mm-435 Subject: Re: Jennifer Jill Herman - September 27th 1978 > WARNING: Read below before even thinking about responding to this > twit.Why do you not follow the advise?home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: [OT] Re: Jennifer Jill Herman - September 27th 1978> ...> > WARNING: Read below before even thinking about responding to this> > twit.> Why do you not follow the advise?Dik,Does this strike you as a raging case of unmedicated schizophrenia? === Subject: Re: Tricks>let A = { a0,.. aj } be a collection of decimal digets,>and n = a0 + a1 10 + ... + aj 10^j> For every p there exist an f_p : A^j->N so thatfor every (positive) prime p ...> f_p < n> n = f_p (mod p)f_p is a function, it's not a number.Thus explain f_p < n and n = f_p (mod p)Do you mean for all x = (x1,x2,..xj) in A^j f_p(x) < n; n = f_p(x) (mod p) ?Proof: For all x in A^j, let f_p(x) = n-p> For every prime there exists a function that takes the decimal digits of> a natural number n and outputs a number smaller than n and congruent with> n modulus p.> I cant be more specific than this. Does or does not this generalised> relation(function) exist and is anything know anything about it, i.e. my> original question. Can it even be proved? === Subject: Re: Tricks let A = { a0,.. aj } be a collection of decimal digets,> and n = a0 + a1 10 + ... + aj 10^j> For every p there exist an f_p : A^j->N so that> for every (positive) prime p ...Precisely> f_p < n> n = f_p (mod p)> f_p is a function, it's not a number.Is this a notation problem? I mean N in f_p : A^j->N to be all naturalnumbers, so the functions input is one or more of the decimal digits andit's output is natural number.> Thus explain f_p < n and n = f_p (mod p)The meaning of all this is to make it simpler to see if a given numberis divisible by a given prime. So if f_p >= n it would be rather pointless.> Do you mean for all x = (x1,x2,..xj) in A^j> f_p(x) < n; n = f_p(x) (mod p) ?Precisely> Proof: For all x in A^j, let> f_p(x) = n-pHehe. Well that was proof allright.. Not a very practial one for findingf_p for a given p though but still a proof.. === Subject: More analysis helpHello all,Yet another plea for help from an analysis student. I've solved similarproblems but this one's giving me fits. Here's the statement:Let 0 < a_1 <= a_2 <= ... <= a_k for fixed values a_n. Let b_n = (a_1^n +a_2^n + ... + a_k^n)^(1/n). Show that the limit, as n tends to infinity, ofb_n is equal to a_k.In previous problems there was a sort of subtle trick involving certainassumptions and showing that two sequences were equivalent. But I can't, forthe life of me, figure this one out. Any hints/suggestions/tips/prods/&c.?Much in your debt as usual. === Subject: Re: More analysis help> Let 0 < a_1 <= a_2 <= ... <= a_k for fixed values a_n. Let b_n = (a_1^n +> a_2^n + ... + a_k^n)^(1/n). Show that the limit, as n tends to infinity, of> b_n is equal to a_k.a_k <= b_n <= n^(1/n) a_k -> a_k as n->oo === Subject: Re: More analysis help>Let 0 < a_1 <= a_2 <= ... <= a_k for fixed values a_n. Let b_n = (a_1^n +>a_2^n + ... + a_k^n)^(1/n). Show that the limit, as n tends to infinity, of>b_n is equal to a_k.> a_k <= b_n <= n^(1/n) a_k -> a_k as n->ooHow can it be so simple? I was coming at it from the squeezing anglebut, for some reason, never looked at a_k as a lower bound. Doh! === Subject: Partial Sums by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9CDOJh22072;The sum issum[i=0,n-1](n-i)(n-i-1) = sum[k=1,n]k(k-1) = sum[k=1,n]k^2 -sum[k=1,n]k= n(n+1)(2n+1)/6 - n(n+1)/2. === Subject: Buffon's Needle Problem (PLEASE HELP!) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9D3rMf14504;Any help on this problem will be GREATLY appreciated! Please reply.Here is the problem: If there are two parallel lines on a table acertain distance apart and a needle of length (l) is dropped, what isthe probability that the needle will come to rest between the lineswithout touching either of them?My teacher assures me that solving this problem IS possible, evenwitht te limited info. === Subject: Re: Buffon's Needle Problem (PLEASE HELP!)> Any help on this problem will be GREATLY appreciated! Please reply.> Here is the problem: If there are two parallel lines on a table a> certain distance apart and a needle of length (l) is dropped, what is> the probability that the needle will come to rest between the lines> without touching either of them?evaluate:>-14 - (-3)>do not use brackets in your answer Hint: subtracting a negative number (-3) is the same as doing what to a positive number (3)?Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAddress munging may or may not reduce the spam you get; it surelyreduces the number of useful answers you get. http://www.cs.tut.fi/~jkorpela/usenet/laws.html === Subject: Confusing natural logarithms by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9EHSFF19524;I'm just wondering, is it possible to solve this equation without theuse of some number crunching software?4.551 = X + 2e^(-0.5X)We only have one variable here, but I just can't seem to isolate it. Avvfter computing away, i got a rough value of X being equal to 4.32Any help with this or any resources you could send me to would begreatly appreciated. Thank you. === Subject: Re: Confusing natural logarithms>I'm just wondering, is it possible to solve this equation without the>use of some number crunching software?>4.551 = X + 2e^(-0.5X)>We only have one variable here, but I just can't seem to isolate it. Neither can I. I don't believe a closed-form solution exists.>Avvfter computing away, i got a rough value of X being equal to 4.32My TI-89 gives numerical solutions 4.3204 and -2.528.If you graph 2e^(-.5x) (exponential decay curve) and 4.551-x (straight line), you'll see they cross in two points, which tells you there are two solutions.Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAddress munging may or may not reduce the spam you get; it surelyreduces the number of useful answers you get. http://www.cs.tut.fi/~jkorpela/usenet/laws.html === Subject: mode of a group of numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9ENqjQ15360; If no number in the group appears more than once there is no mode. === Subject: Why me? