mm-467 === Subject: : Re: Illegal to do research on cryptography?> I love this line on the webpage :))> Our proprietary encryption algorithms are regularly changed to provide> the highest level of security.> If you had a proper encryption algorithm you would not have to change it> regularly to provide a high level of security.If the change is automated, including automated generation of purchase webpage with javascript coder / decoder, than this change is just part of thealgorithm.Also I could decide to not show nor use the decrypt function on the webpage. This will increase the true security but decrease the securityfeeling thus resulting in fewer sales. Overall I *might* theoretically getmore fraud (by showing the decrypt function) but this loss is offset byincreased sales. In a business, security is not everything.> you can always setup a self-signed certificate and convince your clients> to trust it. HTTPS would provide you with strong encryption, key> management, and server authentication, none of them is provided by your> current homegrown 'encryption' algorithm.> Vincent Granville, Ph.D.> -- Lauri> Algorithm from webpage sources:> function crypt(string) {> var len=string.length> var intCarlu> var carlu> var newString=e> if ((string.charCodeAt(i)!=101)&&(len>0)) {> for (var i=0; i intCarlu=string.charCodeAt(i)> rnd=Math.floor(Math.random()*7)> newIntCarlu=30+10*rnd+intCarlu+i-48> if (newIntCarlu<48) { newIntCarlu+=50 }> if (newIntCarlu>=58 && newIntCarlu<=64) { newIntCarlu+=10 }> if (newIntCarlu>=90 && newIntCarlu<=96) { newIntCarlu+=10 }> carlu=String.fromCharCode(newIntCarlu)> newString=newString.concat(carlu)> }> return newString> } else {> return string> }> }> function decrypt(string) {> var len=string.length> var intCarlu> var carlu> var newString=> if (string.charCodeAt(i)==101) {> for (var i=1; i intCarlu=string.charCodeAt(i)> carlu=String.fromCharCode(48+(intCarlu-i+1)%10)> newString=newString.concat(carlu)> }> return newString> } else {> return string> }> }Vincent Granville, Ph.D.Strategy Architect & FounderData Shaping Solutions, LLChttp://www.datashaping.com=== === Subject: : how to calculate the following questions?[img]http://www.geocities.com/i141802596/Image1.txt[ /img][img]http://www.geocities.com/i141802596/Image2.txt[/img] [img]http://www.geocities.com/i141802596/Image3.txt[/img][img] http://www.geocities.com/i141802596/Image4.txt[/img][img]http: //www.geocities.com/i141802596/Image5.txt[/img]=== === Subject: : Re: How to Solve this nonlinear ODE system> everybody! I have a system of nonliear ODE as what follows:> dx/dt=A1*y^2+B1*y+C1> dy/dt=A2*X^2+B2*x+c2> x(t=0)=M0, X(t=infinity)=Mf;> y(t=0)=N0, y(t=infinity)=Nf;> A1,A2,B1,B2,C1,C2, M0,Mf,N0,Nf are constants;> Could anybody give some suggestions about how to get a closed> form of x(t), y(t)? You reply will be highly appreciated!Firstly, you can shift x and y (i.e. let x' = x + a for constant a,and similarly for y) to obtain a system as above with B_1 = B_2 = 0.Integrating both sides of (a2*x^2 + c2)dx/dt = (a1*y^2 + C1)dy/dt,gives a result which is trivial if a1 = 0 and/or a2 = 0.Otherwise you can scale x and y to express the result in the forma.(x^3 + 3x) + b.(y^3 + 3y) + c = 0, where a, b, c are constants(with a and b known, in terms of the a_i and b_i, and c determinedfrom boundary conditions, i.e. -c = a.(x_0^3 + x_0) + b.(y_0^3 + y_0).Plugging x, y = T.u(t) + x_0, T.v(t) + y_0 back in the solutiongives (with numeric constants d_i, e_i): (a.u^3 + b.v^3).T^2 + (3a.x_0.u^2 + 3b.y_0.v^2).T + (3a.x_0^2.u + 3b.y_0^2.v) = 0and with: P = u/v Q = 2.(a.P^3 + b).T/v + (3a.x_0.P^2 + 3b.y_0)this becomes: (3a.x_0.P^2 + 3b.y_0)^2 - (a.P^3 + b)(3a.x_0^2.P + 3b.y_0^2) = Q^2If the LHS is positive for any real value of P then there is a realbirational transformation of P, Q which reduces the condition tothe following with numeric A, B: P'^3 + A.P' + B = Q'^2and thus P', Q' and hence P, Q and hence (scaling u, v by taking T = 1)u, v can be parametrized in terms of p(t) and dp/dt where p(t) is theWeierstrass function.----------------------------------------------------- ----------------------John R Ramsden (jr@adslate.com)---------------------------------------------- -----------------------------Eternity is a long time, especially towards the end. Woody Allen=== === Subject: : Re: How well do the first n terms of Maclaurin's expansion approximate a function?|> i guess you're only doing the case where x0 times?||Not sure how you got that idea. I only assumed x0, x1, ..., xn are |distinct; I see no problem allowing one of xj's to be 0. I haven't thought |about degenerate cases.that's what i was talking about, that you didn't discuss thedegenerate cases. (in the non-degenerate case x0 Well, as a hint for this sort of problems (and they are the standard> describe the generated by ), one can approach it> by considering all things that will ->definitely<- be in there. To do> that, consider the operations you want to close your set > { {n} : n in N}> under: finite unions, complementations, differences. Think about the> result of doing each of those; take all you get, and apply it again;> take all you get, and apply it again, etc. See if you can describe> them. Or at least, come up with a large inclusive class, all of whose> elements will have to be in the (in this case, any> not-necessarily-sigma field that contains the set). Once you have a> very large class that includes everything you can think of that will> ->definitely<- be there, see if you can show that it already is an> object of the desired kind. If you can, you're done. If you cannot,> you might be able to see what sort of elements you were missing...> -- ==================> Arturo Magidin> magidin@math.berkeley.edu=== === Subject: : Re: maximal graph <0404081358220.8230-100000@gandalf.math.ukans.edu> I was wondering if anybody would be kind enough to help me with this> problem.> Your professor refuses to help?> I'm an undergraduate student taking a graph theory course, and one> problem my professor gave me has me dumbfounded. It goes:> Describe graphs, G, that are maximal in the following sense, G has> the property that any added edge will decrease its diameter. I know> that a complete graph does not have this property> Do you really think you can add an edge to a complete graph without> decreasing its diameter? Where are you going to put it?> because diam(Kn)=1 for all n.> Even when n = 1?> Other than that, I don't know where to go. Any help would be> appreciated.> How about a complete graph minus an edge? How about a path?> I know you can't add an edge to a complete graph. That's what I> said a complete graph does NOT have this property!!! That is:> you can't add an edge to a complete graph and decrease it's> diameter.You mean, the property you're interested in is, you can add an edgeto it and decrease the diameter? That's *not* what you said in youroriginal post.> You can't add an edge to a complete graph and have it remain> simple!!!! I may be just a lowly undergrad student, but give me a> little credit!I gave you credit for stating the problem correctly the first time.Now, what is the exact wording? What book is it from?=== === Subject: : Re: maximal graph> I was wondering if anybody would be kind enough to help me with this> problem.> Your professor refuses to help?> I'm an undergraduate student taking a graph theory course, and one> problem my professor gave me has me dumbfounded. It goes:> Describe graphs, G, that are maximal in the following sense, G has> the property that any added edge will decrease its diameter. I know> that a complete graph does not have this property> Do you really think you can add an edge to a complete graph without> decreasing its diameter? Where are you going to put it?> because diam(Kn)=1 for all n.> Even when n = 1?> Other than that, I don't know where to go. Any help would be> appreciated.> How about a complete graph minus an edge? How about a path?>I know you can't add an edge to a complete graph. That's what I said>a complete graph does NOT have this property!!! That is: you can't>add an edge to a complete graph and decrease it's diameter. You can't>add an edge to a complete graph and have it remain simple!!!! I may>be just a lowly undergrad student, but give me a little credit!Uh, maybe you should relax a little - if someone says something youdon't understand it might be that you should ask for an explanationinstead of starting out with the explamation points.If you know that you can't add an edge to a complete graph thenyou should also realize that this means that a complete graph_does_ have the stated property. The property in question saysthis: If you add an edge you decrease the diameter. That'strue for a complete graph, precisely _because_ you can't addan edge.=== === Subject: : Re: maximal graph> I was wondering if anybody would be kind enough to help me with this> problem.> Your professor refuses to help?> I'm an undergraduate student taking a graph theory course, and one> problem my professor gave me has me dumbfounded. It goes:> Describe graphs, G, that are maximal in the following sense, G has> the property that any added edge will decrease its diameter. I know> that a complete graph does not have this property> Do you really think you can add an edge to a complete graph without> decreasing its diameter? Where are you going to put it?> because diam(Kn)=1 for all n.> Even when n = 1?> Other than that, I don't know where to go. Any help would be> appreciated.> How about a complete graph minus an edge? How about a path?I know you can't add an edge to a complete graph. That's what I saida complete graph does NOT have this property!!! That is: you can'tadd an edge to a complete graph and decrease it's diameter. You can'tadd an edge to a complete graph and have it remain simple!!!! I maybe just a lowly undergrad student, but give me a little credit!=== === Subject: : Re: Non-Noether Conserved Quantity?>In playing around with Lagrangians, I found a general case>in which there exists a conserved quantity that doesn't in>any obvious way correspond to a Noetherian symmetry. Is there>some way to view the following as a symmetry of the Lagrangian?>Let the Lagrangian L(x,v_x,y,v_y) have the following form:> L = A(x,v_x) + B(x) 1/2 v_y^2 - C(y)/B(x)>Then the following quantity is conserved (by motion that>obeys the Lagrange equations of motion)> Q(y,v_y) = 1/2 (B(x) v_y)^2 + C(y)But Q is not obviously a momentum or any other Noetherian>conserved quantity.>To see that Q is conserved, start witht the Lagrange>equations of motion for v_y:> d/dt (@L/@v_y) = @L/@y>(where @ means partial derivative)>In our particular case, this becomes> d/dt (B(x) v_y) = - (d/dy C(y))/B(x)>Now multiply both sides by B(x) v_y:> B(x) v_y d/dt (B(x) v_y) = - (d/dy C(y)) v_y>The left side can be rewritten as> d/dt (1/2 (B(x) v_y)^2)>The right side can be rewritten as> - d/dt(C(y))>So we have> d/dt (1/2 (B(x) v_y)^2 + C(y)) = 0>So Q(y,v_y) = 1/2 (B(x) v_y)^2 + C(y)>is a conserved quantity, but it doesn't>obviously correspond to any symmetry.> Those conserved quanties have> known for ages. > Well, then why don't you tell us what they are physically?> But the only> quanities *physics* considers> to be conserved are> square law symmetries,> i.e. gravity wells.> What on earth are square law symmetries?> What have they to do with gravity wells?> And who ever proposed that gravity wells are conserved?> For that matter, ask him why gravity wells and square law> symmetries are the same thing -- or in other words, ask him to learn> the difference between i.e. and e.g.. There is no difference between i.e and e.g. Since they were both invented by dead Philosophers. And Gravity wells are obviously identically the same thing as sqaure laws given that the only people who believe they exist are Einstonians and their dead relatives, the Sagonoids.=== === Subject: : Re: Non-Noether Conserved Quantity?> For that matter, ask him why gravity wells and square law> symmetries are the same thing -- or in other words, ask him to learn> the difference between i.e. and e.g..> There is no difference between i.e and e.g. > Since they were both invented by dead> Philosophers.I have investigated some of your earlier posts to consider thepossibility that this is really an incredibly dead-pan bit of ironyrather than a willful stupidiy, but my investigations remainedinconclusive.Let's assume you are serious: I.e. and e.g. are abbreviations for different latin phrases, and theymean different things, and if you can't be bothered to learn what theymean you have no business using them in the first place, but shouldconfine yourself to standard English; unless you really think lardingyour prose with random abbreviations increases its savor?Howsthat?I can't determine if you are troll, an ironist or a fool, but nomatter which set you are an e.g. of, you lie in the intersection withthe boring.=== === Subject: : Fermat's Factoring MethodGiven a composite, c = pq, where p and q are prime, for |p - q| < c^(1/4),how long might one expect Fermat's factoring method to take to factor c?Russell=== === Subject: : Inverse Laplace Transform - Check my answerCan someone check my answer to the following problem.Find the inverse laplace transform of:F(s) = 100s / (s^2 + 4)(s^2 + 2s + 10)Factoring (s^2 + 4) I got s = 2jFactoring (s^2 +2s + 10) I got s = -1+3jFor M1Can someone check my answer to the following problem.>Find the inverse laplace transform of:>F(s) = 100s / (s^2 + 4)(s^2 + 2s + 10)>Factoring (s^2 + 4) I got s = 2j>Factoring (s^2 +2s + 10) I got s = -1+3j>For M1For M2My final answer is:>F(t) = 13.87 sin (2t + 56.31) + 14.62 E^-t sin (3t - 15.26)>I would really appreciate if someone could verify this>answer for me.Just to give you something to check against, here is Maple output:invlaplace(100*s / ((s^2 + 4)*(s^2 + 2*s + 10)),s,t);150/13*cos(2*t)+100/13*sin(2*t)-150/13*exp(-t)*cos(3 *t)-350/39*exp(-t)Where the first two terms obviously come from the s^2+4 and the othertwo from the s^2+2s+9. Use partial fractions.Why you express your answers with decimals instead of exactly withfractions?=== === Subject: : Re: Inverse Laplace Transform - Check my answer> Just to give you something to check against, here is Maple> output: > invlaplace(100*s / ((s^2 + 4)*(s^2 + 2*s + 10)),s,t);> 150/13*cos(2*t)+100/13*sin(2*t)-150/13*exp(-t)*cos(3*t)-35> 0/39*exp(-t) > Where the first two terms obviously come from the s^2+4> and the other two from the s^2+2s+9. Use partial> fractions. > Why you express your answers with decimals instead of> exactly with fractions?> --LynnBecause I used a calculator and did the problem usingRectangular > Polar conversions using imaginary numbers andnone of the answers came out as real numbers.http://www.webzila.com=== === Subject: : Re: Inverse Laplace Transform - Check my answer> Why you express your answers with decimals instead of> exactly with fractions?> --Lynn>Because I used a calculator and did the problem using>Rectangular > Polar conversions using imaginary numbers and>none of the answers came out as real numbers.You will never understand what is going on if you use a calculator forproblems such as this. Break it up into partial fractions, do the workmanually, and you will know your answer is correct and understand why.IMHO calculators should not be used for problems like this until afteryou understand how to work them. Sure, they can save a lot of time,but they don't enhance understanding.=== === Subject: : checkingHave you taken the Laplace transform of your answer to see if you getthe original function?=== === Subject: : Re: functions that halt===>Subject: Re: functions that halt>Message-id: <75akk1-qkh.ln1@lexi2.athghost7038suus.netAs it is, I'd like for someone to recode CollatzIterations() using>nothing but fixed-count for loops. (The count can be any>integer computable from the input parameter, of course.)import gmpydef collatz(x): a = gmpy.mpz(x) t = 0 while a>1: f = gmpy.scan1(a,0) if f>0: t += f a = a/2**f else: t += 1 a = a*3 + 1 return tdef big_bang(core_radius): core_iterations = len(gmpy.digits(core_radius,2)) nebula_radius = 0 for j in xrange(core_iterations): nebula_radius += 1.585 nebula_iterations = 0 for j in xrange(int(nebula_radius)): nebula_iterations += 2.410 return core_iterations*2 + int(nebula_iterations)*3b = big_bang(2**10000 - 1)c = collatz(2**10000 - 1)print the big_bang calculation:, bprint actual Collatz run:, cprint (float(b)/float(c)-1)*100,% errorthe big_bang calculation: 134588 actual Collatz run: 1344040.13690068748 % errorClose enough for engineering.>-- >#191, ewill3@earthlink.net>It's still legal to go .sigless.MensanatorAce of Clubs=== === Subject: : Re: functions that haltIn sci.math, Mensanator === Subject: : Re: functions that halt>Message-id: <75akk1-qkh.ln1@lexi2.athghost7038suus.net> As it is, I'd like for someone to recode CollatzIterations() using>nothing but fixed-count for loops. (The count can be any>integer computable from the input parameter, of course.)> import gmpy> def collatz(x):> a = gmpy.mpz(x)> t = 0> while a>1:I said *fixed* count; this is a mu-loop. Nice try, though. :-)> f = gmpy.scan1(a,0)> if f>0:> t += f> a = a/2**f> else:> t += 1> a = a*3 + 1> return t> def big_bang(core_radius):> core_iterations = len(gmpy.digits(core_radius,2))> nebula_radius = 0> for j in xrange(core_iterations):> nebula_radius += 1.585> nebula_iterations = 0> for j in xrange(int(nebula_radius)):> nebula_iterations += 2.410> return core_iterations*2 + int(nebula_iterations)*3> b = big_bang(2**10000 - 1)> c = collatz(2**10000 - 1)> print the big_bang calculation:, b> print actual Collatz run:, c> print (float(b)/float(c)-1)*100,% error> the big_bang calculation: 134588> actual Collatz run: 134404> 0.13690068748 % error> Close enough for engineering.Heh.>-- >#191, ewill3@earthlink.net>It's still legal to go .sigless.> Mensanator> Ace of Clubs#191, ewill3@earthlink.netIt's still legal to go .sigless.=== === Subject: : Re: functions that halt===> === Subject: : Re: functions that halt>Message-id: In sci.math, Mensanator> === Subject: : Re: functions that halt>Message-id: <75akk1-qkh.ln1@lexi2.athghost7038suus.net> As it is, I'd like for someone to recode CollatzIterations() using>nothing but fixed-count for loops. (The count can be any>integer computable from the input parameter, of course.)> import gmpy> def collatz(x):> a = gmpy.mpz(x)> t = 0> while a>1:>I said *fixed* count; this is a mu-loop. Nice try, though. :-)This wasn't inteded to be the fixed loop function, it is onlyhere for comparison.> f = gmpy.scan1(a,0)> if f>0:> t += f> a = a/2**f> else:> t += 1> a = a*3 + 1> return tThe big_bang() function is the fixed loop form. Of course, it doesn'tactually solve CollatzIterations, it just estimates it, much liken/(log n) estimates the prime counting function.And it only works for Mersenne numbers (2**n - 1). Fermat numbers(2**n + 1), for example, require a different function.> def big_bang(core_radius):> core_iterations = len(gmpy.digits(core_radius,2))> nebula_radius = 0> for j in xrange(core_iterations):> nebula_radius += 1.585> nebula_iterations = 0> for j in xrange(int(nebula_radius)):> nebula_iterations += 2.410> return core_iterations*2 + int(nebula_iterations)*3> b = big_bang(2**10000 - 1)> c = collatz(2**10000 - 1)> print the big_bang calculation:, b> print actual Collatz run:, c> print (float(b)/float(c)-1)*100,% error> the big_bang calculation: 134588> actual Collatz run: 134404> 0.13690068748 % error> Close enough for engineering.>Heh.>-- >#191, ewill3@earthlink.net>It's still legal to go .sigless.> -->-- >#191, ewill3@earthlink.net>It's still legal to go .sigless.MensanatorAce of Clubs=== === Subject: : g(x+y)=g(x)g(y),then..Let g be a function on R to R which is not identically zero and whichsatisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then(a)g is continuous at every point of R if and only if it is continuousat one point of R.(b)g(x)>0 for all x in R.(c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.(d)The function g is strictly increasing, is constant, or is strictlydecreasing according as g(1)>1, g(1)=1, or 0Let g be a function on R to R which is not identically zero and which>satisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then>(a)g is continuous at every point of R if and only if it is continuous>at one point of R.>(b)g(x)>0 for all x in R.>(c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.>(d)The function g is strictly increasing, is constant, or is strictly>decreasing according as g(1)>1, g(1)=1, or 0I want to know how to show part(d) of the questioin.I'm a little surpised that there have been so few responses.Unless g is continuous, (d) is surely false.I suggest that you instead try to prove that the conditions in (d)would imply that g is continuous. In that case, the existence ofnon-continuous solutions would show that (d) is in general false.-----BEGIN PGP SIGNATURE-----Version: GnuPG v1.2.4 (SunOS)iD8DBQFAeWd8vmGe70vHPUMRAn/hAJ9Ry8e+ wQpk9K1wn9DXyhdHxv137ACffjbZ53G+6LM+jPh6CBvElYzbCjo==/+4Q----- END PGP SIGNATURE-----=== === Subject: : Re: g(x+y)=g(x)g(y),then..> Let g be a function on R to R which is not identically zero and which> satisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then> (a)g is continuous at every point of R if and only if it is continuous> at one point of R.> (b)g(x)>0 for all x in R.> (c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.> (d)The function g is strictly increasing, is constant, or is strictly> decreasing according as g(1)>1, g(1)=1, or 0 I want to know how to show part(d) of the questioin.For one thing, any such continuous function can be show to have the form f(x) = e^(b*x), for some real number b, from which (b), (c) and (d) follow trivially.=== === Subject: : Re: g(x+y)=g(x)g(y),then..> Let g be a function on R to R which is not identically zero and which> satisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then> (a)g is continuous at every point of R if and only if it is continuous> at one point of R.> (b)g(x)>0 for all x in R.> (c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.> (d)The function g is strictly increasing, is constant, or is strictly> decreasing according as g(1)>1, g(1)=1, or 0 I want to know how to show part(d) of the questioin.> For one thing, any such continuous function can be show to have the form > f(x) = e^(b*x), for some real number b, from which (b), (c) and (d) > follow trivially.But we don't know if g is continuous..=== === Subject: : Re: g(x+y)=g(x)g(y),then..The purpose of this whole procedure is showing that g(x)=a^x for some a, sowe cannot use that fact, this is a problem for Bartle's The Elements ofReal Analysis> Let g be a function on R to R which is not identically zero and which> satisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then> (a)g is continuous at every point of R if and only if it is continuous> at one point of R.> (b)g(x)>0 for all x in R.> (c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.> (d)The function g is strictly increasing, is constant, or is strictly> decreasing according as g(1)>1, g(1)=1, or 0 I want to know how to show part(d) of the questioin.> For one thing, any such continuous function can be show to have the form> f(x) = e^(b*x), for some real number b, from which (b), (c) and (d)> follow trivially.=== === Subject: : Re: g(x+y)=g(x)g(y),then..> The purpose of this whole procedure is showing that g(x)=a^x for some a, so> we cannot use that fact, this is a problem for Bartle's The Elements of> Real Analysis> Let g be a function on R to R which is not identically zero and which> satisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then> (a)g is continuous at every point of R if and only if it is continuous> at one point of R.> (b)g(x)>0 for all x in R.> (c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.> (d)The function g is strictly increasing, is constant, or is strictly> decreasing according as g(1)>1, g(1)=1, or 0 I want to know how to show part(d) of the questioin.> For one thing, any such continuous function can be show to have the form> f(x) = e^(b*x), for some real number b, from which (b), (c) and (d)> follow trivially.but we don't know if g is continuous!=== === Subject: : Re: g(x+y)=g(x)g(y),then..> The purpose of this whole procedure is showing that g(x)=a^x for some a, so> we cannot use that fact, this is a problem for Bartle's The Elements of> Real Analysis>Let g be a function on R to R which is not identically zero and which>satisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then>(a)g is continuous at every point of R if and only if it is continuous>at one point of R.>(b)g(x)>0 for all x in R.>(c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.