). Have dinner around 6.Then , it varies how my day ends. That was 6th grade. In 7th garde,my Math tutoring moved to 7Am till 8:30 in the morning and so I had tocatch the school bus when it came around the senodn time by walking toa friend's house on the next block. In 8th grade, it was after school.So on.Now, say that I was spoiled or privileged. But notice that therewasn't extracurricular activites sponsored by school like during myolder siblings time. They went to Private schools (founded duringBritish time) which did not exist anymore in my time. That's becausethere was no budget for it except for short term ones, which I joinedsome times.I did learn swimming one summer at a private place. But before 7thgrade, i.e when we were not old enough to be rebellious, in summer, mymother would teach us.With college education during my time, unlike my oldest sister's time,I would not qualify to be considered privileged. I wasn't complainingabout that.About your telling me I suggest that you begin to work to build an infrastructure., I tell you this: stay withinthe context of diuscussion and ... ..do not act so dumb not to know that there are countries wherepeople are not free to do (build) their infrastructure. But mostly,just stay within the context, will you?>> Housework is a very good lesson in >> eliminating some choices of employment. >Not for those who is left with no time to study when it is not just>helping around the house with chores.> Excuse me? Housework has ample opportunity for study. Since it> takes no thinking (or not much) to vacuum or wash a floor, one> can use their brains to learn at the same time. >I would set a book> up, read a paragraph, and think about it while doing other work.> If I still hadn't got it, I would read it again and continue doing> the physical work. When I figured it out, I would read the next> part of the text. In the computer biz, we called this flavor> of doing many things at once, timesharing. Well..I never was so keen to study like a bookworm. In fact, only oneof my siblings was. But I happened to pay attention during lectures(and when my mother taught us) and that was mostly enough for me. Myhobby was to read all kinds of books - there happened to be many booksaround the house brought by my much older brothers, cousins, etc. -that are not related to my school work.I was about 10 or 11 when I saw that book, the Biography of Peron(translated version). I advanced from reading story books to suchbooks though I didn't really understand all of it. There were manysuch books (my oldest brother was keen on politics ) and other kindsof books. Also, my next door neighbor was the president of the office- I don't know how to traslate it - where all books get sensoredbefore they get published. So I had access to those books, cartoons,etc. he brought home since their children and us were very close.Going back to the context, ...the point was that I was NOT talking about me. I was talking aboutthe kids from the poor family. Do you know how poor family lives in3rd world countries? Obviosuly not.>>It also gives one a fallback>> option for employment.> The curious will find a way to satisfy their itch. A good resource>> is^Wwas public libraries. The curious also have to learn how to avoid>> spoon-feeding or they will lose that precious commodity.> Public libraries? You seem to think that all countries have good>public libraries.> This thread was talking about US schools and such.No it was not. It was about Asians. > All countries may not have 100% good public libraries. Ours> sucks (and it's in the USA). If you have no access to any> library, go buy a text. If you have no money to buy a text,> barter with a professor (or somebody who does have books)> to clean his/her house in exchange for a couple of hours> access to his books every week.For cripe's sake, I can see that there is no point in explaining toyou. You are a typical avergae American, who doesn't know much aboutbeyond your border. Sad! Very sad! > Get a job at your country's university. Now you can even talk> to people who are learning; I used to sop up lots and lots of> knowledge just by listening to people talking about their work.> If the country is in an industrial stage, get a job at a factory.> Lots of physics, mechanics, efficiency studies, chemistry, math,> accounting, management, etc. can be learned just by noticing> how the company does its biz. Again..GEE!!!!!!!!!!> You are a troll.> Not more than you are.> I may be a troll, but I'm one that can figure out how to get> something done with no resources. It's called work.I am very sad for you that . In fact, Josh halpern's reply was enough. I shouldn't even havebothered to reply to you. You were so way out of the context of thetopic in discussion.> /BAH> Subtract a hundred and four for e-mail. === Subject: : Re: Uncle Al is Sadistic . charset=iso-8859-1>When you run a race, you don't automatically award first>place to every runner.> There were six hundred places. What is the difference between the 599th> person in that class and the 601st in terms of ability? The wonderprof> simply abdicated any responsiblity for educating the students. Quite> common.> Exactly. That is why a rational society only cares about win, place,> and show rather than compassionately about a huge pool of> incompetents.> Uncle AlI wouldn't insist too strongly on your idea, Al. The treatment of theincompetents has to be cared about by society, otherwise they,or many of them, will come to your house and tell you to beat itand kick your ass out without thought nor regrets.I sympathize with you that things should be as simple as you say.But it just isn't in the cards. If it were that straight forward thenthe solution would have been found long ago and would have been practiced all over the place. Utopia would be at hand.hanson === Subject: : Re: Usenet Posting Guide? Well, I wasn't talking *just* about programming computers. Primarily I> meant that I'd hoped that everyone would get so far into the programming> mindset that they'd approach everyday life as a sort of programming> exercise. It's much easier to deal with people who approach everything> in a series of logical steps and who don't introduce extraneous issues.Approach everything in a series of logical steps and don't introduceextraneous issues?Doesn't sound like the programmers I know. Certainly doesn't soundlike the programmers I've been.-- [T]here's no point in telling any of you what mathematicians I'm inemail contact with, just like there's no point in going into detailabout my contacts in a major news organization. [...] [P]olitedisinterest is what I've found. --JSH on his important contacts. === Subject: : Re: Uncle Al is Sadistic .> Littlemanwearingbigboypants whined:> Exactly. That is why a rational society only cares about win, place,> and show rather than compassionately about a huge pool of> incompetents. > Not necessarily incompetent, but folks of middling ability, as most of > us are. Remembers it is the grunts who provide the ballast to keep the > ship stable in the water. It is the general labor force that make it > possible for capitalists to be rich (in addtion to the genius of the > capitalist).> On a desert island an enterprising capitalist will survive quite nicely, > but he will not be able to build an empire unless he can have it done by > Friday.> We all have some part to play.It pays to remember that drones won't work for vinegar.> I'm trying to find an e-version of Church's On the Concept of a Random> help?> gimma === Subject: : Re: Why is math so difficult for some people?> Similarly, Cramer's proof of the Levy-Cramer theorem that> the sum of two independent random variables is not normal> unless they both are is probably the only easy proof, but> it likewise obscures the ideas. Any time characteristic> functions are used to prove a probability theorem, the> concepts are not even present.>Do you happen to know of a probabilistic proof of the Cramr-Lvy >theorem?There are entropy proofs. I am not up on all the available proofs, but I doubt there can be a fullyprobabilistic proof. -- 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: (765)494-6054 FAX: (765)494-0558 === Subject: : Re: division by general integer using register shifts>You're in trouble. Distributivity works for multiplication but>it doesn't for division.> n/(a+b) /= n/a + n/bYou're right. In fact, the characteristic properties areAxiom 1: a/(b/c) = c/(b/a),Axiom 2: a/(a/b) = b,Axiom 3: (a+b)/c = a/c + b/c.>Use> n/(a+b) = 1/(a/n + b/n) = 1/ [(1/n)(a + b)]>Twice use a fast reciprocal algorithm and multiplication shifting.In fact, n/(a+b) = z/(z/(n/(a+b))) by Axiom 2 = z/((a+b)/(n/z)) by Axiom 1 = z/(a/(n/z) + b/(n/z)) by Axiom 3and different z's other than 1 may be more useful.Other properties that follow are: (a/c)/(b/c) = c/(b/(a/c)) = c/(c/(a/b)) = a/b (a/b)/(a/c) = c/(a/(a/b)) = c/b a/(b/b) = b/(b/a) = a (a/a)/(b/c) = c/(b/(a/a)) = c/b a/(b/(c+d)) = (c+d)/(b/a) = c/(b/a) + d/(b/a) = a/(b/c) + a/(b/d) a/a = b/(b/(a/a)) = b/(a/(a/b)) = b/brespectively by Axioms 1,1,2; 1,2; 1,2; 1,(1,2); 1,3,1-twice & 2,1,2.> If you saw> (c_1 x + 7)(c_2 x + 7)( c_3 x + 2) => 49(x^3 + 5x^2 + 3x + 2)> with the c's algebraic integers, I think few of you would have a> problem realizing that only two of the c's have 7 as a factor.> But, of course, you're looking at *functions* of x, as you have> f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x,> so I could also write it as> (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 2) = 49(x^3 + 5 x^2 + 3x + 2).> Notice that dividing both sides by 49 gives> (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 2) = x^3 + 5 x^2 + 3x + 2> as long as you're in a ring where 7 is not a factor of 22.> I want to emphasize that point as notice there's only *one* way to> divide through by 49 if 7 is not a factor of 22.> Usually you can *see* the other factors of 7, but I want you to> abstract, and generalize.> Please pay careful attention to that example.OK. Let's follow through (paying careful attention). The example you gavehas solutions for the constants c_1, c_2 and c_3 as follows:c_1 = 4.45887 - 9.4089Ic_2 = 4.45887 + 9.40789Ic_3 = 0.452072Hence, they are *not* functions of 'x'. It doesn't add anything to rewrite'c_1 x' as 'f_1(x)' since it only consists of a constant times 'x'.Conveniently, if you set 'x' to zero, the equation you gave reduces to:(7)(7)(2) = 49*2 = (7)(7)(2),which is all very tidy.But if you set 'x' to 1, the equation becomes,(c_1 + 7)(c_2 + 7)(c_3 + 2) = 49*11 = (7)(7)(11)and, in view of the values of the 'c's,c_1 + 7 = 11.45887 - 9.4089Ic_2 + 7 = 11.45887 + 9.40789Ic_3 + 2 = 2.452072and we have:(11.45887 - 9.4089I)(11.45887 + 9.4089I)(2.452072) = (7)(7)(11)How do the factors distribute here?You have demonstrated a plausible partitioning for the case 'x' = 0, butyou suggest that your solution holds for all 'x'. If so, you should beable to explain what goes where for the case 'x' = 1, or any other value.Since the objections posted by your critics specifically challenge thegeneralization from the case x = 0 to all 'x', this is a crucial point.Please address it.--There are two things you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com === Subject: : Re: Uncle Al is Sadistic .> ..do not act so dumb not to know that there are countries where> people are not free to do (build) their infrastructure. But mostly,> just stay within the context, will you?Awwwwwww. Where's the fun in that?> For cripe's sake, I can see that there is no point in explaining to> you. You are a typical avergae American, who doesn't know much about> beyond your border. Sad! Very sad!Farm life and computer room, in fact.>You are a troll.>>Not more than you are.