mm-4529 === alt.math.recreational === Subject: Re: Bug in Mathematica 6 - N - 13 (MeijerG, invalid divergence, regression bug) posting-account=ubyIWAkAAABW-OTbVB1QiN1oZlu0qUgw CLR 2.0.50727),gzip(gfe),gzip(gfe) BB> it's been fixed for a while. For a while? ...................................................... {$Version,$ReleaseNumber} {6.0 for Microsoft Windows (32-bit) (June 19, 2007),1} f = MeijerG[{{1, 1}, {}}, {{2, 2}, {}}, 1]; N[f] ComplexInfinity ...................................................... > (**************************************************************) If the same bugs exist through numerous software releases, > I think that is valuable public information. It just should not happen. -- Brad Cooper (**************************************************************) Our little demo continues..... Hello again from the VM machine > which hopefully soon will not be ignored by CAS manufacturers. This example demonstrates YET ANOTHER case of bad defects in > Wolfram Research Quality Assurance process. (***************************************************************) f = MeijerG[{{1, 1}, {}}, {{2, 2}, {}}, 1]; > FunctionExpand[f] > æ æ æ æ æ æ æ æ æ æ æ æ 1/6 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ(* OK *) > N[f] (***************************************************************) Mathematica 6.0 æ æ æ æ ComplexInfinity æ æ æ æ <------------ BUG Mathematica 5.2 æ æ æ æ 0.166667 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ OK Mathematica 4.2 æ æ æ æ 37.417 æ æ æ æ æ æ æ æ æ<------------ BUG Mathematica 3.0 æ æ æ æ -219.948 æ æ æ æ æ æ æ æ<------------ BUG (***************************************************************) Let there be a better Computer Algebra System (Mathematica 7?), Vladimir Bondarenko VM and GEMM architect > Co-founder, CEO, Mathematical Director Try it in the next maintenance release... it's been fixed for a while. Bhuvanesh, > Wolfram Research- Hide quoted text - - Show quoted text - === Subject: Re: LOG(3) / LOG(2) Prove that LOG (3) / LOG(2) is irrational. Proceeding indirectly, assume that log(3)/log(2) is rational, i.e., >let log(3)/log(2) = p/q with p and q integers and q>0. >Then, p log(2) = q log(3) >So log(2^p) = log(3^q) >So 2^p = 3^q. >But this is impossible because the left side is even 2^0 is even? 2^(-17) is even? > Be a bit more careful with your hypotheses. Didn't you see my hypothesis that q>0? Of course, that implies that > p>0 also since both log(2) and log(3) are positive. Then it's a question of what you put into the proof explicitly, and what you leave to the reader to supply for herself. I figure that someone who can't prove on her own that log 3 / log 2 is irrational may not notice the implication p > 0, so I'd put it in and not leave anything to chance. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: How to prove the continiuity of the function. > Cp is Cantor set in real analysis. m is Lebesgue measure. x .81ü [ 0, 1]. m( [0,x] .81À ( [0,1] - Cp ) ) > f (x) = ------------------------------------------------------- > m( [0, 1] - Cp ) f (0) =0 , f (1)=1, f: [0,1] --->[ 0,1] How to prove the continuity of f ? I think it is m(Cp)=0 and therefore m([0,x].81À([0,1]-Cp)) = m([0,x].81À([0,x]-Cp)) = m([0,x]-Cp) = m([0,x]) and m([0,1]-Cp) = 1 Thus f(x) = m([0,x]) = x. f = id is continuous. Hero Wanders === Subject: Re: How to prove the continiuity of the function. posting-account=d9DpDwoAAADlGCWGhrCHDkiN-0F85Exg MathPlayer 2.10b; Avant Browser; .NET CLR 1.1.4322; .NET CLR 2.0.50727; TheWorld),gzip(gfe),gzip(gfe) Cp is Cantor set in real analysis. m is Lebesgue measure. x Áæ [ 0, 1]. m( [0,x] Á.83 ( [0,1] - Cp ) ) > f (x) = ------------------------------------------------------- > m( [0, 1] - Cp ) f (0) =0 , f (1)=1, f: [0,1] --->[ 0,1] How to prove the continuity of f ? I think it is m(Cp)=0 and therefore m([0,x]Á.83([0,1]-Cp)) = m([0,x]Á.83([0,x]-Cp)) = m([0,x]-Cp) = m([0,x]) > and m([0,1]-Cp) = 1 > Thus f(x) = m([0,x]) = x. f = id is continuous. Hero Wanders If Cp is general cantor set. 0 <= m(Cp) < 1. How to prove the continuity. === Subject: Re: How to prove the continiuity of the function. > Cp is Cantor set in real analysis. m is Lebesgue measure. x '.be [ 0, 1]. > m( [0,x] '.83 ( [0,1] - Cp ) ) > f (x) = ------------------------------------------------------- > m( [0, 1] - Cp ) > f (0) =0 , f (1)=1, f: [0,1] --->[ 0,1] > How to prove the continuity of f ? > I think it is m(Cp)=0 and therefore > m([0,x]'.83([0,1]-Cp)) = m([0,x]'.83([0,x]-Cp)) = m([0,x]-Cp) = m([0,x]) > and m([0,1]-Cp) = 1 > Thus f(x) = m([0,x]) = x. > f = id is continuous. > Hero Wanders If Cp is general cantor set. 0 <= m(Cp) < 1. How to prove the >continuity. If E is a subset of [x, x+delta] then m(E) <= delta. ************************ David C. Ullrich === Subject: Re: How to prove the continiuity of the function. posting-account=d9DpDwoAAADlGCWGhrCHDkiN-0F85Exg MathPlayer 2.10b; Avant Browser; .NET CLR 1.1.4322; .NET CLR 2.0.50727; TheWorld),gzip(gfe),gzip(gfe) > Cp is Cantor set in real analysis. m is Lebesgue measure. x '.be [ 0, 1]. > æ æ æ æ æ æ æm( [0,x] '.83 ( [0,1] - Cp ) ) > f (x) = æ------------------------------------------------------- > æ æ æ æ æ æ æm( [0, 1] - Cp ) > f (0) =0 , f (1)=1, f: [0,1] æ--->[ 0,1] > How to prove the continuity of f ? > I think it is m(Cp)=0 and therefore > m([0,x]'.83([0,1]-Cp)) = m([0,x]'.83([0,x]-Cp)) = m([0,x]-Cp) = m([0,x]) > and m([0,1]-Cp) = 1 > Thus f(x) = m([0,x]) = x. > f = id is continuous. > Hero Wanders If Cp is general cantor set. æ æ0 <= m(Cp) < 1. How to prove the >continuity. If E is a subset of [x, x+delta] then m(E) <= delta. ************************ David C. Ullrich- ñ[Thorn].b2¯±È[Ca pitalOGrave][YAcute]îÌë[CapitalA DoubleDot][Times].85 - - ìïæ.beñ[YAcute] îÌ[Micro]Äë[C apitalADoubleDot][Times].85 - === Subject: Re: Numbering of years. CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) NetApp/5.6.2R1D28) > I am truly an admirer of the West, but a Democratic, æAtheistic, > Scientific and Artistic West. > We cannot survive otherwise! Survive as exactly what? === Subject: Re: Numbering of years. posting-account=6eXpYAoAAABj5bAPUCJz60RPJj70DBtk Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > I am truly an admirer of the West, but a Democratic, Atheistic, > Scientific and Artistic West. > We cannot survive otherwise! Survive as exactly what? As a Civilization! Start a new thread! Sigge === Subject: Re: Numbering of years. posting-account=n4TzyQkAAADLWxrRHqyiUZ-1SZdOB4vv Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) Hero >Actually my name sounds greek too, but is frisian. The Frisians, an ethnic group of northwestern Europe inhabiting an > area known as Frisia. That is a location to which the Greco-Roman culture spread, so you >yourself are likely a product of that culture, as well as a product of a >gene pool that has its home base in northwestern Europe. So Science were traded for the tin, amber and salt of these > northwestern regions? Okay, that's unfair, but: Let me only refer to math: .... Budala! Hero is an English word with Greek Etymology! As a proper name > it a name for a woman (Hero and Leander, you know??). And the mathematician Hero of Alexandria was female too. And in frisia my name must come from Greek origin. And Hera was a female god too. What was attributed to her eyes as a charming expression: she had eyes like a...? > But I understand > from your texts (the only thing you are in usenet is just a series of > texts) that you are really Turkish, in the cultural meaning surely > maybe even in the usual meaning. So i'm a female turk of frisia with a greek name, found out by Sigge, the scientist. Is this about me or what? I suspected from your first input that you are of lower intelligence! > Now you open your mouth and simply confirm my suspicion. You disclose a scientific illiteracy that > disqualifies you from further discussion. Are you just an insolent child? Sigge Now hear one scientist of ancient greece, Plato about the godin Athena.She was born at the banks of Triton lake in Lybia, as the godin of Neith, with a temple in Sais. Black Athena. Bye, bye Hero PS Riace warriors were brown or black too. htp://www.northernlightstudio.com/patinalec.php === Subject: Re: Numbering of years. posting-account=6eXpYAoAAABj5bAPUCJz60RPJj70DBtk Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > Hero > Actually my name sounds greek too, but is frisian. The Frisians, an ethnic group of northwestern Europe inhabiting an > area known as Frisia. That is a location to which the Greco-Roman culture spread, so you >yourself are likely a product of that culture, as well as a product of a >gene pool that has its home base in northwestern Europe. So Science were traded for the tin, amber and salt of these >northwestern regions? Okay, that's unfair, but: Let me only refer to math: .... Budala! Hero is an English word with Greek Etymology! As a proper name > it a name for a woman (Hero and Leander, you know??). And the mathematician Hero of Alexandria was female too. And in frisia my name must come from Greek origin. And Hera was a female god too. > What was attributed to her eyes as a charming expression: she had eyes > like a...? But I understand > from your texts (the only thing you are in usenet is just a series of > texts) that you are really Turkish, in the cultural meaning surely > maybe even in the usual meaning. So i'm a female turk of frisia with a greek name, > found out by Sigge, the scientist. Is this about me or what? I suspected from your first input that you are of lower intelligence! > Now you open your mouth and simply confirm my suspicion. You disclose a scientific illiteracy that > disqualifies you from further discussion. Are you just an insolent child? Sigge Now hear one scientist of ancient greece, Plato about the godin > Athena.She was > born at the banks of Triton lake in Lybia, as the godin of Neith, with > a temple in Sais. > Black Athena. > Bye, bye > Hero > PS Riace warriors were brown or black too. > htp://www.northernlightstudio.com/patinalec.php When I saw Hero alone as a signature I thought you took the greek word heros (omega) that has been adopted as hero in English. The female name Hero is mainly known from the tragic love story Hero-Leander that illiterate people like you would call it Turkish story, since it is played in what is Turkey today. (I think that Alma-Tadema, your great Ancestor, if you really are Frisian, would laugh at you!) There is a third Hero the Alexandrian scientist! His name is Heron (omega again) but it is transliterated as Hero in English, exactly the same way as Platon becomes Plato. Now as a homework, instead of reading stupid fantasy books, learn Hero's method of computing square roots to any desired accuracy. And then compare his method with the Newton-Raphson method. When you say Bernal in a context like ours, you mean of cource Black Athena. You do not even read my input! A few inputs back, it was I who took up the Bernal case. With utmost contempt towards scholars who have nothing to say and they invent fanciful, sensational theories njust to make a living/name. In bernals case it was Herostratic (another idea came up, change your name to Herostratos). He was pulverized along with all Afrocentrists in Black Athena Revisited. A very bad thing was that even Bush's FM Condom Rice is an Afrocentric Idiot. In 1987, I was shocked by the title Black Athena and even more by the subtitle The fabrication of Ancient Greece..... For a person like me who approached Ancient Greece through Mathematics and Science it was unimaginable that somebody would even think about massive transfers from Egypt to Greece! The two Civilizations are so antithetical that it is obvious to any person of moderate intellectual capacity that this could not be the case. But Bernal, much the same way like you, imagined black armies in Kurdistan, because no one said there were no black armies in Kurdistan! A fine way of argumentation! Start a new thread, budala! Sigge PS. Riace Warriors are neither brown or black (are you a mixture of races?). They are Bronzes found in Riace at the bottom of the sea (as ALL bronzes from Ancient Greece that survived the religious fanaticism). They are according to experts the pinnacle of Sculpture (modern artists cannot