>(d)The function g is strictly increasing, is constant, or is strictly>decreasing according as g(1)>1, g(1)=1, or 0I want to know how to show part(d) of the questioin.> For one thing, any such continuous function can be show to have the form> f(x) = e^(b*x), for some real number b, from which (b), (c) and (d)> follow trivially.> but we don't know if g is continuous!We know, in accord with part (a), that if g is discontinuous anywhere then it is discontinuous everywhere.One can show, along the lines of (xc), that if g(a) = b, then for any rational number q, g(q*a) = b^q, which makes g continuous on (the restriction of domain to) the set of rational multiples of a.Thus, given a Hamel basis of the reals as a vector space over the field of rationals, the values of g at the points of that Hamel basis are necessary and sufficient to determine g uniquely.=== === Subject: : Re: g(x+y)=g(x)g(y),then..Virgil We know, in accord with part (a), that if g is discontinuous anywhere> then it is discontinuous everywhere.> One can show, along the lines of (xc), that if g(a) = b, then for any> rational number q, g(q*a) = b^q, which makes g continuous on (the> restriction of domain to) the set of rational multiples of a.> Thus, given a Hamel basis of the reals as a vector space over the field> of rationals, the values of g at the points of that Hamel basis are> necessary and sufficient to determine g uniquely.I know g(q*a)=g(a)^q for all rational number qbut I don't know how to show it is also true for all real number qCould you tell me more?I've never heard about Hamel basis before, thank you for telling me!I read the definition from the following webpage, but I still can't solvethe question by using it.http://mathworld.wolfram.com/HamelBasis.html=== === Subject: : Re: g(x+y)=g(x)g(y),then.. smallmilk.bbs@wretch.twbbs.org Virgil We know, in accord with part (a), that if g is discontinuous anywhere> then it is discontinuous everywhere.> One can show, along the lines of (xc), that if g(a) = b, then for any> rational number q, g(q*a) = b^q, which makes g continuous on (the> restriction of domain to) the set of rational multiples of a.> Thus, given a Hamel basis of the reals as a vector space over the field> of rationals, the values of g at the points of that Hamel basis are> necessary and sufficient to determine g uniquely.> I know g(q*a)=g(a)^q for all rational number q> but I don't know how to show it is also true for all real number q> Could you tell me more?The whole point is that while g(q*a)=g(a)^q must hold for all rational q, even when g is discontinuous, it will only hold for all real q when g is continuous.> I've never heard about Hamel basis before, thank you for telling me!> I read the definition from the following webpage, but I still can't solve> the question by using it.> http://mathworld.wolfram.com/HamelBasis.html=== === Subject: : Re: g(x+y)=g(x)g(y),then..> Let g be a function on R to R which is not identically zero and which> satisfies the function equation g(x+y)=g(x)g(y) for all x,y in R, then> (a)g is continuous at every point of R if and only if it is continuous> at one point of R.> (b)g(x)>0 for all x in R.> (c)g(0)=1. If a=g(1),then a>0 and g(r)=a^r for all r in Q.> (d)The function g is strictly increasing, is constant, or is strictly> decreasing according as g(1)>1, g(1)=1, or 0 I want to know how to show part(d) of the questioin.For (d), use (c). You muyst be assuming g is continuous at this point, right?=== === Subject: : Re: Recurrence for Squares and S_r Type Sequences ** First I recall the S_r sequences ** Second I introduce R_r sequences ** Third ask about R-friends and S-friends**** First ****> {a(n)} is called S_r sequence iff> a(0)=0, a(1)=1, a(2)=r, (1) and> a(n-1)*a(n+1)=(a(n)-1)^2 (2)> Examples:> S_4 = A000290 the squares> S_5 = A004146 Alternate Lucas Numbers - 2> S_7 = A054493 A Pellian-related sequence> S_8 = A001108 a(n)-th triangular number is a square> S_9 = A049684 F(2n)^2 where F() = Fibonacci numbers> S_20 = A049683 a(n)=(L(6n)-2)/16, L=Lucas Sequence> S_36 = A001110 Both triangular and square> S_49 = A049682 a(n)=(L(8n)-2)/45, L=Lucas sequence> I notice that further connections will be revealed, if we> look for other sequences, whose squares are of S_r type.Some progress has been achieved in the meantime. (S_6 is nowin the OEIS :-), it's A092184.Let us define S(x) = (x-1)^2. Then recurrence (2) is simplya(n-1)*a(n+1) = S(a(n)).**** Second ****If we use function R: R(x) = x^2-1, then we get a morebasic family of integer sequences R_r, starting 0,1,r,r^2-1,...with recurrence a(n-1)*a(n+1) = R(a(n)).It is nice and easy to see that S_{r^2} = {R_r}^2.(See R-friens and S-friend below in Third.)The sequences of R_r type were already in the OEIS (except for r=21):R_2 = A000000 the natural numbersR_3 = A001906 bisection of Fibonacci sequenceR_4 = A001353 Number of spanning trees in 2 X n gridR_5 = A004254 m=7 of Wolfdieter Lang's m-familyR_6 = A001109 a(n)^2 is a triangular numberR_7 = A004187 a(n)=F(4n)/3 F=Fibonacci numbersR_8 = A001090 a(n) = U(n-1,4) Chebyshev polynom 2. kindR_9 = A018913 a(n) = U(n-1,9/2), author R. K. GuyR_10 = A004189 a(n) = U(n-1,5), author Neil SloaneR_11 = A004190 Chebyshev or generalized Fibonacci sequence (m=13)R_12 = A004191 a(n) = U(n-1,6), author Wolfdieter Lang (m=14)R_13 = A078362 A Chebyshev S-sequence with diophantine propertyR_14 = A007655 Related to Fleenor-Heronian trianglesR_15 = A078364 G.f.: 1/(1-15*x+x^2) m=17 in Lang's m-familyR_16 = A077412 Chebyshev U(n,x) polynomial evaluated at x=8R_17 = A078366 A Chebyshev S-sequence with diophantine propertyR_18 = A049660 a(n)=F(6n)/8 F=Fibonacci numbersR_19 = A078368 G.f.: 1/(1-19*x+x^2) Chebyshev U(n-1,19/2)R_20 = A075843 99*a(n)^2 + 1 is a square (G. V. Richardson)R_21 = A092499 Sequence R_21: 0,1,21,...A,B,C,... A*C=B^2-1R_22 = A077421 Chebyshev sequence U(n,11) with diophantine propertyR_4 has an interesting relationship to sqrt(3) (see A001906):For n>0, ratios a(n+1)/a(n) may be obtained as convergentsof the continued fraction expansion of 2+sqrt(3)**** Third ****A=a(n) and B=a(n+1) are /R-friends/ in the following sense: A divides R(B)=B^2-1 and B divides R(A)=A^2-1 (R)Computing R-friends A < B up to A = 7000, I always found that theybelonged to some R_r sequence. I.e. we always have (A^2-1)/B integer.Isn't that funny?For /S-friends/ we define: A and B are S-friends iff A divides S(B)=(B-1)^2 and B divides S(A)=(A-1)^2 (S)I like the following Lemma: If A and B are R-friends, then A^2 and B^2 are S-friends.Proof: A and B R-friends --> A|(B^2-1) and B|(A^2-1) --> A^2|(B^2-1)^2 and B^2|(A^2-1)^2 --> A^2 and B^2 S-friends.I will check, whether S-friends always belong to some S_r type sequence.**** Conclusion ****These investigations are a lot of fun because they connect differentinteger sequences ...A,B,C,... via simple A*C=R(B) or A*C=S(B),making them members of R- and S-families. There are connectionsbetween the families too, since the square of R_r is S_q with q=r^2.While R_r sequences are quite complete listed by Chebyshevs U(n,x) oras members of the m-family of W. Lang, S_r sequences are more likeindividuals up to now. S_r sequences with non-square r seem to beespecially nice.=== === Subject: : Re: Recurrence for Squares and S_r Type Sequences> R_8 = A001090 a(n) = U(n-1,4) Chebyshev polynom 2. kind> R_9 = A018913 a(n) = U(n-1,9/2), author R. K. Guy> R_10 = A004189 a(n) = U(n-1,5), author Neil SloaneAha, the R_r family is best explained by Chebyshev's U-functionand Horadam's W-family w(a,b;p,q). So I started to learn moreabout these functions. Here is something which seems like amistake in Maple's documentation:Maple says: If the first parameter is not equal to a non-negative integer, then this is the analytic extension of ChebyshevU and is such that: ChebyshevU(mu,z) = hypergeom([-mu,mu+2],[3/2],(1-z)/2)Could someone please confirm that this should read If the *second* parameter is not equal to a non-negative integer ...and that the equation should be ChebyshevU(mu,z) = (mu+1)*hypergeom([-mu,mu+2],[3/2],(1-z)/2)This last line is just a guess, because it works well e.g. forR. K. Guy's sequence R_9 = A018913 = 0,1,9,80,711,6319,56160,...We have ChebyshevU(4,9/2) = 6319but hypergeom([-4,4+2],[3/2],(1-9/2)/2) = 1263.8.Multiplication gives the correct value: (4+1)*1263.8 = 6319.=== === Subject: : Basic Calculus questionA ship is to make a voyage of 100km at a constant speed of v kmh. The running cost of the ship is (0.8v^2+2000/v) per hour, and thisis to be kept to minimum. i) Write down the time taken to go 100km at constant speed of vkmhii) Hence write down the total cost, c, of travelling 100km at vkmh iii) By considering DC/DV find the speed which keeps the cost of thejourney to a minimum.(Give your answer to the nearest kmh)iv) Find the minimum cost of the voyage In revision for my exams I was wondering if maybe someone couldexplain this question for me to enhance my understaning of steps totake when applying calculus. Thanx any help would be greatlyappreciated.=== === Subject: : Re: Basic Calculus question> A ship is to make a voyage of 100km at a constant speed of v kmh.> The running cost of the ship is (0.8v^2+2000/v) per hour, and this> is to be kept to minimum.> i) Write down the time taken to go 100km at constant speed of vkmh> ii) Hence write down the total cost, c, of travelling 100km at vkmh> iii) By considering DC/DV find the speed which keeps the cost of the> journey to a minimum.(Give your answer to the nearest kmh)> iv) Find the minimum cost of the voyage> In revision for my exams I was wondering if maybe someone could> explain this question for me to enhance my understaning of steps to> take when applying calculus. Thanx any help would be greatly> appreciated.First, can you do i) ?=== === Subject: : Re: Basic Calculus questioni) T = 100/v hii) total cost = ( 0.8 v^2 +2000/v) T = 80 v + 200,000/(v^2) iii) dc/dv = 80 - K*200,000 / v^3max set dc/dv = 0 (slope=0 at max and min points, graph the curve first andsee it)80-K*200,000/ v^3 = 0v^3 = K*200,000 / 80v = cube root of a constant (K*200,000 / 80)This looks like homework so I left in K you can find it, and check if iii)is rightThis is differential calculus-Chu Ito> A ship is to make a voyage of 100km at a constant speed of v kmh.> The running cost of the ship is (0.8v^2+2000/v) per hour, and this> is to be kept to minimum.> i) Write down the time taken to go 100km at constant speed of vkmh> ii) Hence write down the total cost, c, of travelling 100km at vkmh> iii) By considering DC/DV find the speed which keeps the cost of the> journey to a minimum.(Give your answer to the nearest kmh)> iv) Find the minimum cost of the voyage> In revision for my exams I was wondering if maybe someone could> explain this question for me to enhance my understaning of steps to> take when applying calculus. Thanx any help would be greatly> appreciated.> First, can you do i) ?=== === Subject: : Re: Resistance to Change> Very true, if the proposed great improvement is just silly.> No, quite the opposite. If a proposal is patently without merit, then> it is very easy to point out the flaws and avoid the impression that> the above gives: that the person making this response has no valid> objection and is just avoiding the question.> Well if it was that easy, why didn't the author do it themselves?Must not have been flawed in the first place.> Now if you *suggested* a meathod would improve on the status quo, AND> you were able to back it up, having done a bit of research and then> came here using our language to describe a theroem you want this body> to accept, then you're going to get a better response. I think this> doesn't have anything to do with resistance to change as much as it> has to do with the thought rude people are treated rudely.Ok, read these and tell me what you think:http://www.arxiv.org/html/cs.lo/0003071http:// www.mathpreprints.com/math/Preprint/CharlieVolkstorf/ 20021008.1/1If you'd like, I'll post a summary. And I'll certainly answer anyquestions or criticisms that you might have - with technicalresponses, not anger or sarcasm. Just tell me if it's worthwhile andwho has ever done it before, ok?Charlie VolkstorfCambridge, MA=== === Subject: : Re: Resistance to Change> . . ., then you are apt to get a much different response, one that:> (1) lacks mathematical content, (2) is emotional, e.g. expresses> anger or sarcasm, and (3) is resistant to making any meaningful> analysis or comments.> There is a strong prior belief that it is very unlikely that the> solution to a famous problem such as P=NP will be solved by a> previously unknown internet poster. Evenmoreso, there is an almost as> strong presumption that such an attempt will contain quite basic> errors. The response is an expression of irritation with someone who> attempts to tackle an elephant with a pea shooter. The thought is,> Do you really think somebody who knows essentially nothing is going> to stumble over a solution to a problem that someone much better> prepared might spend their life on? They find the chutzpah excessive> to the point of being insulting.1. I didn't say anything about famous unsolved problems ofmathematics. As you quoted above, I am talking about improvements tofundamental aspects of theoretical computer science. These are notspecific problems that have been publicized, for which people havecontributed to the quest for a solution for many years (e.g. byproving less general theorems) or for which awards have been offered. Many amateurs have proposed solutions to FLT, but few have proposedways to represent computer programs or theorems from the Theory ofComputation and how they are derived.I am not talking about solving long standing problems of mathematics. I am talking about making fundamental changes to how concepts arerepresented and problems are solved, and showing how improvements canthus be realized.2. You are making numerous assumptions that fall outside of myquestion: The poster is previously unknown, the proposal has basicerrors, the proposal is a pea shooter, he knows essentiallynothing, the person responding is better prepared and has spent theirlife on the problem. In fact, you are merely constructing atautology, equivalent to: If a smart person works on a problem for along time and doesn't solve it, do you think a dumb person working onit for a short while probably will?. These assumptions andcircumstances are not part of what I describe.I also fail to see why there should be a correlation between aperson's fame and presumed abilities. That seems to be defending theold boy's club principle of political clout taking the place ofcompetency.3. The reference to being irritated and feeling insulted is an exampleof what I was describing as being inappropriate and nonproductive. Ifsomeone claims to be as skilled as you say, then they are able toadvanced programming text years ago, which has been discussed in printa number of times, referred to as a standard text on the subject, andhas been used by others to teach classes on the subject. When I am athonored. If it is particularly simple or contains errors, I amreminded that my work is better than many other people's. And if theychallenge my conclusions, I eagerly enter into a technical debate, inwhich I either reaffirm the validity of my work or learn something new- both positive results.Honestly, I think it is really a question of emotional maturity. Atechnical response is appropriate. An emotional response isinappropriate and counter productive. It avoids the question at handaltogether, prolongs the debate, stands in the way of progress andmakes me wonder why there is such resistance to change. I think thatperhaps you have contributed, at one level at least, to answering thatquestion, for which I thank you.Charlie VolkstorfCambridge, MA=== === Subject: : Re: Resistance to Changesays...>[...]>But the fact that some geniuses were laughed at does not imply that all who are>laughed at are geniuses. They laughed at Columbus, they laughed at Fulton, they>laughed at the Wright brothers. But they also laughed at Bozo the Clown.> Hmmm... I have the strong impression that it was Martin Gardner, in> Fads and fallacies in the name of science who said something like> that... I'll have to check my copy at home over the weekend...Arturo,Go to amazon.com and search for also laughed at Bozo and you willget five hits. The second one is Carl Sagan's Broca's Brain.If you have a user account there, you can click on the page 75link and it will take you to a scanned page where the quote canbe found.The other four hits are quote attributions to Sagan.Christer EricsonSony Computer Entertainment, Santa Monica=== === Subject: : Re: Resistance to Change> The most recent example that I have witnessed was: It really _is_> quite an accomplishment when you stop to think about it. All those> people studying recursive this and that all these years, and nobody> has ever realized that the notion can be expressed in a formal> notation! I mean it's positively inspiring. It's going to allow> people to give rigorous proofs of all those facts that they've just> been waving their hands at.> Sounds like Ullrich. :-)> The response lacks mathematical content and is fraught with sarcasm.> Right.So he does that a lot, huh?> On the other hand, if you subtract the sarcasm there's actually> *a message* in his saying.> Let me try to help you:Sure! I'm always open to learning from any source.> Think about it: All those people studying recursive this and> that all these years, and nobody has ever realized that the> notion can be expressed in a formal notation?> Hard to believe..., actually to c l a i m that would be utter> nonsenseYou seem to be putting forth the proposition that if one believes aproposal to be silly nonsense, then you need not justify thatopinion. But, as I posted elsewhere in this thread, doesn't thatrisk,(1) the possibility of being wrong (as so many have been in the past),which would be uncovered if you attempted to justify your opinion(which, after all, should be easy of something so patently wrong),and,(2) allowing unscrupulous people to use this as a tactic to avoidjustifying their rejection of a proposal that they actually know to bevalid and worthwhile?> (You'd better check the relevant literature!)Of course. But it's not something you check once. It's somethingthat I've been studying for years. In any case, what resources do youuse in general? And what have you discovered to affirm or refute theassertion that these notions and proofs concerning computability havebeen formalized? (Now you seem to be advocating my approach of makingtechnical, rather than emotional and unsubstantiated, responses! :)> [Of course, a formal treatment is]> going to allow people to give rigorous proofs of all those> facts that> [already have been proved using standard mathematical arguments/methods.> But actually there's nothing WRONG with the usual/standard approach,> mathematician did it for more than 2400 years, and they STILL do it -> quite successfully, btw.]Are you saying that you see no value to formal proofs within theTheory of Computation? Or, should I say, more generally, whatadvantages do you see in formalizing these proofs?Also, note what Ullrich was referring to, in his statement: Of coursesomething like I(f) would make more sense than f(I) here to which Ipointed out, among other problems, I use Predicate Calculus wffs. The syntax is not changed. I make a simple extension to thesemantics. You use Predicate Calculus plus a function of a functionand lambda calculus notation. That is unnecessarily complex anddoesn't allow the existing machinery of manipulations to PredicateCalculus wffs since you no longer have Predicate Calculus wffs.In other words, I stay very much within fundamental Mathematics,whereas Ullrich's proposal unnecessarily goes outside of this simpleframework. (He never did respond to my question as to why I(f) wouldmake more sense than f(I). But, as you mentioned above,unsubstantiated statements are characteristic of his postings.)> If you are interested in this topic, you might read> When is a proof? by Keith Devlin> http://www.maa.org/devlin/devlin_06_03.html> for a starter.logically correct or a convincing argument. I would say that thoseare two independent characteristics (predicates.) Hopefully theargument is both, and it is quite unfortunate when it is convincingand not logically correct! I think that a proof can be formal orinformal and still be correct. And any correct proof can beformalized, simply by considering a collection of similar proofs thatincludes it and developing a formal representation of them.BTW Coincidentally, I saw that URL (I believe linked from a mathemail address that he gives, on 6/9/03:I would say that a proof (or refutation) is the evaluation of anexpression whose value is true or false, i.e., a predicate. Ifthis evaluation is rigorous, then it is the execution history of aprogram with an output that is always true or false. For example,the proof of FLT is the evaluation of the logical expression(predicate calculus wff) that states that it is true. (Fordefinitions of these terms, see the relevant literature.)Charlie VolkstorfIn other words, a proof is an evaluation of an expression, such as1+(2 x 3) except that the final result is TRUE or FALSE, as in 1+(2 x3) > 5. If the value is TRUE (FALSE), then we have proved (refuted)it. As far as what constitutes an evaluation goes, I sometimes thinkof that as being based on convention: Addition is defined as thefunction that some particular Turing Machine computes, and the onlything that we can say about addition is what can be expressedregarding that program. But that's just my opinion. (I received noresponse from Devlin. :( But I guess he gets lots of fanmail.)Charlie VolkstorfCambridge, MA> F.=== === Subject: : Re: Map distribution from plane to sphere>Hi all,>I have a question regarding the tranformation of a probability density>distribution (pdf) taken at a plane to the surface of a sphere.>An exampel : >A distribution is uniform (isotropically) distributed on a x,y-plane>with marginal pdf's>p(x) = 1; p(y) = 1;>On a sphere, the marginal pdf for the isotropic distribiution is>p(theta) = 1/2 sin(theta) according to>mathworld.wolfram.com/SpherePointPicking.html>My question: >For a pdf on the plane that has a Gaussian as marginal pdf in x>direction and a uniform distribution in y direction, how do I map such>a distributon onto the sphere ?>I know that the Fisher distribution on the sphere is the analogue to>the Gaussian in the plane, but I have no idea if I can perfoirm a>transformation and how.It seems to me you need more details in your specification. In the case of a uniform distribution, the specification is clear: Surface area is well-defined, and you want the measure of a region proportional to its surface area.What exactly do you expect of your partial Gaussian distributuion? I expect that you want it uniform on strips parallel to the equator. But strips or wedges from one pole to the other are of finite length, whereas the support of a Gaussian distribution is the entire line. Do you want a truncated Gaussian? Or would the projection of a plane onto the sphere (as one does with the complex plane) with the distribution of independent normals on the plane satisfy your needs?Stephen J. Herschkorn herschko@rutcor.rutgers.edu=== === Subject: : Re: Cardinality Adjunct Assistant Professor at the University of Montana.>Correction and new questions: >1.- I just want to know if the class of all sets having fix>cardinality is a set...>or if there exist a set A sucht that for each cardinality x there>exist a set a in A such that |a| = x . >... Ok.. What is a Proper class??Intuitively, a collection too big to be a set. Things like allsets. As somoneone pointed out, the only case of both your questionsthat has an affirmative answer is when the fixed cardinality is 0,as then the collection of all sets of cardinality 0 is the set whoseonly element is the empty set.=============== === Subject: : Re: Cardinality>Correction and new questions: >1.- I just want to know if the class of all sets having fix>cardinality is a set...>or if there exist a set A sucht that for each cardinality x there>exist a set a in A such that |a| = x . >... Ok.. What is a Proper class??> Intuitively, a collection too big to be a set. Things like all> sets. As somoneone pointed out, the only case of both your questions> that has an affirmative answer is when the fixed cardinality is 0,> as then the collection of all sets of cardinality 0 is the set whose> only element is the empty set.> -- ==================> Arturo Magidin> magidin@math.berkeley.eduOk... I understand this.... but, what about the oder questions? Imean, the questions that i made but with total order insted ofwell order any idea?