>I may be a troll, but I'm one that can figure out how to get>something done with no resources. It's called work.> I am very sad for you that . > In fact, Josh halpern's reply was enough. I shouldn't even have> bothered to reply to you. You were so way out of the context of the> topic in discussion.Now that you've figured it out......? === Subject: : Re: Decimal Expansion> If m contains the digits 2, 5, 1 consecutively. This is an old math contest32/127 = .2519685039... === Subject: : Ideals & LatticesHi everyone,I need some examples of a _prime ideal_ for an infinite lattice.TIA === Subject: : Re: ELLIPTICAL INTEGRAL PROBLEM> I have a problem in solving the following integrals> which falls in to the form of elliptical integrals.> K = LImits 0-PI/2 Integral of 1/sqrt(1-k^2sin^2Theta) w.r.t Theta.> E = Limits 0-PI/2 Integral of sqrt(1-k^2sin^2Theta) w.r.t Theta.> Please help in finding the values of K and EThese are definitions of the complete elliptic integrals K and E. Theyare not elementary functions of k. So what values do you want?table for E(k) from Maple: E(0.0) = 1.570796327 E(0.1) = 1.566861942 E(0.2) = 1.554968546 E(0.3) = 1.534833465 E(0.4) = 1.505941612 E(0.5) = 1.467462209 E(0.6) = 1.418083394 E(0.7) = 1.355661136 E(0.8) = 1.276349943 E(0.9) = 1.171697053 E(1.0) = 1. === Subject: : Re: Uncle Al is Sadistic .> By the time I saw your response to my other post, I have already>replied to this post of yours. I would not have otherwsie.>Put it in your think head. You cannot demand; you can only request.> HUH?!Did you have to use the word 'f***ing'? You could have asked What countries were they from?> /BAH> Subtract a hundred and four for e-mail. === Subject: : Re: Key Core Error Argument> If y is not 0, you can't use the constant term tricks that depend> on y> being 0.> That's stupid. The constant term once found is distinguished by being> constant.> All the trick is doing is finding it.> So let me give you the example that I've used elsewhere which is> a_1(x) + 7, which has a constant term that is 7. Do you understand> what it means for it to be constant?> Here's a test.> If I have x=11, then I necessarily have a_1(11) + 7, right?> Notice the constant term is STILL there, do you understand?> Now then, if I now divide P(11) by 49, what should the constant term> be?> If you answer honestly I'll be shocked.> Let me try again. (a_1(y) + 7)/x is not the same as 7/x, unless a_1(y)> = 0.> Ok, I can go from there Richard Henry, and note that necessarily> a_1(y) has *some* factor in common with 7, right?> Why?Because a_1(y) + 7 has a factor in common with 7, right? > So let's call that factor f, now dividing 49 from P(y) will divide f> off from a_1(y) + 7, understand?> No. Why?Look at my previous answer Richard Henry.> Now then, you have a_1(y)/f + 7/f, and if f does not equal 7, what> does that tell you about the *constant* term Richard Henry?> I do not understand the question. Please rephrase it.If the constant term of a_1(y) + 7 is 7, since a_1(0) = 0, and now youdivide through by some number I've called f, what is the constant termof the new expression?> Necessarily, you have 7/f left as the constant term for a factor of> P(11)/49, and if 7/f does not equal 1, you have a contradiction.> I don't follow your logic to say necessarily.Well I rephrased above so maybe that has changed, has it RichardHenry? > Understand?> No.> The trick is to FIND the constant term, as you know that it's not> affected by the value of x, and it sits there like a rock, unaffected> by the value of x, and none of your protestations against mathematical> reality will change that fact.> The constant term you refer to is found by zeroing out the other terms by> setting x to zero. When x is not zero, and those other terms are not> necessarily zero, you can not always extend the properties found for the> constant term to the entire expression.The property of the constant term is being constant because it's anumber.For instance, with f(x) = x+ 2, the constant term is 2, as notice thatat x=0, f(0) = 2, but the constant term just sits there like a rock,as you vary x, do you understand that Richard Henry?So you can't assume that if you change x the constant term changes,understand?James Harris === Subject: : Modelling the movement of an electro mechanical device. I have a servo system, which moves through a certain angle, depending on theamount of voltage applied to it's input.The response of the movement from when the control signal is input, untilthe armature actually reaching the desired position varies as the deviceoperates through different angles. This can be measured over a range ofangles and frequencies.Does anyone have any suggestions as to what route I should be looking totake if I wanted to produce a mathematical model of the device, so that itis possible to predict the movement of the device?Many thanksJD === Subject: : Re: parallelizability of manifolds||>> Surely the simplest way to understand the concept of parallelizability|>> is this: the n-manifold M is parallelizable if and only if there exist|>> n vectorfields V_i on M with the property that, at each point x of M,|>> the n vectors V_i(x) are a basis of the tangent space of M at x.|>> No imagination required.|>> |>Ah, that's interesting! So why are there no such vector fields for S^2?||There's not even the *beginning* of such a sequence of vectorfields|for S^2: no matter what the vectorfield V_1 on S^2, there is always |some point x of S^2 at which V_1(x) does not belong to any basis of |the tangent space of S^2 at x. That's just a (seemingly more|complicated, but ultimately worthwhile) way of saying that every|(continuous!) vectorfield on S^2 has at least one zero.you should at least mention something about hedgehogs or coconutshere.-- [e-mail address jdolan@math.ucr.edu] === Subject: : Re: Two questions in propositional logic>[...]>2. Let C be a set of propositions. We say C is a chain if for every p, q >in C either p proves q or q proves p (and not both). If the set of >primitive propositions is allowed to be uncountable, can there be an >uncountable chain?>>I doubt it. Say p < q if p |- q but not q |- p; then your chain is>>totally ordered by <. First, either there exists an increasing>>sequence with at least two upper bounds>This is presumably under the assumption that C is uncountable?> Yes.>>or a decreasing sequence with at least two lower bounds>>(by the traditional proof that any sequence of reals contains>>a monotone subsequence: Say you have elements p_a, indexed by the>>countable ordinals. Say a is dominant if p_a > p_b for all b > a.>>Either there are uncountably many dominant a or not. If there>>are uncountably many dominant a then the p_a with a dominant>>give a cofinal decreasing set, while if there are only countably>>many dominant a then the dominant a are bounded above>>and you get a cofinal increasing subset.) So, changing the>>notation and replacing all the wffs by their negations if>>needed, you have> p_1 < p_2 < ... < q_1 < q_2.>>It seems clear to me that this implies q_1 must be a>>tautology (it seems like this is by compactness or>>something, but I can't quite prove it this second)>>and then q_2 is weaker than a tautology, contradiction.>>???>Unless you're using some extra property about the p_i, q_i in saying >this that I'm not seeing > No.>(which is very possible, on account of it being >Early here and I just got up. :), that needn't be true.>For example take the following sequence:>p_3, p_3 v p_4, p_3 v p_4 v p_5 ... , p_1 v p_3 , p1 v p_2 v p_3.> ??? Is it true that p_3 v p_4 |- p_1 v p_3 ? I don't think so...Gah. You're right. I was trying to construct that counterexample from memory way too early in the morning and it didn't work. :)What it is *meant* to say is the following:p_1 ^ p_3, p_1 ^ (p_3 v p_4), ... p_1, p_1 v p_2If you're interested, John's proof went roughly as follows:> (Sorry if there's something below I should have replied> to that I'm appearing not to notice - you've forced me to> stop reading at this point..)> [snipped with eyes shut]Heh. No problem. :) There wasn't anything below it except a sketch of John's proof and my possible proof approach based loosely on what he did.David(E-mail address spam-blocked in the obvious way) === Subject: : Re: Two questions in propositional logicAnyway, as I said in the other post I think I've got this sorted out now >(although I'll want to write my own proof of it before I'm fully happy >with it).>If you're interested, John's proof went roughly as follows:>Define an equivalence relation on C by p ~ q if there is a bijection f >from the set of primitive propositions to itself, such that q = p with >all the p_i in p replaced with f(p_i).>He then showed that there can be at most one element of each equivalence >class in C, and that there are countably many equivalence classes >greater than p (because you can take some countable subset of the whole >set of primitive propositions and represent every formula in C as a >formula using only this countable subset and the primitives that appear >in p).>He then went on to show there were only countably many elements less >than p in a similar manner. I pointed out that this was rather easier, >as you could just reverse the chain by negation and use the previous >result.> I found this the most natural approach. However (and IBL will probably> kill me for saying anything at all about this sheet in public) you> should also be aware there is a one word answer to this question. That> is, there is one particular word (familiar to everyone) which> instantly makes this question a triviality. I'll leave you to guess> what it is.Hmm....Oh hell. Is it the one-word that you always use to prove obvious statements? It is, isn't it... ::sketches proof in his head:: Excuse me while I go sulk.(To anyone reading this who doesn't understand that line, it's a Leaderism. :)> By the way you didn't ask about the last question (if the set of> primitive propositions is allowed to be uncountable is it true given a> set S of propositions that you can find an independent set of> propositions equivalent to S); does that mean you've solved it? I> still don't have it, one year after taking the course. No hints> please!Well... yes and no. I thought I had a solution. There was a tiny problem in it contained right at the end which I think I can fix (but it may be a way bigger problem than I thought it was. :)That being said, my solution to the last question was only one line. My solution to the previous question did not assume countability. I was lazy and didn't feel like proving it twice...David(E-mail address spam-blocked in the obvious way) === Subject: : Re: Two questions in propositional logicUnless you're using some extra property about the p_i, q_i in saying >this that I'm not seeing (which is very possible, on account of it being >Early here and I just got up. :), that needn't be true.>For example take the following sequence:>p_3, p_3 v p_4, p_3 v p_4 v p_5 ... , p_1 v p_3 , p1 v p_2 v p_3.> Pointed out a few minutes ago that this is not a counterexample.