create such things). There are some efforts by the lackeys of a certain establishment that works against the Western Civilization to diminish their Beauty and place them on equal footing along other artifacts from inferior cultures/civilizations. As for my pseudonym, to satisfy your curiosity, I am physically superior, was very well-trained, and my wife (an artist) called me Riace Warrior because of my resemblance to them. I was very much flattered and kept it as a name! Idem === Subject: Re: Computability posting-account=GlvObgoAAADb2DcNXTHND2RnLzl0sF3t Tablet PC 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) >We've digressed enough. What of the proposition that o(ab) mod b =3D max (o(a) mod m / o(b) mod m, o(b) mod m / o(a) mod m)? That's an interesting proposition in algebra. To me, it is... I assume you mean, if o(x.m) is the order of x mod m where (x,m) = 1, > o(ab,m) = max(o(a,m)/o(b,m), o(b,m)/o(a,m)). æBut that's absurd: what about > cases where neither o(a,m)/o(b,m) nor o(b,m)/o(a,m) is an integer? æ Well, that's valid and I located several counterexamples. Let me be more clear: If we have o(a) mod m and o(b) mod m and want o(ab) = max ( o(a)/o(b), > o(b)/o(a) ), then what conditions must we apply to a, b, and m to make > this true? Anybody? I think that a invertible mod a, gcd(phi(a),phi(b),phi(m))>2, and a,b,m pairwise > coprime will suffice for the conditions. Those are the conditions > derived from FLT with variables x, y, and z instead of a, b, and m. > Does anyone have any other suggestions? Those sound like a lot of conditions but then FLT and MA is my > favorite topic. Doug OK, I'll own that. I left out the simpler conditions I mention in my do not suffice, but the question remains: Of the properties derivable from x^p + y^p = z^p using modular arithmetic, which are the minimal set of properties from which it can be shown there are no such triples, using (er, mostly?) the theorems of modular arithmetic and number theory? And is the above about max relevant to that, or a coincidence only? Doug === Subject: Re: Computability posting-account=GlvObgoAAADb2DcNXTHND2RnLzl0sF3t Tablet PC 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > You claimed to have checked it up to z = 256, but apparently you > missed the following counterexample: æ (x,y,z) = (74,129,143) which has signature (6,3,3). Wow. At first I thought this was a candidate triple, because for this triple (x,y,z) the following are indeed all true: x < y < z < x+y (x,y) = (y,z) = (z,x) = 1 o(x/y,z)/2 = o(z/x,y) = o(z/y,x) = p in Po, the odd primes ( o = 3 here) but then I went back and checked more rigorously and checked probablilities, too. at probability = about 29%, x < y < z < x+y at probability = about 21%, (x,y) = (y,z) = (z,x) = 1 independently at probability = << 1%, o = odd prime independently of the above at probability so far zero, (x/y)^p == -1 but only in combination with the above 3. and that last is the test for which this fails. As one might expect, with such a tiny probability that might slip through, so yes, while it has the signature, it doesn't pass that one test: (x/y)^p == 143 - 105 == -38, not -1 which would be 142 mod 143 This comes from the first derivation I made in modular arithmetic, way back, gosh, it was mentioned near the beginning of this thread, and in x^p + y^p = z^p we have x^p + y^p == 0 mod z, and so x^p == -(y^p) mod z, and (x^p)/(y^p) == -1 mod z, and (x/y)^p == -1 mod z, where /y means the multiplicative inverse of y in the reduced residues of z, that is, in what my text calls Zz*. Darn. For a moment, there, I thought I'd finally be able to drop this search! Only around 8 times as many triples to check to limit = 256, but sooo many to limit = 65535. Dang it, Chip Eastham, you rascal, what were we/I thinking!? signature. Well done! (74, 129, 143) Really, that is good work. To limit = 143, my program checked 1,415 triples coprime and with the inequality, from 343,000 generated triples. Wow. Doug P.S. The smallest triple with the inequality and the coprimality is the classic (3,4,5)... P.P.S. I think another condition is x + y == z mod p. I am working to understand that one.... === Subject: Re: Computability You claimed to have checked it up to z = 256, but apparently you > missed the following counterexample: æ (x,y,z) = (74,129,143) which has signature (6,3,3). > signature. Well done! (74, 129, 143) Really, that is good work. To limit = 143, my program > checked 1,415 triples coprime and with the inequality, from 343,000 > generated triples. Wow. > P.P.S. I think another condition is x + y == z mod p. I am working to > understand that one.... Two things here: We have x^p + y^p = z^p or z | x^p + y^p, and then we have in my P.P.S. that x+y = z mod p. Where did I get that? Well, in AA we learned that for prime modulus p, x^p + y^p = (x+y)^p mod p, The Student's Equality. (Students in algebra frequenetly write this in a non-modular form; that is a mistake.) So: x^p + y^p = z^p (x + y)^p == z^p mod p (x + y)^p - z^p == 0 mod p (x + y - z)^p == 0 mod p x + y - z == 0 mod p or p | x + y - z Agreed? Doug x === Subject: Re: Computability > You claimed to have checked it up to z = 256, but apparently you > missed the following counterexample: > æ (x,y,z) = (74,129,143) which has signature (6,3,3). > signature. Well done! > (74, 129, 143) Really, that is good work. To limit = 143, my program > checked 1,415 triples coprime and with the inequality, from 343,000 > generated triples. Wow. > P.P.S. I think another condition is x + y == z mod p. I am working to > understand that one.... Two things here: We have x^p + y^p = z^p or z | x^p + y^p, and then we >have in my P.P.S. that x+y = z mod p. Where did I get that? Well, in >AA we learned that for prime modulus p, x^p + y^p = (x+y)^p mod p, The >Student's Equality. (Students in algebra frequenetly write this in a >non-modular form; that is a mistake.) So: x^p + y^p = z^p >(x + y)^p == z^p mod p >(x + y)^p - z^p == 0 mod p >(x + y - z)^p == 0 mod p >x + y - z == 0 mod p or >p | x + y - z Agreed? Yes, although it's easier than that. By Fermat's little theorem, a^p = a mod p, hence x^p = x mod p y^p = y mod p z^p = z mod p so x^p + y^p = z^p => x + y = z mod p quasi === Subject: Re: Computability posting-account=GlvObgoAAADb2DcNXTHND2RnLzl0sF3t Tablet PC 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) You claimed to have checked it up to z = 256, but apparently you > missed the following counterexample: æ (x,y,z) = (74,129,143) which has signature (6,3,3). (x/y)^p == 143 - 105 == -38, not -1 which would be 142 mod 143 > If (x/y)^p == -1 then (x/y)^2p == 1 and o(x/y,z) = 2p But we lose information when we write o(x/y,z) = 2p. That doesn't specify that (x/y)^p == -1. (x/y)^p could be congruent to x for all we know! It just doesn't say. combination of x and y mod z that has an order of p or of 2p. Let's see... x^p + y^p = z^p x^p + y^p == 0 mod z x^p == -(y^p) mod z x^p / -(y^p) == 1 mod z 1/(-1) * (x/y)^p == 1 mod z.... These are calculations in abstract algebra, I think. I'd better ask for help here, but you see the point? There would be only the inequality, without loss of generality, and the signature, but the signature would have four parts, not just three. The part about coprimality might be implied by the existence of the various signatures. So it would really boil down to just the signature. Aside: I am looking at the conjoint and disjoint probabilities of the inequality, the coprimality, and the signature, applying Bayesian statistics. It's not going well because I only have had stats I, and we just brushed by Bayes. Doug === Subject: Re: Computability >If (x/y)^p == -1 then (x/y)^2p == 1 and o(x/y,z) = 2p But we >lose information when we write o(x/y,z) = 2p. That doesn't >specify that (x/y)^p == -1. (x/y)^p could be congruent to x for >all we know! It just doesn't say. Right. Let me try to restate your conjecture ... For greater simplicity, I won't bother with the order function. Instead, I'll just use the divisibility relation. Dougster's conjecture: There do not exist positive integers x,y,z such that (1) x < y < z < x+y (2) x,y,z, are pairwise coprime (3) z - y is not a multiple of x (4) For some prime p > 2, x | z^p - y^p y | z^p - x^p z | x^p + y^p Remarks: (1) For p = 3, your conjecture holds for z <= 1000. (2) As you've previously noted, if a proof of your conjecture could be had, that would yield an instant proof of FLT, however the known truth of FLT does not appear to yield a proof of your conjecture. (3) As far as trying to prove your conjecture, I would start with a fixed prime, for example p = 3. quasi In the future, for new posts, please start the subject line with a prefix of -- . By using this convention, such posts can be quickly === Subject: Re: Computability posting-account=oTDIagkAAACTxHurtPutBWvNQS8ZCNO9 Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) If (x/y)^p == -1 then (x/y)^2p == 1 and o(x/y,z) = 2p But we >lose information when we write o(x/y,z) = 2p. That doesn't >specify that (x/y)^p == -1. (x/y)^p could be congruent to x for >all we know! It just doesn't say. Right. Let me try to restate your conjecture ... For greater simplicity, I won't bother with the order function. > Instead, I'll just use the divisibility relation. Dougster's conjecture: There do not exist positive integers x,y,z such that (1) x < y < z < x+y (2) x,y,z, are pairwise coprime (3) z - y is not a multiple of x > I haven't been following this closely but... By (1), 0 < (z-y) < x, so (z-y) already cannot be a multiple of x. Perhaps you meant x is not a multiple of z - y? > (4) For some prime p > 2, x | z^p - y^p > y | z^p - x^p > z | x^p + y^p > I note in passing that z - y | z^p - y^p. === Subject: Re: Computability >If (x/y)^p == -1 then (x/y)^2p == 1 and o(x/y,z) = 2p But we >lose information when we write o(x/y,z) = 2p. That doesn't >specify that (x/y)^p == -1. (x/y)^p could be congruent to x for >all we know! It just doesn't say. > Right. > Let me try to restate your conjecture ... > For greater simplicity, I won't bother with the order function. > Instead, I'll just use the divisibility relation. > Dougster's conjecture: > There do not exist positive integers x,y,z such that > (1) x < y < z < x+y > (2) x,y,z, are pairwise coprime > (3) z - y is not a multiple of x I haven't been following this closely but... By (1), 0 < (z-y) < x, so (z-y) already cannot be a multiple of x. Perhaps you meant x is not a multiple of z - y? > (4) For some prime p > 2, > x | z^p - y^p > y | z^p - x^p > z | x^p + y^p I note in passing that z - y | z^p - y^p. Sure, but that's an identity. The other conditions (with (3) now removed) give true restrictions on (x,y,z). However, in any case, I just posted a counterexample, so the conjecture fails. quasi === Subject: Re: Computability >If (x/y)^p == -1 then (x/y)^2p == 1 and o(x/y,z) = 2p But we >lose information when we write o(x/y,z) = 2p. That doesn't >specify that (x/y)^p == -1. (x/y)^p could be congruent to x for >all we know! It just doesn't say. Right. Let me try to restate your conjecture ... For greater simplicity, I won't bother with the order function. >Instead, I'll just use the divisibility relation. Dougster's conjecture: There do not exist positive integers x,y,z such that (1) x < y < z < x+y (2) x,y,z, are pairwise coprime (3) z - y is not a multiple of x (4) For some prime p > 2, x | z^p - y^p > y | z^p - x^p > z | x^p + y^p Remarks: (1) For p = 3, your conjecture holds for z <= 1000. (2) As you've previously noted, if a proof of your conjecture could be >had, that would yield an instant proof of FLT, however the known truth >of FLT does not appear to yield a proof of your conjecture. (3) As far as trying to prove your conjecture, I would start with a >fixed prime, for example p = 3. quasi In the future, for new posts, please start the subject line with a >prefix of -- . By using this convention, such posts can be quickly Well, it appears that your conjecture fails. Here's a counterexample: (x,y,z) = (43, 638, 659) with p = 7. quasi === Subject: Re: Computability posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) You claimed to have checked it up to z = 256, but apparently you > missed the following counterexample: (x,y,z) = (74,129,143) which has signature (6,3,3). Wow. At