=== === Subject: : Re: what to calculate them?En el mensaje:44jg70p334rm6adstet7qt3s4apj337m15@4ax.com, escribi.97:> [IMG]http://www.geocities.com/i141802596/Image1.txt[/IMG]> [IMG]http://www.geocities.com/i141802596/Image2.txt[/IMG]> [IMG]http://www.geocities.com/i141802596/Image3.txt[/IMG]> Two hints: (i) if you take a jpg file and rename it .txt before> posting it won't work. (ii) when you post something on a web> site you should always check that you can see what you> posted before publishing the url.> ************************> If you delete [/IMG], the links work. By example.http://www.geocities.com/i141802596/Image1.txtBut it seems a complete sheet of homework ...Ignacio Larrosa Ca.96estroA Coru.96a (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com=== === Subject: : Re: what to calculate them?>En el mensaje:44jg70p334rm6adstet7qt3s4apj337m15@4ax.com,> escribi.97:> [IMG]http://www.geocities.com/i141802596/Image1.txt[/IMG]> [IMG]http://www.geocities.com/i141802596/Image2.txt[/IMG]> [IMG]http://www.geocities.com/i141802596/Image3.txt[/IMG]> Two hints: (i) if you take a jpg file and rename it .txt before> posting it won't work. (ii) when you post something on a web> site you should always check that you can see what you> posted before publishing the url.> ************************>If you delete [/IMG], the links work. By example.>http://www.geocities.com/i141802596/Image1.txtYour browser is working different from mine. I diddelete the IMG, and got a page of text, consistingof the binary contents of a jpg.Hmm, tried IE(shudder) just now, and it displays itas a jpg. I don't think that it _should_ do so -there's nothing about the file that marks it as ajpg image except that the _contents_ appearjpgish... >But it seems a complete sheet of homework ...Yeah, there's also that.=== === Subject: : Re: what to calculate them?>[IMG]http://www.geocities.com/i141802596/Image1.txt[/IMG ]>[IMG]http://www.geocities.com/i141802596/Image2.txt[/IMG] >[IMG]http://www.geocities.com/i141802596/Image3.txt[/IMG]Two hints: (i) if you take a jpg file and rename it .txt before posting it won't work. (ii) when you post something on a website you should always check that you can see what youposted before publishing the url.=== === Subject: : Re: What experiment should have gone to Mars> -----BEGIN PGP SIGNED MESSAGE-----> I know, John Kerry & his butt-buddy Teddy the it took> me more than twelve hours to find a telephone Kennedy.> They're both demoslutic crack-whores who belong on Mars.The bible you claim to love so much teaches tolerance, understanding andforgiveness. Jesus spoke in favour of the outcasts of society, of the weak,of the lost. Your posts show you to have a profound bigotry and intoleranceagainst those very people who, were Jesus alive today, he would defend. MaryMagdelene was a whore. Have you actually read the bible? Your behaviour isdeeply unchristian, and an embarrassment to all good followers of the princeof peace.The one good thing about your posts to the newsgroups is that they displayyou as the ignorant, foul-minded, nobody that you really are. Now you maystop, since all of usenet is aware of this by now, so no more purpose can beserved by the continued promulgation of your drivel.=== === Subject: : Re: What experiment should have gone to Mars> -----BEGIN PGP SIGNED MESSAGE-----> I know, John Kerry & his butt-buddy Teddy the it took> me more than twelve hours to find a telephone Kennedy.> They're both demoslutic crack-whores who belong on Mars.> The bible you claim to love so much teaches tolerance, understanding and> forgiveness. Jesus spoke in favour of the outcasts of society, of theweak,> of the lost. Your posts show you to have a profound bigotry andintolerance> against those very people who, were Jesus alive today, he would defend.Mary> Magdelene was a whore. Have you actually read the bible? Your behaviouris> deeply unchristian, and an embarrassment to all good followers of theprince> of peace.> The one good thing about your posts to the newsgroups is that they display> you as the ignorant, foul-minded, nobody that you really are. Now you may> stop, since all of usenet is aware of this by now, so no more purpose canbe> served by the continued promulgation of your drivel.Oh but he wants people to be angry and respond to him, that's what Trollsdo!O'=== === Subject: : Re: What experiment should have gone to Mars.> Oh but he wants people to be angry and respond to him, that's what Trolls> do!That's not what a troll is. A troll attempts to deceive someone into making a foolof themselves. You've redefined the word to mean anything that doesn'tconform to your opinions or sensibilities.> O'=== === Subject: : Re: What experiment should have gone to Mars> .> Oh but he wants people to be angry and respond to him, that's whatTrolls> do!> That's not what a troll is. A troll attempts to deceive someone intomaking a fool> of themselves. You've redefined the word to mean anything that doesn't> conform to your opinions or sensibilities.Are you a newbie? You obviously haven't been posting long if you trulybelieve that. Talk to Rudolph_X if you don't believe me. A troll's successis measured in the length of time his thread is carried on. The hotter thediscussion, the bigger the thread, the greater the tribute to the Troll.There are all sorts of Trolls. How about the one who thinks he has theultimate answer and blathers on about it ad infinitum, usually based uponhis or her own ignorance. I already knew that would happen, blah blah blah I predicted.... My many theories on this subject..... You morons will never figure it out.... You people JUST DON'T GET IT.... You people are enslaved to conventional knowledge, you don't neededucation, just use my new and improved theory and everything is easilyseen...The intent being to call attention to himself - perhaps to validate his ownviews, perhaps to just annoy people or perhaps because he's a nutcase.Motivations are hard to read through the many layers of jibberish.O'> O'=== === Subject: : Re: What experiment should have gone to Mars>.>Oh but he wants people to be angry and respond to him, that's what> Trolls>do!>That's not what a troll is. A troll attempts to deceive someone into> making a fool>of themselves. You've redefined the word to mean anything that doesn't>conform to your opinions or sensibilities.> Are you a newbie? You obviously haven't been posting long if you truly> believe that. Talk to Rudolph_X if you don't believe me. A troll's success> is measured in the length of time his thread is carried on. The hotter the> discussion, the bigger the thread, the greater the tribute to the Troll.> There are all sorts of Trolls. How about the one who thinks he has the> ultimate answer and blathers on about it ad infinitum, usually based upon> his or her own ignorance.> I already knew that would happen, blah blah blah> I predicted....> My many theories on this subject.....> You morons will never figure it out.... > You people JUST DON'T GET IT.... > You people are enslaved to conventional knowledge, you don't need> education, just use my new and improved theory and everything is easily> seen...Perhaps yours is a misinterpretation of many of those arguments. While true there is the psychotic breed of trolls, such as Min, OTOH there are others who see bonafide errors in judgment on the part of the mainstream. That you can't distinguish between those arguments is no reflection on their accuracy. I've never seen an argument claiming that education is silly or unnecessary, though I've seen several outlining the reasons that portions of that education amount to little more than hype and religiosity, which BTW is simply a fact of life and actually covered as a subject of interest in other fields of study. If not for discussions about this aspect there would be little else to discuss here, because equations stand on their own merit. OTOH, if you want to discuss mathematical procedures and concepts, then go to sci.math. Arguing against those who correctly note your delusions only supports the validity of their sentiments. Your ad hominem above belies your inability to defend against them by arguing from sound premises. If you don't get it, you don't get it, and an attempt to belittle your accusers by snipping such statements out of context, as though a lengthy argument didn't precede that statement, is childishness, and this tactic prey's upon the similar tendencies of your target audience. Unfortunately there will be those who mindlessly think that you actually made a case with those simple words, which in itself reveals the herd mentality of those claiming to be the objective crowd.Or, IOW, you are the troll, Min is just crazy. He has an excuse, what's yours?Richard Perry> The intent being to call attention to himself - perhaps to validate his own> views, perhaps to just annoy people or perhaps because he's a nutcase.> Motivations are hard to read through the many layers of jibberish.> O'>O'=== === Subject: : Re: What experiment should have gone to MarsComments: This message probably did not originate from the above address. It was automatically remailed by one or more anonymous mail services. You should NEVER trust ANY address on Usenet ANYWAYS: use PGP !!! Get information about complaints from the URL belowX-Remailer-Contact: http://80.65.224.85/POL/ In case my abuse address is unreachable: It is because it has been flooded by , please contact Min is just crazy.CLEAR SKIES! (BRAWHH...HA HA HA HA!)^^^^^ ^^^^^Daniel Joseph Min-----BEGIN PGP SIGNATURE-----iQA/AwUBQHrKvpljD7YrHM/ nEQJbLQCfasyGBlPILHg9nTdQ4neazaU3oeYAn1k5TpbS10hTb1r69UDL38d7QI +O=GOp/-----END PGP SIGNATURE-----=== === Subject: : Re: What experiment should have gone to Mars> -----BEGIN PGP SIGNED MESSAGE-----> Translation: Just because secular academia's much-lauded> Copernican Revolution is completely disproven by scien-> tific facts, then the only thing left for idiots like me> to do is call Min crazy, since none of us can possibly> refute his incontrovertible evidence that he's presented:You haven't actually disproved anything because there is nothing todisprove. All Copernicus did was publish a book that suggested, correctly,that the sun is the centre of the solar system. Your enormous tirade againstthis does not even address this simple fact, instead waffling on ludicrouslyat the straw man claim that science claims that people before Copernicuscouldn't make basic observations of the sky, which of course they could.Your argument isn't wrong, as such, it's just completely irrelevant.Next: Min's proof that water freezes at 0 degrees C.It's interesting that you claim an interest in scientific facts since itseems your whole belief system is based on accepting only that for whichthere is *no* scientific evidence; God, astrology, the moon hoax and soon, while denying that for which there is evidence, basic cosmology, forinstance. Where are your scientific proofs of your astrological predictionsfor instance?Ian=== === Subject: : Re: What experiment should have gone to Mars> .> Oh but he wants people to be angry and respond to him, that's what> Trolls> do!> That's not what a troll is. A troll attempts to deceive someone into> making a fool> of themselves. You've redefined the word to mean anything that doesn't> conform to your opinions or sensibilities.> Are you a newbie? You obviously haven't been posting long if you truly> believe that. Talk to Rudolph_X if you don't believe me. A troll'ssuccess> is measured in the length of time his thread is carried on. The hotterthe> discussion, the bigger the thread, the greater the tribute to the Troll.> There are all sorts of Trolls. How about the one who thinks he has the> ultimate answer and blathers on about it ad infinitum, usually based upon> his or her own ignorance.> I already knew that would happen, blah blah blah> I predicted....> My many theories on this subject.....> You morons will never figure it out.... > You people JUST DON'T GET IT.... > You people are enslaved to conventional knowledge, you don't need> education, just use my new and improved theory and everything is easily> seen...> The intent being to call attention to himself - perhaps to validate hisown> views, perhaps to just annoy people or perhaps because he's a nutcase.> Motivations are hard to read through the many layers of jibberish.I think the strict definition of a troll is somebody who posts purely withthe intention of creating humbug; posting gardening is crap to a gardeninggroup for instance, especially if the person has no interest one way or theother in gardening, so I don't know if Min's execrable effusions countstrictly as trolling, at least the vaguely on topic ones (astronomy,astrology, etc).It seems Min is driven by a desperate need for attention and totallymisplaced belief in himself; he clearly believes himself to be a significantintellectual force, a great thinker that society in its foolishness hasoverlooked, despite the overwhelming evidence to the contrary. I daresaythat once one has concluded that everybody else, everywhere, is wrong, it'seasy for a mindset such as his to develop. The sad thing is that anyone whohas read his wordy dissertations can see that it's very unlikely that he iscapable of ever creating anything of worth to others, so one can onlypresume that he can only get worse.What's most annoying (and here one has to admit a certain grudgingadmiration) is his subversion of the Google Groups system, to turn it intohis publishing house. If only he would actually write something that'sworth reading... May as well also add here that what is most sickening aboutthe little twerp is his recent posts urging military action by others (hisuse of the general we as in we are going to kick ass meaning otherpeople as in other people will have to go fight while I sit here safe incyberspace spouting bilge) while the Great Min chooses to fight only thatmost fearsome of enemies... astronomers...One suspects that he is a very lonely, isolated individual, and in thatsense is to be pitied.Ian=== === Subject: : Re: What experiment should have gone to Mars> The bible you claim to love so much teaches tolerance, understanding and> forgiveness. Jesus spoke in favour of the outcasts of society, of the weak,> of the lost. Your posts show you to have a profound bigotry and intolerance> against those very people who, were Jesus alive today, he would defend. Mary> Magdelene was a whore. Have you actually read the bible? Your behaviour is> deeply unchristian, and an embarrassment to all good followers of the prince> of peace.The story of Jesus is clear proof that no good deed will go unpunished.Bob Kolker=== === Subject: : What experiment should have gone to MarsComments: This message probably did not originate from the above address. It was automatically remailed by one or more anonymous mail services. You should NEVER trust ANY address on Usenet ANYWAYS: use PGP !!! Get information about complaints from the URL belowX-Remailer-Contact: http://80.65.224.85/POL/ In case my abuse address is unreachable: It is because it has been flooded by , please contact They could haveCLEAR SKIES! (BRAWHH...HA HA HA HA!)^^^^^ ^^^^^*Min's Notes On The Cowardice & Abject Incompetence Of Modern-Day Astronomers:*Min's $0.00 (zero cents, was cash $10,000 U.S.) Reward For Orthodox Astronomers:-----BEGIN PGP SIGNATURE-----iQA/AwUBQHmZqJljD7YrHM/ nEQIcHACgtDlwoCAcvgg32cpbxM9lgR4wVLkAoPLXTJwje3Y4KilA1FsW4n7E9M lk=mfxS-----END PGP SIGNATURE-----=== === Subject: : Re: Sqaure Root QuestionEn el mensaje:%V_dc.13953$Y45.4191@fe2.texas.rr.com,> Can Sqrt[5 + 2Sqrt[3]] be written in the form kSqrt[a] + jSqrt[b],> where k and j are rationals, and a and b are integers? I'm trying to> simplify an expression where this comes up, and I know sometimes> expressions of the form Sqrt[s+rSqrt[t]] can be transformed, but I> don't know necessary and sufficient conditions so that I can tell in> this case.Squaring,2sqrt(3) + 5 = a*k^2 + 2k*j*sqrt(a*b) + b*j^2 ==5 = a*k^2 + b*j^2sqrt(3) = k*j*sqrt(a*b) ===> 3 = k^2*j^2*a*b ==b*j^2 = 3/(a*k^2)Let t = a*k^2. Then5 = t + 1/t ===> t^2 - 5t + 1 = 0 ===> t = (5 +/- sqrt(21))/2If a is integer and k rational, t must b rational, but it isn't ...Ignacio Larrosa Ca.96estroA Coru.96a (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com=== === Subject: : Re: Sqaure Root Question> Can Sqrt[5 + 2Sqrt[3]] be written in the form kSqrt[a] + jSqrt[b], where k> and j are rationals, and a and b are integers?No. The polynomial satisfied by sqrt(5+2*sqrt(3)) isx^4 - 10*x^2 + 13 . According to Maple, this has Galois group D(4)(dihedral group, order 8). But k*sqrt(a)+j*sqrt(b) willsatisfy an equation with Galois group at worst Klein 4-group(abelian, order 4).> I'm trying to simplify an> expression where this comes up, and I know sometimes expressions of the form> Sqrt[s+rSqrt[t]] can be transformed, but I don't know necessary and> sufficient conditions so that I can tell in this case.> Zach=== === Subject: : determining transition matrixI have A, a column vector of dimension (5x1) and B, a column vector ofdimension (5x1). I would like to determine a matrix P such that PA = B. Iknow that it is impossible to determine it uniquely, but I have anadditional 5 equations that relate the elements on the rows of P (it is atransition matrix, so they must sum to 1). The way I see it, I have 10equations, 25 unknowns, so if I randomly generate 15 elements of A, I candetermine the other 10 by way of those 10 governing equations.Doing this, however, in practice, is giving me a large headache. I wasn'tabout to do it by hand, but the mathematical package I'm using, Maple,doesn't seem to like the problem in any form I give it. I've triedexplicitly setting the randomly generated elements, letting it determinethem, etc., but I'm not coming up with anything that works, and occasionallyI'm getting an error indicating a singular matrix, which I know isn't true,because I haven't even defined the entries of P yet.Any hints, suggestions? Am I missing something in the theory that would makethis impossible? If you were to go about doing this, how would you?=== === Subject: : Re: determining transition matrix> I have A, a column vector of dimension (5x1) and B, a column vector of> dimension (5x1). I would like to determine a matrix P such that PA = B. I> know that it is impossible to determine it uniquely, but I have an> additional 5 equations that relate the elements on the rows of P (it is a> transition matrix, so they must sum to 1). The way I see it, I have 10> equations, 25 unknowns, so if I randomly generate 15 elements of A, I can> determine the other 10 by way of those 10 governing equations.> Doing this, however, in practice, is giving me a large headache. I wasn't> about to do it by hand, but the mathematical package I'm using, Maple,> doesn't seem to like the problem in any form I give it. I've tried> explicitly setting the randomly generated elements, letting it determine> them, etc., but I'm not coming up with anything that works, andoccasionally> I'm getting an error indicating a singular matrix, which I know isn'ttrue,> because I haven't even defined the entries of P yet.> Any hints, suggestions? Am I missing something in the theory that wouldmake> this impossible? If you were to go about doing this, how would you?Problem solved. Though A and B are exogenously determined, Maple may stilltry to solve for them because... it doesn't know that. There are tworelationships I didn't let it know, that the sum of the elements of A = 1and that of B = 1, with which it was able to get the solution. Hope none ofyou worked on this for too long.=== === Subject: : Fourier analysis on groups questionAre there any practical applications for Harmonic/Fourier analysis ongroups other than R or R/Z? Is Harmonic analysis ever really used on groupsother than R or R/Z? (I suppose anyone could study Harmonic analysis ongroups other than R or R/Z, but is it useful in any way other than theadvancement of Math?)Moshe=== === Subject: : Re: Fourier analysis on groups question> Are there any practical applications for Harmonic/Fourier analysis on> groups other than R or R/Z? Is Harmonic analysis ever really used on groups> other than R or R/Z? (I suppose anyone could study Harmonic analysis on> groups other than R or R/Z, but is it useful in any way other than the> advancement of Math?)Harmonic analysis on R^n (although you may not see that as different).Spherical harmonic expansions essentially are Harmonic analysis on the compact group SO(3). Analysis on the Poincare group is also quite important.--Dan Grubb=== === Subject: : Re: Fourier analysis on groups question> Are there any practical applications for Harmonic/Fourier analysis on> groups other than R or R/Z? Is Harmonic analysis ever really used on> groups> other than R or R/Z? Fast Fourier Transform, and Fourier analysis on finite cyclic groups?e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ietel: +353-86-2336090, +353-1-2842366s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland=== === Subject: : Re: PROOF that numbers are countableIn sci.logic, |-|erc:> oo> ____|mn> / /_/ / _> / K-9/ /_/ - www.YeOldeCoffeeShoppe.com -> /____/_____> --------------> RIGHT!>> So you can see why I say rationals COVERS the number line. I don't need> to define Cover look up a dictionary if all the terms are deliberately miss mashed.>> Yes, you mean as I suspected that the rationals are DENSE in the> reals, not that they cover the reals.>> By map I mean draw each point/element/number from the set onto the number line.> Number line is obviously the real number line.>> To you I need the *set* itself to mark1/3, to me I just have to map the *infinite* number of> points and I'll get there. Much like 0.33.. recurring infinitely reaches 1/3.>> Ah not entirely true, I said you either need a set, or a collection of> pointers, my collection of pointers is your map. I still don't see> what this has to do with the constructable number, and essentially I> fail to see the point of this line of argumentation. What exactly are> you trying to prove here?>> There is certainly no point on the number line between the infinite set of marks made> by S and 1/3!! As #marks ->oo, Smarks -> 1/3. And here we allow the> theoretical construct of infinite marks.>> You've now introduced two terms you haven't used before, #marks ->oo> ans Smarks -> 1/3 as well as the idea of infinite marks. I don't see> how you're going from the power set to these objects. You must more> clearly define what you mean here. So far all it looks to me that> you've proven nothing, except that that subset of the rational numbers> is countable.>> I'm going from these infinite objects, to the powerset, to the real.>> The number made from 1/2 + 1/4 + 1/8 +... = 1.>> Can you describe a process using this sequence that marks the> position 1 on the number line?> Yes:> S = {1/2, 3/4, 7/8, 15/16, ... }> will have the lim sup of 1. If one prefers, one can write S as:> S = {1/2,> 1/2 + 1/4,> 1/2 + 1/4 + 1/8,> 1/2 + 1/4 + 1/8 + 1/16,> ...}.> So if you draw a point at each of these numbers.> 1/2> 3/4> 7/8> ..> you would draw the number 1?> number 1 has been referenced via the infinite set of points!Not quite. The limit of the partial sums is the number 1.In any event, the points cluster to the left side of 1.> So this set> {0,> 0.1, 0.2, 0.3, 0.4, ..., 0.9,> 0.01, 0.02, ... 0.09, 0.11, 0.12, ..., 0.19, ... 0.91, 0.92, ... 0.99,> 0.001, 0.002, ..., 0.009, 0.011, 0.012, ... 0.019, ... 0.999,> 0.0001, 0.0002, ..., 0.0009, 0.0011, 0.0012, ... 0.0019, ... 0.9999,> ...}> would completely cover the number line, color it in, no gaps!That is correct, if every element in the set is surrounded byan open interval of any (fixed) width.> Herc#191, ewill3@earthlink.netIt's still legal to go .sigless.=== === Subject: : Re: PROOF that numbers are countable oo ____|mn / /_/ / _ / K-9/ /_/ - www.YeOldeCoffeeShoppe.com -/____/_____--------------> RIGHT!> So you can see why I say rationals COVERS the number line. I don't need> to define Cover look up a dictionary if all the terms are deliberately miss mashed.> Yes, you mean as I suspected that the rationals are DENSE in the> reals, not that they cover the reals.> By map I mean draw each point/element/number from the set onto the number line.> Number line is obviously the real number line.> To you I need the *set* itself to mark1/3, to me I just have to map the *infinite* numberof> points and I'll get there. Much like 0.33.. recurring infinitely reaches 1/3.