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9F0GKP17624;I thought I told you not to post anything. You really have a lot ofnerve pal. I'm not sure what I did to you, besides feel sorry for youdown before I run into you, it would be best for both of us.Michael === Subject: Re: Why me?> I thought I told you not to post anything. You really have a lot of> nerve pal. I'm not sure what I did to you, besides feel sorry for you> down before I run into you, it would be best for both of us.Sorry, it won't happen again === Subject: limit of a product by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9FCQxo00385;Hi everyone,I`m trying to prove that the limit of a product is the product of thelimits.That is, iflim(x->a) f(x) = L and lim(x->a) g(x) = M, thenlim(x->a) f(x)g(x) = [lim(x->a)f(x)][lim(x->a)g(x)] = LMSo we have to find a d>0 such that for for every e>0:|f(x)g(x)-LM| < e whenever |x-a| < dfor sums of functions you can easily use the triangle inequality toprove the limit of a sum is the sum of the limits,but there is no such inequaltiy for products.It did occur to me that if we choose d=min{d1,d2}, whered1 is the number such that |x-a| < d1 ==> |f(x)-L| |g(x)-M| I`m trying to prove that the limit of a product is the product of the> limits.> That is, if> lim(x->a) f(x) = L and > lim(x->a) g(x) = M, then> lim(x->a) f(x)g(x) = [lim(x->a)f(x)][lim(x->a)g(x)] = LM> So we have to find a d>0 such that for for every e>0:> |f(x)g(x)-LM| < e whenever |x-a| < d> for sums of functions you can easily use the triangle inequality to> prove the limit of a sum is the sum of the limits,> but there is no such inequaltiy for products.> It did occur to me that if we choose d=min{d1,d2}, where> d1 is the number such that |x-a| < d1 ==> |f(x)-L| and d2 is the number such that |x-a| < d2 ==> |g(x)-M| then we have> |f(x)-L||g(x)-M| < e^2> if we can find a delta such that the above product is smaller than> e^2,> then surely there is a delta such that the product is smaller> than e (didn`t prove it though), but that only shows the limit of the> products exists, not> tat is is equal to the product of the limits..> Can anyone please give me some hints or tips on how to prove this?It would be surprising to me if you are supposed to do this on your own without having seen something like what William suggested. What hints or limit proofs have you seen so far? === Subject: Re: limit of a product> I`m trying to prove that the limit of a product is the product of the> limits.> That is, if> lim(x->a) f(x) = L and> lim(x->a) g(x) = M, then> lim(x->a) f(x)g(x) = [lim(x->a)f(x)][lim(x->a)g(x)] = LMHint: |f(x)g(x) - LM| <= |f(x)g(x) - f(x)M| + |f(x)M - LM| === Subject: isomorphic proof by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9FFmb214444;I am trying to prove that the set of linear transformations from V toV are isomorphic to an n x n matrix with real number entries. I knowI have to prove that M (the mapping) is well defined. And that it isa linear transformation. And that it is one-to-one and onto. I'mhaving trouble deciding how to state that it is one-to-one and onto. The first two parts I'm okay on. === Subject: Re: isomorphic proof> I am trying to prove that the set of linear transformations from V to> V are isomorphic to an n x n matrix with real number entries. I know> I have to prove that M (the mapping) is well defined. And that it is> a linear transformation. And that it is one-to-one and onto. I'm> having trouble deciding how to state that it is one-to-one and onto. > The first two parts I'm okay on. I'm sure you meant to say the set of linear transformations from V to Vwhere V is a real linear space of dimension n is isomorphic to R^(n,n).I'm also sure that your text defines one-to-one and onto. You may evenknow that for finite dimensional spaces with the same dimension,one-to-one and onto are equivalent (you only need to show one of them).Do you know about kernels?I gather that you want some hints in regard to your particular M. Howabout telling us how you defined M? === Subject: Laplace Transforms by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9FJ1GV28673;Hola guys, how do you go about transforming L{f(t)} for f(t) =e^t(sinh(t)). do you take the transforms of each and multiply them byeach other?JC === Subject: maths by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9FJ1NM28776;hello there i would like some help on my decimals and my percents yestomorow i have a test so could u write me back please === Subject: GCD of polynomials i have a few problems of the kind: Given two polynoms f and g that have at least one common root, find all the roots for f and g.The problem is not in how to find the roots but when I'm applyingEuclides alg. to find the gcd. In several cases the last remainder is akonstant but this constant is ignored and the previous remainder(thelast one containing x) is used as gcd. Is this correct? Musn't the gcdbe the last non-zero remainder for polynomials as well? === Subject: Re: GCD of polynomials Visiting Assistant Professor at the University of Montana.> i have a few problems of the kind: Given two polynoms f and g that have> at least one common root, find all the roots for f and g.>The problem is not in how to find the roots but when I'm applying>Euclides alg. to find the gcd. In several cases the last remainder is a>konstant but this constant is ignored and the previous remainder(the>last one containing x) is used as gcd. Is this correct?If, when you apply the Euclidean algorithm, you get a nonzero constantas a remainder, then the two polynomials have no commonroots. (Assuming you are working over the reals or the rationalnumbers or the complex numbers).> Musn't the gcd>be the last non-zero remainder for polynomials as well? Yes. Can you give an example where th elast remainder is a constantbut this constant is ignored?