first I thought this was a candidate triple, because for this > triple (x,y,z) the following are indeed all true: > Darn. For a moment, there, I thought I'd finally be able to drop this > search! Only around 8 times as many triples to check to limit = 256, > but sooo many to limit = 65535. Dang it, Chip Eastham, you rascal, > what were we/I thinking!? It was Oct. 24, 2006. I was only egging you on, out of the sort of misguided optimism that led to my previous encouragement of FLT seekers! It's been a year since I looked at this... much cobwebs to dust off my notes. === Subject: Help with a trig problem.. I have to solve this: [sqrt]x-4 + [sqrt]x+4 = 2[sqrt]x-1 I know the answer is simple, I've just been breaking my head over it :lol: any help is appreciated. === Subject: Re: Help with a trig problem.. <14285991.1200271069050.JavaMail.jakarta@nitrogen.mathforum.org>, > .... > [sqrt]x-4 + [sqrt]x+4 = 2[sqrt]x-1 > .... I know it's not easy to type mathematical formulae in ascii, but your notation [sqrt]a+b probably means sqrt(a + b). If so, then you're asking about sqrt(x - 4) + sqrt(x + 4) = 2.sqrt(x - 1). Squaring both sides of the equation should give you something with only one square root instead of three. Get that square root all by itself on one side of the equation, then square again. You should then have no square roots left, and finish the solution quite easily. If you get stuck, post another message showing how much you've managed to do. (Incidentally this is algebra, not trigonometry as in your title.) Ken Pledger. === Subject: Re: Help with a trig problem.. >I have to solve this: [sqrt]x-4 + [sqrt]x+4 = 2[sqrt]x-1 I know the answer is simple, I've just been breaking my head over it :lol: any help is appreciated. Firstly, it's algebra, not trig. Secondly, please try to write your mathematical expressions correctly. Presumably, the equation you intended is the following ... sqrt(x-4) + sqrt(x+4) = 2*sqrt(x-1) If so, the solution is routine. After all, what does it take to get rid of an isolated square root term? Square both sides, of course. Ok, so try it. Square both sides. The right side simplifies nicely, while the left side still ends up with a radical term. No problem, just isolate that term and square both sides again. Voila! Of course, make sure to test any solutions found, so as not to inadvertently accept an extraneous root. quasi === Subject: Re: Help with a trig problem.. === Subject: today latest technology posting-account=ejuv2woAAABptnlTVi8ZpjM9uWEsPBvX SV1),gzip(gfe),gzip(gfe) today latest technology **************************************** http://padmagirl.blogspot.com **************************************** === Subject: Material Science and Engineering an Introduction 7th Ed. by Callister posting-account=rKPskgoAAADzgydxXwui4zUVbJiqNJkW AppleWebKit/523.10.6 (KHTML, like Gecko) Version/3.0.4 Safari/523.10.6,gzip(gfe),gzip(gfe) Can anyone send me the solutions manual to: Material Science and Engineering an Introduction 7th Ed. by Callister Kareem === Subject: An example of the discovery function of proof A new addition to my homepage is a link to JavaSketchpad sketches which allow users the ability to drag and experience dynamic geometry on the internet. Or go directly to http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm My homepage also contains the usual bi-monthly mathematical/mathematics education quote, e-Newsletter with news about conferences, etc., and cartoon at the bottom. Michael === Subject: complete solutions manual for (Elementary Linear algebra by anton 9th) posting-account=3NVHhAoAAADBR74PaUwWt9BY6cRu_vzn .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.2),gzip(gfe),gzip(gfe) does anyone have a the complete solutions manual for (Elementary LInear Algebra by anton 9th) pls email me to mark.limon03@gmail.com === Subject: Re: SOLUTION MANUALS CHEAP!!! posting-account=3NVHhAoAAADBR74PaUwWt9BY6cRu_vzn .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.2),gzip(gfe),gzip(gfe) > I am selling solution manuals for $5.00. I only accept payments > through PAYPAL. Please do not post back on this site, send me an > email > if you are interested and I will reply with instructions. You will > have your solutions manual within 24 hours, USUALLY WITHIN THE HOUR!! dc9...@comcast.net I have the following solutions manuals: Single Variable Calculus Early Trancendentals (5th Edition) // > Anderson, Cole, Drucker Principles and Applications of Electrical Engineering (5th Edition) // > Giorgio Rizzoni Principles and Applications of Electrical Engineering 4th Edition) // > Giorgio Rizzoni Mechanics of Materials (3rd Edition... same as 4th Edition, just > different numbers) // Beer, Johnston, Dewolf Fundamentals of Modern Manufacturing (3rd Edition) // Groover Thermodynamics: an engineering approach (5th Edition) // Cengel, > Boles Thermodynamics: an engineering approach (6th Edition) // Cengel, > Boles Design of Machinery (3rd Edition) // Norton Physics for Scientists and Engineers with Modern Physics (3rd Edition > Volume 11) // Giancoli Introduction to Fluid Mechanics (6th Edition) // Fox Introduction to Fluid Mechanics (5th Edition) // Fox Shigley's Mechanical Engineering Design (8th edition) // Budynas Fundamentals of Heat and Mass Transfer (6th Edition) // F.P.Incropera > and D.P.DeWitt Probability, Random Variables and Stochastic Processes (4th > Edition) // Papoulis, Unnikrishna Pillai Analytical Mechanics (7th Edition) // Fowles Fundamentals of Engineering Thermodynamics // Moran, Shapiro Electric Machinery Fundamentals (4th Edition) // Chapman Digital Image Processing (2nd Edition) æ// Gonzalez Classical Electrodynamics (2nd Edition) // Jackson, Kasper, Wijk Communication System Engineering (2002) // Proakis Advanced Engineering Mathematics (8th Edition) // Kreyszig Calculus (5th Edition) // Stewart Introduction to Linear Algebra (3rd Edition) // Strang Physics (5th Edition)// Halliday, Resnick, Krane Applied Statistics and Probability for Engineers (3rd Edition) // > Montgomery, Runger Communication Systems (4th Edition) // Haykin Digital Signal Processing: A computer Based Approach (1st Edition) // > Mitra Econometric Analysis (5th Edition) // Greene Electric Machinery (6th Edition) // Fitzgerald, Kingsley, Uman Elementary Mechanics & Thermodynamics (2000) // Norbury Engineering Fluid Mechanics (7th Edition) // Crowe, Elger, Roberson Fundamentals Of Fluid Mechanics (3rd Edition) Fundamentals of Fluid Mechanics (4th Edition) Signals and Systems (2nd Edition) // Oppenheim and Wilsky Signals and Systems (2nd Edition) // Haykin > Goldstein Classical Mechanics (2nd Edition) // Reid Chemical and Engineering Thermodynamics (3rd Edition) // Wiley Engineering Fluid Mechanics (7th Edition) // Elger, Crowe Probability Random Variables and Stochastic Processes (4th Edition) // > Papoulis, Pillai Communication System Engineering (2nd Edition) // Proakis, Salehi Physics for Scientists and Engineers (6th Edition) // Serway, Jewett Microwave Engineering (3rd Edition) // Pozar Engineering Electromagnetics (6th Edition) // Hayt, Buck Elementary Differential Equations and Boundary Value Problems (7th > Edition) // Diprima, Boyce Solved and Unsolved Problems in Number Theory (2nd Edition) // Shanks Fundamentals of Physics 1,2,3,4 (4th Edition) CALCULUS Early Transcendentals æ(7th Edition) // Davis Applied Statistics and Probability for Engineers (3rd Edition) // > Montgomery, Runger Calculus (10th Edition) Volume 1,2// Thomas Control Systems Engineering // Nise Engineering Electromagnetics (6th Edition) // Buck, Hayt I have others that may not be listed, please email with other > requests. Again, my email is dc9...@comcast.net Email me what solutions you want and i will reply back with > instructions for PAYPAL payment hi sir i want the solutions manual for the ff. (Elementary differential equations and boundary value problems 7th by Boyce and Diprima) is that a complete solutions manual. sci.physics === Subject: Re: The Tidal Warming of Earth (as of 12,500~12,700 BP) posting-account=nf79RwoAAABXjvy5ztMzmPxgY1WGoktI Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) On behalf of lord BONZO (of which I can in part [<90%] agree with), and otherwise of those unable to think the least bit outside of their cozy mainstream status quo or bust kind of box; Here's that good two part AGW or ET Global Warming Quiz Part one: What would our geophysical planetology and environment be like if never having a moon, thereby our not having such a seasonal tilt and subsequently dealing with roughly 1/3 the tidal issues as contributed by Earth's orbital and rotational physics related to just our world interacting with the sun? Part two: What would change about the above geophysical and thermal dynamic environment upon Earth once getting impacted by an icy proto- moon, of such an encounter creating and sustaining our seasonal tilt, and subsequently having been orbited by the likes of what's currently existing as our physically dark moon? - BTW, each of these considerations are relatively straight forward and by rights should be rather simple supercomputer simulations, of which our NASA has a really good one of 2048 fast CPUs along with all the necessary software, those multiple terabytes of memory and essentially unlimited data storage. - Here's the real GW kicker. The moon is essentially our GW nemesis, all because of those pesky regular laws of physics and the fact of Earth being at least 98.5% fluid to those terrific gravity/tidal forces, and by all the rights of such tidal physics is where you simply can not have such terrific tidal influx without friction taking place, not to mention the added secondary/recoil worth of IR contributed by our physically dark moon. Next is the human induced AGW factor that's worth 10~25% of the thermal dynamic global budget, depending on how you'd care to add such things up. Earth w/o moon and without nearly as much seasonal tilt would have been a very icy world from time to time, especially without those nifty interactive tidal benefits of what has been moving terrestrial stuff (inside and out) by double the amount caused by the solar tidal interaction. Any way you'd care to argue this one, our moon is representing a great deal of orbital and thus unavoidable tidal energy that is not going to waste or somehow vanishing into thin air(sort of speak), without leaving its mark. As much as you'd like to think that our moon has always been with us, there is simply no such objective science that has that theory nailed down. On the other hand, as of 12,500 some odd years ago, there is at least good subjective evidence that put this glancing and lithobraking encounter of Earth with our icy proto-moon is every bit as real as are the antipode generated mountains opposite the Arctic ocean basin at the point of contact. - Brad Guth sci.physics === Subject: Re: The Tidal Warming of Earth (as of 12,500~12,700 BP) posting-account=_1tR4QoAAADW-tqNvDBN3zFr5IlCNzSj 5.1),gzip(gfe),gzip(gfe) sci.physics === Subject: Re: The Tidal Warming of Earth (as of 12,500~12,700 BP) posting-account=nf79RwoAAABXjvy5ztMzmPxgY1WGoktI Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) And this is yet another prime example of the very best of eyeball and company. No wonder those weird genetic mutations are the Usenet norm around here. - Brad Guth === Subject: Random permutations posting-account=DWxXYAoAAAB6f2Qst5yWt6dpm0LYlq_k Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) The foll. code is Floyd's algo for generating a random permutation of size M from the range 1 to N. initialize sequence S to empty for J = N-M+1 to N do T=Rand(1,J) if T is not in S then prefix T to S else insert J in S after T Why is prefixing needed ? How do you prove that this generate uniform random permutation ? i.e. all permutations are equally likely === Subject: compact space === Subject: Re: compact space > Is it possible to define a topology on N (the set of natual numbers) > s.t. to make it a _compact_ space? Take the topology for which the open sets are: 1) All finite subsets of N of which 1 is not a member. 2) All subsets of N which contain a set of the form {1, n, n + , n + 2, n + 3, ...