> Ah not entirely true, I said you either need a set, or a collection of> pointers, my collection of pointers is your map. I still don't see> what this has to do with the constructable number, and essentially I> fail to see the point of this line of argumentation. What exactly are> you trying to prove here?> There is certainly no point on the number line between the infinite set of marks made> by S and 1/3!! As #marks ->oo, Smarks -> 1/3. And here we allow the> theoretical construct of infinite marks.> You've now introduced two terms you haven't used before, #marks ->oo> ans Smarks -> 1/3 as well as the idea of infinite marks. I don't see> how you're going from the power set to these objects. You must more> clearly define what you mean here. So far all it looks to me that> you've proven nothing, except that that subset of the rational numbers> is countable.> I'm going from these infinite objects, to the powerset, to the real.> The number made from 1/2 + 1/4 + 1/8 +... = 1.> Can you describe a process using this sequence that marks the> position 1 on the number line?> Yes:> S = {1/2, 3/4, 7/8, 15/16, ... }> will have the lim sup of 1. If one prefers, one can write S as:> S = {1/2,> 1/2 + 1/4,> 1/2 + 1/4 + 1/8,> 1/2 + 1/4 + 1/8 + 1/16,> ...}.> So if you draw a point at each of these numbers.> 1/2> 3/4> 7/8> ..> you would draw the number 1?> number 1 has been referenced via the infinite set of points!> Not quite. The limit of the partial sums is the number 1.> In any event, the points cluster to the left side of 1.This doesn't make sense, I'm not dividing by infinity or something like thatwhere we NEED to take the limit to have a defined value. I'm stipulatingwhat region on the number line is plotted by the INFINITE set.Its very similar to integration, we sum the areas of smaller and smaller regionsusing that limit definition of integration, and increase the number of regions towardsinfinity. The total 'area' converges to a real value, we then don't say the area under thecurve F(x) is *approached by* F(x), we realise the area IS that function.0.999.. doesn't form a point to the left side of 1, and neither does the infiniteset of plotted points {0.9, 0.99, 0.999, ..}. If you want to declare to the leftof 1 then you have to specify some quantitative difference between the infinite set and 1.The infinite sequence forms the same point as 0.9.., it just plots values along theway from 0.9 to 1.> So this set> {0,> 0.1, 0.2, 0.3, 0.4, ..., 0.9,> 0.01, 0.02, ... 0.09, 0.11, 0.12, ..., 0.19, ... 0.91, 0.92, ... 0.99,> 0.001, 0.002, ..., 0.009, 0.011, 0.012, ... 0.019, ... 0.999,> 0.0001, 0.0002, ..., 0.0009, 0.0011, 0.0012, ... 0.0019, ... 0.9999,> ...}> would completely cover the number line, color it in, no gaps!> That is correct, if every element in the set is surrounded by> an open interval of any (fixed) width.dont bull me saying yes you are wrong. Does the powerset of this set containmembers to specify all irrationals?You need some basic priming on why 0.9.. = 1. 0.9.. = 0.9 + 0.09 + 0.009 ..Plot those points on the number line then tell me 0.9.. is to the left of 1.Herc=== === Subject: : Re: PROOF that numbers are countableIn sci.logic, |-|erc:> oo> ____|mn> / /_/ / _> / K-9/ /_/ - www.YeOldeCoffeeShoppe.com -> /____/_____> --------------> RIGHT!> So you can see why I say rationals COVERS the number line. I don't need>> to define Cover look up a dictionary if all the terms are deliberately miss mashed.> Yes, you mean as I suspected that the rationals are DENSE in the>> reals, not that they cover the reals.> By map I mean draw each point/element/number from the set onto the number line.>> Number line is obviously the real number line.> To you I need the *set* itself to mark1/3, to me I just have to map the *infinite* number> of>> points and I'll get there. Much like 0.33.. recurring infinitely reaches 1/3.> Ah not entirely true, I said you either need a set, or a collection of>> pointers, my collection of pointers is your map. I still don't see>> what this has to do with the constructable number, and essentially I>> fail to see the point of this line of argumentation. What exactly are>> you trying to prove here?> There is certainly no point on the number line between the infinite set of marks made>> by S and 1/3!! As #marks ->oo, Smarks -> 1/3. And here we allow the>> theoretical construct of infinite marks.> You've now introduced two terms you haven't used before, #marks ->oo>> ans Smarks -> 1/3 as well as the idea of infinite marks. I don't see>> how you're going from the power set to these objects. You must more>> clearly define what you mean here. So far all it looks to me that>> you've proven nothing, except that that subset of the rational numbers>> is countable.>> I'm going from these infinite objects, to the powerset, to the real.>> The number made from 1/2 + 1/4 + 1/8 +... = 1.>> Can you describe a process using this sequence that marks the> position 1 on the number line?>> Yes:>> S = {1/2, 3/4, 7/8, 15/16, ... }>> will have the lim sup of 1. If one prefers, one can write S as:>> S = {1/2,> 1/2 + 1/4,> 1/2 + 1/4 + 1/8,> 1/2 + 1/4 + 1/8 + 1/16,> ...}.> So if you draw a point at each of these numbers.> 1/2> 3/4> 7/8> ..>> you would draw the number 1?> number 1 has been referenced via the infinite set of points!> Not quite. The limit of the partial sums is the number 1.> In any event, the points cluster to the left side of 1.> This doesn't make sense, I'm not dividing by infinity or> something like that where we NEED to take the limit to> have a defined value. I'm stipulating what region on the> number line is plotted by the INFINITE set.There is no region on the number line plotted by the (countable)set of points specified above. It's merely a set, specified procedurally.The range of the set is a proper subset of the half-open interval [1/2, 1).1 is the sup (or is it lim sup? I never can remember Rudin's notations)of the set. However, 1 is not in the set, although one can use theset to define 1 (through a variant of Dedekind cuts or Cauchy sequences).> Its very similar to integration, we sum the areas of smaller and> smaller regions using that limit definition of integration, and> increase the number of regions towards infinity. The total 'area'> converges to a real value, we then don't say the area under the> curve F(x) is *approached by* F(x), we realise the area IS that function.Riemannian integration has some limitations. Try integrating the functionf(x) = 0 if x is a rational numberf(x) = x if x is an irrational numberover the interval [0,1]. (It turns out to be 1/2, but one has to use Lebesgueintegration to see why.)> 0.999.. doesn't form a point to the left side of 1,0.999... is a representation of a number that happens to equal 1.This has been repeatedly proven (and probably is in a FAQ somewhere).> and neither does the infinite set of plotted points> {0.9, 0.99, 0.999, ..}.That particular set is another procedurally-specified entity.Briefly, that set is the set of all items (1 - 10^(-n)), where n isa natural number. This set therefore is countable.> If you want to declare to the left of 1 then you have to specify> some quantitative difference between the infinite set and 1.How about {0.8, 0.88, 0.888, ...} ? That defines 8/9, which isdefinitely less than 1.As for differences between infinite sets and numbers, you'll haveto be more specific as to the meaning of such an operation.> The infinite sequence forms the same point as 0.9.., it just> plots values along the way from 0.9 to 1.The points in that set are dense around 1; for any e > 0, Ican find a point p in the set such that |p - 1| < e.>> So this set>> {0,> 0.1, 0.2, 0.3, 0.4, ..., 0.9,> 0.01, 0.02, ... 0.09, 0.11, 0.12, ..., 0.19, ... 0.91, 0.92, ... 0.99,> 0.001, 0.002, ..., 0.009, 0.011, 0.012, ... 0.019, ... 0.999,> 0.0001, 0.0002, ..., 0.0009, 0.0011, 0.0012, ... 0.0019, ... 0.9999,> ...}>> would completely cover the number line, color it in, no gaps!> That is correct, if every element in the set is surrounded by> an open interval of any (fixed) width.> dont bull me saying yes you are wrong. Does the powerset of> this set contain members to specify all irrationals?Within the interval (0,1), yes (since that particular set's elementsare restricted thereto). However, Cantor has proven that card(P(S)) > card(S).In any event, one has to be careful how one specifies draw. The set is,of course, dense, which presumably is what you meant.> You need some basic priming on why 0.9.. = 1. 0.9.. = 0.9 + 0.09 + 0.009 ..> Plot those points on the number line then tell me 0.9.. is to the left of 1.All points in that set are to less than 1. That does not mean thesup of the set is less than 1.> Herc#191, ewill3@earthlink.netIt's still legal to go .sigless.=== === Subject: : Re: PROOF that numbers are countableWe'll end it there, i've used 10 different words over 20 posts tosay the infinite set of rationals plots the irrational, and every word youdefine differently. If you don't want to face your flaw I can't make you.Herc=== === Subject: : Re: PROOF that numbers are countableIn sci.logic, |-|erc:> We'll end it there, i've used 10 different words over 20 posts to> say the infinite set of rationals plots the irrational, and every word you> define differently. If you don't want to face your flaw I can't make you.I take it the flaw is that mathematicians think the interval [0,1)uncountable, and you can prove differently?Or am I misconstruing your argument?As it is, it's quite clear that all of the reals mankindcan ever write down explicitly are finite -- and we'llnever get all of them; we can't even get the infinitefractions. Hence notations such as 1/3, pi = 3.14159...,and sqrt(2) * pi, which are talking about numbers,but not explicitly expanding them.And those form a finite set as well, which means one hasto go into the realm of generators (e.g., all numbers ofthe form N * pi are transcendental, where N is a nonzerointeger, is -- or should be -- a provable result). Unlessone gets into meta-generators, the set of generators isalso finite.(Bear in mind that if I were to draw a circle the sizeof what would have been the SuperConducting Supercollider(about 87 km in circumference), and use only 20 digits ofpi, I'd be off by about the width of an atomic nucleus.So all of this stuff about infinite numbers isn'thorribly useful unless one wants to talk about somethinglike random digit generation.)As it is, I think you've attempted to prove that:[Ai] [Ej] [F(i,j) = g(j)]where g is constructed such that g(j) = 5 if F(j,j) = 4, and 4 otherwise.For most function arrays F this should be relatively simple.However,[Ej] [Ai] [F(i,j) = g(j)]is quite different, and false by construction.> Herc#191, ewill3@earthlink.netIt's still legal to go .sigless.=== === Subject: : Re: PROOF that numbers are countable oo ____|mn / /_/ / _ / K-9/ /_/ - www.YeOldeCoffeeShoppe.com -/____/_____--------------> {0,> 0.1, 0.2, 0.3, 0.4, ..., 0.9,> 0.01, 0.02, ... 0.09, 0.11, 0.12, ..., 0.19, ... 0.91, 0.92, ... 0.99,> 0.001, 0.002, ..., 0.009, 0.011, 0.012, ... 0.019, ... 0.999,> 0.0001, 0.0002, ..., 0.0009, 0.0011, 0.0012, ... 0.0019, ... 0.9999,> ...}> would completely cover the number line, color it in, no gaps!> Herc> You mean color in the points, or color in the INTERVALS between> points? Once again you're stating the property of DENSITY of the> rationals as a subset of the reals, not that these numbers are colored> in. Take a square who's sides are length one, and construct it on your> number line, then take it's diagonal, obviously this line has a> length, and should be on your number line, But the square root of two> is NOT covered by the set you've defined. No matter how close you get,> you never reach the square root of two. Are you trying to argue that> the real number line doesn't contain any irrational numbers? Unless> you had some special meaning for the word color it in then your> argument is wrong.I'm not trying to INDEX sqrt(2) or any irrational with a single element of the above set,I'm showing that that irrational is plotted on the number line by an infinite sequence of numberscountained in the above set.You DO reach sqrt(2) if you use an infinite sequence. Plotting EVERY point from aninfinite set is exactly the same accuracy as plotting the irrational.Take a powerset element of the full finite digit set above, it contains one number from each row.Q = {0.7, 0.70, 0.707, ...} make it 1/sqrt(2)This infinite sequence represents 1/sqrt(2) exactly.Remember I'm plotting the infinite set of points. These are basic properties ofinfinite sequences you all ignore when the topic turns to counting, you think aninfinite set of *elements* (from Q) doesn't represent *the set* (Q) when you agreedthe infinte set is equivalent to an irrational. result, you're saying take ALL the rational numbers of the form 0.ab....cd0000.. and take the power set of all of those numbers. Now every number is contained as an element of that subset. So the process you're using here to get the numbers from SET 3 is take the power set, and the numbers we're talking about are elements. We do this with the completed set of 0.ab....cd0000.. numbers and we can represent every number on the real line. I do indeed agree that the Power Set of these types of numbers represent the reals and do cover the reals by my definition and yours, whatever yours is. All that's left for you to do, is demonstrate that the power set of these numbers is computable and can be stored on your tape.You agreed this far :I do indeed agree that thePower Set of these types of numbers represent the reals and do coverthe reals by my definition and yoursYou agreed the real number line is covered, now you say its only density thatI have demonstrated?? The powerset is just a query over the complete setof finite length numbers, if every powerset element plots EVERY REALthe the base table must plot them also.Herc=== === Subject: : Re: PROOF that numbers are countable> I'm not trying to INDEX sqrt(2) or any irrational with a single element of the above set,> I'm showing that that irrational is plotted on the number line by an infinite sequence of numbers> countained in the above set.> You DO reach sqrt(2) if you use an infinite sequence. Plotting EVERY point from an> infinite set is exactly the same accuracy as plotting the irrational.> Take a powerset element of the full finite digit set above, it contains one number from each row.> Q = {0.7, 0.70, 0.707, ...} make it 1/sqrt(2)> This infinite sequence represents 1/sqrt(2) exactly.> Remember I'm plotting the infinite set of points. These are basic properties of> infinite sequences you all ignore when the topic turns to counting, you think an> infinite set of *elements* (from Q) doesn't represent *the set* (Q) when you agreed> the infinte set is equivalent to an irrational.> result, you're saying take ALL the rational numbers of the form> 0.ab....cd0000.. and take the power set of all of those numbers. Now> every number is contained as an element of that subset. So the process> you're using here to get the numbers from SET 3 is take the power> set, and the numbers we're talking about are elements. We do this> with the completed set of 0.ab....cd0000.. numbers and we can> represent every number on the real line. I do indeed agree that the> Power Set of these types of numbers represent the reals and do cover> the reals by my definition and yours, whatever yours is. All that's> left for you to do, is demonstrate that the power set of these numbers> is computable and can be stored on your tape.> You agreed this far :> I do indeed agree that the> Power Set of these types of numbers represent the reals and do cover> the reals by my definition and yoursRight, but remember the Power Set is NOT a set of numbers! It's acollection of SETS. The rational numbers in your original set do NOTcover the reals. Again you're blurring the distinction between a set,and it's elements. Only the completed SET can cover the line, NOT theelements. Do see you see how the number 0.1 is different from the set{0.1}? They're different object, and you must make specific referenceto exactly which object covers the line, you can't blur them into oneidea. > You agreed the real number line is covered, now you say its only density that> I have demonstrated?? The powerset is just a query over the complete set> of finite length numbers, if every powerset element plots EVERY REAL> the the base table must plot them also.> HercYes I'm still saying it's only density you're demonstrating, and whatgives it away is your colour in the line metaphor. How can you provethat no matter how far I zoom in, and no matter how much I scrutinizethe line, that there really IS a number of the form you're talkingabout? How can I be sure that you CAN actually approach the realnumber Pi with a sequence of rationals, I mean, you need to prove aneighborhood of rational points in that sequence converge to thatpoint. You are doing this, by instead constructing the explicitsequence neccesary. The most natural way to prove this is to provedensity, and show that no matter how you select a bound , say e, thatyou can construct a rational between the real number e, and the numberthe set is supposed to be. You can say this another way, prove thatgiven any two real numbers, a and b, that you can construct a rationalnumber r, such that a < r :> oo> ____|mn> / /_/ / _> / K-9/ /_/ - www.YeOldeCoffeeShoppe.com -> /____/_____> -------------->> {0,> 0.1, 0.2, 0.3, 0.4, ..., 0.9,> 0.01, 0.02, ... 0.09, 0.11, 0.12, ..., 0.19, ... 0.91, 0.92, ... 0.99,> 0.001, 0.002, ..., 0.009, 0.011, 0.012, ... 0.019, ... 0.999,> 0.0001, 0.0002, ..., 0.0009, 0.0011, 0.0012, ... 0.0019, ... 0.9999,> ...}>> would completely cover the number line, color it in, no gaps!>> Herc> You mean color in the points, or color in the INTERVALS between> points? Once again you're stating the property of DENSITY of the> rationals as a subset of the reals, not that these numbers are colored> in. Take a square who's sides are length one, and construct it on your> number line, then take it's diagonal, obviously this line has a> length, and should be on your number line, But the square root of two> is NOT covered by the set you've defined. No matter how close you get,> you never reach the square root of two. Are you trying to argue that> the real number line doesn't contain any irrational numbers? Unless> you had some special meaning for the word color it in then your> argument is wrong.> I'm not trying to INDEX sqrt(2) or any irrational with a single> element of the above set, I'm showing that that irrational is> plotted on the number line by an infinite sequence of numbers> countained in the above set.sqrt(2)/2, maybe, but you're correct; there is a subset that defines such.> You DO reach sqrt(2) if you use an infinite sequence. Plotting EVERY> point from an infinite set is exactly the same accuracy as plotting> the irrational.Depends on how one defines plot. However, one can extract fromthe power set a subset that defines 1/sqrt(2).> Take a powerset element of the full finite digit set above, it> contains one number from each row.> Q = {0.7, 0.70, 0.707, ...} make it 1/sqrt(2)> This infinite sequence represents 1/sqrt(2) exactly.> Remember I'm plotting the infinite set of points.Which will take you until doomsday, but never mind that. :-)> These are basic properties of infinite sequences you all ignore> when the topic turns to counting, you think an infinite set of> *elements* (from Q) doesn't represent *the set* (Q) when you agreed> the infinte set is equivalent to an irrational.A proper subset of elements in a set does not define the set, it'smerely a proper subset of elements. Of course, the counterexampleproves the rule -- but only because the proper subset selectedcan be used to infer a generation rule.In other words, given a proper subset description such as{0, 0.1, 0.2, ..., 0.9, 0.01, 0.02, ..., 0.09, 0.11, 0.12, ..., 0.99, 0.001, 0.002, ..., 0.009, 0.011, 0.012, ..., 0.999, ...}one can usually determine how the set is generated; in this case,this set is {0} union the set of all numbers M / 10^(-K),0 < M < 10^K, K > 0 both integers, M is not a multiple of 10.> result, you're saying take ALL the rational numbers of the form> 0.ab....cd0000.. and take the power set of all of those numbers. Now> every number is contained as an element of that subset. So the process> you're using here to get the numbers from SET 3 is take the power> set, and the numbers we're talking about are elements. We do this> with the completed set of 0.ab....cd0000.. numbers and we can> represent every number on the real line. I do indeed agree that the> Power Set of these types of numbers represent the reals and do cover> the reals by my definition and yours, whatever yours is. All that's> left for you to do, is demonstrate that the power set of these numbers> is computable and can be stored on your tape.The obvious implementation has a fatal flaw:func BadPowerSet(Set S) returns Set{ int k; Set PS = EmptySet; for(k=0;;k++) { Set SS = EmptySet; int i, j; for(i = 1, j = 0; i < k; i *= 2, j++) { if(logand(k,i) != 0) SS = SS union {S.pick(j))}; } PS = PS union {SS}; } return PS;}The returned set of this routine does not contain theelement S (after all, S is an element of its own power set)if card(S) is not finite. (That diagonal argument again!)And that's assuming this routine ever returns anyway, as ithas an infinite loop.> You agreed this far :> I do indeed agree that the> Power Set of these types of numbers represent the reals and do cover> the reals by my definition and yours> You agreed the real number line is covered, now you say its only density that> I have demonstrated?? The powerset is just a query over the complete set> of finite length numbers, if every powerset element plots EVERY REAL> the the base table must plot them also.Define covered. If each point in Q is plotted using an open ball,then the real line is covered, as Q is dense everywhere. I'm not evensure the open ball needs to have a fixed width.Of course, if every point in Q is plotted using a mere*point*, there will be gaps -- sqrt(2), pi, and e, toname three. They are strange gaps, to be sure, as nothingbigger than a point can squeeze between them (because Qis dense) -- but gaps they are, all the same.> Herc#191, ewill3@earthlink.netIt's still legal to go .sigless.=== === Subject: : Re: Factoring idea, off Mwhat's that theorem about coprimality,wherein ax + by has to equal one? I have forgotten how that works, nor do I know,how it might relate to the given polynomial. > | x^2 + ax + by + y^2 = z^2, > How about x = 2, y = 2, z = 3? I don't think there there are any> integers a and b that make x^2 + ax + by + y^2 = z^2 true in that> case. --Give Earth a Trickier Dick Cheeny -- out of office, after GIGA years.http://www.benfranklinbooks.com/http://www.rand.org/ publications/randreview/issues/rr.12.00/http:// members.tripod.com/~american_almanac=== === Subject: : Re: Factoring idea, off M> what's that theorem about coprimality,> wherein ax + by has to equal one?> I have forgotten how that works, nor do I know,> how it might relate to the given polynomial.You can use the Euclidean algorithm to find gcd(x,y), then reverse it toexpress the gcd as a linear combination of x and y, so that for someintegers a and b, ax + by = 1, if x and y are coprime.> | x^2 + ax + by + y^2 = z^2,> How about x = 2, y = 2, z = 3? I don't think there there are any> integers a and b that make x^2 + ax + by + y^2 = z^2 true in that> case.> --Give Earth a Trickier Dick Cheeny -- out of office, after GIGA years.> http://www.benfranklinbooks.com/> http://www.rand.org/publications/randreview/issues/rr.12.00/> http://members.tripod.com/~american_almanac=== === Subject: : Re: Factoring idea, off MI mean, since you may know what it is,can you please give us a definition? I mean, it might be fun to try them out, iff ... andregardless of what you're trying to do with them, herein. > What in Hell is a Tautological Space? --Give Earth a Trickier Dick Cheeny -- out of office, after GIGA years.http://www.benfranklinbooks.com/http://www.rand.org/ publications/randreview/issues/rr.12.00/http:// members.tripod.com/~american_almanac=== === Subject: : Re: Factoring idea, off M> Now I rather arbitrarily set a conditional statement, which is> x^2 + ax + by + y^2 = z^2,> which is basically a way for me to pick any x, y and z that I want,> since there will always exists integers 'a' and 'b' that will allow> that to work.If x,y are even and z is odd, then you have just proven even=odd. What'snext? Proving 0=1?I think I can predict where James is going to go with this one. He willinvent an undefined set of numbers (probably a subset of the rationals)and claim that using these natural fractions he can quickly factor anynumber. When we point out that his technique basically amounts to saying 63= (63/2)x(2), he will claim 1) we are too stupid to understand hisgroundbreaking discovery and 2) we are brilliantly figuring out a way toconvince the math world to oppress him.