> Of course what I said is wrong - a counterexample is> p_1 & (p_2), p_1 & (p_2 v p_3), ... p_1.Oops. Should have read this post before my last one. That's exactly the counterexample I had in mind when I made the (wrong) counterexample this morning, I was just half asleep and misremembering. :)> (Realized I wanted to show that the intersection of a strictly> decreasing sequence of open subsets of the Cantor set had> empty interior. Realized that that was false. Now, I bet> that there's no strictly decreasing omega_1-sequence of> subsets of the Cantor set, but never mind, the proof you> already have is simpler.)Are you sure that the cantor set approach works? Also, do you mean {0,1}^X for some possibly uncountable X rather than the cantor set? (which is homeomorphic when X is countable). Because I tried that, and I couldn't come to the conclusion that you could get the association to work both ways - you could get a clopen subset of {0, 1}^X for every proposition, but I wasn't sure you could go the other way... it looked like if you chose some sufficiently nasty clopen set it wouldn't correspond to a proposition. (I didn't prove that you couldn't, but it looked like it was going to go badly wrong if I tried to prove that you could, so I gave up on that approach).David(E-mail address spam-blocked in the obvious way) === Subject: : Re: Uncle Al is Sadistic .> 3 Africans I met in Computer science department in the last 4> years were way above average in thier programming skills in> the midst of Chinese and Indian grad students who are the> overwhelming majority in that department.> You will always find bright smart people in just about any > naturally> occuring group of humans. Race is nothing. Culture is everything.> Bob Kolker> That's my point.> What point? Culture Shmulture....that is simplistic politically > correct> heurist bull of the first kind. >No, it is not.>> If it were that simple > It is THAT simple where I grew up. >Children in the poor neighborhood with their parents making ends meet>>and the children having to do the house work like an adult and start>>working while they themslevs are children (rather than play or study,>>do not get a chance to test their brain since there is no books (not>>referring to school text books and notes) lying around the house.> That's a lot of bull. House work (or farm work) is so boring>> one has nothing else to do but think. It also gives one an incentive>> to go to school and study real hard so that one can get a job that>> doesn't involve either. > Afriend of mine from Wisconsan said justt hat.> He had to get up and helped his father milk the cows in the farm and>it was boring for him. He became a programmer. It was possible for him>because he is the this US of A.> A country which was built by people. Nobody gave them anything;> they did the work themselves. Since you want us to believe that> you are from an underprivileged country that contains no books,> no schools, no business opportunities, I suggest that you begin> to work to build an infrastructure. However, since you're complaining> about having to do housework from dawn to dusk, I conclude that> there exist people who are able to hire other people to clean.> That implies that they make the money they pay those housecleaners> in some other industry than housecleaning. Thus, that country> has opportunities available for thousands.> Can you read well? > When did I ever say I had to work from dawn to dusk? First of all, I> wasn't talking about myself. Secondly, here is how my day started.> Get up, brush my teeth and wash my face. Have breafast which was> already on the the table. Get ready for school. Grab my lunch box and> go wait for school bus which arrives around 8:20Am.> School starts at 9:15 and releaseed at 3:15. > School bus brought me home. Change my clothes. Have some snack and> wait for my Math tutor (not everyday; in that case, I did my Math> homework, the only homework I was keen to do). Have dinner around 6.> Then , it varies how my day ends. That was 6th grade. In 7th garde,> my Math tutoring moved to 7Am till 8:30 in the morning and so I had to> catch the school bus when it came around the senodn time by walking to> a friend's house on the next block. In 8th grade, it was after school.> So on.> Now, say that I was spoiled or privileged. But notice that there> wasn't extracurricular activites sponsored by school like during my> older siblings time. They went to Private schools (founded during> British time) which did not exist anymore in my time. That's because> there was no budget for it except for short term ones, which I joined> some times.> I did learn swimming one summer at a private place. But before 7th> grade, i.e when we were not old enough to be rebellious, in summer, my> mother would teach us.> With college education during my time, unlike my oldest sister's time,> I would not qualify to be considered privileged. I wasn't complaining> about that.> About your telling me I suggest that you begin> to work to build an infrastructure., I tell you this: stay within> the context of diuscussion and ...> ..do not act so dumb not to know that there are countries where> people are not free to do (build) their infrastructure. But mostly,> just stay within the context, will you?>> Housework is a very good lesson in >> eliminating some choices of employment. >Not for those who is left with no time to study when it is not just>helping around the house with chores.> Excuse me? Housework has ample opportunity for study. Since it> takes no thinking (or not much) to vacuum or wash a floor, one> can use their brains to learn at the same time. >I would set a book> up, read a paragraph, and think about it while doing other work.> If I still hadn't got it, I would read it again and continue doing> the physical work. When I figured it out, I would read the next> part of the text. In the computer biz, we called this flavor> of doing many things at once, timesharing.> Well..I never was so keen to study like a bookworm. In fact, only one> of my siblings was. But I happened to pay attention during lectures> (and when my mother taught us) and that was mostly enough for me. My> hobby was to read all kinds of books - there happened to be many books> around the house brought by my much older brothers, cousins, etc. -> that are not related to my school work.> I was about 10 or 11 when I saw that book, the Biography of Peron> (translated version). I advanced from reading story books to such> books though I didn't really understand all of it. There were many> such books (my oldest brother was keen on politics ) and other kinds> of books. Also, my next door neighbor was the president of the office> - I don't know how to traslate it - where all books get sensored> before they get published. So I had access to those books, cartoons,> etc. he brought home since their children and us were very close.> Going back to the context, ...> the point was that I was NOT talking about me. I was talking about> the kids from the poor family. Do you know how poor family lives in> 3rd world countries? Obviosuly not.>>It also gives one a fallback>> option for employment.> The curious will find a way to satisfy their itch. A good resource>> is^Wwas public libraries. The curious also have to learn how to avoid>> spoon-feeding or they will lose that precious commodity.> Public libraries? You seem to think that all countries have good>public libraries.> This thread was talking about US schools and such.> No it was not. It was about Asians. I was mistaken with another thread when I said No... but still yourresponse was to my post about the poor Asian kids having to work toomuch too early in life.Also, you failed to understand poor over there means not enoughfood, and in extreme cases, not enough wood or charcoal to make afire to cook, etc.> All countries may not have 100% good public libraries. Ours> sucks (and it's in the USA). If you have no access to any> library, go buy a text. If you have no money to buy a text,> barter with a professor (or somebody who does have books)> to clean his/her house in exchange for a couple of hours> access to his books every week.> For cripe's sake, I can see that there is no point in explaining to> you. You are a typical avergae American, who doesn't know much about> beyond your border. Sad! Very sad!I meant to type ... much about things beyond ...> Get a job at your country's university. Now you can even talk> to people who are learning; I used to sop up lots and lots of> knowledge just by listening to people talking about their work.> If the country is in an industrial stage, get a job at a factory.> Lots of physics, mechanics, efficiency studies, chemistry, math,> accounting, management, etc. can be learned just by noticing> how the company does its biz.> Again..GEE!!!!!!!!!!> You are a troll.> Not more than you are.> I may be a troll, but I'm one that can figure out how to get> something done with no resources. It's called work.> I am very sad for you that . > In fact, Josh halpern's reply was enough. I shouldn't even have> bothered to reply to you. You were so way out of the context of the> topic in discussion.> /BAH> Subtract a hundred and four for e-mail. === Subject: : Re: Uncle Al is Sadistic .> It is THAT simple where I grew up. [= Culture is everything per Bob]> Children in the poor neighborhood with their parents making ends meet> and the children having to do the house work like an adult and start> working while they themslevs are children (rather than play or study,> do not get a chance to test their brain since there is no books (not> referring to school text books and notes) lying around the house.> This has nothing to do with culture. It's poverty, which is pancultural.> May be this is a bit too simplistic scenario to apply in every parts> of the world but I would say that it would apply to most parts of the> world.> No, it does not apply because it has nothing to do with culture.> There is no culture which says We are Povertarians, we are> poor and proud of it, because POVERTY is our way of life> No such country or culture exists. Poverty is pancultural.> then the problem [if it would be of cultural origin] would have> been solved a long time ago. Painlessly.> Care to elaborate?> ... because, everybody everywhere would adopt, copy and> assimilate into such a wonderful (nonexisting) utopian culture.> But it does not exist. So any elaboration aint gonna help you.> But one can easily see the problem being enhanced and aggravated> at every step of its emergence and development because, basically...> What was thoughtless came by easy.> What was easy became habit.> What were habits became tradition.> What were traditions became culture.> What was culture became religion.> What is religion becomes thoughtless.> Here, think about this hexliner above. The reasons for your> unhappiness are in there. That's how I see the world.> You may see it differently then me. That's cool by me.> End of story.> Diversity will always be with us, whether we like it or not.> again:> ahahahaha...........ahahahahansonI don't know what happened to my response post where I said In somesituations, the poverty and culture get intertwined to the extent thatit becomes one, the culture.I do not want to spell it out but I hope you know which group I amtalking about. Also that group was forcefully formed as a group,despite the different oiginal cultures among them, if you get what Imean. === Subject: : Re: Uncle Al is Sadistic .amanda grava .88 la saucisse et au marteau:[SNIP]Could you *please* the lines you are not responding to. Otherwise, thisis very hard and unpleasant to read and, besides, it takes more time todownload (and it's significative with a low-speed modem).-- Nicolas === Subject: : Re: Factorization dispute> I have a result that shows a problem with a definition that> mathematicians have used for over a hundred years,No, you don't. Your claim is false.> and rather than> face the result which follows from rather basic algebra mathematicians> are being pussies and running like scared cowards from the result.They are challenging your (false) claim. No one is running, except in thesense that anyone would run from a maniac flinging manure.