} for some natural _n_. It is compact and metrisable. Jose Carlos Santos === Subject: Re: compact space > Is it possible to define a topology on N (the set of natual numbers) > topology does not work. > Is it possible to give N a compact Hausdorff topology? Exercise. Give an example of a countable, compact Hausdorff space. === Subject: Re: compact space > Is it possible to define a topology on N (the set of natual numbers) > topology does not work. > Is it possible to give N a compact Hausdorff topology? Consider {0,1} with the discrete topology: it is compact Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable product) equipped with the product topology. It is compact Hausdorff by Tychonoff's theorem. Any bijection between {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N. pg. === Subject: Re: compact space Is there a countable subset of the real line that is compact? What is the relevance of that to your problem? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: compact space >Is there a countable subset of the real line that is compact? >What is the relevance of that to your problem? Nice hint. quasi === Subject: Re: compact space > Is it possible to give N a compact Hausdorff topology? > Consider {0,1} with the discrete topology: it is compact > Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable > product) equipped with the product topology. It is compact Hausdorff > by Tychonoff's theorem. Any bijection between > {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N. do you mean that the product {0,1}x{0,1}x{0,1}x... (countable product) is countable itself ? === Subject: Re: compact space > do you mean that the product {0,1}x{0,1}x{0,1}x... > (countable product) is countable itself ? Oopps sorry :-). I found a compact Hausdorff topology on R, not on N. And {0,1}+{0,1}+{0,1}+... (disjoint sum) does not work either. It is countable but not compact anymore. pg. === Subject: Re: compact space > Is it possible to give N a compact Hausdorff topology? > Consider {0,1} with the discrete topology: it is compact > Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable > product) equipped with the product topology. It is compact Hausdorff > by Tychonoff's theorem. Any bijection between > {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N. do you mean that the product {0,1}x{0,1}x{0,1}x... >(countable product) is countable itself ? Good point. In fact, it's clearly _uncountable_. quasi === Subject: Re: compact space > Is it possible to define a topology on N (the set of natual numbers) > topology does not work. > Is it possible to give N a compact Hausdorff topology? Consider {0,1} with the discrete topology: it is compact > Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable > product) equipped with the product topology. It is compact Hausdorff > by Tychonoff's theorem. Any bijection between > {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N. Well, yes. But is is more natural to take, for instance {0} U { 1/n | n natural } with its natural topology. It is a compact Hausdorff space. No need to use Tychonoff's theorem here. Jose Carlos Santos === Subject: Re: compact space > Is it possible to define a topology on N (the set of natual numbers) > topology does not work. > Is it possible to give N a compact Hausdorff topology? > Consider {0,1} with the discrete topology: it is compact > Hausdorff. Consider the product {0,1}x{0,1}x{0,1}x... (countable > product) equipped with the product topology. It is compact Hausdorff > by Tychonoff's theorem. Any bijection between > {0,1}x{0,1}x{0,1}x... and N yields a compact Hausdorff topology on N. Well, yes. Not well :-/. Here is another example which should work (I wanted to give a conceptual example). Consider N with the discrete topology. It is locally compact. I am a little bit confused between French and English terminology: I mean Hausdorff and any point admits a compact Hausdorff neighbourhood. Then consider the Alexandroff one-point compactification. The space N u {oo} is still countable (no mistake this time). Then using a bijection between N u {oo} and N, one obtains a compact Hausdorff topology on N. pg. === Subject: Re: compact space > Is it possible to define a topology on N (the set of natual numbers) > topology does not work. > Is it possible to give N a compact Hausdorff topology? Then consider the Alexandroff one-point compactification. The >space N u {oo} is still countable (no mistake this time). Then >using a bijection between N u {oo} and N, one obtains a >compact Hausdorff topology on N. That works perfectly, quasi === Subject: Re: compact space Actually, the _trivial_ topology, where the open sets are just the empty set and the full space, does work. Perhaps you meant that the _discrete_ topology doesn't work. quasi === Subject: Re: compact space >Actually, the _trivial_ topology, where the open sets are just the >empty set and the full space, does work. Perhaps you meant that the _discrete_ topology doesn't work. Some ideas to try ... (1) What if you take a topology on N which has only finitely many open sets? Wouldn't it automatically be compact? (2) How about cofinite sets? Consider covers of N using such sets. quasi === Subject: Re: compact space Is it possible to define a topology on N (the set > of natual numbers) s.t. to make it a _compact_ space? Actually, the _trivial_ topology, where the open > sets are just the >empty set and the full space, does work. Perhaps you meant that the _discrete_ topology > doesn't work. Some ideas to try ... (1) What if you take a topology on N which has only > finitely many open > sets? Wouldn't it automatically be compact? (2) How about cofinite sets? Consider covers of N > using such sets. quasi === Subject: Dense vs. Continuous I am reading through Elementary Real Analysis by Thompson, Bruckner & Bruckner. It is well-written, user friendly, and free! They discuss the fact that the rational numbers are dense and describe the concept thusly: The rational numbers are dense. They make an appearance in every interval; there are no gaps, no intervals that miss having rational numbers. And later, they state ...every real is as close as we please to a rational... Finally, they mention: For theoretical reasons this fact is of great importance too. It allows many arguments to replace a consideration of the set of real numbers with the smaller set of rationals. Now intuitively, I understand that the real numbers includes the rational and the irrational and so it seems to make sense to say in the third quoted sentence It allows many arguments to replace a consideration of the set of real numbers with the smaller set of rationals (the set of real numbers is larger than the rationals). Ok, the rationals are countably infinite but dense and the real numbers are uncountably infinite. But if, looking only at the rational numbers, we can always find an infinity of rational numbers between any two given rational numbers, how can we say that the set is not continuous? Yes, it is missing the irrationals and is a smaller set than the real numbers, but from the definition of dense, there does not seem to be any holes in the set. I think that I understand the meaning of dense, from the author's description, but perhaps I am not correctly understanding what the term implies. Any thoughts? Alan p.s. if anybody is looking for a free Analysis textbook, they authors have both an elementary and a graduate level text available for free download here: http://classicalrealanalysis.com/download.aspx === Subject: Re: Dense vs. Continuous posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) On 14 Jan, 04:20, The poster formerly known as Colleyville Alan > how can we say that the set is not continuous? That would be a category mistake: sets cannot be continuous; functions can be continuous. Victor Meldrew I don't believe it! === Subject: Re: Dense vs. Continuous posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > how can we say that the set is not continuous? > That would be a category mistake: sets cannot be continuous; > functions can be continuous. There is the notion of a linearly ordered set (and a linear order) being continuous, although the more common term for this property (now at least; older books often used continuous) is completeness (the linear order notion, not the the Cauchy sequence notion). However, I don't know if the book in question uses continuous in the discussion at hand or if this is the original poster's injection of the term. (I have the book, and I'm somewhat familiar with it, but it's at home and I'm not.) Dave L. Renfro === Subject: Re: Dense vs. Continuous posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > I have the book, and I'm somewhat > familiar with it, but it's at home and I'm not. Oh, I fogot. It's freely available in digital form now. So let me rephrase this as I'm not sufficiently motivated in resolving the issue of whether the authors used continuous in the context of describing sets to pursue the matter. Dave L. Renfro === Subject: Re: Dense vs. Continuous > On 14 Jan, 04:20, The poster formerly known as Colleyville Alan > how can we say that the set is not continuous? That would be a category mistake: sets cannot be continuous; > functions can be continuous. Ok. I'm getting there slowly but surely. So we go back to Calculus II and the definition of a continuous function to see whethere a given function is continuous. http://mathworld.wolfram.com/ContinuousFunction.html More concretely, a function in a single variable is said to be continuous at point if 1. f(x_o) is defined, so that x_o is in the domain of f. 2. lim x->x_o exists for x in the domain of f. 3. lim x->x_o = f(x_o) If I say that x is an element of the rationals rather than the reals, then the 2nd and 3rd parts would be satisfied for a function like y = f(x) = x. But the first part, ...f(x_o) being defined so that x_o is in the domain of f... does not seem to hold. Yet the function f(x) = x would appear to me to be continuous if you can get infinitely close to any real even though the irrationals are not included. Does it make any sense at all to think of the rationals as a domain for a function or should functions only be considered in terms of the real number system and thereby eliminating all of this confusion on my part :) === Subject: Re: Dense vs. Continuous <478b5c46$0$4967$4c368faf@roadrunner.com> posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) On 14 Jan, 12:57, The poster formerly known as Colleyville Alan > More concretely, a function in a single variable is said to be continuous > at point if Phrases like function in a single variable are a bit A-Level :-( > Does it make any sense at all to think of the > rationals as a domain for a function Of course! > or should functions only be considered > in terms of the real number system ABSOLUTELY NOT!!! A function can have *any* set as domain, and *any* set as codomain. If one has a function f: A -> B and the sets A and B have topologies on them then there is a notion of continuity of the function f. Victor Meldrew I don't believe it! === Subject: Re: Dense vs. Continuous >I am reading through Elementary Real Analysis by Thompson, Bruckner & >Bruckner. It is well-written, user friendly, and free! They discuss the fact that the rational numbers are dense and describe the > concept thusly: The rational numbers are dense. They make an appearance in every > interval; there are no gaps, no intervals that miss having rational > numbers. And later, they state ...every real is as close as we please to a rational... Finally, they mention: For theoretical reasons this fact is of great importance too. It allows > many arguments to replace a consideration of the set of real numbers with > the smaller set of rationals. > Now intuitively, I understand that the real numbers includes the rational > and the irrational and so it seems to make sense to say in the third > quoted sentence It allows many arguments to replace a consideration of > the set of real numbers with the smaller set of rationals (the set of > real numbers is larger than the rationals). Ok, the rationals are countably infinite but dense and the real numbers > are uncountably infinite. But if, looking only at the rational numbers, > we can always find an infinity of rational numbers between any two given > rational numbers, how can we say that the set is not