=== === Subject: : Set of orthogonal matricesI found the following exrcise: Show that the set of all n x northogonal matrices is a compact subset of (R^n)^n. That's all theexercise said. (I'm not sure, but, since to prove such propositionit's fundamental to have a norm or a topology in (R^n)^n, I take itthe norm of a matrix is defined as the square root of the sum of thesquares of it's terms - an extension of the definition of norm inR^n).I'm starting to think this proposition is false. It seems to me theset of orthogonal matrices doesn't need to be bounded with respect tothe norm I cited. But I'm really stuck. any help is welcome.Amanda=== === Subject: : Re: Set of orthogonal matrices> I found the following exrcise: Show that the set of all n x n> orthogonal matrices is a compact subset of (R^n)^n. That's all the> exercise said. (I'm not sure, but, since to prove such proposition> it's fundamental to have a norm or a topology in (R^n)^n, I take it> the norm of a matrix is defined as the square root of the sum of the> squares of it's terms - an extension of the definition of norm in> R^n).> I'm starting to think this proposition is false. It seems to me the> set of orthogonal matrices doesn't need to be bounded with respect to> the norm I cited. But I'm really stuck. any help is welcome.Aren't the column vectors of an orthogonal matrix unit vectors? And aren't there only n of them? So let's see, that would make the norm of an orthogonal matrix what exactly? And then there's closedness ...=== === Subject: : Re: Hey guys, isolated singularities?> When are singularities of analytic functions necessarily isolated?Look up the definition of isolated singularity.=== === Subject: : Re: Hey guys, isolated singularities?> When are singularities of analytic functions necessarily isolated?> Look up the definition of isolated singularity.To both of you:yes, obviously isolated singularities are isolated.I thought z was an isolated singularity of f if f was analytic in adeleted neighborhood of z but not at z.By simply saying singularity, I mean that f is not analytic at z,but it is analytic at a point in every neighborhood of z.Aren't there any non-isolated singularities of analytic functions? Forexample, the one I gave, sin 1/z, has a singularity at 0 (this is nota common definition?) and this singularity is not isolated.After all, why would isolated singularities be called isolated ifthere wasn't any other kind?=== === Subject: : Re: Hey guys, isolated singularities?> When are singularities of analytic functions necessarily isolated?> Look up the definition of isolated singularity.>To both of you:>yes, obviously isolated singularities are isolated.What Wade actually meant to say was Look up the definition of pole and essential singularity.Seriously: Your questions about poles and essentialsingularities _are_ answered by just knowing thedefinition, and trying to deal with concepts byguessing what they mean without learning theofficial definition is not going to work.>I thought z was an isolated singularity of f if f was analytic in a>deleted neighborhood of z but not at z.>By simply saying singularity, I mean that f is not analytic at z,>but it is analytic at a point in every neighborhood of z.If that's what you mean by singularity then no, singularitiesneed not be isolated. But here's an important question:_is_ that the _actual_ definition of singularity in someactual reference, or is it just a definition you made up?(Nothing wrong with making up your own definitions,people have to do so all the time. But before expectingothers to _know_ what definition you have in mind youneed to at least _state_ the definition, if you just madeit up.)>Aren't there any non-isolated singularities of analytic functions? For>example, the one I gave, sin 1/z, has a singularity at 0 (this is not>a common definition?) and this singularity is not isolated.>After all, why would isolated singularities be called isolated if>there wasn't any other kind?>-Greg=== === Subject: : Re: Hey guys, isolated singularities?> When are singularities of analytic functions necessarily isolated?> Look up the definition of isolated singularity.>To both of you:>yes, obviously isolated singularities are isolated.> What Wade actually meant to say was Look up the > definition of pole and essential singularity.> Seriously: Your questions about poles and essential> singularities _are_ answered by just knowing the> definition, and trying to deal with concepts by> guessing what they mean without learning the> official definition is not going to work. That doesn't work though, since nobody has yet shown that *essential* singularity is a concept. That is obvious though since the only thing Feynmannoids are more clueless about than singulatities are the concept of essential. >I thought z was an isolated singularity of f if f was analytic in a>deleted neighborhood of z but not at z.>By simply saying singularity, I mean that f is not analytic at z,>but it is analytic at a point in every neighborhood of z.> If that's what you mean by singularity then no, singularities> need not be isolated. But here's an important question:> _is_ that the _actual_ definition of singularity in some> actual reference, or is it just a definition you made up?> (Nothing wrong with making up your own definitions,> people have to do so all the time. But before expecting> others to _know_ what definition you have in mind you> need to at least _state_ the definition, if you just made> it up.)>Aren't there any non-isolated singularities of analytic functions? For>example, the one I gave, sin 1/z, has a singularity at 0 (this is not>a common definition?) and this singularity is not isolated.>After all, why would isolated singularities be called isolated if>there wasn't any other kind?>-Greg> ************************> === === Subject: : Re: Hey guys, isolated singularities?> When are singularities of analytic functions necessarily isolated?> Look up the definition of isolated singularity.> To both of you:> yes, obviously isolated singularities are isolated.> I thought z was an isolated singularity of f if f was analytic in a> deleted neighborhood of z but not at z.> By simply saying singularity, I mean that f is not analytic at z,> but it is analytic at a point in every neighborhood of z.> Aren't there any non-isolated singularities of analytic functions? For> example, the one I gave, sin 1/z, has a singularity at 0 (this is not> a common definition?) and this singularity is not isolated.> After all, why would isolated singularities be called isolated if> there wasn't any other kind? Beacuse only physicists and chemists really seem to believe that the *Big Bang* is a singularity. Many over people believe that it's just a common mistaking of energy and computers for recursive philosophy. > -Greg=== === Subject: : Re: Hey guys, isolated singularities?Gregory Magarshak escribi.97:> When are singularities of analytic functions necessarily isolated?> Look up the definition of isolated singularity.> To both of you:> yes, obviously isolated singularities are isolated.> I thought z was an isolated singularity of f if f was analytic in a> deleted neighborhood of z but not at z.> By simply saying singularity, I mean that f is not analytic at z,> but it is analytic at a point in every neighborhood of z.> Aren't there any non-isolated singularities of analytic functions? For> example, the one I gave, sin 1/z, has a singularity at 0 (this is not> a common definition?) and this singularity is not isolated.You meant here 1/sin(1/z) ...> After all, why would isolated singularities be called isolated if> there wasn't any other kind?But I haven't seen anywhere before a definition of singularitiy. Only ofisolated singularity.It is pretty obvius tha 1/sin(1/z) has isolated singularities in z =1/(k*pi) with k integer, and that in zero it isn`t analytic. Then it is anon-isolated singulariy, according with your definition.Ignacio Larrosa Ca.96estroA Coru.96a (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com=== === Subject: : Re: Hey guys, isolated singularities?> But I haven't seen anywhere before a definition of singularitiy. Only of> isolated singularity.Often we see a theorem that the radius of convergence for the Taylor seriesof an analytic function reaches to the nearest singularity. For this, you cannotrestrict it to isolated singularities.=== === Subject: : Re: Hey guys, isolated singularities?> But I haven't seen anywhere before a definition of singularitiy. Only of> isolated singularity.> Often we see a theorem that the radius of convergence for the Taylor series> of an analytic function reaches to the nearest singularity. For this, you cannot> restrict it to isolated singularities.Good point. For example a natural boundary has non-isolatedsingularities.Suppose we take my definition of singularity as valid (i.e. thefunctino is not analytic at the zbut every neighborhood of z containspoints at which the function is analytic). Then I have the followingquestions:A pole is defined as a zero of h(x) when f(x) = g(x)/h(x) and f, g, hare all analytic. Poles are isolated singularities by definition, buteven if that wasn't in the definition they'd be isolated, right?Because h(x) has to have a zero of a finite order.What does the monodromy theory say regarding the different types ofsingularities? If I continue a function analytically along twodifferent paths around a singularity, I may or may not get twodifferent values at the same point depending on the path I take. Is myguess correct that this only happens when the singularity is a branchpoint? Not when it is an essential singularity or a pole? I say thisbecause the set of all analytic continuations forms a Riemann surface,and the function is multiple-valued like that iff. the point is abranch point.So, if I continue a function analytically 'round a pole or essentialsingularity I will get the same value every time, right?(I looked on wikipedia and they define singularity as a point wheresomething breaks down or doesn't exist. They aren't the authority onthis, but I think that is pretty reasonable. They have a link to apage called Singularity Theory which talks about manifolds.)=== === Subject: : Re: Hey guys, isolated singularities?> But I haven't seen anywhere before a definition of singularitiy. Only of> isolated singularity.> Often we see a theorem that the radius of convergence for the Taylor series> of an analytic function reaches to the nearest singularity. For this, you cannot> restrict it to isolated singularities.>Good point. For example a natural boundary has non-isolated>singularities.>Suppose we take my definition of singularity as valid (i.e. the>functino is not analytic at the zbut every neighborhood of z contains>points at which the function is analytic). Then I have the following>questions:>A pole is defined as a zero of h(x) when f(x) = g(x)/h(x) and f, g, h>are all analytic. Poles are isolated singularities by definition, but>even if that wasn't in the definition they'd be isolated, right?>Because h(x) has to have a zero of a finite order.>What does the monodromy theory say regarding the different types of>singularities? If I continue a function analytically along two>different paths around a singularity, I may or may not get two>different values at the same point depending on the path I take. Is my>guess correct that this only happens when the singularity is a branch>point? Not when it is an essential singularity or a pole? You're using the word function in two different senses here.It's a _function element_ that one applies analytic continuationto - it's actual _functions_ that have poles, etc. The questionreally makes no sense as far as I can see.For example, one might talk about continuing log(z) along a pathenclosing the origin. But it doesn't make any sense to ask whetherlog(z) has a pole or an essential singularity at the origin, becausethere simply is no function log(z) defined near the origin!(Honest: If we did say that log had some sort of singularityat the origin it would be an isolated singularity, right? There'sno singularities elsewhere... Now is that isolated singularitya pole or an essential singularity? You can tell the differenceby looking at the Laurent series, but log(z) simply doesn't_have_ a Laurent series!Cuz log is not a function. Now, the principal branch, oftenknown as Log, _is_ a function, but you can't analyticallycontinue Log along a path enclosing the origin...)>I say this>because the set of all analytic continuations forms a Riemann surface,>and the function is multiple-valued like that iff. the point is a>branch point.>So, if I continue a function analytically 'round a pole or essential>singularity I will get the same value every time, right?>(I looked on wikipedia and they define singularity as a point where>something breaks down or doesn't exist. They aren't the authority on>this, but I think that is pretty reasonable. They have a link to a>page called Singularity Theory which talks about manifolds.)>-Greg=== === Subject: : Re: Hey guys, isolated singularities?> But I haven't seen anywhere before a definition of singularitiy. Only of> isolated singularity.>Often we see a theorem that the radius of convergence for the Taylor series>of an analytic function reaches to the nearest singularity. It's often stated that way in informal contexts - have you seen itstated that way in a carefully written book? (And did the authorinclude a definition of singularity?)>For this, you cannot>restrict it to isolated singularities.=== === Subject: : Re: Hey guys, isolated singularities?>I've got this question,>When are singularities of analytic functions necessarily isolated?Huh? Isolated singularities are necessarily isolated...>I think poles always have to be isolated (since f(z) = g(z)/h(z), so>1/f(z) has must have isolated zeroes else it will be all zero -- and>by the way, what's the convention for 1/h(z) if h(z)=0 identically in>a domain D? is 1/h(z) infinity? hehe)It's true that poles are isolated singularities. Not for thereason you give - poles are isolated singularities_by definition_. (Def: A pole of f is an isolated singularityz_0 such that f(z) -> infinity as z->z_0.)>But what about essential singularities? For example,They're also isolated singularities, by definition. An essentialsingularity is an isolated singularity which is not a pole andnot removable.>sin (1/z) has a sequence of zeroes with a limit at the origin>but sin (1/z) is not analytic at the origin so it has the right to>be not identically zero>but what about 1/(sin (1/z)) ... we get an essential singularity, at>the origin, right?No.>Am I correct in saying that essential singularities are always not>isolated?No, exactly backwards.> (e.g. because of Picard's theorem)Huh? Exactly what does Picard's theorem state?>-Greg=== === Subject: : Re: textbooks - limits & continuity> The sections on limits and continuity using epsilon-delta proofs in the> undergraduate Calculus textbooks are very short. Does some book exist which> goes into more deeper detail that someone trying to study at the lower division> undergraduate level can understand?> Spivak, Calculus> Alternatively, are these epsilon-delta> proofs really not too important at this lower division undergraduate level of> study?> Proofs are considered not too important for engineering and science students. Proofs are considered important for engineering students. But, Calculus is usually put on the zero budget by most engineers.> Examing such proofs more thoroughly should maybe wait until more> advanced coursework?> There is probably a course for juniors or seniors that does this.=== === === Subject: : Re: Product of these fractions never a power of 2 ?The only solutions that I have found, with each denominator : a lessthan 394, b less than 871 and c less than 890{a,b,c,exponent}{2,22,469,5}{2,23,245,5}{2,25,133,5}{ 2,28,85,5}{2,29,77,5}{2,35,53,5}{2,37,49,5}{4,5,13,5}{5,5,8,5} === === Subject: : Re: Product of these fractions never a power of 2 ?Am 11.04.04 04:20 schrieb Jason Rodgers:> The only solutions that I have found, with each denominator : a less> than 394, b less than 871 and c less than 890> {a,b,c,exponent}> {2,22,469,5}> {2,23,245,5}> {2,25,133,5}> {2,28,85,5}> {2,29,77,5}> {2,35,53,5}> {2,37,49,5}> {4,5,13,5}> {5,5,8,5}Nice! It is interesting, that in each solution at leastone element id even (and one is 5, and one is a multiple of 7,if the solutions are greater than a certain limit btw)Do you search by brute force or do you have someoptimized algorithm?=== === Subject: : Re: Product of these fractions never a power of 2 ? 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21:> The only solutions that I have found, with each denominator : a less> than 394, b less than 871 and c less than 890> {a,b,c,exponent}> {2,22,469,5}> {2,23,245,5}> {2,25,133,5}> {2,28,85,5}> {2,29,77,5}> {2,35,53,5}> {2,37,49,5}> {4,5,13,5}> {5,5,8,5}to search for them, with no inherent limitations, and it produced the same output very quickly.To explain the program, first note that the product (3+1/a)(3+1/b)(3+1/c) is strictly between 27 and 3.5**2 < 43, so the only power of two possible is 2^5=32. Then, if a>=b>=c, (3+1/a)**3 and (3+1/a)(3+1/b)**2 must both be greater than 32, which can be used to terminate the loops searching for a and b. The loop for c can similarly be terminated because the product monotonically shrinks until it is less than 32, unless it will never get less than 32 (because 3(3+1/a)(3+1/b) >= 32) in which case there is no point in searching for c.Here is the program, in Python:def count(n=0): while True: yield n n += 1for a in count(2): anum = 3*a+1 if anum**3 // a**3 < 32: break for b in count(a): bnum = 3*b+1 if (anum * bnum**2) // (a * b**2) < 32: break if (anum * bnum * 3) // (a * b) < 32: for c in count(b): cnum = 3*c+1 if (anum*bnum*cnum) // (a*b*c) < 32: break if anum*bnum*cnum == 32*a*b*c: print a,b,c http://www.ics.uci.edu/~eppstein/=== === Subject: : Re: Product of these fractions never a power of 2 ? 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21:> To explain the program, first note that the product > (3+1/a)(3+1/b)(3+1/c) is strictly between 27 and 3.5**2 < 43, so the > only power of two possible is 2^5=32. Then, if a>=b>=c, (3+1/a)**3 and > (3+1/a)(3+1/b)**2 must both be greater than 32, which can be used to > terminate the loops searching for a and b. The loop for c can similarly > be terminated because the product monotonically shrinks until it is less > than 32, unless it will never get less than 32 (because 3(3+1/a)(3+1/b) >= 32) in which case there is no point in searching for c.Of course, that inequality should be a<=b<=c. http://www.ics.uci.edu/~eppstein/ === === Subject: : Re: Product of these fractions never a power of 2 ?Am 11.04.04 06:07 schrieb David Eppstein:>To explain the program, first note that the product >(3+1/a)(3+1/b)(3+1/c) is strictly between 27 and 3.5**2 < 43, so the >only power of two possible is 2^5=32. Then, if a>=b>=c, (3+1/a)**3 and >(3+1/a)(3+1/b)**2 must both be greater than 32, which can be used to >terminate the loops searching for a and b. The loop for c can similarly >be terminated because the product monotonically shrinks until it is less >than 32, unless it will never get less than 32 (because 3(3+1/a)(3+1/b) >= 32) in which case there is no point in searching for c.> Of course, that inequality should be a<=b<=c. algorithm. I'll have a look at your python-program.=== === Subject: : Population ModelFor a = -0.0001 and b = 0.0338 I have a differential equation whichturns out to be a very poor model of the US population over the last 200years. So bad that I'm questioning my solution.1/P.dP/dt = b + atSeparating the variables1/P.dP = (b + at).dtlnP = bt + 1/2at^2 + CP(t) = e^(bt + 1/2at^2 + C)P(t) = e^(bt + 1/2at^2).e^CWhere P0 = e^C = 3.9 when t = 0So P(t) = P0.e^(bt + 1/2at^2)P(t) = 3.9e^(.0338t - .00005t^2)Any comments?Phil Holman=== === Subject: : Re: Population Model>For a = -0.0001 and b = 0.0338 I have a differential equation which>turns out to be a very poor model of the US population over the last 200>years. So bad that I'm questioning my solution.>1/P.dP/dt = b + atMaybe you'd get better results with b/(1 + a t^2). It has a nicerbehaviour qualitatively: always growing, near exponential when t is small, and tending to a finite limit as t -> infinity.=== === Subject: : Re: Population Model> For a = -0.0001 and b = 0.0338 I have a differential equation which> turns out to be a very poor model of the US population over the last 200> years. So bad that I'm questioning my solution.> 1/P.dP/dt = b + at> Separating the variables> 1/P.dP = (b + at).dt> lnP = bt + 1/2at^2 + C> P(t) = e^(bt + 1/2at^2 + C)> P(t) = e^(bt + 1/2at^2).e^C> Where P0 = e^C = 3.9 when t = 0> So P(t) = P0.e^(bt + 1/2at^2)> P(t) = 3.9e^(.0338t - .00005t^2)> Any comments?I don't see anything wrong with your math, but your exponent will go tonegative infinity as t grows to infinity, so that you'll see your populationeventually shrink to 0 after a while of growth. What is it that motivatesthis behavior in your model? Is it the result of a regression on the actualdata?=== === Subject: : Re: Population Model> For a = -0.0001 and b = 0.0338 I have a differential equation which> turns out to be a very poor model of the US population over the last200> years. So bad that I'm questioning my solution.> 1/P.dP/dt = b + at> Separating the variables> 1/P.dP = (b + at).dt> lnP = bt + 1/2at^2 + C> P(t) = e^(bt + 1/2at^2 + C)> P(t) = e^(bt + 1/2at^2).e^C> Where P0 = e^C = 3.9 when t = 0> So P(t) = P0.e^(bt + 1/2at^2)> P(t) = 3.9e^(.0338t - .00005t^2)> Any comments?> I don't see anything wrong with your math, but your exponent will goto> negative infinity as t grows to infinity, so that you'll see yourpopulation> eventually shrink to 0 after a while of growth. What is it thatmotivates> this behavior in your model? Is it the result of a regression on theactual> data?This confirms that the model is bad. The equation was based on a linearregression of the population data. There are more accurate methods whichI'll get to later e.g. 1/P.dP/dt = b + aP but the original solutionwhich I posed is so poor it makes some of subsequent assignmentquestions worthless.Phil Holman=== === Subject: : Re: Population Model> For a = -0.0001 and b = 0.0338 I have a differential equation which> turns out to be a very poor model of the US population over the last> 200> years. So bad that I'm questioning my solution.> 1/P.dP/dt = b + at> Separating the variables> 1/P.dP = (b + at).dt> lnP = bt + 1/2at^2 + C> P(t) = e^(bt + 1/2at^2 + C)> P(t) = e^(bt + 1/2at^2).e^C> Where P0 = e^C = 3.9 when t = 0> So P(t) = P0.e^(bt + 1/2at^2)> P(t) = 3.9e^(.0338t - .00005t^2)> Any comments?> I don't see anything wrong with your math, but your exponent will go> to> negative infinity as t grows to infinity, so that you'll see your> population> eventually shrink to 0 after a while of growth. What is it that> motivates> this behavior in your model? Is it the result of a regression on the> actual> data?> This confirms that the model is bad. The equation was based on a linear> regression of the population data. There are more accurate methods which> I'll get to later e.g. 1/P.dP/dt = b + aP but the original solution> which I posed is so poor it makes some of subsequent assignment> questions worthless.Ought to perhaps do hypothesis testing on it. Maybe you'll be able to acceptthe hypothesis at some reasonable level of significance that a is notdifferent from 0, so that you can solve the model 1/P.dP/dt = b, which isthe exponential growth model I think most people are familiar with. I can'timagine that with population data that you won't be able to draw thisresult, as noisy as it probably is.=== === Subject: : Re: continuous function on a bounded set===> === Subject: : Re: continuous function on a bounded set> Every compact, connected, locally connected metric space X> has a continuous, onto function g:[0,1]->X. For [0,1]^2, there> are well-known examples (Peano curves). Mazurkiewicz and Hahn> have credit for the general result.That's close to the idea of the proof, yes.> Hurrah. Any other details?The first good thing to know is that a compact, connected metric spacewith exactly two non-cut points (i.e. points p where X{p} is connected)is homeomorphic to the unit interval. This is tricky to prove.Now show that the space in question is arcwise connected by intersecting chainsof open sets with compact closures between two points and using the resultin the previous paragraph (this bypasses your concerns with convergence). Then, take a continuous map from the Cantor set onto X (an exercise in itself) and link up images of the endpoints using arc connectedness. This will give f from [0,1] onto X.