> Some posters, not professional mathematicians from what I've gathered,> are at least trying to stand and fight, but because what I have is a> math proof, their claims are necessarily irrational.Your proof is flawed. It has been thoroughly refuted many times. Thatmakes the challenges rational, and your defense irrational.> I need you to stand up for the truth, here and now.Been there, done that. You ignore, or simply repost.> Let's chase these mathematician cowards down, and make them face the> music...and dance?> After all physics had its challenges and physicists faced them, while> mathematicians seem to believe that they can just ignore problems.What mathematicians seem to believe that they can just ignore problems?I've seen no evidence of that.You, on the other hand, have a consistentprior record of ignoring counter-examples and refutations of yourarguments.> Let's take 'em out.> James Harris> http://mathforprofit.blogspot.com/Take 'em out? What do you mean by that? Are you back to your previousthreat of calling out the military or the CIA and the FBI, or are youadvocating the formation of a private militia?--There are two things you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com === Subject: : Re: Factorization dispute> Let's take 'em out.> James Harris> Take 'em out? What do you mean by that? Are you back to your previous> threat of calling out the military or the CIA and the FBI, or are you> advocating the formation of a private militia?I think James wants his vast silent audience to treat mathematicians todinner and a movie. I suspect that there's an ulterior motive. === Subject: : Re: Uncle Al is Sadistic .>>Sure, the I don't fail you you fail you bull. He didn't>>educate anyone either. He didn't go out and get more>>resources for more students.>When you run a race, you don't automatically award first>place to every runner.> There were six hundred places. What is the difference between the 599th> person in that class and the 601st in terms of ability? The wonderprof> simply abdicated any responsiblity for educating the students. Quite> common.> Exactly. That is why a rational society only cares about win, place,> and show rather than compassionately about a huge pool of> incompetents. With your approach, my sister would have never become a physician ifshe had not come to US. Back home, there's only once chance to getadmitted to medical school. She missed it by a shortage of .3 in herscore.Years later in US, she went to Med School in Georgia. === Subject: : Re: Probability of a Run>Your formula>u(m + 1) = u(m) + (1 - u(m - n)) (1 - p) p^n>is deducible from mine, for m > n. Just calculate P(n,m+1)-P(n,m) in>my notation, swap P(n,m) to u(m) and q to p and you'll get the same. I>had just about realized this couldn't be the difficult part of the>problem!>I'm slightly puzzled as to why you said I want to calculate the>probability of this if you know the answer, unless you're not a kid>with a TI-83 or a home computer!> Glad to explain! I am working on the subject of betting strategies for> my web site at> http://www.cybcity.com/ranmath/start.htm> A resort city close to me has been inundated by gambling casinos. If> everyone knew as much math as I do or as you do or had studied> psychology under B. F. Skinner as I have, they wouldn't gamble, yet> they do. I am trying to educate them. In a recent paper, Edward O.> expectation for a game with fixed expectation in the slightest, yet I> have found a counterexample. I have found a betting strategy that will> make the player's expectation WORSE, so I feel that the last word on> this subject has not yet been spoken.> To calculate the player's expectation for game + strategy one must> take into account the probability of ruin and for the case of the> player who uses a Martingale betting strategy it is necessary to refer> to the Theory of Runs. It is a deep and difficult subject, or at least> it has been up to now. The above recurrence relation is not mine. I> got it from Burnside and Uspensky. My own attempts to derive such> things generally lead to wrong answers.> I have the same mathematical machinery as Robert Israel does but he> knows how to use it better and even has helped me with it. I don't> feel I can tell my people that all they have to do is to expand this> complicated rational function in powers of s and the coefficients will> be the required probabilities. Robert Israel can do it and maybe I can> do it but they can't do it. I want to give them something they can> use.> I am studying everything you write but am a little behind. I have> gotten as far as your formula>P(n,m) = 0 if m < n,> = q^n.(1 + (1-q).( (m-n) - sum(i=n..m-n-1, P(n,i) ) ).>The logic is simple. Either the first n trials are losses or the first> trial is a success and then we have n losses or we have no runs of n> or more in i trials followed by a success and then n losses (for i => 1.. m-n-1). I think this enumerates all the possibile combinations> without duplication.> and have the following observation:> P(n,m) appears to be defined in terms of P(n,n) for the range n > m > 2n+2 and there seems to be no definition of P(n,n) except a circular> one. Setting m = n, the limits of the summation are n to -1 and the> first term in the sum is P(n,n). Presumably the other terms would have> the value 0 because m < n.> I construe the character between (m-n) and sum in your formula as> a minus sign. I assume that your formula is not intended to be> executable and will study your programming code. Have you checked the> output against results obtained in other ways?I have no difficulty believing my explanations are difficult tounderstand. My father was a very good teacher but I never inheritedany of those skills.beginning to think I was back at schoool.I'll answer your last question first. The only results I could obtainwere hand calculations for very small examples, the simple case wheren=1 and m is any value and quite a few results from large simulations.I did say I'd tried to make sure the formula wasn't ludicrous before Iput forward any reply.When I said Burnside and Uspensky's formula is deducible from mine, Ididn't make it clear that my definition is also deducible from their's(i.e. they are equivalent and I have not discovered anything which wasnot already known).In my definition the summation is intended to be 0 if the lower limitexceeds to upper limit as is the case for P(n,n).Although I have seen languages in which you could execute the functiondefinitions, this wasn't really the intention here. Besides, it wouldnot be an efficient algorithm if it worked directly from thedefinition. The first program does work using this definition, you'llbe reassured to know.By writing SP(n,m) = 0 if n > m = sum(i=n..m, P(n,i)) otherwisewe can get rid of the summations.Basically we calculate P(n,n) and SP(n,n), P(n,n+1) andSP(n,n+1)...P(n,j) {using SP(n,j-n-1) which we have alreadycalculated} and SP(n,j)...P(n,m)We don't really need all the values of SP at any given moment and asold habits die hard I economised on the space. Unfortunately, it makesa simple algorithm look complicated.The second algorithm works by undoing the recursive part of thefunction definition. I'll rewrite the new definition as there wereerrors in the ranges for parts of the definition. I'll also replace(1-q) by p just to save space.P(n,m)= 0, if m < n= q^n(1+p((m-n))), if m < 2n+1= q^n(1+p((m-n)-q^n((m-2n)+p/2((m-2n-1)(m-2n))))), if m < 3n+2= q^n(1+p((m-n)-q^n((m-2n)+p/2((m-2n-1)(m-2n)/2)-q^n((m-3n-1)(m- 3n))+p/3((m-3n-2)(m-3n-1)(m-3n))))))),if m < 4n+3...This has been reorganised for calculation purposes as P(n,m)= 0, if m < n= q^n(1+p(m-n)), if m < 2n+1= q^n(1+p(m-n)(1-q^n(m-2n)/(m-n)(1+p/2(m-2n-1)))), if m < 3n+2= q^n(1+p(m-n)(1-q^n(m-2n)/(m-n)(1+p/2(m-2n-1)(1-q^n(m-3n-1)(m- 3n)/(m-2n)/(m-2n-1)(1+p/3(m-3n-2)))))),if m < 4n+3...The things to notice are that the new definition is exact. The approxin the function name in the code merely means that the expression forthe m < 20n+19 case is similar to that for the case m < 21n+20 and youdon't really have to go all the way down to the bottom of theexpression to evaluate it accurately.My remarks on cancellation error problems might make more sense if youcompare this algorithm with that for 1-exp(-x) which can be calculatedasx(1-x/2(1-x/3(1-x/3(1-x/4...))))This formula is ok until x gets too large and then its startsreturning nonsense.It might be sensible to limit how big m can be - basically if j (thenumber of levels inside the brackets it starts the expression from) istoo big you probably won't get a sensible answer. You might also haveto wait a while due to an inefficient implementation of mnfunc forlarge values of i - I struggle to pick sensible names!The code I've produced is in VBA because I find spreadsheetsconvenient for holding disorganised calculations. The code is prettysimple so I don't think it will be hard to translate the code into alanguage of your choice.Ian Smith === Subject: : Re: Factorization dispute> It turns out that I can isolate the current dispute easily enough by> focusing on the factorizations:> Consider [snipped]> Crank Information> http://www.crank.net/harris.html> http://www.crank.net/usenet.html> http://www.google.com/search?q=harris+site%3Awww.crank.net> http://www.google.com/search?q=%22james+harris%22+site% 3Ausers.pandora.beReaders should please check out *all* the links Sam the Worm listed.As for the math, notice that with(5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22no other factorization works as long as 7 is not a factor of 22.That's because I've isolated the factors of 22, which is obvious byinspection.James Harrishttp://mathforprofit.blogspot.com/ === Subject: : 3 x 3 matrix / eigenvalueIf A is a nonsingular 3 x 3 matrix with nonnegative entries, then why must Ahave a positive real eigenvalue?Mike === Subject: : Re: Difficult social problem> It looks like I'm swinging at tissue paper with a sledgehammer when it> comes to getting acceptance of my work, as while I can get initial> contact with mathematicians they tend to run as soon as I give them> enough information to realize the implications of my work and that I> am correct.> Mr. Harris,> even if your work were correct (and it is not), it would be as awe inspiring> as my morning piss after waking up.And people that's math society.That's how math people *really* are, so forget the movies.Keep that image of a math person taking his morning piss.James Harris === Subject: : Re: Does compact+continuum connected+locally connected==>pathwise connected?>>Does compact+continuum connected+locally connected>>==> pathwise connected?>>Continuum connected means that any two points of the space lie>>in a continuum (= compact connected set).> Usually continuums are Hausdorff, but as that was omitted...Out of curiousity, what was the countable cofinite example?> Any infinitely countable set with the cofinite topology> isn't path connected, hence counterexample.> Finite set with cofinite topology, tho not path connected,> isn't counterexample.> A set with cardinality > c and> the cofinite topology is path connected> Thus not counterexample.>Let U be the first uncountable ordinal. Give X = U x [0, 1) +>{infinity} the lexographic order on U x [0, 1) and have infinity>greater than all other elements. Then give it the order topology.Itis complete and densely ordered, with endpoints, so compact>It is a linear continuum, so connected. In particular any interval>in it is connected, and the intervals generate the topology, so it>is locally connected.>> path p from a to b in linear order S, a /= b> ==> [a,b] order isomorphic [0,1]>all you have to do is you have to show that from such a path you can>construct a path which is strictly increasing. I don't see>immediately how to do that though, if the path is something stupid>with infinitely many local maxima / minima. Oh well,.. Is there asimple proof?