continuous? > What do you mean by the set is not continuous? If you mean that the set of rationals is not a path connected subset of the real numbers--then that is correct. > Yes, it is missing the irrationals and is a smaller set than the real > numbers, but from the definition of dense, there does not seem to be any > holes in the set. I think that I understand the meaning of dense, > from the author's description, but perhaps I am not correctly > understanding what the term implies. Any thoughts? Alan p.s. if anybody is looking for a free Analysis textbook, they authors have > both an elementary and a graduate level text available for free download > here: > http://classicalrealanalysis.com/download.aspx === Subject: Re: Dense vs. Continuous reply-type=response >I am reading through Elementary Real Analysis by Thompson, Bruckner & >Bruckner. It is well-written, user friendly, and free! > They discuss the fact that the rational numbers are dense and describe > the concept thusly: > The rational numbers are dense. They make an appearance in every > interval; there are no gaps, no intervals that miss having rational > numbers. > And later, they state > ...every real is as close as we please to a rational... > Finally, they mention: > For theoretical reasons this fact is of great importance too. It allows > many arguments to replace a consideration of the set of real numbers with > the smaller set of rationals. > Now intuitively, I understand that the real numbers includes the rational > and the irrational and so it seems to make sense to say in the third > quoted sentence It allows many arguments to replace a consideration of > the set of real numbers with the smaller set of rationals (the set of > real numbers is larger than the rationals). > Ok, the rationals are countably infinite but dense and the real numbers > are uncountably infinite. But if, looking only at the rational numbers, > we can always find an infinity of rational numbers between any two given > rational numbers, how can we say that the set is not continuous? What do you mean by the set is not continuous? If you mean that the set > of rationals is not a path connected subset of the real numbers--then that > is correct. Ok, I was assuming that the rationals were not continuous since they were missing the irrational numbers. The square root of two is between one and two so that point would be missing if one used the rationals rather than the reals. But perhaps my assumption was wrong; are the rationals a contiuous set? The definition of dense seems to me to imply that they are, but I assumed that only the reals were continuous and that if one were looking at, say, the graph of a function it would be plotted along the real number line (i.e. the standard Cartesian grid). So although we can get infinitely close to, say, the square root of two, that exact value is not defined in the set of rational numbers and it would seem that the graph of the function would have an infinitely small hole at that point and thus not be continuous. As I say, I am not certain what dense implies. Does it imply contiuity? Do we ignore the missing irrationals or is that a poor way to think about it? I only have studied Calculus and so stuff like this is new to me. === Subject: Re: Dense vs. Continuous >I am reading through Elementary Real Analysis by Thompson, Bruckner & >Bruckner. It is well-written, user friendly, and free! > They discuss the fact that the rational numbers are dense and describe > the concept thusly: > The rational numbers are dense. They make an appearance in every > interval; there are no gaps, no intervals that miss having rational > numbers. > And later, they state > ...every real is as close as we please to a rational... > Finally, they mention: > For theoretical reasons this fact is of great importance too. It allows > many arguments to replace a consideration of the set of real numbers with > the smaller set of rationals. > Now intuitively, I understand that the real numbers includes the rational > and the irrational and so it seems to make sense to say in the third > quoted sentence It allows many arguments to replace a consideration of > the set of real numbers with the smaller set of rationals (the set of > real numbers is larger than the rationals). > Ok, the rationals are countably infinite but dense and the real numbers > are uncountably infinite. But if, looking only at the rational numbers, > we can always find an infinity of rational numbers between any two given > rational numbers, how can we say that the set is not continuous? What do you mean by the set is not continuous? If you mean that the set > of rationals is not a path connected subset of the real numbers--then that > is correct. Ok, I was assuming that the rationals were not continuous since they were > missing the irrational numbers. The square root of two is between one and > two so that point would be missing if one used the rationals rather than > the reals. But perhaps my assumption was wrong; are the rationals a > contiuous set? You haven't told us what it means for a set to be continuous. Until that is given a precise meaning, no one can answer. > The definition of dense seems to me to imply that they > are, but I assumed that only the reals were continuous and that if one were > looking at, say, the graph of a function it would be plotted along the real > number line (i.e. the standard Cartesian grid). So although we can get > infinitely close to, say, the square root of two, that exact value is not > defined in the set of rational numbers and it would seem that the graph of > the function would have an infinitely small hole at that point and thus not > be continuous. As I say, I am not certain what dense implies. Does it imply contiuity? Do > we ignore the missing irrationals or is that a poor way to think about it? > I only have studied Calculus and so stuff like this is new to me. === Subject: -- Series Convergence I thought i would try my luck posting this one again with the new filter: I am trying to discuss the convergence of the following series, but I'm getting nowhere fast: S = sum(n=0,oo) (4*n+1) / [( b + 2*n*(b+c) )*n!*Gamma(1/2 - n) ] * z^(2*n) *P(2*n,cos(Y)) for 0<=Y<=pi where P(n,x) is the legendre polynomial, b and c are >0 and real. my guess is that |z| must be <1, but does this need to hold for all Y? my particular interest is about the point |z|=1. my attempt would be the following... Consider the same series of S however with absolute values. Since |P(n,x)|<=1, for -1<=x<=1 each term in the series is then less than or equal to a_n = |(4*n+1) / [( b + 2*n*(b+c) )*n!*Gamma(1/2 - n) ] |*| z^(2*n)| . The ratio test then shows that this is convergent for |z|<1, and so the series is absolutely convergent there. Then I consider |a_n+1/a_n| for |z|=1, and prove the bound |a_n+1/a_n| > (4n+5)*(1+2n)*n / (2*(4n+1)*(1+n)^2). I can then solve this recurrence relation and get |a_n| > 2*(1+4n)Gamma(1/2+n)/(5*n^2*Sqrt(pi)*Gamma(n)) * |a_1|. Now Gamma(1/2+n)/Gamma(n) = sqrt(n)-8/sqrt(n) +O(n^-3/2) as n->oo. and so the terms of the series behave as 8/(5*sqrt(pi*n)) + O(n^-3/2) and so by the limit comparison theorem, the series diverges and therefore by comparison the series of |a_n| diverges. Therefore the series is not bsolutely convergent for |z|=1. But is it convergent? and if so, for what values of Y? cheers moth === Subject: Re: Type Theory TT posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > This link is not working. Try this link here: http://math.boisestate.edu/~holmes/ Then scroll down to: New: my book to be available on-line in a new edition There should be a link to the PS and PDF versions of the NF book by Randall Holmes. === Subject: Re: complex number polar form arithmetic <20080113155221.383$DD@newsreader.com> posting-account=ncjJCAoAAAB1ZUB1tGBAFi5il88_MM_d 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I'm suspecting that I could use this formula for the > magnitude: > |z1|+|z2|=3Dsqrt(|z1|^2+|z2|^2-2*|z1|*|z2|*cos(theta)) theta can be calculated this way: >Pi-arg(z1)+arg(z2) >right? I did some other tests and I think it could be done this way: addition: >|z1|+|z2|=3Dsqrt( |z1|^2+|z2|^2 + 2*|z1|*|z2|*cos(theta) ) >where theta=3Darg(z1)-arg(z2) actually intended to write |z1 + z2| instead. > I'm going to give an answer for addition, leaving subtraction for you to do > yourself. Given two complex numbers in polar form, you want to know how to add them > directly obtaining an answer in polar form. For convenience, let's denote a complex number in polar form, r*e^(i*t), as > the ordered pair (r; t). Then (r1; t1) + (r2; t2) = ( sqrt( r1^2 + r2^2 + 2 r1 r2 cos(t2 - t1) ); > atan( r1 sin(t1) + r2 sin(t2), r1 cos(t1) + r2 cos(t2) ) ) It seems that you already had obtained the correct result for the modulus > of the answer. For the argument of the answer, note that I used a function > of the form atan(y, x), instead of the common inverse tangent. The function > I used is also called atan2 in some computer languages. For example, see > in the vicinity of items > (9) and (10). David W. Cantrell- Hide quoted text - - Show quoted text - z3=z1+z1: r3= ( sqrt( r1^2 + r2^2 + 2 r1 r2 cos(t2 - t1) ) t3= atan( r1 sin(t1) + r2 sin(t2), r1 cos(t1) + r2 cos(t2) ) ) z3=z1-z1: r3= ( sqrt( r1^2 + r2^2 - 2 r1 r2 cos(t2 - t1) ) t3= atan( r1 sin(t1) - r2 sin(t2), r1 cos(t1) - r2 cos(t2) ) ) I think this is correct but I'm just asking to be sure. === Subject: Re: complex number polar form arithmetic posting-account=ncjJCAoAAAB1ZUB1tGBAFi5il88_MM_d 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I'm suspecting that I could use this formula for the > magnitude: > |z1|+|z2|=3Dsqrt(|z1|^2+|z2|^2-2*|z1|*|z2|*cos(theta)) > theta can be calculated this way: > Pi-arg(z1)+arg(z2) > right? I did some other tests and I think it could be done this way: addition: >|z1|+|z2|=3Dsqrt( |z1|^2+|z2|^2 + 2*|z1|*|z2|*cos(theta) ) >where theta=3Darg(z1)-arg(z2) actually intended to write |z1 + z2| instead. > I'm going to give an answer for addition, leaving subtraction for you to do > yourself. Given two complex numbers in polar form, you want to know how to add them > directly obtaining an answer in polar form. For convenience, let's denote a complex number in polar form, r*e^(i*t), as > the ordered pair (r; t). Then (r1; t1) + (r2; t2) = ( sqrt( r1^2 + r2^2 + 2 r1 r2 cos(t2 - t1) ); > atan( r1 sin(t1) + r2 sin(t2), r1 cos(t1) + r2 cos(t2) ) ) It seems that you already had obtained the correct result for the modulus > of the answer. For the argument of the answer, note that I used a function > of the form atan(y, x), instead of the common inverse tangent. The function > I used is also called atan2 in some computer languages. For example, see > in the vicinity of items > (9) and (10). David W. Cantrell- Hide quoted text - - Show quoted text - z3=z1+z1: > æ r3= ( sqrt( r1^2 + r2^2 + 2 r1 r2 cos(t2 - t1) ) > æ t3= atan( r1 sin(t1) + r2 sin(t2), r1 cos(t1) + r2 cos(t2) ) ) z3=z1-z1: > æ r3= ( sqrt( r1^2 + r2^2 - 2 r1 r2 cos(t2 - t1) ) > æ t3= atan( r1 sin(t1) - r2 sin(t2), r1 cos(t1) - r2 cos(t2) ) ) I think this is correct but I'm just asking to be sure. > - Show quoted text - and ... z3=z1*z2: r3= r1 * r2 t3= (t1 + t2)%2pi z3=z1/z2: r3= r1 / r2 t3= (t1 - t2)%2pi === Subject: Re: complex number polar form arithmetic >I'm suspecting that I could use this formula for the >magnitude: >|z1|+|z2|=3D3Dsqrt(|z1|^2+|z2|^2-2*|z1|*|z2|*cos(theta)) > theta can be calculated this way: > Pi-arg(z1)+arg(z2) > right? > I did some other tests and I think it could be done this way: > addition: > |z1|+|z2|=3D3Dsqrt( |z1|^2+|z2|^2 + 2*|z1|*|z2|*cos(theta) ) > where theta=3D3Darg(z1)-arg(z2) you actually intended to write |z1 + z2| instead. >I'm going to give an answer for addition, leaving subtraction for you >to do yourself. Given two complex numbers in polar form, you want to know how to add >them directly obtaining an answer in polar form. For convenience, let's denote a complex number in polar form, >r*e^(i*t), as the ordered pair (r; t). Then (r1; t1) + (r2; t2) = ( sqrt( r1^2 + r2^2 + 2 r1 r2 cos(t2 - t1) ); >atan( r1 sin(t1) + r2 sin(t2), r1 cos(t1) + r2 cos(t2) ) ) It seems that you already had obtained the correct result for the >modulus of the answer. >For the argument of the