> Yes, I'm amazed Hahn & Mazurkiewicz could get it to work.> I'm still beset with the details, wherein doesth not the devil reside?Always. And yes, the corollary is that a path connected Hausdorff spaceis arc connected.--Dan Grubb=== === Subject: : Re: Antidiagonal, InfinityI agree that the composition of functions is a function.The point is that in equational form, the function is described as acomposition of function f and function g, h = f o g. I'm interestedin a simple counterexample of a function between a closed interval andall reals that is in equational form that is not a composition of twofunctions, with f having a different domain than g.About the infinitesimals vis-a-vis the standard reals, this scalarinfinitesimal iota is within a model of the real numbers where iota isdefined as the least positive real number and through deduction aboutthe known properties of the real numbers, and as well the understoodlimitations of such a definition within a model of the reals, iota canbe determined to fulfill or satisfy certain properties ascribed to animmediate neighbor of zero in this model of the reals, one of severalwithin the Finlayson numerical model, contiguous reals. The models ofreals are to be as interchangeable as possible with the standard andvariously non-standard models of reals as used to describe thecharacteristics of the real numbers, dimensionless scalar values usedto compare by sign and magnitude two scalar values.For example, if I integrate a function, I can manually perform variouseasy univariate antidifferentiations, I use the standard model ofreals and expect the result to match the geometric analog of area. Inmy limited knowledge of the reals as hyperreals, an embedding withinthe reals, I am content to assume that the answer is the same. Idemand of my models that the correct answer results.About the compositions, I'm trying to consider how the naturals aresimilar to a compact set, and have much to learn because I am bereftof advanced graduate instruction which generally leads to very solidunderstanding of that aspect of topology.I don't care to bother you about the leading zeros. Leading zeros,'nuff said? I think the binary case is sufficient, and hope toextend that as I am still working on treatments of powerset mappings,within consideration of foundations, Finlayson set theory. I claimit. There are enough problems with iota and nested intervals, andperhaps even other results along the same lines or different ones.If Borel and Mr. Combinatorics have a tug of war, with each pullingequally, and the flag on the rope is in the middle, at the halfwaypoint, what happens to the rope?Please keep in mind that you're on top of your game. Me, I don't evenhave a solid comprehension of number theory or multivariate analysisyet. Where that is so, there are vast compendiums and the leadingedges of research available right on your screen, and the publiclibrary can request almost any published document in the world foryou.It's always one more.=== === Subject: : Re: Antidiagonal, Infinity> I agree that the composition of functions is a function.> The point is that in equational form, the function is described as a> composition of function f and function g, h = f o g. I'm interested> in a simple counterexample of a function between a closed interval and> all reals that is in equational form that is not a composition of two> functions, with f having a different domain than g.Counterexample of what? Any function from a closed interval onto R can be factored into the compostion of a function from the closed interval onto the corresponding open interval followed by a function from that open interval onto R, so your search for that particular Holy Grail is futile.> About the infinitesimals vis-a-vis the standard reals, this scalar> infinitesimal iota is within a model of the real numbers where iota is> defined as the least positive real number And iota is on a par with four sided triangles and unicorns. No such number exists for the standard reals nor for any of the various hyperreals either. > About the compositions, I'm trying to consider how the naturals are> similar to a compact set, and have much to learn because I am bereft> of advanced graduate instruction which generally leads to very solid> understanding of that aspect of topology.Anyone competent to earn a bachelor's degree in mathematics will have learnt enough topology as an undergraduate to sort that out.> I don't care to bother you about the leading zeros. Leading zeros,> 'nuff said? I think the binary case is sufficient, It seems to be sufficient to bemuse Ross, but it is not sufficient to disprove Cantor. and hope to> extend that as I am still working on treatments of powerset mappings,> within consideration of foundations, Finlayson set theory. I claim> it. Keep it, as it is of no use to anyone else.There are enough problems with iota and nested intervals, and> perhaps even other results along the same lines or different ones.> If Borel and Mr. Combinatorics have a tug of war, with each pulling> equally, and the flag on the rope is in the middle, at the halfway> point, what happens to the rope?May we hope that it is looped aroung the Finlayson neck?> Please keep in mind that you're on top of your game. Me, I don't even> have a solid comprehension of number theory or multivariate analysis> yet. Among other things.> Where that is so, there are vast compendiums and the leading> edges of research available right on your screen, and the public> library can request almost any published document in the world for> you.> It's always one more.=== === Subject: : Re: Antidiagonal, Infinity> I agree that the composition of functions is a function.> The point is that in equational form, the function is described as a> composition of function f and function g, h = f o g. I'm interested> in a simple counterexample of a function between a closed interval and> all reals that is in equational form that is not a composition of two> functions, with f having a different domain than g.> About the infinitesimals vis-a-vis the standard reals, this scalar> infinitesimal iota is within a model of the real numbers where iota is> defined as the least positive real number and through deduction about> the known properties of the real numbers, and as well the understood> limitations of such a definition within a model of the reals, iota can> be determined to fulfill or satisfy certain properties ascribed to an> immediate neighbor of zero in this model of the reals, one of several> within the Finlayson numerical model, contiguous reals. Run-on sentence notwithstanding, there is no smallest positive real in the standard model of the reals. You can prove that by noting that for any e>0, 0 < e/2 < e.=== === Subject: : Re: Antidiagonal, Infinity> Is it not possible to map a closed finite interval to an open infinite> interval without compositionally mapping to an open finite interval? > I've overlooked obvious counterexamples before.Even functions as simple asw f(x) = m*x + b are compositions of even simpler functions, so I would guess that there is no such function which cannot be resolved into a compositions of simpler function, But I di not see the point of your question.A composition of functions is as much a function as the ones of which it is composed.> I asked that question to lead to the thought that the notion of a> necessarily implicit composition of functions is not so far-fetched in> describing a function between the closed interval and the infinite> interval (with the point at infinity being an open endpoint). Where> that is so, I hope to draw a parallel for you that the notion of> mapping between the naturals and a finite interval and compositionally> to an infinite interval is not ridiculous or legerdemain, or> unprecedented.There are lots of functions from the naturals to any non-trivial interval, but no surjections. Any attempt to find a surjection is doomed to failure.> I hope to so prove it myself.> Please do say if you came to that concept, I think you did.I am not sure which concept you are talking about, but it does not appear to be one I am liable to come to, at least until I am in my dotage.> One concept that has come to fore in this and surrounding discussions> is that of the infinitesimal.Not in connection with the standard reals!> It is by no means a new concept, for> nearly as long as there was the concept of the infinite there was that> of the infinitesimal, they are flip sides of a coin. Where the> infinite is treated institutionally these days as transfinite or> disconnected ordinal, I hope to help reestablish it as a scalar. The> non-standard there was once quite standard. It may so be again, in> rigorous new ways.It requires enough serious work to understand non-standard analysis that I suspect Ross will never contribute anything to it.> What do you think about the leading zero(e)s in the matrix of list> element expansions? Why are not there obvious mappings with simple> transformations between a closed interval and the set of all reals?If by simple, you mean continuous, because closed finite intervals are compact but the set of reals is not.=== === Subject: : Re: Why is it ln on the calculator?> High School! I am in the 10th grade and right now in Honors Algebra> 2! We are currently working on Logarithms and their functions! My> question to anyone who reads this is : Why does our calculator have> the LN key when it stands for natural logarithms? Would it not be> easier to put the letter in order like NL. I have been wondering this> for a while now and searching on the internet for some kind of answer> that satisfies me. I have yet to come across such an answer which is> why I am writing this to everyone who wishes to read it. If you too> are wondering the answer to this simple mystery my class and I have> come across please inform me of any information you may have. Also,> if anyone does know the answer please feel free to email me and let> me in on the little secret I am anxiously awaiting!!The L comes first to remind you it is a logarithm. Since there are two other logarithms commonly used (base 2 and base 10), the N follows to remmind you it is the (N)atural logarithm. Log is conventionally is used for base 10 logs and log_sub 2 is or lg2 is used for base two logs. I do not know of anyother system of logs in use although any number > 1 can be a base.The short answer is the same as Tevyeh gave in -Fiddler on the Roof-. Tradition! Tradition!Bob Kolker=== === Subject: : Re: Why is it ln on the calculator? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3B1oEn11185;> Hi! I am a student from Alexander Central High School! I am in the 10th> grade and right now in Honors Algebra 2! We are currently working on> Logarithms and their functions! My question to anyone who reads this is>:> Why does our calculator have the LN key when it stands for natural> logarithms?> Real Mathematicians write log rather than ln.>Why do you say that? 'ln' is less subject to confusion anyway, because the>base is always e. log is often used (e.g. on TI calculators) for base ten>logarithms.>My calculus book (Leithhold _The Calculus With Analytical Geometry_, 6e)>uses ln for the natural logarithm throughout.>So, I am lead to conclude that what you are saying is nonsense...>No offense, of course.>Jonathan Are you lead to conclude that there is no mathematics beyond calculus? It is, in fact true, that mathematicians (who, nowadays, work with mathematics that was developed after 1635) use log to mean natural logarithm.=== === Subject: : matrix questionI have A,B,C,D four matrixthen what's the determinant of [A B;C D]and where I can find all the matrix operationsfor example like the derivative of the determinant of one matrix.=== === Subject: : first sylow theoremlet G be a finite group and let |G|=(p^n)*m where n>=1and where p does not divide m. thenG contains a subgroup of order p^i for each i where 1<= i <=n------------------------------next writing is proof by book.We know G contains a subgroup of order p by Cauch's theorem.We use an induction argument and show that the existence ofa subgroup of order p^i for i N[H]/H is the canonical homomorphism, thenr^(-1)[K] = {x in N[H] | r(x) in K} is a subgroup of N[H] andhence of G.this subgroup contains H and is of order p^(i+1)------------------------------um.....i can't understand |r^(-1)[K]| = p^(i+1) of final sentence.let me relieve.....please. thank you.Your respectfully , fool hot-girl=== === Subject: : Re: first sylow theorem> We know G contains a subgroup of order p by Cauch's theorem.As a matter of interest, is there a simple way to prove this?e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ietel: +353-86-2336090, +353-1-2842366s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland=== === Subject: : Re: first sylow theorem> We know G contains a subgroup of order p by Cauch's theorem.>As a matter of interest, is there a simple way to prove this?Yes. Another Proof of Cauchy's Group TheoremJames H. McKayThe American Mathematical Monthly, Vol. 66, No. 2. (Feb., 1959), p. 119.Stable URL: http://links.jstor.org/sici?sici=0002-9890%28195902%2966%3A2% 3C119%3AAPOCGT%3E2.0.CO%3B2-F=== === Subject: : Re: first sylow theorem> let G be a finite group and let |G|=(p^n)*m where n>=1> and where p does not divide m. then> G contains a subgroup of order p^i for each i where 1<= i <=n> ------------------------------> next writing is proof by book.> We know G contains a subgroup of order p by Cauch's theorem.> We use an induction argument and show that the existence of> a subgroup of order p^i for i of order p^(i+1).> Let H be a subgroup of order p^i .> since i we know p divides (N[H]:H) by (N[H]:H) = (G:H) (mod p).> Since H is a normal subgroup of N[H], we can form N[H]/H,> and we see that p divides |N[H]/H|.> By cauchy's theorem, the factor group N[H]/H has a subgroup> K which is of order p.> if r : N[H] -> N[H]/H is the canonical homomorphism, then> r^(-1)[K] = {x in N[H] | r(x) in K} is a subgroup of N[H] and> hence of G.> this subgroup contains H and is of order p^(i+1)> ------------------------------> um.....i can't understand |r^(-1)[K]| = p^(i+1) of final sentence.If x is in K and y in N[H] is such that r(y) = x, then r^{-1}({x}) ={h.y | h in H}, and this set has p^i elements. So, the set r^{-1}(K),which is the disjoint union of all sets r^{-1}(x) with x in K, has(cardinal of K) x p^i elements, that is, it has p^{i+1} elements.Jose Carlos Santos=== === Subject: : Re: Basic question please explain wont take long> Hi I am 17 years of age and a first year student of higher maths past> required national level this question is rattling my brains any help> given, will be greatly appreciated thank you.> The sum of Sn of the first n terms of an arithmetic progression is> given by Sn=3n+2n^2.> i)Write down the first and second terms of the progression and find a> formula for the Kth termS1 = 3*1+2*1^2 = 5S2 = 3*2+2*2^2 = 14Sk+1 = 3*(k+1)+2*(k+1)^2 = 3*k+3+2*k^2+2*2*k+2*1 = Sk + 4*k + 5Sk = Sk-1 + 4*k + 1Hope this helps.=== === Subject: : Intriguing Principle of Inclusion-Exclusion Problem..HELP PLEASE![Question]Hello.I came upon this problem and I have been intrigued since...it deals with the Principle of Inclusion-Exclusion (PIE):A group of 100 students took SAT II's in Chinese, English, and Math. Among them, 80 took Chinese, 70 took English, 60 took Math, and 50 took Physics. What is the smallest number of students who took all four exams? [Difficulty]I don't understand where I should start in minimizing the number of students who took all four exams.[Thoughts/Scratch Work]I found a similar question from the 1983 AHSME:The probability that event A occurs is 3/4. The probability that event B occurs is 2/3. Let P be the probability that both A and B occur. What is the smallest interval necessarily containing p?The solution is something like this:P(A or B)=P(A)+P(B)-P(A and B)P=P(A and B)=P(A)+P(B)-P(A or B)=3/4+2/3-P(A or B)at the most, P(A or B)=1 and at the least P(A or B)=max{P(A),P(B)}=3/4So 3/4+2/3-1= [Question]> Hello.I came upon this problem and I have been intrigued since...it> deals with the Principle of Inclusion-Exclusion (PIE):> A group of 100 students took SAT II's in Chinese, English, and Math.> Among them, 80 took Chinese, 70 took English, 60 took Math, and 50> took Physics. What is the smallest number of students who took all> four exams?> [Difficulty]> I don't understand where I should start in minimizing the number of> students who took all four exams.> [Thoughts/Scratch Work]> I found a similar question from the 1983 AHSME:> The probability that event A occurs is 3/4. The probability that> event B occurs is 2/3. Let P be the probability that both A and B> occur. What is the smallest interval necessarily containing p?> The solution is something like this:> P(A or B)=P(A)+P(B)-P(A and B)> P=P(A and B)=P(A)+P(B)-P(A or B)> =3/4+2/3-P(A or B)> at the most, P(A or B)=1 and at the least P(A or B)=max{P(A),P(B)}> =3/4> So 3/4+2/3-1= or 5/12= My problem is that I don't know how to apply this method to the> above problem since it contains 4 elements.> Here is my equation that I have so far from using the PIE:> Let #X denote the number of elements in set X> #(A or B or C or D)= (#(A)+#(B)+#(C)+#(D))-(#(A and B)+#(A and C)+#> (A and D)+#(B and C)+#(B and D)+#(C and D))+(#(A and B and C)+#(A> and B and D)+#(A and C and D)+#(B and C and D))-#(A and B and C and> D)> Please correct me if the above is wrong ;)=== === Subject: : Re: Intriguing Principle of Inclusion-Exclusion Problem..HELP PLEASE!> 60 took math, 50 took physics. Therefore at least 10 must have taken both.> Let the 70 who took English NOT include these 10. In this scenario, no> students took all three, therefore no students took all four.> So the answer is zero.> Unless there is something about SAT II that I don't understand - like there> is a minimum (eg three) number of exams you have to take - whatever the SATs> are, we don't have them where I live ... Most places don't have them. Since they are a scholastic test, not a math test. And you can always prove in an SAT that 0% of the Mathematics, English, and Chinese Teachers took a History Test.=== === Subject: : um.....continuous question.lim f(x) =x->alim f(a+h)h->0um.....is it always true?? (without regard to continuity of f)--------------------------problem)suppose thatf:R->R , f(x+y) = f(x) + f(y) for all x,y in Rlet f(x) continuous at x = cshow that f is continuous on R.----------------------------i can induce that f is continuous at x=0so, for each e>0 , there is a d>0 such that|x-0| |f(x)-f(0)|0 , there is a d>0 such that|x-y| |f(x-y)| |f(x) - f(y)| continuous.----------------------------but if uppermost question is true,for all a in R,lim f(x) =x->alim f(a+h) = f(a) + lim f(h) = f(a)h->0thus, f is continuous on R.this is a motive that ask a question.------------------------------other problem)let f(x) be defined for [0,2]f(x) = x , x in rationalf(x) = x^2 , x in irrationalshow that f is continous at x=1i can show by e-d method.but if if uppermost question is right, can i use it??so,lim f(x) =x->1lim f(1+h) = 1h->0thus , f is continuous at x=1that's right??advice...please...thank you sir.~=== === Subject: : Re: um.....continuous question.hot-girl a .8ecrit dans le message de> lim f(x) => x->a> lim f(a+h)> h->0> um.....is it always true?? (without regard to continuity of f)> --------------------------um.....Yes ans NoOf course it's always true IF it make sense. If either limit exists, theother too and they are equals.But if these limits does not exist, you cannot say they are equal. You haveto use adherence values instead.> problem)> suppose that> f:R->R , f(x+y) = f(x) + f(y) for all x,y in R> let f(x) continuous at x = c> show that f is continuous on R.> ----------------------------> i can induce that f is continuous at x=0> so, for each e>0 , there is a d>0 such that> |x-0| |f(x)-f(0)| thus, for each e>0 , there is a d>0 such that> |x-y| |f(x-y)| |f(x) + f(-y)| |f(x) - f(y)| thus, f is uniformly continuous => continuous.> ----------------------------> but if uppermost question is true,> for all a in R,> lim f(x) => x->a> lim f(a+h) = f(a) + lim f(h) = f(a)> h->0> thus, f is continuous on R.> this is a motive that ask a question.> ------------------------------Yes that works.> other problem)> let f(x) be defined for [0,2]> f(x) = x , x in rational> f(x) = x^2 , x in irrational> show that f is continous at x=1> i can show by e-d method.What is e-d ?Just use the gluing limit theorem :If lim (x->a, x in A) f(x) =l and lim (x->a, x in B) f(x) =l thenlim (x-> a, x in A U B) f(x) =l> but if if uppermost question is right, can i use it??> so,> lim f(x) => x->1> lim f(1+h) = 1> h->0True but the difficulty remains. f(1+h) = 1+h or (1+h)^2.> thus , f is continuous at x=1Why ?> that's right??> advice...please...thank you sir.~=== === Subject: : Re: um.....continuous question.> lim f(x) => x->a> lim f(a+h)> h->0> um.....is it always true?? (without regard to continuity of f)Yes provided that the above limit actually exist.=== === Subject: : Re: what is the inverse of a monomial>what kind of object is 1/(a+bx) >can it be reduced to a simpler form >kiran chandra>university of kerala>A minor point: a + bx is a binomial, not a monomial.> Second minor point, 1/(ax+b) is usually called the RECIPROCAL of ax+b, > rather than the inverse.Certainly, 1/(ax + b) is always the reciprocal of (ax + b); whether onecalls it the inverse would likely depend on context. For instance, ifthe context were one of functions, then the use of the term inverse would be inappropriate. However, if the context were in an extensionof the ring of polynomials, then inverse is entirely natural.> The function f(x) = ax+b will have an INVERSE whenever a <> 0, and then > the inverse will be g(x) = (x-b)/a.> The reciprocal of ax+b is about as simple as it can get. Alternate > representations will be more complicated rather than less complicated.No controversy on those points.=== === Subject: : Re: what is the inverse of a monomial> what kind of object is 1/(a+bx) > can it be reduced to a simpler form I can't be reduced. It can be changed to pole-zero form. 1/(a+bx) = z/(x-p) z=1/b, p=-a/b.=== === Subject: : Re: what is the inverse of a monomialit's also a special case fractional bilinear ,also known as Moebius functions with complex variable,as used in projective planar geoetry andin conformal mapping on the sphere. > 1/(a + bx)> as> (1/a) 1/(1 + bx/a),> and express the second factor as> 1/(1 + bx/a) = 1 - bx/a + (bx/a)^2 - (bx/a)^3 + ...> = sum((-bx/a)^j, j=0, ..., infinity).> leading to this:> 1/(a + bx) = sum( (-bx)^j / a^(j+1) , j=0, ..., infinity). http://tarpley.net=== === Subject: : Re: what is the inverse of a monomial> what kind of object is 1/(a+bx) > can it be reduced to a simpler form > kiran chandra> university of kerala> A minor point: a + bx is a binomial, not a monomial.Second minor point, 1/(ax+b) is usually called the RECIPROCAL of ax+b, rather than the inverse.The function f(x) = ax+b will have an INVERSE whenever a <> 0, and then the inverse will be g(x) = (x-b)/a.The reciprocal of ax+b is about as simple as it can get. Alternate representations will be more complicated rather than less complicated.=== === Subject: : Re: Need help with partial differential equation> It just occurs to me that I can't write proper titles and I haven't supplied> the boundary conditions:Since the PDE is 2nd order in x and 1st order in t, you are allowedtwo conditions (e.g., x = 0 *and* x --> oo) in x and *only one* condition (e.g., *either* t = 0 *or* t --> oo) in t. > c(0,t) = C (constant) for all t> c(x,0) = 0 for all t > 0> c(x,t) = 0 for x, t >> 0 (x is semi-infinite)Doesn't look right. Let's tryTwo boundary conditions in x:c(0,t) = 1 for t > 0c(x,t) --> 0 for t > 0 as x --> + oo.and one initial condition in t:c(x,0) = 0 for x > 0.> Specifically, I'm looking for an explicit solution, which I've found for the> equation written in its original form:> dc/dt = D(d2c/dx2 + a*(dc/dx))> where a is a constant.Nice name.Since the equation is homogeneous, the coefficients are constant, and the domain is semi-infinite, try a continuous superposition of separable terms of the formc(x,t) = Int(0,oo){ dh C(x,t,h) }whereC(x,t,h) = T(t,h) *X(x,h).Substituting c(x,t) = T*X yieldsT'/T = (D*X +D*a*X')/X = constant in (x,t) = -h==> T = exp(-h *t).Which doesn't satisfy the initial condition. Defineg = sqrt{ (a^2/4)-h/D }, Then X = exp(-a*x/2)*[ p(h)*cosh(g*x) + (q(h)/g)*sinh(g*x) ],for h <= D*a^2/4 and X = exp(-a*x/2)*[ p(h)*cos(|g|*x) + (q(h)/|g|)*sin(|g|*x) ],for h >= D*a^2/4. Now,X(0,h) = p(h)andX(x,h) --> 0 as x --> oo provided h > 0.However, I don't see how the initial condition can be satisfied by the superposition integral as long a h is real.Perhaps taking h to be complex might do the trick.Hope this helps.