> path p from a to b in linear order S, a /= b> ==> [a,b] order(iso)morphic [0,1]> proof:> wlog a < b; retract r:S -> [a,b]; rp:[0,1] -> [a,b] surjection> rp([0,1]) connected, convex; a,b in rp([0,1); [a,b] subset rp([0,1)> [a,b] = rp([0,1]); [a,b] continuous image separable continuum> [a,b] separable multi-point linear continuum; [a,b] ordermorphic [0,1]> That last step isn't a step, it's a theorem based upon the theorem> a countable dense linear order sans end pts is ordermorphic Q.Ah yes, of course. Clever, and way more elegant than my initial attempts at a proof. :)>> path connected linear order S ==> |S| <= c>Hmm. This one I'm less convinced by. I'm tentatively willing to>believe it to be true - certainly it looks intuitively like it might>well be right - but do you have some sort of reference I could check>for that?> Put some end points on it and apply the above. ;-) If you can> do that continuously, let me know, it'll save us the hassle ofBut... that doesn't work. Obviously. X is a counterexample, as the long-line is path-connected. (Of course that's not a counterexample to the original statement, it just shows that proof can't work)>but I'd like to see a proof of that.> Reaching into the jumble jungle of my notes and> grabing a choice handful of random coherence:> path connected linear order S ==> |S| <= c.> proof by assuming c < |S|> some s in S with c < |(.,s]| or wlog c < |[s,.)|> If { xi | s < s_xi } has last element beta:> c < |[s,.)| = |[s,s_beta]| <= c which cannot be> Thus let beta = lim { xi | s < s_xi }> c < |[s,.)| = |/{ [s,s_xi] | xi < beta }| <= c|beta|; c < |beta|> note: from 1st theorem |[s,s_xi]| = c> omega_1 <= c < |beta| <= beta; some path p from s to s_(omega_1)> [s,s_(omega_1)] homeomorphic [0,1]; omega_1 embeds R, not so!Ok. Makes sense, I think... I'll have to look over it later in a bit more detail.David(E-mail address spam-blocked in the obvious way) === Subject: : Re: Applications of mathematics>Suppose you were to tell senior highschool students about applications>of mathematics that would be interesting and understandable to them.>What applications would you talk about?>Knot theory and sex, because sex is the only thing interesting to them.That reminds of a quote from Louis-Ferdinand C.8eline's Voyage au boutde la nuit found on the Mathematical Quotations Server(http://math.furman.edu/~mwoodard/mquot.html):---------- -------------------------------------------------------------- ----------------------------Entre le p.8enis et les math.8ematiques... il n'existe rien. Rien! C'estle vide.[Between the penis and mathematics there is nothing. Nothing! Thevoid!]----------------------------------------------------- -----------------------------------------------John Mitchell === Subject: : Re: parallelizability of manifolds> There's not even the *beginning* of such a sequence of vectorfields> for S^2: no matter what the vectorfield V_1 on S^2, there is always> some point x of S^2 at which V_1(x) does not belong to any basis of> the tangent space of S^2 at x. That's just a (seemingly more> complicated, but ultimately worthwhile) way of saying that every> (continuous!) vectorfield on S^2 has at least one zero. And that,> in turn, follows (after building the appropriate machinery relating> differential topology to algebraic topology) from the fact that the> Euler characteristic of S^2 is non-zero.>Indeed, algebraic topology seems to be an extremely powerful tool: S^2 is>simply connected, so every continuous image of it also is; in contrast,>R^2{(0,0)} is not. Realizing that this gives the proof for S^2 was a>pretty cool heureka-moment! ...I'm not sure your eureka! moment was authentic. For example, S^3 isboth simply connected and parallelizable.John Mitchell === Subject: : Re: Uncle Al is Sadistic .>>Genius isn't a monopoly. You consider individuals as individuals, one>>by one. However, you mostly won't mine diamonds out of anything but>>lamproite and kimberlite, plus placer deposits. If resources are>>limited you invest where the odds are demonstrated best. Only a fool>>*continues* to commit a majority of resources in barren ground. You>>can dig down five miles deep and you won't find one natural diamond in>>New Jersey.>The top 20% of any university graduating class wasn't the top 20% of>>SAT scores on admission. OTOH, the top 20% of SAT scores on admission>>are substantially over-represented in the top 20% of the graduating>>class - especially in subjects containing objective truth. A>>university seeking to fill its sciences, engineering, math,>>computer... departments would be insane to choose diversity (racial>>quotas) over objective performance. Look in sci.physics. What do you>>do to an idiot to fix it? Education isn't relevant.>If you want metal you mine ore and dispose of the gangue matter of>>course. You sure as Hell don't dig granite if you want copper. It's>>a waste of granite, too.>>UC/Irvine intensely, Liberally, enthusiastically courted nigger and>>spics - guaranteed admission no matter what and full free ride plus>>fat perqs. UC/Irvine intensely, Liberally, enthusiastically excluded>>Asians and Jews. I visit the campus. The real subjects are almost>>exclusively populated by Asians and Jews with astounding credentials>>and demonstrated abilties. Your average White kid looks around and>>drops those courses after the first lecture. Don't entertain>>hallucinations of diversity in linear algebra courses. The>>University of California Regents can't for the life of them figure out>>how Asians and Jews can universally do so well.> That' just because of all California is filled with morons, > not just U.C. Since Linear Algrebra was invented in > ancient Eqypt, most us are still trying to figure why *any*> mathematicians anywhere believe that they have the credentials > necessary to grade California Home Economics Homework, > nevermind anything that appears to be politics > that idiots at Harvard invented, rather than a > run down compsci dump like U.C.> It's always useful to check assumptions against reality> http://math.sfsu.edu/hsu/workshops/treisman.htmlenough to make a comment when I read the following:<<<<<<> I have noticed (in grad school) that was exactly how the Chinesestudents functioned.What I also noticed was that they would not take in those fromoutside their ethnic group unless it is a Malay Chinese who is more aChinese than a Malay.It obviously was politics, at least in that particular grad school.Indian students functioned similarly too. Vietnamese too are a strong group but they are at the undergardprograms (at the school mentioned above).May be the reason most vietnamese are not in grad school is that theylive here while the Chinese come here as foreign studnets and usuallyjoin grad programs (better chance to get student visa).> josh halpern === Subject: : Re: Modelling the movement of an electro mechanical device. >I have a servo system, which moves through a certain angle, depending on the>amount of voltage applied to it's input.>The response of the movement from when the control signal is input, until>the armature actually reaching the desired position varies as the device>operates through different angles.What do you means by operates through different angles?> This can be measured over a range of angles and frequencies.Frequencies of what? The input signal? How does that affect themovement?>Does anyone have any suggestions as to what route I should be looking to>take if I wanted to produce a mathematical model of the device, so that it>is possible to predict the movement of the device?>Many thanks>JD === Subject: : Re: Does compact+continuum connected+locally connected==>pathwise connected?Countable cofinite space counter exampleContent-transfer-encoding: 8bit>Continuum connected means that any two points of>the space lie in a continuum (= compact connected set).> Usually continuum means compact connected Hausdorff.Yes. But I posted quickly, and neglected to mention I was thinking ina METRIC SPACE.--Ron Bruck === Subject: : computation of a seriesis there any formula to compute the following series {sum{a_j,j=0,j=k-1})^k for real a_j's? === Subject: : Re: Usenet Posting Guide?> With all respect, dear man, you strike me as that type more enamored> with process rather than function. Most of us out here have far better> things to do than screw around for weeks configuring newsreaders and> the like, even if we were so inclined, which most of us clearly are> not. I suppose we could memorize the phone book, too, but would that> help us communicate our ideas any better?> Some of us prefer to view the forest rather than count the trees down> there. Better view, too.> Well, I find the technical aspects of how USENET functions much more> interesting than most of the discussions that take place on it. By the> same token, I am much more interested in the hardware and operating system> software of the systems I administer than in any of the applications> for which they are used.> Once I hoped that the growing popularity of personal computers meant> that nearly everyone would learn to think like programmers. It never> occurred to me that, sadly, the opposite would happen: That computers> would be designed to be used by people who *can't* program them.Dear man, that is all fine and good, but you'd do well to examine yourthinking here a little more closely. You're seeing the world from avery narrow vantage point. Technology and inventive genius exist toserve those who use and take advantage of them, not merely those whodesign and invent, otherwise you might just as easily make theargument that the world was a far better place when mostly carbuilders and those who tinkered all day with Model T Fords were outthere driving on the roads. Certainly the roads were safer then, andthat those driving them generally knew their cars inside and out, butthat ignores the fundamental purpose behind that of automobiles, whichis to facilitate transportation.The same exact argument can be made for telephones, television,airplanes and a whole host of other modern conveniences we now takefor granted and which represent fundamental shifts in the way wecommunicate and get around. They were all once the domain of a selectfew tinkerers and inventors who had absolutely no idea of what theywere about to unleash on the world.With all respect, sir, I can guess your politics right across theboard. Just for openers, you're a Sierra Club, save-the-whales type.Gotcha, huh? === Subject: : Re: Factorization dispute> It turns out that I can isolate the current dispute easily enough by> focusing on the factorizations:> Consider> (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 b_3(x) + 22) = > 300125 x^3 - 18375 x^2 - 360 x + 22> where even by inspection you can see that the constant terms are> separated out, so that you have 1(1)(22) = 22, the constant term of> the polynomial.> I'll add that at x=0, a_1(0) = a_2(0) = b_3(0) = 0.> Notice, > (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 again you see that the constant terms match as now you have> 7(7)(22) = 1078, which is again the constant term of the polynomial.> If 22 does not have 7 as a factor, the former factorization is the> *only* allowed way for 49 to divide through.> (For more detail, like what the a's are, see> http://mathforprofit.blogspot.com/> where more is explained.)> I have a result that shows a problem with a definition that> mathematicians have used for over a hundred years, and rather than> face the result which follows from rather basic algebra mathematicians> are being pussies and running like scared cowards from the result.