answer, note that I used a >function of the form atan(y, x), instead of the common inverse >tangent. The function I used is also called atan2 in some computer >languages. For example, see > in the vicinity of >items (9) and (10). z3 = z1+z1: > r3 = ( sqrt( r1^2 + r2^2 + 2 r1 r2 cos(t2 - t1) ) > t3 = atan( r1 sin(t1) + r2 sin(t2), r1 cos(t1) + r2 cos(t2) ) ) You intended to type z3 = z1+z2 above and z3 = z1-z2 below. > z3 = z1-z1: > r3 = ( sqrt( r1^2 + r2^2 - 2 r1 r2 cos(t2 - t1) ) > t3 = atan( r1 sin(t1) - r2 sin(t2), r1 cos(t1) - r2 cos(t2) ) ) I think this is correct but I'm just asking to be sure. It looks good to me. You're welcome. > and ... z3=3Dz1*z2: > r3=3D r1 * r2 > t3=3D (t1 + t2)%2pi z3=3Dz1/z2: > r3=3D r1 / r2 > t3=3D (t1 - t2)%2pi Yes. David alt.magick === Subject: The Casimir Effect - Metal Bender posting-account=ULPuKwgAAABiFOIZLVSqwHfL36Wkpdbn Gecko/20061010 Firefox/2.0,gzip(gfe),gzip(gfe) Can someone please help me understand this mechanism ? http://u2r2h-documents.blogspot.com/2008/01/911-wtc-towers-to-dust-mechanism .html === Subject: Re: The Casimir Effect - Metal Bender Can someone please help me understand this mechanism ? http://u2r2h-documents.blogspot.com/2008/01/911-wtc-towers-to-dust-mechanism . html > http://www.mazepath.com/uncleal/sunshine.jpg http://www.mazepath.com/uncleal/analysis.jpg -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/lajos.htm#a2 === Subject: Re: The Casimir Effect - Metal Bender | | Can someone please help me understand this mechanism ? | | http://u2r2h-documents.blogspot.com/2008/01/911-wtc-towers-to-dust-mechanism .html | No, nobody can help you. Your mother can wipe your arse for you but you have to for yourself. === Subject: Re: The Casimir Effect - Metal Bender Can someone please help me understand this mechanism ? http://u2r2h-documents.blogspot.com/2008/01/911-wtc-towers-to-dust-mechanism .html > yes, it's magick. blame The Goddess if you disagree. === Subject: Re: JSH: The conspiracy of silence posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 2.0.50727; CNPVersion2 - Congoo NetPass),gzip(gfe),gzip(gfe) Why are nobody responding to JSH posts? > Is the ban still in effect? He has nothing new to say. æBecause his congruences > will (eventually) provide a factorisation, he remains > convinced that they are an efficient way to break RSA, > that he need neither prove this nor demonstrate it, > and that there is a sci.math conspiracy to conceal it. > Well, actually it's not that he need not prove it, it's that he CANNOT prove it (if he could, proving it would be a Very GOOD Thing), because it just does not work! === Subject: determining if a function is periodic posting-account=Gt-ffQoAAAAJl9zUa7Q7iwGkJH-TT-Gt Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) I was trying to prove whether cos(10*pi*t)+sin(5*pi*t+1) is periodic. When substituting t = t + T on the LHS and leaving t = t on the RHS, I got to this point: cos(10*pi*(t+T))+sin(5*pi*(t+T)+1) = cos(10*pi*t)+sin(5*pi*t+1) I have trouble seeing how I can simply this equation and solve for T. Is it safe to say I can't really solve for T at this point? === Subject: Re: determining if a function is periodic >I was trying to prove whether cos(10*pi*t)+sin(5*pi*t+1) is periodic. >When substituting t = t + T on the LHS and leaving t = t on the RHS, I >got to this point: cos(10*pi*(t+T))+sin(5*pi*(t+T)+1) = cos(10*pi*t)+sin(5*pi*t+1) I have trouble seeing how I can simply this equation and solve for T. You don't have to solve the above equation for T. What you want is to determine if there exists some nonzero constant T for which the above equation is an _identity_, valid for all real values of t. Assume such a T exists. Let f(t) = cos(10*pi*(t+T))+sin(5*pi*(t+T)+1)-cos(10*pi*t)-sin(5*pi*t+1) Then f must be identically zero. Here's a rough outline of a strategy to either determine T or prove that no such T exists ... (1) If f is identically zero, so is f^(n) where f^(n) denotes the n'th derivative of f. Thus, f^(1), f^(2), f^(3), ... are all identically zero. (2) Don't stop at just 3 derivatives -- do one more. (3) Compare the form of f and f^(4). Very similar, right? But not exactly the same. If you look carefully, you can write f and f^(4) in the forms f = (a-c) + (b-d) f^(4) = r*(a-c) + s*(b-d) where r,s are constants, and a,b,c,d are appropriately chosen expressions. Looking at the actual value of r,s, you'll see that r is not equal to s. (4) Thus you get a system of 2 equations in the 2 unknowns (a-c), (b-d). (a-c) + (b-d) = 0 r*(a-c) + s*(b-d) = 0 Since r is not equal to s, when you solve for a-c and b-d, you get a - c = 0 and b - d = 0 or equivalently, a = c and b = d. (5) What does a = c says about T? What does b = d say about T? quasi === Subject: Re: determining if a function is periodic posting-account=VR0DOgoAAADggPTteFeA2AkmHNhjcrDV Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) I was trying to prove whether cos(10*pi*t)+sin(5*pi*t+1) is periodic. >When substituting t = t + T on the LHS and leaving t = t on the RHS, I >got to this point: cos(10*pi*(t+T))+sin(5*pi*(t+T)+1) = cos(10*pi*t)+sin(5*pi*t+1) I have trouble seeing how I can simply this equation and solve for T. You don't have to solve the above equation for T. What you want is to determine if there exists some nonzero constant T > for which the above equation is an _identity_, valid for all real > values of t. Assume such a T exists. Let f(t) = > cos(10*pi*(t+T))+sin(5*pi*(t+T)+1)-cos(10*pi*t)-sin(5*pi*t+1) Then f must be identically zero. Here's a rough outline of a strategy to either determine T or prove > that no such T exists ... (1) If f is identically zero, so is f^(n) where f^(n) denotes the n'th > derivative of f. Thus, f^(1), f^(2), f^(3), ... are all identically > zero. (2) Don't stop at just 3 derivatives -- do one more. (3) Compare the form of f and f^(4). Very similar, right? But not > exactly the same. If you look carefully, you can write f and f^(4) in > the forms f = (a-c) + (b-d) f^(4) = r*(a-c) + s*(b-d) where r,s are constants, and a,b,c,d are appropriately chosen > expressions. Looking at the actual value of r,s, you'll see that r is > not equal to s. (4) Thus you get a system of 2 equations in the 2 unknowns (a-c), > (b-d). (a-c) + (b-d) = 0 r*(a-c) + s*(b-d) = 0 Since r is not equal to s, when you solve for a-c and b-d, you get a - c = 0 and b - d = 0 or equivalently, a = c and b = d. (5) What does a = c says about T? What does b = d say about T? quasi Seems like a lot of work to me. If t increases by 2, then 5*pi*t+1 increases by 2*pi, so sin(5*pi*t+1) is unchanged. Also, cos(10*pi*t) is unchanged. It follows that 2 is a period (of course there is a smaller one). === Subject: Re: determining if a function is periodic posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/20071213 Fedora/2.0.0.10-3.fc8 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > I was trying to prove whether cos(10*pi*t)+sin(5*pi*t+1) is periodic. > When substituting t = t + T on the LHS and leaving t = t on the RHS, I > got to this point: cos(10*pi*(t+T))+sin(5*pi*(t+T)+1) = cos(10*pi*t)+sin(5*pi*t+1) I have trouble seeing how I can simply this equation and solve for T. > Is it safe to say I can't really solve for T at this point? The period is 2/5... Just expand the sin of a sum, and think about what is the period of a function f(t) + g(t) when f is periodic of period p, g is periodic of period q, and q is a multiple integer of p. -- m === Subject: Re: determining if a function is periodic > I was trying to prove whether cos(10*pi*t)+sin(5*pi*t+1) is periodic. > When substituting t = t + T on the LHS and leaving t = t on the RHS, I > got to this point: cos(10*pi*(t+T))+sin(5*pi*(t+T)+1) = cos(10*pi*t)+sin(5*pi*t+1) I have trouble seeing how I can simply this equation and solve for T. > Is it safe to say I can't really solve for T at this point? Try this first: Is cos(2*Pi*t) + sin(Pi*t) periodic? ---------- sci.math, write -- at the beginning of your subject line. Set up a === Subject: piecewise function arithmetic posting-account=Gt-ffQoAAAAJl9zUa7Q7iwGkJH-TT-Gt Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) Is it possible to add up two piecewise functions or is it on a case by case basis? === Subject: Re: piecewise function arithmetic >Is it possible to add up two piecewise functions or is it on a case by >case basis? It depends. If the domain intervals for the pieces are the same, then yes, you can add the corresponding pieces. However in general, you would have to separate the domain into new intervals, using as separating points, the union of the separating points of each of the domains of each function. Then on each new interval, you can add the corresponding pieces. Try an example yourself to see the effect. quasi In the future, for new posts, please start the subject line with a prefix of -- . By using this convention, such posts can be quickly === Subject: latest technology posting-account=3wROkgkAAAAApaz_6Ojvbw60CVeYqSUA SV1),gzip(gfe),gzip(gfe) latest technology **************************************** http://padmagirl.blogspot.com **************************************** === Subject: Re: -- quasi's average distance problem in R^2 > quasi made several conjectures about finite sets K in the plane > and points x in R^2 that minimize ( sum_{y in K} d(x,y) )/| K| > (minimum/(infimum of) average distance to K. If all points in K lie on some line, this reduces to a 1-dimensional > problem, and with an even number of (distinct) points in K, > from what I remember there is a minimum average distance reached > on some interval of positive length. If no line contains all points of K, I don't think anything > definite was concluded here, as contrasted with the squared-distance > problem. What were the conclusions with the squared-distance problem? Curious .. Han de Bruijn === Subject: Re: Complex Spin group and covering <5urbc4F1i4u98U1@mid.individual.net> posting-account=CcRsGwoAAAALoO8-Sevd7m81tsLMVZeb Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > To me it appears there is a contradiction between the book > 'Supersymmetry for mathematicians' by Varadarajan and the book Spin > Geometry by Lawson & Michelsohn. Varadarajan claims (page 193 > that the Spin group for complex vector space V is a double cover of > SO(V). > In the book by Lawson it is claimed (formula 2.28) that it is a 4- > My question is now, who is right? (Or, something that's perhaps more > probable, why am I wrong in seeing a contradiction?) > anticipation! > I don't see a contradiction. They are talking about different things. > Varadarajan is talking about the complex Lie group SO(n,C), which is the > group of those linear maps from C^n into C^n which preserve the bilinear > form q:C^n ---> C^n defined by > q(z 1,...z n) = sum k(z k)^2. > Lawson and Michelsohn are talking about the group SO(V,q), where V is a > finite-dimensional vector field over what they call a spin field. Also, > they assume that they are working with a non-degenerate quadratic form > q . But the specific quadratic form defined above *is* degenerate (when > the field is the complex field and dim(V) > 1), since it corresponds to > the bilinear form B defined by > B((z 1,...z n),(w ,...,w n)) |-> sum k z k*w k > and B((1,i,0,...,0),(1,i,0,...,0)) = 0. So, then the spin group is a double cover of SO(V) when V is a complex > vector space with a degenerate quadratic form. And it is a 4-sheeted > cover when V is a complex field with a non-degenerate quadratic form ? > Is that correct? The spin group is a double cover of SO(V) when V is a complex vector > space with the specific degenerate quadratic form mentioned in my > previous post. On the other hand, every quadratic form over C^n is > degenerate when n > 1. > Jose Carlos Santos Ok, clear, then one last check: Lawson is talking about V being a complex vector space with non-degenerate quadratic form? And in that case the spin group is a 4-sheeted covering of SO(V), correct? === Subject: Re: Complex Spin group and covering > To me it appears there is a contradiction between the book > 'Supersymmetry for mathematicians' by Varadarajan and the book Spin > Geometry by Lawson & Michelsohn. Varadarajan claims (page 193 > that the Spin group for complex vector space V is a double cover of > SO(V). > In the