> Mucho appreciated for any help you might care to provide for solving this> PDE:> dc/dt = D*(d2c/dx2)+s*(dc/dx)> where c(x,t) and D, s are constants. As you may have guessed, this is a> form of the Nernst-Planck equation, but rewritten to express the second> (electric) term directly in terms of the drift speed s. D is of course the> diffusion coefficient.=== === Subject: : smallest positive integer m such that 2^3^4^5^...^n == m mod nThis sequence, Sloane's A092188, was recently suggested by John HortonConway, to NJA Sloane. It looks hard to compute, but is actuallyrelatively easy.To give the first few examples: 2 == 2 mod 2 2^3 == 2 mod 3 2^(3^4) = 2^81; 2^81 == 4 mod 4, so 2^3^4 == 4 mod 4Beyond that it appears very hard: 2^(3^(4^5)) = 2^(3^1024). What is2^(3^1024) modulo 5?To work out terms in this sequence, you have to find a new modulus forthe given base and modulus, that can be used to reduce the exponent.For example, consider 2^P % 10 (where % is the remainder function).This is just the last digit of the powers of 2. As P increases, 2^P %10 goes through the values 2,4,8,6, and then repeats. So (2^P % 10)depends on (P % 4). In this case, the exponent modulus for base 2mod 10 is 4. No matter how big the power is, as long as you can figureout what it is mod 4, then you can figure out what 2^P is mod 10. Sowe reduce 2^P % 10 to P % 4, simplifying the problem by removing oneexponent from the tower, and replacing one modulus (10) with another(4).It is important to note that the pattern B^P % M does not necessarilystart right away -- the first few powers of a base might not fit intothe pattern. Just to make sure, we should look go through at least 2Mpowers of B to find the pattern.Algorithm: To compute (B^P) % M, where base B and modulus M are normalnumbers, and P is a normal number or a power tower of 2 or moreterms: If P is 1, return B % M Otherwise P is a normal number greater than 1, or a power tower: Calculate B^0 % M, B^1 % M, B^2 % M, through B^M % M, thencontinue until you find the least N such that N>M and B^N == B^M modM. Then (N-M) is the length of the cycle for powers of B mod M; callthis C. If P is a power tower, invoke this algorithm recursively to getthe value of P % C. If P is a normal number, P % C is easy tocalculate. If P is smaller than M, B^P % M is one of the valuesalready examined to determine C. Otherwise, use P % C to determinewhich term in the series [B^M % M, B^(M+1) % M, ...] is equal to B^P %M.Using this method, the terms are easy to calculate. There is only onerecursive call, so the amount of computation does not increasedramatically with each level of the power tower -- but the loop tofind the pattern for B^P % M increases with M, so the Nth term takesabout O(N^2) calculations. For the first 100 terms I get: 2, 2, 4, 2, 2, 1, 8, 8, 2, 2, 8, 5, 8, 2, 16, 2, 8, 18, 12, 8, 2, 16, 8, 2, 18, 26, 8, 11, 2, 2, 32, 2, 2, 22, 8, 31, 18, 5, 32, 2, 8, 27, 24, 17, 16, 8, 32, 43, 2, 2, 44, 45, 26, 2, 8, 56, 40, 47, 32, 33, 2, 8, 64, 57, 2, 5, 36, 62, 22, 60, 8, 1, 68, 2, 56, 57, 44, 8, 32, 80, 2, 2, 8, 2, 70, 11, 24, 16, 62, 57, 16, 2, 8, 37, 32, 70, 92, 35, 52, ...More terms at http://home.earthlink.net/~mrob/pub/math/nu-sequences.htmlIf someone can verify, let us know. Don't let the power towers dauntyou (-:- Robert Munafo mrob.com=== === Subject: : Re: smallest positive integer m such that 2^3^4^5^...^n == m mod n> This sequence, Sloane's A092188, was recently suggested by John Horton> Conway, to NJA Sloane. It looks hard to compute, but is actually> relatively easy.> To give the first few examples:> 2 == 2 mod 2> 2^3 == 2 mod 3> 2^(3^4) = 2^81; 2^81 == 4 mod 4, so 2^3^4 == 4 mod 4When n > 2, 2^n = 0 = 4 (mod 4)> Beyond that it appears very hard: 2^(3^(4^5)) = 2^(3^1024). What is> 2^(3^1024) modulo 5?2^4n = 1 (mod 5); 3^1024 = (-1)^1024 = 1 (mod 4)2^(3^1024) = 2 (mod 5)2^3 = 2 (mod 6)2^3^n = 2^(3*3^(n-1)) = 2^3^3^(n-1) = 2^3^(n-1) (mod 6)2^3^4^5^6 = 2 (mod 6)2^3 = 1 (mod 7)2^3^n = 2^(3*3^(n-1)) = 2^3^3^(n-1) = 1^3^(n-1) = 1 (mod 7)8 is very simple.When n >= 3, 2^n = 0 = 8 (mod 8)2^3 = -1 (mod 9)2^3^n = 2^(3*3^(n-1)) = 2^3^3^(n-1) = (-1)^3^(n-1) = -1 (mod 9)2^3 = -2 (mod 10)2^3^n = 2^(3*3^(n-1)) = (2^3)^3^(n-1) = (-2)^3^(n-1) (mod 10) = (-2)^(3*3^(n-2)) = ((-2)^3)^3^(n-2) = 2^3^(n-2) (mod 10)2^3^4^5^6^7^8^9^10 = -2 = 8 (mod 10)2^10n = 1 (mod 11)3^4 = 1 (mod 10)3^4^n = (3^4)^4^(n-1) = 1 (mod 10)2^3^4^n = 2 (mod 11)=== === Subject: : Implicit differentiationHi thereSuppose that the equation F(x,y,z) = 0 implicity defines each of thethree variables x, y, and z as functions of the other two: x=h(y,z),y=g(x,z),z=f(x,y). If F is differentiable and Fx, Fy, and Fz, are all nonzero,show that (dz/dx)(dx/dy)(dy/dz) = -1 (where dz/dx - is the partialderiv. etc)Ok well it WOULD be easy because we know that dz/dx = - Fx/Fz, etc.Multiply these together and you get -1.But I don't understand something: if we implicity differentiate F withrespect to x for example, we get (dF/dx)(dx/dx) + (dF/dy)(dy/dx) + (dF/dz)(dz/dx) = 0Now obviously dx/dx = 1. But WHY can we say dy/dx = 0 ?! Surely y isnow a function x?(In the case where F(x,y,z) and z=f(x,y) and x and y are not given asfunctions of each other then certainly dy/dx = 0 and from this we getthe formuladz/dx = -Fx/Fz).Any help?in general :)R=== === Subject: : Re: Implicit differentiation>Suppose that the equation F(x,y,z) = 0 implicity defines each of the>three variables x, y, and z as functions of the other two: x=h(y,z),>y=g(x,z),>z=f(x,y). If F is differentiable and Fx, Fy, and Fz, are all nonzero,>show that> (dz/dx)(dx/dy)(dy/dz) = -1 (where dz/dx - is the partial>deriv. etc)>Ok well it WOULD be easy because we know that dz/dx = - Fx/Fz, etc.>Multiply these together and you get -1.>But I don't understand something: if we implicity differentiate F with>respect to x for example, we get> (dF/dx)(dx/dx) + (dF/dy)(dy/dx) + (dF/dz)(dz/dx) = 0If all of your d's are partial derivatives, then the preceding equationis false. Take for example F = xyz - 1. Your equation would say (yz)(1) + (xz)(-1/(zx^2)) + (xy)(-1/(yx^2)) = 0or xyz = 2If the d's in dF/dx, etc. are partials and the d's in dy/dx, etc. aretotal derivatives, then the equation is true. Then the dy/dx equals thepartial when we set dz/dx = 0 to hold z constant. Likewise, dz/dx isthe partial when we set dy/dx = 0 to hold y constant.>Now obviously dx/dx = 1. But WHY can we say dy/dx = 0 ?! Surely y is>now a function x?No it is not. When computing dz/dx, z is a function of x and y, so wehold y constant.>(In the case where F(x,y,z) and z=f(x,y) and x and y are not given as>functions of each other then certainly dy/dx = 0 and from this we get>the formula>dz/dx = -Fx/Fz).=== === Subject: : Re: Implicit differentiation>Suppose that the equation F(x,y,z) = 0 implicity defines each of the>three variables x, y, and z as functions of the other two: x=h(y,z),>y=g(x,z),>z=f(x,y). If F is differentiable and Fx, Fy, and Fz, are all nonzero,>show that> (dz/dx)(dx/dy)(dy/dz) = -1 (where dz/dx - is the partial>deriv. etc)>Ok well it WOULD be easy because we know that dz/dx = - Fx/Fz, etc.>Multiply these together and you get -1.>But I don't understand something: if we implicity differentiate F with>respect to x for example, we get> (dF/dx)(dx/dx) + (dF/dy)(dy/dx) + (dF/dz)(dz/dx) = 0> If all of your d's are partial derivatives, then the preceding equation> is false. Take for example F = xyz - 1. Your equation would say> (yz)(1) + (xz)(-1/(zx^2)) + (xy)(-1/(yx^2)) = 0> or> xyz = 2> If the d's in dF/dx, etc. are partials and the d's in dy/dx, etc. are> total derivatives, then the equation is true. Then the dy/dx equals the> partial when we set dz/dx = 0 to hold z constant. Likewise, dz/dx is> the partial when we set dy/dx = 0 to hold y constant.>Now obviously dx/dx = 1. But WHY can we say dy/dx = 0 ?! Surely y is>now a function x?> No it is not. When computing dz/dx, z is a function of x and y, so we> hold y constant.>(In the case where F(x,y,z) and z=f(x,y) and x and y are not given as>functions of each other then certainly dy/dx = 0 and from this we get>the formula>dz/dx = -Fx/Fz).> Rob Johnson take out the trash before replyingBut what really asking is this:>Suppose that the equation F(x,y,z) = 0 implicity defines each of the>three variables x, y, and z as functions of the other two: x=h(y,z),>y=g(x,z),>z=f(x,y). If F is differentiable and Fx, Fy, and Fz, are all nonzero,>show that> (dz/dx)(dx/dy)(dy/dz) = -1 (where dz/dx - is the partial>deriv. etc)How do you prove that? I need more explanation than dz/dx=-Fx/Fz <-how do we know this?R=== === Subject: : Re: Does differentiability imply continuity of partial derivatives?> Hi there> Given some function z=f(x,y). Does the fact that f is differentiable> at (a,b) imply that fx and fy (partial derivatives) are continuous at> (a,b)?> Not even for functions of one variable does this hold. One counterexample is> f(x) = x^2 * sin(1/x). This is differentiable everywhere, even at x=0, but> the derivative does not have a limit as x->0 and hence is not continuous at> x=0.BTW, I assume you mean to define f(x) above as a piecewise function, that is, f(x) = 0 when x = 0, and f(x) =x^2*sin(1/x) when x not = 0.So that f'(x) = 0 (x=0), and f'(x) = -sin(1/x) + (1/x)^2* cos(1/x) (x not = 0).So f'(x) is not continuous at 0 since -sin(1/x) doesn't approach anything as x->0, and f'(0) = 0.Just checking,R=== === Subject: : Re: Does differentiability imply continuity of partial derivatives?> Hi there>> Given some function z=f(x,y). Does the fact that f is differentiable> at (a,b) imply that fx and fy (partial derivatives) are continuous at> (a,b)?> Not even for functions of one variable does this hold. One counterexample is> f(x) = x^2 * sin(1/x). This is differentiable everywhere, even at x=0, but> the derivative does not have a limit as x->0 and hence is not continuous at> x=0.> -Michael.>BTW, I assume you mean to define f(x) above as a piecewise function, that is, >f(x) = 0 when x = 0, and f(x) =x^2*sin(1/x) when x not = 0.>So that f'(x) = 0 (x=0), and f'(x) = -sin(1/x) + (1/x)^2* cos(1/x) (x not = 0).>So f'(x) is not continuous at 0 since -sin(1/x) doesn't approach anything as >x->0, and f'(0) = 0.f'(x) = 2xsin(1/x) - cos(1/x) for x<>0. 2xsin(1/x) -> 0 as x->0, butcos(1/x) has no limit as x->0.>Just checking,>R=== === Subject: : Chinese remainder theoremCould someone help me a little with this problem, because I am stuckShow that there are exactly gcd(m , n) pairs of numbers (a , b) with 0 =< a < m, 0 =< b < n, so that:x == a (mod m)x == b (mod n)has a solutionx = a + mr = b + nr, is a solution, which means thatmr - nr = b - a=> gcd(m , n) | b - aBut how can I conclude that there are (m , n) solutions?=== === Subject: : Re: Chinese remainder theorem>Show that there are exactly gcd(m , n) pairs of numbers (a , b) with >0 =< a < m, 0 =< b < n, so that:>x == a (mod m)>x == b (mod n)>has a solution>x = a + mr = b + nr, is a solution, which means that>mr - nr = b - a>=> gcd(m , n) | b - a>But how can I conclude that there are (m , n) solutions?In the ranges you have specified, there is exactly one solution to theequations you have above. Unless m and n are relatively prime, you cannot conclude that there are gcd(m,n) solutions.=== === Subject: : Complex ring endomorphismsHow many complex ring endomorphisms h:C -> C other thanh(x + iy) = 0; h(x + iy) = x + iy; h(x + iy) = x - iy ?----=== === Subject: : Re: Complex ring endomorphisms> How many complex ring endomorphisms h:C -> C other than> h(x + iy) = 0; h(x + iy) = x + iy; h(x + iy) = x - iy ?Why don't you start by reading the answers to the threadcomplex automorphism, which was started by you more than ayear ago?Jose Carlos Santos=== === Subject: : concordant functionsLet's say that the functions f(x) and g(x) are concordant in M, iff(x)-f(y) and g(x)-g(y) have the same sign for any x,y in M. My question is: If f(x) and g(x) are concordant on some closedsubset C of M, can we extend g|C to some function g' so that f and g'are concordant in M? Here M is a topological space and all functions are real-valued andcontinuous.=== === Subject: : Re: concordant functions>Let's say that the functions f(x) and g(x) are concordant in M, if>f(x)-f(y) and g(x)-g(y) have the same sign for any x,y in M.> My question is: If f(x) and g(x) are concordant on some closed>subset C of M, can we extend g|C to some function g' so that f and g'>are concordant in M?> Here M is a topological space and all functions are real-valued and>continuous.No. Suppose M = R, C = (-infinity,-1] union [1,infinity),f(x) = 1/x for |x| >= 1, f(x) = x for |x| <= 1,g(x) = 1/x for x <= -1, g(x) = 1+1/x for x >= 1.Note that f and g|C are concordant on C, and f(x) = f(1/x) for x <> 0. If g' and f were concordant on M, we'd also need g'(x) = g'(1/x). But if g' was an extension of g|C, g' would be discontinuous at 0 (limits 0 fromthe left and 1 from the right). === === Subject: : Re: concordant functions>Let's say that the functions f(x) and g(x) are concordant in M, if>f(x)-f(y) and g(x)-g(y) have the same sign for any x,y in M.You might want to clarify whether 0 and 1 have the same sign> My question is: If f(x) and g(x) are concordant on some closed>subset C of M, can we extend g|C to some function g' so that f and g'>are concordant in M?Don't know, but I suspect the answer is yes. I'd start by noting thefollowing: GIven F : MxM -> R, there exists f such that F(x,y) = f(x) - f(y) if and only if F(x,x) = 0 and F(x_1,y) - F(x_2, y) isindependent of y. (Given an F satisfying these conditions, definef(x) = F(x, y_0). Then f(x) - f(y) = F(x, y_0) - F(y, y_0)= F(x,y) = F(y,y) = F(x,y).)So I'd try to show that G : CxC -> R can be extended to anappropriate F...> Here M is a topological space and all functions are real-valued and>continuous.> Simeon=== === Subject: : Differential equation...(x+1)(y*y'-1)=y^2Someone please point me to the solution... I cannot find it for 2 days :((=== === Subject: : Re: Differential equation...write z=y^2 and w=1+xthen z'-2z/w=2or(z/w^2)'=2/w^2this you can integrate> (x+1)(y*y'-1)=y^2> Someone please point me to the solution... I cannot find it for 2 days :((=== === Subject: : Re: * V Interesting geometry problem *> First square center is at (0,0) and is not tilted. > Second square center is at (h,k) and is tilted theta CCW.Sorry, one can even include a non-zero values of k as well ... x2=x Cos[t]- y Sin[t]+h ; y2=x Sin[t]+y Cos[t]+k ; Overlappings possible only if sqrt(h^2+k^2) < =(1+sqrt(2))a .=== === Subject: : Re: * V Interesting geometry problem *> I need a formula or an algorithm in terms of three numbers:> h,k,theta, call theta as t for simplicity.HINT: Without loss of relative generality,put k = 0 (You are anywayrotating second square through t with respect to first).By Resultants or Elimination find intersection points solve thefollowing second degree equations for pairs of straight lines, finding a maximum of 8 real points of intersection. a=1/2 for unitsquare.Intersection points (x,y) exist for zeros of of square1 ,square2 in :square1= Sqrt[(a^2-x^2) ( a^2-y^2)] ; x2=x Cos[t]- y Sin[t]+h ; y2=x Sin[t]+y Cos[t];square 2=Sqrt[(a^2-[x2-h]^2) ( y2^2-a^2)];which can be also used to include only overlapping areas. solve by hand or symbolically. Seggregate real intersection pointsinto four sets. Tag on to any one set , labeling with respect tocorner of a rotating square. Interstingly, the conrer of the squareand intersection points pass through a circle as square corners have90 degrees each.We can identify the diameter and its center pointwhich is equidistant from 4 important points. Use Heron's modifiedformula, sqrt[(s-a) (s-b) (s-c) (s-d)] to compute area.There is more to the relative postioning of squares than just findingarea of common cyclic quadrilateral. HTH=== === Subject: : Geometric QuestionIf I have a like-sided triangle with sides of 8 units, how can I calculate the largest possible radius for a circle that would fit inside of this triangle, with centre in the middle of the triangle?I have tried these approaches:Area (of triangle) = (base * height) --------------- 2A = (8 * 6,9) --------- 2A = 27,6Area (of circle) = pi r^2pi x^2 < 27,6------ ---- pi pix^2 < 8,7854 /------ /------ / / / x^2 < / 8,7854x < 2,964Circumference(or whatever it is called in English, it's that thing that you get by adding the sides of a figure)(of triangle) = s + s + sC = 8 * 3C = 24Circumference (of circle) = 2 pi r2 pi x < 24------ ----- 2 pi 2 pix < 3,8197=== === Subject: : Re: Geometric Question> If I have a like-sided triangle with sides of 8 units,> how can I calculate the largest possible radius for a> circle that would fit inside of this triangle, with> centre in the middle of the triangle?The largest circle in a triangle has its center at theintersection of the lines, which bisect the angles.The center M divides these lines such that 2/3 arebetween M and the respective corners.These lines are identical to the heights in the case ofthe equilateral triangle.The height is h = (side/2)*sqrt(3) = 4*sqrt(3) becauseyou have side = 8 units. The distance between M and anyside is h/3, because it is the part between M and theside and thus h minus the 2/3 of h between C and thecorner.This distance h/3 then is the radius you are looking for.So the answer is: Radius = h/3 = 4*sqrt(3)/3 = 2.309 units.=== === Subject: : Re: Geometric Question* Rainer Rosenthal> This distance h/3 then is the radius you are looking for.> So the answer is:> Radius = h/3 = 4*sqrt(3)/3 = 2.309 units.That's what I found too, but I got thinking. Given an arbitrarytriangel, it seems obvous that the question on finding the largestcircle fitting inside this triangel is well-defined. However, how canone prove that it is a well-defined question?Next, it seems also obvious that this largest cirlce would be one thathas all the three sides of the triangel tangent to the circle, but howcan this be proved?Third, given a triangel with sides a,b and c, what is the radius ofthe largest inscribed circle? http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92=== === Subject: : Re: Geometric Question> * Rainer Rosenthal>This distance h/3 then is the radius you are looking for.>So the answer is:> Radius = h/3 = 4*sqrt(3)/3 = 2.309 units.> That's what I found too, but I got thinking. Given an arbitrary> triangel, it seems obvous that the question on finding the largest> circle fitting inside this triangel is well-defined. However, how can> one prove that it is a well-defined question?> Next, it seems also obvious that this largest cirlce would be one that> has all the three sides of the triangel tangent to the circle, but how> can this be proved?If a circle C is drawn in the interior of a triangle T, and C fails tomeet one of the sides of T, then one can draw a circle C' that containsC in its interior, again within the interior of the triangle T. Thatcircle C' will have area strictly greater than the area of C.> Third, given a triangel with sides a,b and c, what is the radius of> the largest inscribed circle?Don't know that one.Dale=== === Subject: : Re: Geometric Question> Third, given a triangel with sides a,b and c, what is the radius of> the largest inscribed circle?> Don't know that one.But I do :-))Consider the triangle ABC with a=BC, b=CA and c=AB as usual.The in-circle touches a, b and c at points U, V, W.We have AV=AW, BW=BU and CU=CV since the center of thein-circle lies on the bisectors of the angles.And we have AW+BW+BU+CU+CV+AV = a+b+c.Let r be the radius of the in-circle. You can dissect thetriangles area A into 6 triangles with height r and sidesequal to the 6 sections AV, AW, BW, BU, CU, CV. This givesyou A = AV*r/2 + AW*r/2 + ... + CV*r/2 = r*(AV+AW+...)/2.This is to say A = r*(a+b+c)/2 = r*s, where s is halfthe circumference of the triangle.This same s is used in the famous formula of Heron, whichgives the area of the triangle in terms of its sides:A^2 = s(s-a)(s-b)(s-c). But we know A^2 = (r*s)^2 fromthe construction above. Thus we have for the radius rof the in-circle: r^2 = (s-a)(s-b)(s-c)/sWell, let's check this with the original problem, wherewe already know the result: r = 4*sqrt(3)/3.All sides are a=b=c=8 units long, so s = (8+8+8)/2 = 12 andr^2 = (s-a)(s-b)(s-c)/s = 4*4*4/12 = 4^2/3 = 4^2*3/3^2, i.e.r = 4*sqrt(3)/3, confirming the result.=== === Subject: : Re: Geometric Question* Rainer Rosenthal> This same s is used in the famous formula of Heron, which> gives the area of the triangle in terms of its sides:> A^2 = s(s-a)(s-b)(s-c). But we know A^2 = (r*s)^2 from> the construction above. Thus we have for the radius r> of the in-circle:> r^2 = (s-a)(s-b)(s-c)/sBeautiful.(However, I must work with Heron's formula for a bit.) http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92=== === Subject: : Re: Geometric Question* W. Dale Hall> That's what I found too, but I got thinking. Given an arbitrary> triangel, it seems obvous that the question on finding the largest> circle fitting inside this triangel is well-defined. However, how can> one prove that it is a well-defined question?> Next, it seems also obvious that this largest cirlce would be one> that> has all the three sides of the triangel tangent to the circle, but how> can this be proved?> If a circle C is drawn in the interior of a triangle T, and C fails to> meet one of the sides of T, then one can draw a circle C' that contains> C in its interior, again within the interior of the triangle T. That> circle C' will have area strictly greater than the area of C.Yes, I can see this too, but I am not sure it does satisfy my formalrequirements. Perhaps I miss something like:1. If a circle and a line share some points in the plane, then either the line is tangent to the circle or the circle has points on both sides of the line. (To put it clumsily.)2. If a circle crosses any of the sides in a triangle, it is not inside the triangle.3. If a circle is inside a triangle with distance d from the side AB, you can make a new circle with the center d/2 closer to AB and radius R=. The new radius R is strictly larger than r.4. The largest largest incribed circle is one that touches all sides in the triangle. http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92=== === Subject: : Would like to use network flow to prove Menger's and the Konig-Egervary TheoremsHello math folks,I would like to use network flows to prove the Menger's (1.) andKonig-Egervary (2.) Theorems.1. Use network flows to prove Menger's Theorem for nonadjacent vertices ingraphs.Menger's Theorem: If x, y are vertices of a graph G and xy is not an elementof the edges of G, then the minimum size of an x, y-cut equals the maximumnumber of pairwise internally disjoint x, y-paths.Lambda (x, y) is the maximum number of internally disjoint pathsKappa (x, y) is the minimum size of an x, y-cut.A. Show that Kappa (x, y) >= Lambda (x, y). This would seem to bestraightforward. The minimum size of an x, y-cut is at least as large as themaximum number of internally disjoint x, y-paths. You can simply remove onevertex from each x, y-path to get a minimum cut.B. This is where I think that the need for network flows comes in. Show thatLambda (x, y) >= Kappa (x, y).TIP 1: The hint for the problem indicates that a network model for theproblem of internally disjoint paths in a digraph D can be created byreplacing each vertex v with two vertices v- and v+ that inherit theentering and exiting edges at v. By adding an edge of unit capacity, from v-to v+, we obtain the effect of limiting flow through v to one unit. Byputting very large capacity on the edges that were in D, we ensure that aminimum cut will count only edges of the form v- v+.TIP 2: To obtain a network model for the problem of edge-disjoint paths in agraph G, we must permit flow to pass either way in an edge. This can be doneby replacing each edge uv with two directed edges uv and vu. When thenetwork sends unit flow in both directions, in effect the edge is not beingused at all.I understand the concepts of global flow and cuts in a graph, but don'tunderstand the last two paragraphs, and how to maximize the network flowusing the two TIPs above. I don't understand how to implement thesuggestions for local flow and cuts.2. Konig-Egervary Theorem: The maximum size of a matching is the same sizeas a vertex cover number if a graph G is bipartite.Diana=============================================== ======God made the integers, all else is the work of man.L. Kronecker, Jahresber. DMV 2, S. 19.