> Some posters, not professional mathematicians from what I've gathered,> are at least trying to stand and fight, but because what I have is a> math proof, their claims are necessarily irrational. That's not really true. The people with brains stopped fighting mathematicians and their holistic continuum, the day after they invented it. Since it's still well-known throughtout the universe that they're the only people who believe that there are an infinite number of dimensions in a universe that quite obviously only has three.> I need you to stand up for the truth, here and now. It's still impossible to stand up in a mathematics class, since they're only people who actually believe in infinite value logic. And if you did stand up, all they would do is start chanting some some sort of political spirituals about the evil of things that aren't positive.> Let's chase these mathematician cowards down, and make them face the> music.> After all physics had its challenges and physicists faced them, while> mathematicians seem to believe that they can just ignore problems. That's not true. The only statement that has ever universally about science is that mathematicians do G's, physicists do E's, and the people with brains do brainy things. Fission is old, Fusion is new, the NSA is cold, and the U.N. is Blue.> Let's take 'em out.> James Harris> http://mathforprofit.blogspot.com/ === Subject: : Re: computation of a series> is there any formula to compute the following series> {sum{a_j,j=0,j=k-1})^k > for real a_j's?Nothing especially nice, no.For a general field (in particular for real numbers) [ sum{j = 1 to m} a_j ] ^n = Sum { x in {1, ..., m}^n } Product {i = 1 to n} a_(x_i)Prove it by induction basically. It's not especially difficult. This doesn't simplify to anything nice in your case, although it might for some specific sequence a_iDavid(E-mail address spam-blocked in the obvious way) === Subject: : Re: 3 x 3 matrix / eigenvalue> If A is a nonsingular 3 x 3 matrix with nonnegative entries, then why must> A have a positive real eigenvalue?This is not true. For example1 0 00 0 10 1 0has eigenvalues 1, 1, and -1. However, it is true if you add the restriction that the matrix be triangular because then the eigenvalues are the diagonal entries.Have a tolerable existence. Eli === Subject: : Re: Usenet Posting Guide?> Some of us prefer to view the forest rather than count the trees down> there> The point was this, back in the days when you had to think before you> posted, those who, to borrow your analogy, visited the forest actually> appreciated it, and contributed to the ecology of the forest, rather than> now, where too many people use the forest as a place to hold their drugand> alcohol parties, while thrashing the forest.It's more like now the forest has been clearcut and paved over to make roomfor a Wal-Mart.-- Visit my blahg site.http://myblahg.blogspot.com/ === Subject: : Re: complex integral.....??> um...... i think that root of 1+(z^3)+(z^5)}] exist between -1 ~ -0.5> thus, f(z) is analysis of |z|<1/2> thus.......i think that answer is 0> it is right??> All the roots of a_n*z^n + a_{n-1}z^(n-1) + ... + a_1*z + a0 verify> 1/(1 + B/|a_0|) < |z| < 1 +A/|a_n|> Where A = max(|a_0|, |a_1|, ..., |a_{n-1}|) and B = max(|a_1|, ...,> |a_{n-1}|, |a_n|)I think it's easier to notice that if |z| <= 1/2, then |z^3 + z^5| <= ..., hence z^3 + z^5 could not = -1. === Subject: : Re: Difficult social problem[...]> Mr. Harris,> even if your work were correct (and it is not), it would be as > awe inspiring as my morning piss after waking up.> And people that's math society.> That's how math people *really* are, so forget the movies.> Keep that image of a math person taking his morning piss.> James HarrisJames, how soon you forget:: David Ullrich is a ing piece of dog.: : I think it's funny that I can call a professor at Oklahoma State: University a ing piece of dog knowing that he'll keep : replying in my threads.: : You see, he has to keep replying pushing the same old lies.: : He's stuck. He's trapped in something that he can't get out of, : so it doesn't matter what I call him, or what I say about him, : he has to come back.: : You see I'm the person who has the correct math argument, so : posters like David Ullrich or Arturo Magidin are *compelled* : to reply out of fear that if they go away, then I'll get some : people who'll pay attention to the truth.: : So David Ullrich, the math professor at Oklahoma State University, : is demeaned by me as the piece of ing dog he is, and he : *has* to keep coming back.: : : James Harris === Subject: : Re: Modelling the movement of an electro mechanical device. > I have a servo system, which moves through a certain angle, depending onthe> amount of voltage applied to it's input.> The response of the movement from when the control signal is input, until> the armature actually reaching the desired position varies as the device> operates through different angles. This can be measured over a range of> angles and frequencies.> Does anyone have any suggestions as to what route I should be looking to> take if I wanted to produce a mathematical model of the device, so that it> is possible to predict the movement of the device?Well, plot (angular) movement versus electrical input as one graph and plotmovement versus time as another plot. If both graphs are straight lines thencalculating movement from electrical input is just a proportion. Thencalculating time from movement is just another proportion.If the graphs are not straight lines you might begin with a statisticalcurve fit... === Subject: : Re: Math dependency logic REVISED> [...]>Of course, you don't deny that if one starts with something true and>apply correct reasoning, then the conclusion must be true. > Well that's progress - for some reason you've decided not to be> entirely stupid today.> So of course I didn't deny that. So that example is totally> irrelevant regarding your (_repeated_) statement that I was> _lying_ when I said nobody was denying that. Which brings> us back to the beginning: If I was lying when I said nobody> was denying that you should produce an example where> someone _has_ denied that. Or admit that you were being> either stupid or dishonest when you said I was lying.> Of course neither of those is going to happen, because> you have more concern for exposing evildoers than for> telling the truth.> But>coming from you these are empty words, for you have demonstrated (on more>than one occasion!) your inability to reason from C1-C4 to Ex~(x=x).> Oops. Back to the irrelvancies...When (with no evidence) *you* accuse someone of faulty reasoning,you expect to be taken seriously. But when glaring instancesof your own imperviousness to logic are cited and referenced, youdismiss these as irrelevant!Are you Rush Limbaugh? Do you do prescription drugs?======================================================== ===========C1 AxAy[x=y -> Az(z in x <-> z in y)]C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] C3 EyAx[x in y <-> Et(x in t) & A] (with y not free inA)ClassificationC4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (WeakExtensionality)Exhibit of proof of Ex~(x=x) from C1-C4 and someone will point outthe error. === Subject: : Re: 3 x 3 matrix / eigenvaluebojdsa$c4rt$1@netnews.upenn.edu...> If A is a nonsingular 3 x 3 matrix with nonnegative entries, then why mustA> have a positive real eigenvalue?Perron-Frobenius theorem. === Subject: : Re: Uncle Al is Sadistic .The LaRouche Show is a weekly audio talk show, broadcast live on theInternet every Saturday, featuring interviews with Lyndon LaRouche,his associates, and special guests. Hosted by Michele Steinberg,Counterintelligence co-director of Executive Intelligence Review, andby Marcia Merry Baker, EIR Economics Intelligence director. Live Broadcast TODAY3:00-4:00 p.m. Eastern Standard Time (2000-2100 UTC) Speaker: William Wertz on Schiller: Poet of Freedom. High-speed audio: Stream Low-speed audio: Stream During the live broadcast, you can ask questions by calling one of thefollowing numbers: > been practiced all over the place. Utopia would be at hand.--ils duces d'Enron!http://larouchepub.com/radio/index.html === Subject: : Factorization dispute, againNotice, (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 you see that the constant terms match as now you have 7(7)(22) =1078, which is the constant term of the polynomial49(300125 x^3 - 18375 x^2 - 360 x + 22).Various people have debated me about what happens when you divide off49, where for some odd reason, some of them seem to believe that youcan havew_1(x), w_2(x), and w_3(x) such that w_1(x) w_2(x) w_3(x) = 49, and(5 a_1(x) + 7)/w_1(x) (5 a_2(x) + 7)/w_2(x) (5 b_3(x) + 22)/w_3(x) = 300125 x^3 - 18375 x^2 - 360 x + 22where the w's vary as x varies, which is a rather naive notion.That's because you can multiply *everything* out, and simplify to get(7/w_1(x)) (7/w_2(x)) (22/w_3(x)) = 22which should be simple enough for all of you.Now those of you who usually work in the field of complex numbers maythink that it's not a big deal, as you may think it doesn't matter ifw_3(x) has some factor factor of 7, despite *seeing* (22/w_3(x)) butyou see, as 22 is coprime to 7 in the ring of algebraic integers, ifw_3(x) isn't coprime to 7, (22/w_3(x)) does not exist in the ring.You know, it's like how in integers 1/2 doesn't exist. It's not aninteger, so it's not in the ring.So you see, my argument is correct and simple, and mathematicians areindeed running from a little gut check in their field. They'repussies too scared to handle the truth.But you should also understand, some people will be able to see that,which is part of my plan. I can let mathematicians destroy themselvesproving they can't be trusted based on what they *see*, while theyforget what they can't see: the wearing down of the mathematicianmystique.James Harrishttp://mathforprofit.blogspot.com/ === Subject: : Re: Uncle Al is Sadistic .> Speaker: William Wertz on Schiller: Poet of Freedom.Schiller, poet of Romantic Dreck. Have your read the words to -Ode an die Freude-?Bob Kolker === Subject: : Re: Proof of Loan Amortization FormulaIn sci.math, Jay:> Great Oracle of Mathmaticians,> I know you guys get tired of guys like me asking for proofs but, this> will be the last time I ask. I think seeing the proof actually helps> me visualize and learn how to apply it in real life. When I look at> the loan amortization formula, I just dont see that link right away.> Could you take some time to explain or recommend a book that has the> P.S Here is the demon below:> L t(1+t)^n> p = _________________> [ (1+t)^n - 1 ]The simplest way of proving it would be to posit the problemin this fashion, perhaps.Assume one has applied for loan of a principal L,paid to him at the start of the loan. He is to payback this loan in n months, at a constant (t * 100) %interest rate (per month). [*] Basically, one pays intereston any outstanding principal at the start of the month.