book by Lawson it is claimed (formula 2.28) that it is a 4- > My question is now, who is right? (Or, something that's perhaps more > probable, why am I wrong in seeing a contradiction?) > anticipation! > I don't see a contradiction. They are talking about different things. > Varadarajan is talking about the complex Lie group SO(n,C), which is the > group of those linear maps from C^n into C^n which preserve the bilinear > form q:C^n ---> C^n defined by > q(z_1,...z_n) = sum_k(z_k)^2. > Lawson and Michelsohn are talking about the group SO(V,q), where V is a > finite-dimensional vector field over what they call a spin field. Also, > they assume that they are working with a non-degenerate quadratic form > _q_. But the specific quadratic form defined above *is* degenerate (when > the field is the complex field and dim(V) > 1), since it corresponds to > the bilinear form B defined by > B((z_1,...z_n),(w_,...,w_n)) |-> sum_k z_k*w_k > and B((1,i,0,...,0),(1,i,0,...,0)) = 0. > So, then the spin group is a double cover of SO(V) when V is a complex > vector space with a degenerate quadratic form. And it is a 4-sheeted > cover when V is a complex field with a non-degenerate quadratic form ? > Is that correct? > The spin group is a double cover of SO(V) when V is a complex vector > space with the specific degenerate quadratic form mentioned in my > previous post. On the other hand, every quadratic form over C^n is > degenerate when n > 1. > Ok, clear, then one last check: Lawson is talking about V being a > complex vector space with non-degenerate quadratic form? And in that > case the spin group is a 4-sheeted covering of SO(V), correct? Forget my previous reply; it was just plain silly. I made a (huge) mistake about the meaning of non-degenerate. The quadratic form q on C^n defined by q(z_1,...z_n) = sum_k(z_k)^2 *is* non-degenerate. The problem lies elsewhere. Varadarajan defines Spin(n,C) (on page 193) as the universal cover of SO(n,C). Then he proves (on page 198) that Spin(n,C) is isomorphic to the group of the invertible elements _x_ of the Clifford algebra Cl(C^n,q) such that x.C^n.x^{-1} is a subset of C^n and x.beta(x) = 1, where beta is the principal antiautomorphism of Cl(C^n,q). On the other hand, Lawson and Michelsohn define Spin(n,C) (on page 18) as the group of invertible elements Cl(C^n,q) generated by those elements of the form v_1 v_2 ... v_n where each v_k belongs to C^n, _n_ is even and q(v_k) = 1 or -1 for each _k_. Therefore, they are not the same groups! If _v_ is an element of C^n such that q(v) = -1, it belongs to the group that Lawson and Michelsohn are talking about, but not the group that Varadarajan is talking about. (However, they would be the same group over the reals; in that case, the possibility q(v) = -1 does not occur.) The group considered by Lawson and Michelson has, so to speak, twice as many elements as the one considered by Varadarajan. Jose Carlos Santos === Subject: what is the probability of transition? posting-account=-3CL3AoAAAB0rDYMAas5GqnYKnCMoXG3 5.1),gzip(gfe),gzip(gfe) hi. would any of you fine intellectuals care to focus your attention on this very important problem? case: headless clones are grown for 5 years in order to harvest organs for transplant. subsequently, entire lower bodies are transplanted (including facial transplants from heads that have had brain development inhibited) every 50 years. to find: the probability of transition from a uniform mass of immortals to intermittent manufacture of healthy babies to replace accidental deaths. is this solvable? do you need further clarification? === Regier > Find all functions f defined on the set of all real numbers with real > values, such that f(x2 + f(y)) = y + f(x)2 for all x, y. I suppose you mean f(x^2 + f(y)) = y + f(x)^2? Please use the standard > '^' when you want to write a power. Anyway, what have you done so far? > Where are you having trouble? If you show your work, it is easier for > people to know how to help you. > f(x^2 + f(0)) = f(x)^2 When f(0) = 0, f(x^2) = f(x)^2 f(x) = x f(x) = -x f(x) = 0 f(x) = 1 f(x) = ?? x^2 + y = y + x^2, yup -(x^2 - y) = y + x^2, nope 0 = y + 0, nope 1 = y + 1, nope When f(0) /= 0 ?? Assume f(x) = ax + b, what do you get for a and b? Assume f(x) = ax^2 + bx + c, what do you get for a,b and c? Assume f(x) = ax^3 + etc. ---- === Regier posting-account=4n0P8QoAAACPj0DJnja1mCT1wUiU6txx Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > Assume f(x) = ax + b, what do you get for a and b? > Assume f(x) = ax^2 + bx + c, what do you get for a,b and c? > Assume f(x) = ax^3 + etc. Not claiming any expertise in this area, I can definitively say that, given f(x^2 + f(y)) = y + f(x)^2, f(x) cannot be a finite polynomial of a degree greater than one. To demonstrate this, look at the simplest case scenario of a polynomial of degree n: f(x) = x^n (x^2 + y^n)^n = y + x^(2n) Notice that y has a term (y^n)^n, or y^(n^2) (in addition to some intermediates of the binomial expansion, of course), on the left-hand- side. The same problem exists in general for any polynomial with a degree greater than 1. I have also found that f(0) cannot be 1, 2 or -1, using the logic that follows: suppose f(0) = c x=0, y=0: f(c) = c^2 x=0, y=c: f(f(c)) = f(c^2) = c^2 + c x=c, y=0: f(c^2+c) = c^4 x=0, y=c^2: f(f(c^2)) = f(c^2+c) = c^2 + f(0)^2 = 2c^2 In a simple map: 0 --> c c --> c^2 c^2 --> c^2 + c c^2 + c --> c^4 c^2 + c --> 2c^2 Notice that c^2 + c is repeated. Since f(x) is a function, it follows that it must be true that: c^4 = 2c^2 c^4 - 2c^2 = 0 c^2(c^2 - 2) = 0 This means f(0) can be 1 of 3 possibilities: 0 or +/-sqrt(2). I imagine that one could keep on going to eliminate even the latter 2 possibilities, although I haven't actually done that. This, in combination with my demonstration that it can't be a polynomial of greater degree than 1, means that the only form of a polynomial that needs to be checked would be: f(x) = mx It can be demonstrated that the only possible value of m is 1, which is the obvious answer that I saw about 2 seconds after first seeing this problem. Of course, this hardly helps for any other type of function, or for infinite polynomials. I don't think exponential functions are a possibility, though, for much the same reason that a polynomial with a degree greater than 1 isn't allowed. I haven't looked at logarithms, 'polynomials' with rational exponents, rational expressions, or trigonometric functions (which, of course, are infinite polynomials). === Regier posting-account=4n0P8QoAAACPj0DJnja1mCT1wUiU6txx Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) Assume f(x) = ax + b, what do you get for a and b? > Assume f(x) = ax^2 + bx + c, what do you get for a,b and c? > Assume f(x) = ax^3 + etc. Not claiming any expertise in this area, I can definitively say that, > given f(x^2 + f(y)) = y + f(x)^2, f(x) cannot be a finite polynomial > of a degree greater than one. To demonstrate this, look at the > simplest case scenario of a polynomial of degree n: f(x) = x^n > (x^2 + y^n)^n = y + x^(2n) Notice that y has a term (y^n)^n, or y^(n^2) (in addition to some > intermediates of the binomial expansion, of course), on the left-hand- > side. The same problem exists in general for any polynomial with a > degree greater than 1. I have also found that f(0) cannot be 1, 2 or -1, using the logic that > follows: Obviously, I narrowed it down quite a bit further in the course of writing my post. ;) > suppose f(0) = c > x=0, y=0: f(c) = c^2 > x=0, y=c: f(f(c)) = f(c^2) = c^2 + c > x=c, y=0: f(c^2+c) = c^4 > x=0, y=c^2: f(f(c^2)) = f(c^2+c) = c^2 + f(0)^2 = 2c^2 In a simple map: > 0 --> c > c --> c^2 > c^2 --> c^2 + c > c^2 + c --> c^4 > c^2 + c --> 2c^2 Notice that c^2 + c is repeated. Since f(x) is a function, it follows > that it must be true that: c^4 = 2c^2 > c^4 - 2c^2 = 0 > c^2(c^2 - 2) = 0 This means f(0) can be 1 of 3 possibilities: 0 or +/-sqrt(2). I > imagine that one could keep on going to eliminate even the latter 2 > possibilities, although I haven't actually done that. This, in > combination with my demonstration that it can't be a polynomial of > greater degree than 1, means that the only form of a polynomial that > needs to be checked would be: f(x) = mx It can be demonstrated that the only possible value of m is 1, which > is the obvious answer that I saw about 2 seconds after first seeing > this problem. Of course, this hardly helps for any other type of function, or for > infinite polynomials. I don't think exponential functions are a > possibility, though, for much the same reason that a polynomial with a > degree greater than 1 isn't allowed. I haven't looked at logarithms, > 'polynomials' with rational exponents, rational expressions, or > trigonometric functions (which, of course, are infinite polynomials). === > Find all functions f defined on the set of all real numbers with real > values, such that f(x2 + f(y)) = y + f(x)2 for all x, y. your time! Nancy Regier I assume you mean f(x^2 + f(y)) = y + f(x)^2 Note that f is one-to-one and onto. What is its inverse? (consider the case x=0). What is f(-x)? ... Eventually you should be able to show f(x+y) = f(x) + f(y) and f(1) = 1. And then since f is increasing, it's determined by its values on the rationals... -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Find all functions f defined on the set of all real numbers with real > values, such that f(x2 + f(y)) = y + f(x)2 for all x, y. your time! Nancy Regier I assume you mean f(x^2 + f(y)) = y + f(x)^2 Note that f is one-to-one and onto. > What is its inverse? (consider the case x=0). > What is f(-x)? > ... > Eventually you should be able to show f(x+y) = f(x) + f(y) > and f(1) = 1. And then since f is increasing I should also mention that this is true because f(x^2) = f(x)^2. , it's determined > by its values on the rationals... -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Holomorphic representation Em 13-01-2008 21:28, publiosulpicio@gmail.com escreveu: > Suppose that f:GL(n+1,C) to GL(V) is an holomorphic representation of > GL(n+1,C) for some finite dimensional complex vector space V. > Define g: gl(n+1,C) to gl(V) such that g(Z)=d/dt f(exp(tZ)) in > t=0 .where exp is the exponential map. > (this is of course a special case of the fact that a representation of > a Lie group induces a representation of its lie algebra). > How to prove that g is linear? Is trivial to prove that g(aZ)=ag(Z), > but I cannot prove that g(Z+W)=g(Z)+g(W). Observe that exp(Z+W) is not > necesserarly equal to exp(Z)*exp(W). > The map _g_ is just the derivative of _f_ at the identity element and, > therefore, a linear map. That _g_ is the derivative of _f_ is a > consequence of the chain rule together with the fact that the derivative > at the identity of the exponential map is the identity. This is for sure the best proof, but how can I prove it directly? I > mean, without using Lie group theory? Can you please tell me exactly _where_ did I use Lie group theory? Jose Carlos Santos === Subject: Re: a < b implies 2^a < 2^b There is a _crucial difference_. Artists, musicians, novelists, actors, >screenwriters, professional athletes, chess players, they all entertain >_other_ people, rather than themselves. There are many fields where artists arguably entertain only each other. So-called Modern Art? Han de Bruijn === Subject: Re: a < b implies 2^a < 2^b > IMO, useful is the most _useless_ word there is > when it comes to describing mathematics. It would be useful to have a mathematical definition of useful. I'm not kidding. It could be something along the lines of being input for functions. The more functions something is input for, the more useful it is. Just thinking wildly .. Any ideas? Han de Bruijn === Subject: Re: a < b implies 2^a < 2^b IMO, useful is the most _useless_ word there is > when it comes to describing mathematics. It would be useful to have a mathematical definition of useful. I'm not kidding. It could be something along the lines of being > input for functions. The more functions something is input for, > the more useful it is. Just thinking wildly .. Any ideas? Han de Bruijn Usefulness is no more relevant to mathematical creativity than it is to poetic creativity. Useful poetry is that which appears in birthday cards. === Subject: Re: a < b implies 