=== === Subject: : A problem of Graph thoery: How many different DAG are there with distinct Transitive Closure?I wonder if anyone can help with the follow problem:How many different acyclic digraphs are there with distinct TransitiveClosure? Here the vertices is labeled.Richard Sun=== === Subject: : Re: Frequentist probability confusion> It was my point, and I'll offer four possible interpretations:> 1. Probability theory only works for finite sets.> 2. The axiom of choice is wrong.> 3. Probability theory does not tell us what kinds of processes> of generating numbers are or aren't possible, and in fact> methods of generating numbers exist which have no corresponding> probability distribution.> 4. The problem is a problem with real arithmetic. Replace> reals by (for example) Conway numbers and consistent probability > distributions can be generated.> I think 3 is the most conservative.> R.You began with a criticism of frequentism. Frequentism is anuts-and-bolts area of mathematics intended for use by physicists,statisticians, and the like. As such, it concerns itself with finiteprocesses, computable numbers, and limits. To use other tools toattempt to invalidate such results is simply irrelevant: there is nopoint in arguing about what tools are permissible, one simple choosesone's arena and proceeds from there. So this area of probabilitytheory indeed places limits on processes of generating random numbers. If you choose to play a different game, you will likely get differentresults. And very well you may do so: probability theory is by nomeans restricted to frequentism.=== === Subject: : Factoring simplificationI've 2^512 * X = MI know X and M. I would like to find p and q (prime numbers) so that p*q=M.M = 13407807929942597099574024998205846127601533213795 05607720946287702503436819662776124055278656993783 12010775909824626384612108955400344285294121039523 749392^512 = 13407807929942597099574024998205846127479365820592 39337772356144372176403007354697680187429816690342 76900318581864860508537538828119465699464336490060 84096X = M - 2^512 note that 13407807929942597099574024998205846127 the first chars are the same.Is there any way to simplificate the problem ?=== === Subject: : Re: Factoring simplification> I've > 2^512 + X = M> I know X and M. I would like to find p and q (prime numbers) so that p*q=M.> M = 13407807929942597099574024998205846127601533213795> 05607720946287702503436819662776124055278656993783> 12010775909824626384612108955400344285294121039523> 74939> 2^512 = 13407807929942597099574024998205846127479365820592> 39337772356144372176403007354697680187429816690342> 76900318581864860508537538828119465699464336490060> 84096> X = M - 2^512 > note that 13407807929942597099574024998205846127 the first chars are the same.> Is there any way to simplificate the problem ?The way these things useually work is that you assume the factors are2^256+p and 2^256+q, with p,q small, and thereforeM=(2^256+p)*(2^256+q) = 2^512 + 2^256*(p+q) + p*q=> X = 2^256*(p+q) + p*qX = 122167393202662699485901433303270338123080784438678488403034403 511045732795976587607457012728087858582978454946290843X/2^256 ~= 10^39so p+q might be ~ 10^39X%2^256 = 177547503701392265973377372298289965909567806123822447140310676 72899388152987 = 23*300041463768385601* 2572796634743855408737253552583086836911046485893492725469So there's no choice of p, q which satisfy the conditions.Therefore, if there does exist such a factorisation, it's not of the form (2^256+p)*(2^256+q) with p, q << 2^256.Unless I've made a booboo somewhere.Phil1st bug in MS win2k source code found after 20 minutes: scanline.cpp2nd and 3rd bug found after 10 more minutes: gethost.cBoth non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL)=== === Subject: : Re: Factoring simplificationSeb escribi.97:> I've> 2^512 * X = M> I know X and M. I would like to find p and q (prime numbers) so that> p*q=M.I suppose that it is 2^512 + X = M> M = 13407807929942597099574024998205846127601533213795> 05607720946287702503436819662776124055278656993783> 12010775909824626384612108955400344285294121039523> 74939> 2^512 = 13407807929942597099574024998205846127479365820592> 39337772356144372176403007354697680187429816690342> 76900318581864860508537538828119465699464336490060> 84096> X = M - 2^512> note that 13407807929942597099574024998205846127 the first chars> are the same.> Is there any way to simplificate the problem ?X = 120670271369483373896166224983852615147192124992128373005140985 666967296618769192155613215115969655964889542672040234123925429 87271912141417021545053709157= 3 * 31 * 243381447109* 533126010665372074896650613978059028866848093780281204030755055 314111963980246892592802602388438081321404376453680712581608687 691774246765461The las factor is a composite number of 141 digits and no factor less than10^14. Then M is not a product of two prime factors.(see http://www.alpertron.com.ar/ECMC.HTM)Ignacio Larrosa Ca.96estroA Coru.96a (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com=== === Subject: : Re - Is Science Converging Towards Truth?Further ruminations on a speculative hypothesis. I'm of the opinion that a key revolutionary shift in physics isimminent, specifically, the realization that the speed of light is notan actual physical barrier.(Once this realization is reached theactual accomplishment of breaking of the barrier is a mere technicalproblem and will be reached soon after.) This of course is of theorder of the greatest scientific revolutions in history and will be onthe order of 100 years after the last great scientific revolution, atleast in physics, that of relativity. My hypothesis is that revolutionary increases in physical knowledgeare occurring at an exponentially faster rate. Then to estimate howfast is the exponential growth we might go back to the last greatrevolutionary advance prior to relativity, Newton's Principia, ca.1700 AD. We would then estimate that prior to that the last greatrevolutionary advance occurred 400 years prior to that, ca. 1300 AD.The natural philosopher I could most associate with this time wasRoger Bacon:Medieval Sourcebook: Roger Bacon: On Experimental Science, 1268 http://www.fordham.edu/halsall/source/bacon2.htmlThe First Scientist: A Life of Roger Baconhttp://www.amazon.com/exec/obidos/tg/detail/-/0786711167/ 104-5357401-0726320 But who could we associate to a revolutionary advance 800 years priorto that, ca. 500 AD? This was during the time of the Dark Ages and Idon't believe any European could be associated to such an advance.Perhaps then, assuming the validity of the hypothesis, it could beamong the Chinese, Hindu, or Arabic thinkers of that time. I'm notfamiliar with the scientific history of those cultures during thosetimes. I understand though Dick Teresi has written a book describingthe ancient scientific discoveries of non-western cultures:Lost Discoveries : The Ancient Roots of Modern Science--from theBabylonians to the Maya.http://www.amazon.com/exec/obidos/tg/detail/-/074324379X/ 104-5357401-0726320 Was there anyone ca. 500 AD who could be described as providing arevolutionary advance in physical knowledge relative to priorunderstanding? Time shift to now. Assuming the hypothesis, the next great revolutionafter the current era's breaking of the light speed barrier will occurabout 50 years later, ca. 2050 AD. Then after that, in 2075 AD. Thenin 2088. Then 2095, 2099, 2101, and 2102. But after that? Pehaps then there will be just minor adjustments ofthe decimal points. Bob Clark********************************************************* *******************=== === Subject: : Is Science Convergening Towards Truth? It has been much noted that scientific revolutions are occurring atfaster andfaster rates. Doesn't this imply that at some point they won't be ableto occurany faster? At that time might we be able to say that we will havereached afinal theory? Interestingly, judging by the past rate at which the scientificrevolutionsoccurred, we might even be able to estimate when such a final theorymight bereached for each scientific discipline. Bob Clark********************************************************* *****************=== === Subject: : Re: Re - Is Science Converging Towards Truth?> Time shift to now. Assuming the hypothesis, the next great revolution> after the current era's breaking of the light speed barrier will occur> about 50 years later, ca. 2050 AD. Then after that, in 2075 AD. Then> in 2088. Then 2095, 2099, 2101, and 2102.> But after that? Pehaps then there will be just minor adjustments of> the decimal points.That's when they'll realise that there is an infinite number of further levels of structure in science, and give up.Paul TownsendI put it down there, and when I went back to it, there it was GONE!Interchange the alphabetic elements to reply=== === Subject: : Re: Re - Is Science Converging Towards Truth? charset=iso-8859-1> Further ruminations on a speculative hypothesis.> I'm of the opinion that a key revolutionary shift in physics is> imminent, specifically, the realization that the speed of light is not> an actual physical barrier.(Once this realization is reached the> actual accomplishment of breaking of the barrier is a mere technical> problem and will be reached soon after.) This of course is of the> order of the greatest scientific revolutions in history and will be on> the order of 100 years after the last great scientific revolution, at> least in physics, that of relativity.I guess right after they realize the earth is flat and not a ball?=== === Subject: : Re: Re - Is Science Converging Towards Truth?> Further ruminations on a speculative hypothesis.> I'm of the opinion that a key revolutionary shift in physics is> imminent, specifically, the realization that the speed of light is not> an actual physical barrier.(Once this realization is reached the> actual accomplishment of breaking of the barrier is a mere technical> problem and will be reached soon after.) Why are you of that opinion? There's a difference between believing in the possibility of the existence of an adjustable mechanism behind light speed and in believing that that it is imminent that it exists.Richard Perry=== === Subject: : cross correlation R_xy(t1,t2) = R_yx(t1,t2)* ?I saw this strange property,R_xy(t1,t2) = complex conjugate of R_yx(t1,t2)where R_xy(t1,t2) is defined as E[x(t1)y(t2)*]I can't see that porperty from the definition.Can anybody see it? === === Subject: : Re: Why do math folks tend to end their letters with cheers?> .... > Nothing to do with mathematics: cheers is very commonly used in the> British Isles (and possible some of the countries that used to be part of> the extinct British empire) in the way you describe. .=== === Subject: : Mobius transformations and circle packingsSuppose A and B are isomorphic packings of 3 circles. What I mean is, saywe have 3 circles that are all tangent to each other in the complex planeand denote the 3 circles plus the triangular region that they bound by A.Now say we have 3 totally different circles that are all tangent to eachother in the complex plane and denote the 3 circles plus the triangularregion that they bound by B. Then by A and B are isomorphic packings I meanthat there is an orientation preserving homeomorphism, denoted f, between Aand B.Let's assume for convenience that the circles in A (denoted P_1, P_2, P_3)and the circles in B (denoted P_1 ', P_2 ', P_3 ') are indexed by the sameset, V = {1,2,3}, and that the isomorphism between them respects theindexing; that is, A = {P_i, i in V}, B = {P_i ' : i in V} and f(P_i) = P_i'.Now, there is a unique mobius transformation that maps the tangency pointsof the 3 circles in A to the 3 tangency points in B. Why does it followthat this mobius transformation maps P_1 to P_1 ', P_2 to P_2 ', and P_3 toP_3 '?????? Also, why is it that this mobius transformation maps thetriangular region bounded in A to the triangular region bounded in B? (Thisrelies on the assumption that f is orientation preserving?)Moshe=== === Subject: : Re: Mobius transformations and circle packings 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21:> Now, there is a unique mobius transformation that maps the tangency points> of the 3 circles in A to the 3 tangency points in B. Why does it follow> that this mobius transformation maps P_1 to P_1 ', P_2 to P_2 ', and P_3 to> P_3 '??????It follows because there is only one triple of circles that can have that set of three tangency points. You can find the centers of the circles by constructing a circle circumscribing the three tangency points, and forming lines tangent to this circle through the three tangency points -- the centers are at the intersections of these three lines. In the limiting case where the three tangency points are collinear, the centers are at the midpoints of each pair of points. http://www.ics.uci.edu/~eppstein/=== === Subject: : Re: heat equation + strange boundary conditions>w_t = a^2.w_xx (heat equation, the indices are the orders of derivation)>my boundary conditions are as such :>w (x, 0)=0 (although it's not a heat diffusion problem, let's say that the>beam is at temp 0 at time 0)>w(0,t)=exp (t / 2.a^2) (one extremity is increasingly heated, notice that w>will be discontinuous in (0,0), which should not be a problem in a theorical>way)You need another boundary condition at some value of x (usually youconsider what happens at the other extremity; perhaps require the limitbe finite as x -> infinity).>hence i pose w(x,t)=X(x).T(t) as usualYou can get solutions u(x,t) = X(x) exp(t/2.a^2) satisfying your condition at x=0, but of course not at t=0. However ifyou write w(x,t) = u(x,t) + v(x,t), you then want v(x,t) to satisfythe pde with boundary conditions v(x,0) = -u(x,0) and v(0,t) = 0,which may be more like what you're used to.=== === Subject: : Question about first-order logicI'm currently studying first-order logic, but I have two questionsthat doesn't seem to be answered in the book I'm using. I hope someonecan spare a moment of their time and briefly explain it :)a) The book starts by defining what are first-order expressions, whatis a model, what does it means that a model satisfies a first-orderexpression, what does it mean for an expression to be valid etc. Thenit goes on to define how to go from some valid expresions to othervalid expressions and proofs that certain sets of expression arevalid. This is used to create a proof system with the properties thata) any expression provable in the system is valid and (amazingly) b)every valid expression has a proof in the system.Now my question is, in all these proofs of completeness and soundnessetc. all kinds of axioms (such as the induction axiom, the axiom ofchoice etc.) are implicitly used along with techniques such asindirect proofs. Now I'm wondering what is the justification for this?I mean, first-order logic should be able to support many kinds ofsets of axioms and the axiom of induction, for instance, certainlydoesn't have to be among them. And we carefully prove that things likeindirect proofs works inside of the system. However when proving thesethings we are - outside of the system - using all kinds of arbitraryaxioms and inference rules (maybe even the rules we are proving)! Somy question is what framework it is that these proofs take place inand where does all these axioms come from and how can we use themright away? And isn't there some kind of logical problem in usingthese rules when studying something that is even more fundamental?b) Exactly how does one define a model? I understand how one cancreate a vocabulary of first-order expressions for the naturalnumbers (it's just a set of sequences of certain symbols). Also, Isee how we can select some of these expressions and call them axiomsand use them inside of the proof system we have constructed. Butexactly how does one specify a model for the natural numbers? Imean, doesn't that in itself require some kind of axioms? I mean, itmust be an infinite construction. So it seems that the specificationof a model will have to take place in some other kind of logicalsystem which confuses me a bit because it seems first-order logicshould be very fundamental?! The book I'm using seems to say somethinglike: Well, 0 should correspond to 0, 1 should correspond to 1 thesuccessor function is simply the function n+1 etc. But how do we knowwhat those objects we are using in the definitions are?!Surfing=== === Subject: : Re: Question about first-order logic> indirect proofs. Now I'm wondering what is the justification for this?> I mean, first-order logic should be able to support many kinds of> sets of axioms and the axiom of induction, for instance, certainly> doesn't have to be among them. And we carefully prove that things like> indirect proofs works inside of the system. However when proving these> things we are - outside of the system - using all kinds of arbitrary> axioms and inference rules (maybe even the rules we are proving)! So> my question is what framework it is that these proofs take place in> and where does all these axioms come from and how can we use them> right away? And isn't there some kind of logical problem in using> these rules when studying something that is even more fundamental?No. The mathematics of the metatheory uses the same plain old logic as any other mathematical theory. If one wants to be heroic, he can take the Intuitionist tack and permit only constructive proof and no use for the law of the excluded middle when dealing with infinite sets. That approach cannot prove as much, but no one argues with its soundness either.If you want to formalize the metatheory then you will run into the same problem in the meta-meta-theory. At some point, starndard mathematical reasoning will be used.Bob Kolker=== === Subject: : Could someone help please.started with this problem because I can't quite get myself to graspwhat I should do. Your help would greatly be appreciated.Question: Let G be a group of all functions f: R -> R with theoperation of addition: (f + g)(x)= f(x) + g(x). If N={f E G:f(1/4)=0}, the prove that N is a normal subgroup of G and G/N isisomorphic to R.=== === Subject: : Re: Could someone help please.> started with this problem because I can't quite get myself to grasp> what I should do. Your help would greatly be appreciated.> Question: Let G be a group of all functions f: R -> R with the> operation of addition: (f + g)(x)= f(x) + g(x). If N={f E G:> f(1/4)=0}, the prove that N is a normal subgroup of G and G/N is> isomorphic to R.Consider the map h: G ---> R defined viah(f) := f(1/4) (for f in G)and note, that h is a homomorphism of groups.Marc=== === Subject: : russian instructional mathematics books?Is mathematics taught in russia any different or better than as taught orunderstood in the rest of the world? Have any special undergraduate level orabove books for mathematics been written in russia or the former soviet union?G C=== === Subject: : Re: russian instructional mathematics books?> Is mathematics taught in russia any different or better than as taught or> understood in the rest of the world? Have any special undergraduate level or> above books for mathematics been written in russia or the former soviet union?Grade for grade Russian schools are more advanced than their American counterpart, at least up to the junior/senior year and graduate school.Bob Kolker=== === Subject: : Re: Infinitesimal generators of the 6x6 symmetric matrices Lie algebra1) In your expression, if I have 6 matrices for the block B, other 6for the block C and other 9 for the block A, the total number ofmatrices is not 6x6x9 ??? How can I construct a 21-element basis?2) I use to write the symplectic sp(6) matrix J as a block-diagonalmatrix with three symplectic forms:/ 0 1 0 0 0 0 | -1 0 0 0 0 0 || 0 0 0 1 0 0 || 0 0 -1 0 0 0 || 0 0 0 0 0 1 | 0 0 0 0 -1 0 /What does it imply on the structure of the basis-matrices? (The tworepresentations should be orthogonally related)Gerardo> Hi to everybody,> I'm a physicist and I need all the 21 6x6 matrices which form a basis> for the Lie Algebra of the 6x6 real symmetric matrices,> or, equivalently, the 21 infinitesimal generators of the Sp(6,R)> symplectic group in the representation of 6x6 matrices (they are> related to the symmetric matrices through matrix multiplication by the> symplectic form).> Please can you tell me a web link (or a book) where I can find them?> Or kindly post me them directly?> It's fairly easy to construct explicitly a basis for the real Lie> algebra sp(6,R) of the the Lie group Sp(6,R).> Let Y be an element of Sp(6,R) and J be the symplectic form, so> J = Y^t J Y.> Take a 1-parameter subgroup {Y(t)} of Sp(6,R) that has Y(0) = 1,> differentiate the above equation, and evaluate at t = 0 to find a> corresponding equation for the elements of the Lie algebra sp(6,R):> 0 = X^t J + J X, (1)> where X = dY/dt evaluated at t = 0.> Write everything in terms of 3x3 blocks:> _ _ _ _> | A B | | 0 1 | > X = | |, J = | |.> |_C D_| |_-1 0_|> Substituting these into (1) gives> _ _ _ _> | -C^t A^t | | C D | > 0 = | | + | |,> |_-D^t B^t_| |_ -A -B_|> so B^t = B, C^t = C, and D^t = -A.> B a 3x3 symmetric matrix gives 6 basis elements; C a 3x3 symmetric> matrix gives 6 basis elements; A an arbitrary 3x3 gives nine basis> elements. This accounts for the 21 element basis of the Lie algebra> sp(6,R). Just put ones and zeros in the appropriate places.> George=== === Subject: : L^p functionsIf f is a L^p function, with p>1, on a open subset A of R^n or on R^n , canI say that f is a L^1 function locally, i.e. on the compact subset of A?=== === Subject: : Re: Isotopic knot diagrams.> [top-posting corrected]>Zhang schreef in bericht> One way one can sometimes see that two knots are not regularly> isotopic, is to compute the writhe. My question is: Does there exist> isotopic diagrams in R^3> having the same writhe but not being regularly isotopic ??>Isn't the writhe computed via the knot group? Because then it would be the>classical example of the connected sums of (a) two trefoil knots and (b) a>trefoil knot and an inverted trefoil-knot. Both have the same knot groups,>proved is listed in Crowell & Fox, I'm to lazy to look it up now).> I think both of you are confused, but in different ways. (On> the other hand, I may not be up on the current use of language,> in particular, on Colin Adams's language in _The Knot Book_.)> The writhe, as I have always understood it, is not an invariant > of (just plain, otherwise unstructureladen) knots in R^3; it is> an invariant of knot *diagrams* in R^2 (and what it's invariant> under is regular homotopy of diagrams, not regular isotopy,> which is not a standard phrase to the best of my knowledge).> It is most certainly not computed via the knot group in the> most obvious sense of that phrase (namely, the one that would > make the rest of Rene's paragraph at least _prima facie_ reasonable). > There's a sense in which it can be computed *in* the knot group, > for indeed the writhe is an integer which is (up to a conventional> sign) equals the image of a certain element of the knot group (to> wit, the diagrammatic longitude) in the infinite cyclic group> that is the abelianization of the knot group. But that's going> an awfully long way around.> To get back to Zhang's original question, I think it must have> been meant to be this question (modulo English style and > increased notation for the sake of clarity):> Do there exist knot diagrams D, D' in R^2 such that > (1) the knots K and K' of D and D' are isotopic in R^3,> (2) the diagrams D and D' have the same writhe, but> (3) D and D' are not regularly homotopic in R^2?> I think the answer to that question is Yes, and I have a feeling> that an example can be constructed out of Milnor's paisley, but> I don't know and am too hungry at the moment to spend any more time> on this now.> Lee RudolphI've been away for some days.Sorry about the confusion I manged to produce. I did mean diagrams inR^2.. typo.. sorry. The terminology I've learned so far is thefollowing: 2 diagramsD and D' in R^2 are called isotopic if there is a sequence ofReidemeister moves( and their inverses) and planar isotopies s.tperforming this sequence on D gives the diagram D'. If we can get fromD to D' without using R1 (and its inverse) then D and D' are regularlyisotopic.Since last I have seen an exercise in a book, where diagrams are drawnon S^2 rather than in R^2. And that exercise is actually to prove thatthe answer to my question is no if diagrams are drawn on S^2... SoI'll have to think about whether there is any essential differencebetween diagrams in R^2 and on S^2.