(We assume that his first payment is a month after theloan, as well. Note that this is a *compound interest*loan, as opposed to a *simple interest* loan, which israrely used nowadays. Another variant is a continuouslycompounded interest loan, which computes the intereston the outstanding loan amount using a formula suchas L * exp(k * ln(1+t)). Still other variants are possible,such as ARMs, which posit a variable t; usually the variationsare such that one can use a table lookup, as they are adjustedevery 6 months or so.)At month 0 his balance sheet (relative to the loan company)looks like:B(0) = LAt month 1, we assume the payment is promptly credited andone has interest on the loan:B(1) = L * (1+t) - PAt month 2:B(2) = (L * (1+t) - P) * (1+t) - PAt month 3:B(3) = ((L * (1+t) - P) * (1+t) - P) * (1+t) - PAt this point you might want to gather terms, as I for onesmell a possible induction hypothesis here:B(3) = L * (1+t)^3 - P * ( (1+t)^2 + (1+t) + 1) = L * (1+t)^3 - P * ((1+t)^3 - 1) / ((1+t) - 1)Let's see if this works for B(4):B(4) = L * (1+t)^4 - P * (1+t) * ((1+t)^3 - 1) / ((1+t) - 1) - P = L * (1+t)^4 - P * ((1+t)^4 - (1+t) + (1+t) - 1) / ((1+t) - 1) = L * (1+t)^4 - P * ((1+t)^4 - 1) / ((1+t) - 1)Oooh!Now let's set up a more formal hypothesis. We assumeB(k) = L * (1+t)^k - P * ((1+t)^k - 1) / ((1+t) - 1)for an integer k, and prove thatB(k+1) = L * (1+t)^(k+1) - P * ((1+t)^(k+1) - 1) / ((1+t) - 1)using similar algebraic manipulation (which I leave to theinterested reader, as it's very similar to my B(3)=>B(4)transition above). We also trivially verify thatB(0) = L * (1+t)^0 - P * ((1+t)^0 - 1) / ((1+t) - 1) = Lso weak induction follows; we now have a closed formfor the balance after the k'th month.The terms of the loan are that after n monthsthe loan is paid off, so B(n) = 0, and thereforeB(n) = 0 = L * (1+t)^n - P * ((1+t)^n - 1) / ((1+t) - 1)orL * (1+t)^n = P * ((1+t)^n - 1) / ((1+t) - 1)or P = L * t * (1+t)^n /((1+t)^n - 1).QEDIn practice, the equality is not quite exact because ofrounding of P to the nearest penny, but the amount ofvariation is at most0.01 * ((1+t)^n - 1) / ((1+t) - 1)regardless of the value of L.If we substitute n = 360 (a 30 year house mortgage) andt = 0.005 (about a 6% per annum rate), we get $10.045 ...,which is miniscule compared to most house loans nowadays;the last payment is increased by at most this amount,or perhaps the mortgage company remits a check.[*] just to confuse things: the payments in many loans are *per month*, but the rate is quoted as an interest rate *per year*, which may complicate the analysis in real life, even for this relatively simple loan.-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: : Re: naive geometry questions> 2) Besides planes and spheres, is there any other surface S such that a> piece of S can be moved around adlibitum while each of its points remains> in contact with S?A surface with the property that I think you're describing is called homogeneous. I think that planes and spheres are the only homogeneous surfaces that can be constructed in three-dimension space. There are others but they can't be constructed in three-dimensional space. For example there are the flat tori and hyperbolic spaces. In three dimensions, a torus has to have a bulging outer area and a caving inner area but in four-dimensions, you can make a torus is flat, that is, it doesn't bulge or cave anywhere. I don't know how to describe hyperbolic space. Maybe someone else in this group can. Since I'm on the topic, does anyone know how many dimensions you need for a Euclidean space in which you can hyperbolic space?Have a tolerable existence. Eli === Subject: : Re: Math dependency logic REVISED> Ain't it touchin' the lengths to which Logikoi will go to bail>> one another out? See Camaraderie of the Experts>I did go to that link. Here's how it begins.>>Lonely, are you?> If not for the way _he_ tends to speculate on people's personal lives> when he can't refute their arguments I'd say this wasn't very nice,> pointing out what a pathetic character he must be, forced to talk to> himself like this in public where everyone can see.>Anyway, the lengths the Logikoi will go to bail each other out of>*what*? Was there something threatening in this thread? > I know _I've_ been terrified of the possible consequences. I mean> of course everything he's said here has been nonsense, but> regardless, what if someone at OSU found out that there was> someone on the internet saying bad things about me?> (Giggle. Worse yet: You may not have noticed, but he often> quotes me saying wild things like everything is equal to> itself. What if someone at OSU found out I was promulgating> that sort of heresy? I can just picture it, once that post-tenure> review he mentioned elsewhere is implemented: There's a> committee meeting in Whitehurst. Ullrich says everything is> equal to itself? Off with his head.)Of more concern to this committee might be your UseNet stalkingof JSH.Searched Groups for JSH OR James OR Harrisauthor:ullrich@math.okstate.edu. Results 1 - 10 of about 2,700.Search took 0.70 seconds.Then again, you could always plead insanity...http://www.fetchfido.co.uk/sound_files/ giggle.wav === Subject: : f(n+1)/f(n)=g(n)?Hiif f(n+1)/f(n)=g(n), is it possible to write the function f(n) on anon-recursive form? That is to say, on the form f(x)=......for some functions g(n) it is easy, like for f(n+1)/f(n)=3, whichgives f=3^x*C, But are there any general formula for any given g(n)? === Subject: : Re: Greek Alphebet> John Savard But in other contexts, the letters of the Greek alphabet are fraught> with meaning!> Thus, alpha comes from aleph, which means Ox.>References please? The _name_ alpha might be a degenerate form of the>Hebrew aleph, but the letter itself started AS the symbol for an ox, in>Greek. It actually comes from the upside down capital alpha, which was the>actual symbol for the head of an ox. (Just picture an upside down A).>Who says it's not the other way around and aleph is not a degenerate form>of alpha?With all due respect to the Greeks, the Semitic alphabet was well established before the Greeks considered usingan alphabet; they started with it, and the Semitic alphabetalready had many versions.-- 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: (765)494-6054 FAX: (765)494-0558 === Subject: : derivate of x^xOk, I've probably asked this question before, but I've forgot the answer...What's the derivate of f(x)=x^x and f(x)=x^(x^(x-1)) ?I guess asking for the integral of these functions would be foolish? === Subject: : Poisson DistributionIn the poission distribution like this: the monthly average number ofairplane crashes is 2.2, can I imply that the average number of airplanecrashes is 4.4 in 2 months?If I use that assumption, I will lead to a different solution to theproblem: the probability of 5 crashes in 2 months, from using Poissonprocess for 1 month and binomial distribution.Thank you for your help-- Khoa Tran === Subject: : Re: Factorization disputebinomial factorizations, a fruitful field of endeavor;what's that one method, Partial Quotients, used in calculus?...I never really got the hang of it, but it was interesting. the funny thing about the Army (and all of the services, andthe NSA etc.) is that they have lots of applied mathematicians ... andgenerals can ask them to look at stuff ... perhaps,they already have looked at monsieur Harris and his ordnances! > Let's take 'em out. > http://mathforprofit.blogspot.com/--ils duces d'Enron!http://larouchepub.com/radio/index.html === Subject: : Re: Math dependency logic REVISED> Ain't it touchin' the lengths to which Logikoi will go to bail> one another out? See Camaraderie of the Experts> I did go to that link. Here's how it begins.> Lonely, are you?Not at all. But don't fail to read Camaraderie of the Experts atIt explains why you Boyz go to such great lengths to fend offassaults on one or another's 'expertise'.But you'll find nothing there about bum-fuzzling. Sorry.> Anyway, the lengths the Logikoi will go to bail each other out of> *what*? Was there something threatening in this thread? Oooh, were> we almost exposed as charlatans and frauds? Dear oh dear, what with> my mortgage[1] and all, we can't have that. === Subject: : Re: Key Core Error Argument[snip long boring discussion, look upthread if interested]> You keep writing a ratio because apparently you think a ratio is more> powerful or mysterious, capable of doing something that it can't.> Now then, if you admit that a_1(x)/w(x) is an algebraic function, it> can be replaced by f(x), and if you admit that 7/w(x) is an algebraic> function, it can be replaced by g(x), so then you have f(x) + g(x).> But the constant term of f(x) + g(x) is 1, so let h(x) + 1 = f(x) +> g(x), to isolate constant terms as before.Which line contains the error?1. The constant term of (h(x) + 1) is 12. (h(x) + 1) = (f(x) + g(x)) (Noting that f(x) = a_1(x)/w(x), g(x) = 7/w(x)3. (h(x) + 1) = (a_1(x)/w(x) + 7/w(x))4. The constant term of a_1(x)/w(x) + 7/w(x) is 1 -William Hughes === Subject: : Re: Routine technique, analysis> Last I remembered setting a variable to 0 to clear it out, knowing> that pulled out terms independent of it was routine in analysis, which> is why a lot of this is funny, ironic, and very, very sad.>I must say, it certainly brings a tear to MY eye.I knew when programming one must be careful using uninitializedvariables in case they contain unexpected values, but didn't realizethe same applied in pure mathematics. Maybe this explains why myalgebra sometimes goes horribly wrong; I've reused x or theta from theprevious calculation and forgotten to clear it out afterwards.Yeah, that's the ticket. I didn't make a mistake, it was the variablesthat somebody else had just used that were out of whack. Honest! === Subject: : My research, publication announcementThere's more to my work than just arguing on Usenet, so I'd like topoint out that my paper Advanced Polynomial Factorization is slatedto be published:See http://www.megasociety.net/NoesisHighlights.htmlThe Mega Foundation is an organization of high IQ people, and I'm gladto be associated with them. To learn further about the organizationyou can use Google, or see:Why a group like the Mega Foundation? http://www.ultrahiq.org/Mega/WhyMega.htmI hope at least some of you will appreciate that often the mostimportant ideas in history have to get past people limited by theirlack of imagination and their prejudices, who act against scientificprogress.What I want you to see is that there's more to me than Usenet, so thatyou can begin to understand that the revolution I'm giving you achance to be a part of is bigger than the small-minded people whocontinually throughout history work to halt progress.Thank you for your time and attention.James Harrishttp://mathforprofit.blogspot.com/ === Subject: : Re: Key Core Error Argumentwow, how embarrassing for you. please,don't tell me how many that I'm on --I'm trying to get out of here, anyway! > I am now at 2710, but I am only around in this newsgroup since> january 1988.--ils duces d'Enron!http://larouchepub.com/radio/index.html === Subject: : Re: Key Core Error Argumentthere is also a proof of the isomorphismof deductive proofs with inductive ones,which may perhaps be amenable to combining the two formsinto a tautology, or necklace. > before the final link (you missed the green, dood !-) --ils duces d'Enron!http://www.wlym.com/covers/7101contents.png === Subject: : Re: Ex(~x=x), counterpart theory, and contingent identity> Well, when you claim that you can define scope so that self-identity >is always implicit, you must deal with a Kantian> possibility--...> The logical determination of a concept by reason> is based upon a disjunctive syllogism, in which the> major premiss contains a logical division (the division> of the sphere of a universal concept), the minor> premiss limiting this sphere to a certain part, and> the conclusion determining the concept by means> of this part.> --Immanuel Kant> Critique of Pure Reason A577/605I'm not sure how this cashes out where the logic of identity isconcerned. One might construe the classical Ax[x=x <-> Ey(x=y)]as characterizing by exclusion the relation between self-identity andidentity-with-something: no self-identical is an identical-with-nothing,and no identical-with-something is a non-self-identical.In respect of the foregoing, ~AxEy(x=y) might be the minor premiseand ~Ax(x=x) the conclusion.> ...--namely, the non-self-identical.> Of course, the concept is still a fiction in Kantian epistemology.> However, it is also the reason for the apparent> complexity of his ideas--he does not trivially assert> self-identity as self-evident. === Subject: : Re: derivate of x^x> Ok, I've probably asked this question before, but I've forgot the answer...> What's the derivate of f(x)=x^x and f(x)=x^(x^(x-1)) ?> I guess asking for the integral of these functions would be foolish?x^x = e^(x*log(x))Now use the chain and product rules. === Subject: : Re: Uncle Al is Sadistic . charset=iso-8859-1> The LaRouche Show is a weekly audio talk show, broadcast live on the> Internet every Saturday, featuring interviews with Lyndon LaRouche,> his associates, and special guests. Hosted by Michele Steinberg,> Counterintelligence co-director of Executive Intelligence Review, and> by Marcia Merry Baker, EIR Economics Intelligence director.> Live Broadcast TODAY> 3:00-4:00 p.m. Eastern Standard Time (2000-2100 UTC)> Speaker: William Wertz on Schiller: Poet of Freedom.> High-speed audio: Stream> Low-speed audio: Stream> During the live broadcast, you can ask questions by calling one of the> following numbers:> been practiced all over the place. Utopia would be at hand.> --ils duces d'Enron!> http://larouchepub.com/radio/index.html === Subject: : Re: My research, publication announcement> There's more to my work than just arguing on UsenetYes, but that's your crowning achievement., so I'd like to> point out that my paper Advanced Polynomial Factorization is slated> to be published:> See http://www.megasociety.net/NoesisHighlights.htmlHow nice that the cranks have gotten together and put up a website. > The Mega Foundation is an organization of high IQ people, and I'm glad> to be associated with them. To learn further about the organization> you can use Google, or see:> Why a group like the Mega Foundation? > http://www.ultrahiq.org/Mega/WhyMega.htm> I hope at least some of you will appreciate that often the most> important ideas in history have to get past people limited by their> lack of imagination and their prejudices, who act against scientific> progress.Not any more, now that we have the Internet . . .> What I want you to see is that there's more to me than Usenet, so that> you can begin to understand that the revolution I'm giving you a> chance to be a part of is bigger than the small-minded people who> continually throughout history work to halt progress.Is there a publication party? Free drinks? Are we all invited?as you didn't label your binomial factorization(binomials being polynomials, and monomials canalso be considered that, I guess, but I dygress),what is the import of transforming a coefficientof a linear variable, x, into a function? I also had New Math in the 3rd grade,so I can empathize with trite Bourbaki-isms! as for rings where 7 is not a factor of 22,isn't it true that in anything where it is a factor,it dyssolves anything of interest? Bud, I know I ain't the first top ask that! > (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 2) = 49(x^3 + 5 x^2 + 3x + 2).> Notice that dividing both sides by 49 gives > (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 2) = x^3 + 5 x^2 + 3x + 2> as long as you're in a ring where 7 is not a factor of 22.> I want to emphasize that point as notice there's only *one* way to> divide through by 49 if 7 is not a factor of 22. > You may see people who reply claiming that the word polynomial has> some significance, as if it's a mystical thing which refutes basic> logic, so if something isn't polynomial it no longer behaves> logically.--ils duces d'Enron!http://larouchepub.com/radio/index.html === Subject: : Re: f(n+1)/f(n)=g(n)?primefinder grava .88 la saucisse et au marteau:> Hi> if f(n+1)/f(n)=g(n), is it possible to write the function f(n) on a> non-recursive form? That is to say, on the form f(x)=......> for some functions g(n) it is easy, like for f(n+1)/f(n)=3, which> gives f=3^x*C, But are there any general formula for any given g(n)?f(n+1) = g(n)g(n-1)....g(0)f(0)That's the best you can do in the general case.-- Nicolas === Subject: : Re: Modelling the movement of an electro mechanical device.> I have a servo system, which moves through a certain angle, depending on the> amount of voltage applied to it's input.> The response of the movement from when the control signal is input, until> the armature actually reaching the desired position varies as the device> operates through different angles. This can be measured over a range of> angles and frequencies.> Does anyone have any suggestions as to what route I should be looking to> take if I wanted to produce a mathematical model of the device, so that it> is possible to predict the movement of the device?> Many thanks> JDIs the device a spring/mass device with torque proportional to voltage?Is it something else? I have in my lab an instrument that is perfect forthis whatever it is if you have an angle transducer (the SRS785 twochannel dynamic analyzer). The down side of this instrument is the$11,000 price tag.Chuck-- ... The times have been, That, when the brains were out, the man would die. ... Macbeth Chuck Simmons chrlsim@earthlink.net === Subject: : Re: Ex(~x=x), counterpart theory, and contingent identity> c37480a7.0311072209.7de21310@posting.google.com>...> (1) states necessary and sufficient conditions for the necessity of> (material) identity:> (1) AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))> If identicals x and y are necessarily self-identical, then--and> only then--is their identity a necessary one. Beyond this,> (1) states nothing more: from (1) it neither follows that> John Correy is necessarily self-identical nor that John> Correy is contingently self-identical--or indeed that John> Correy is self-identical at all. (1) does not say which of> the foregoing is the case. We can sit around and argue about whether you or I are necessarily> self-identical. However, although logicians do argue about such> matters--Who else would bother?--it is not as logicians that they> argue but as metaphysicians, or as pataphysicians, or as what have> you?> So, when I claim that (e.g.) Benjamin Franklin is necessarily> self-identical while the inventor of bifocals is not, my main> warrant for this claim is that if Benjamin Franklin is necessarily> self-identical but the inventor of bifocals is not, then Benjamin> Franklin and the inventor of bifocals are not necessarily identical> (although they are identical). In other words, I take the necessary> self-identity of the former and the contingent self-identity of the> latter to constitute, together with (1), an *explanation* for the> contingent identity of Benjamin Franklin and the inventor of bifocals.> To this you might object that these would be also be contingently> identical if both Benjamin Franklin and the inventor of bifocals were> contingently self-identical. To which I would respond that identities> involving what linguistically oriented analytical philosophers refer> to as rigid designators, are identities whose terms are necessarily> self-identical; whereas identities involving what such philosophers> refer to as non-rigid designators, are identities whose terms are> contingently self-identical. Therefore, granted that I take rigid and> non-rigid designation as the linguistic marks of necessary and> contingent self-identity--putting the cart back behind the horse,> rather than approaching the matter bass ackwards as it is> fashionable to do these days--and granted that I take> Benjamin Franklin and John Correy to be 'rigid' designators> and 'the inventor of bifocals' to be 'non-rigid', I conclude> that the contingent identity of Benjamin Franklin and the inventor> of bifocals has as its basis the contingent self-identity of the> inventor of bifocals, while Benjamin Franklin is necessarily> self-identical.> As to whether physical or mathematical objects are contingently> self-identical or necessarily so, some sort of metaphysical argument> (rather than a logical one) warranting one or the other of these> conclusions would have to be made. I suspect that mathematical> properties are both essential in, and necessary to, mathematical> objects--but this is an intuition and nothing more.> Best regards,> John> PS It won't surprise me if Paul Holbach or G. Frege bring in talk> about 'scope', which I think is only peripherally relevant to> discussions of necessary and contingent identity.> Let me bring in a quote:> One must distinguish between the claim that identity sentences are> contingent and the claim that the identity relation itself is> contingent. For the relation to be contingent, there need to be things> between which it holds merely contingent. For it to be necessary, it> has to be that if the relation obtains between things, it obtains> between those very things of necessity. [...] One can consistently say> that there are contingent identity sentences, though the relation> itself is necessary. Thus one could say that The first> Postmaster-General of the US was the inventor of bifocal lenses. is> contingent and is an identity sentence, but that if we consider the> object, x, which is in fact referred to by the first> Postmaster-General of the US and the object, y, which is in fact> referred to by the inventor of bifocal lenses, it is necessary that> x is identical to y.> [Sainsbury, M. (1995). Philosophical Logic. In A. C. Grayling (Ed.),> /Philosophy. A Guide Through the === Subject: / (pp. 61-122). Oxford: Oxford> University Press. (p. 93)]This sounds so much like what Kripke says, either inIdentity and Necessity or in _Naming and Necessity_, that I hopeSainsbury cited him.Of course, what Kripsbury says represents the Party Line oncontingent identity: There are *statements* of contingentidentity but no instances of contingent identity itself. Oops! Before I forget, let me forestall the inevitable bUllrich-ism:Duh. You're not saying anything *new* you know. 'AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))' is a theorem of standardquantified modal logic with identity. (Giggle) === Subject: : Re: My research, publication announcement> There's more to my work than just arguing on Usenet, so I'd like to> point out that my paper Advanced Polynomial Factorization is slated> to be published:> See http://www.megasociety.net/NoesisHighlights.html> The Mega Foundation is an organization of high IQ people, and I'm glad> to be associated with them. To learn further about the organization> you can use Google, or see:> Why a group like the Mega Foundation?> http://www.ultrahiq.org/Mega/WhyMega.htm> I hope at least some of you will appreciate that often the most> important ideas in history have to get past people limited by their> lack of imagination and their prejudices, who act against scientific> progress.> What I want you to see is that there's more to me than Usenet, so that> you can begin to understand that the revolution I'm giving you a> chance to be a part of is bigger than the small-minded people who> continually throughout history work to halt progress.> Thank you for your time and attention.> James Harris> http://mathforprofit.blogspot.com/Crank Information http://www.crank.net/harris.html http://www.crank.net/usenet.html http://www.google.com/search?q=harris+site%3Awww.crank.net http://www.google.com/search?q=%22james+harris%22+site% 3Ausers.pandora.be === Subject: : [JSH] [Lunatic-Crank-Reposted Nonsense] Re: Factorization dispute, again> So you see, my argument is correct and simple, and mathematicians are> indeed running from a little gut check in their field. They're> pussies too scared to handle the truth.still thinking that pissing out this flawed argument will ever make it true.You are so sad and your so-called math work is even sadder!Happy pissing!