2^a < 2^b Consider the following statement (E) ... >(E): If a,b are cardinal numbers such that a < b, then 2^a < 2^b. >Is (E) true in the standard model (S) of set theory? >Who cares about any standard model? If a and b are whatever numbers, >then (a < b, then 2^a < 2^b) is simply true. >I'm almost certain that you're wrong, but if you think you can prove >it, let's see a proof. Cardinal numbers are not obliged to match your >intuition as to their algebraic relationships. >Here's what is true: >(1) For all cardinals a,b > 2^a < 2^b implies a < b >(2) For all cardinals a,b, > a < b implies 2^a <= 2^b >(3) If we assume the standard axioms of set theory, _plus_ the >Generalized Continuum Hypothesis, then > For all cardinals a,b, > a < b implies 2^a < 2^b. >But based on the discussion so far, it seems likely that the statement > For all cardinals a,b, > a < b implies 2^a < 2^b >is independent of the standard axioms of set theory. >Why don't you say that your default cardinal is an _infinite_ cardinal. >Because, for finite cardinals, (a < b, then 2^a < 2^b) is simply true. >I don't understand the choice of your default, though, because infinite >cardinals IMO do not exist and hence aren't worth considering. Why solve >a problem that simply isn't there in the whole universe? Then HdB's whole universe is too small. Sure. It's just limited to the whole physical universe. If you want to claim that God's creation is small when compared with your virtually unlimited fantasy, be my guest .. Han de Bruijn === Subject: Re: a < b implies 2^a < 2^b > If you want to > claim that God's creation is small when compared with your virtually > unlimited fantasy, be my guest .. If your god's universe is as limited as yours, then your god's creation is too small. Han de Bruijn === Subject: Re: a < b implies 2^a < 2^b > fed this fish to the penguins: Is mathematics an activity of economic value? Is it rewarded? If yes, >then feel the responsability of producing something else than fantasy. >Hey, we're artists, not engineers. >Mathematics can be regarded as the art of thinking abstractly. >Sure it's been applied, and sure it uses the scientific method, but >at heart, it's art, not science. >Are you saying that the quest to achieve high levels of abstract >thought has no value? >Are you saying art has no value to society. >Thus, artists, musicians, novelists, screenwriters, actors, >professional athletes, chess players, philosophers -- all have no >value? Or they do have value, and deserve to be paid, but not >mathematicians? >There is a _crucial difference_. Artists, musicians, novelists, actors, >screenwriters, professional athletes, chess players, they all entertain >_other_ people, rather than themselves. While mathematicians tend to be >quite boring for virtually anybody else than themselves. If mathematics >intends to be an art, it should be enjoyable in the first place, and by >this I mean: not only for mathematicians. But I said: if .. You yourself do not *understand* mathematics, you do not enjoy it, and > from your own personal bias and prejudice, you leap to the conclusion > that no one but mathematicians enjoys mathematics. Wow. HdB does not understand mathematics? Just a _few_ references: http://www.xs4all.nl/~westy31/Electric.html#Irregular http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf http://hdebruijn.soo.dto.tudelft.nl/jaar2006/drievoud.pdf http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/Splines.pdf http://hdebruijn.soo.dto.tudelft.nl/www/grondig/crossing.htm http://hdebruijn.soo.dto.tudelft.nl/jaar2006/kromming.pdf > Never mind reducing the function of art to the entertainment of > others. This is the typical inanity spewed by a complete inability to > make fine distinctions, and joining in the same phrase, say, the > Divine Comedy, and the latest Batman comic strip. HdB does not understand what art is? Just a _few_ references: http://hdebruijn.soo.dto.tudelft.nl/www/muziek/download.htm http://hdebruijn.soo.dto.tudelft.nl/omastijd/index.htm > By God, what an arrogant idiot you are. HdB is arrogant? Meaning that he is uncapable of admiring other people: http://hdebruijn.soo.dto.tudelft.nl/www/muziek/edwin.htm > G. Rodrigues Han de Bruijn === Subject: Re: a < b implies 2^a < 2^b > fed this fish to the penguins: But in any case, many of these theories about things that you think >don't exist give results about things that you surely believe _do_ >exist. >As an analogy, use of sqrt(-1) resolves some difficult problems about >the real numbers. Thus, we should be willing to think outside the box >of reality if only to better understand reality. >Sure. Sqrt(-1) = i = (0,1) , which is a vector in _real_ 2-D space. Now >define the elementary operations (e.g. multiplication) for such vectors >and you're done. You are totally missing the point. The point is that all successful > theories have had the need to recourse to *abstract* mathematical > tools that are removed away from our general experience. Quasi > mentioned the complex numbers, but we could add the Minkowski space > for special relativity or the curvature tensor for General relativity, > physical quantities as operators in Quantum Mechanics with the their > spectrum as the allowed measurable values, the covering group SU(2) to > physical theories has had an absolute need for concepts that are not > borne from ordinary experience, but by *mathematical abstraction*, or, > to repeat Quasi, in order to probe and explain what you see, you had > to imagine the unseen. With all the examples mentioned, there is a path of heuristic reasoning from the seen to the unseen. I've gone such paths while I was a physics student. The unseen does not just come out of the big blue sky. >As a theoretical physicist, I have a decent (non naive) understanding of >reality. I know that (actual) infinity is not in there. But I'll respect >your choice. A theoretical physicist? I have by me the book Quantum Mechanics, second edition, by E. > Merzbacher, from which I learned the subject a few years ago (along > with some others like the Quantum Mechanics by L. Landau). Turning > to chapter 8. entitled Principles of Wave Mechanics, on the start of > the second section we can read: We have learned that every physical quantity F can be represented by > a linear operator, which for convenience is denoted by the same > letter, F. If we go on we also learn that these linear operators are also > self-adjoint - so that their spectrum is real. Self-adjointness is > tied to existence of an inner product in the space. Turning to chapter > 14 we would learn that the spaces where these operators act are > Hilbert spaces, which in general are separable but *not necessarily* > of finite dimension. A separable infinite-dimensional Hilbert space is > as infinite as infinite gets. But it can be considered as the limiting case of a finite vector space. When represented for calculation purposes, you can introduce a cut-off on space dimension and yet have an arbitrarily accurate approximation. The fact that Quantum Mechanical vector spaces are all separable, means that each vector in the infinite space is expressible in the elements of the basis. Which is the same as for finite vector spaces. You would have an argument if you could show me a _useful_ infinite space with QM where the vectors are _not_ expressible in any basis. Because then the latter fact would be clear evidence that there exist _infinite_ vector spaces which can _not_ be regarded as a limiting case of finite vector spaces. With other words: this would prove the applicability of actual infinity for vector spaces. I'm pretty sure that NO such evidence can be found. > We could multiply the examples infinitely (pun intended) drawing from > all sources from Classical Mechanics, Special and General Relativity, > etc. The fact is, is that all the successful theories that model our > universe use highly infinitary mathematical tools. They use moderately infinitary mathematical tools. The infinitary herein can always be looked upon as very large but I don't know and don't care how large but nevertheless finitary. See the example above. > But do not let these simple facts spoil your fantasies. Not fantasies. Knowledge. Han de Bruijn === Subject: Re: a < b implies 2^a < 2^b fed this fish to the penguins: > fed this fish to the penguins: But in any case, many of these theories about things that you think >don't exist give results about things that you surely believe _do_ >exist. >As an analogy, use of sqrt(-1) resolves some difficult problems about >the real numbers. Thus, we should be willing to think outside the box >of reality if only to better understand reality. >Sure. Sqrt(-1) = i = (0,1) , which is a vector in _real_ 2-D space. Now >define the elementary operations (e.g. multiplication) for such vectors >and you're done. You are totally missing the point. The point is that all successful > theories have had the need to recourse to *abstract* mathematical > tools that are removed away from our general experience. Quasi > mentioned the complex numbers, but we could add the Minkowski space > for special relativity or the curvature tensor for General relativity, > physical quantities as operators in Quantum Mechanics with the their > spectrum as the allowed measurable values, the covering group SU(2) to > physical theories has had an absolute need for concepts that are not > borne from ordinary experience, but by *mathematical abstraction*, or, > to repeat Quasi, in order to probe and explain what you see, you had > to imagine the unseen. With all the examples mentioned, there is a path of heuristic reasoning >from the seen to the unseen. I've gone such paths while I was a physics >student. The unseen does not just come out of the big blue sky. > If I understand you right, you agree with me that in tackling the difficult problems that theoretical physics poses, *abstract* mathematical tools were needed time and time again. So what you are decrying is unmotivated abstraction. I can agree to that, but the principle is so general that really affords no refutation, and more importantly, it is useless because it does not give any objective criteria by which to decide which mathematics is good or bad. Many fields of mathematics were investigated as pure mathematics before having any sort of application outside of mathematics itself. >As a theoretical physicist, I have a decent (non naive) understanding of >reality. I know that (actual) infinity is not in there. But I'll respect >your choice. A theoretical physicist? I have by me the book Quantum Mechanics, second edition, by E. > Merzbacher, from which I learned the subject a few years ago (along > with some others like the Quantum Mechanics by L. Landau). Turning > to chapter 8. entitled Principles of Wave Mechanics, on the start of > the second section we can read: We have learned that every physical quantity F can be represented by > a linear operator, which for convenience is denoted by the same > letter, F. If we go on we also learn that these linear operators are also > self-adjoint - so that their spectrum is real. Self-adjointness is > tied to existence of an inner product in the space. Turning to chapter > 14 we would learn that the spaces where these operators act are > Hilbert spaces, which in general are separable but *not necessarily* > of finite dimension. A separable infinite-dimensional Hilbert space is > as infinite as infinite gets. But it can be considered as the limiting case of a finite vector space. >When represented for calculation purposes, you can introduce a cut-off >on space dimension and yet have an arbitrarily accurate approximation. > First, you are not really telling me that that is how physicists do things, are you? Because that is completely false. They simply work in the infinite dimensional space and have no qualms about it. There is no intrinsic rule in QM that says that infinite dimensional state spaces are not possible. In fact, browse through any textbook on QM to the Hydrogen atom. They were there from the start. Second, no, an infinite dimensional space is not just the limiting case of a finite vector space, or more correctly, there is no obvious unique sense in which it is so. In some cases it is intuitively clear how this can be done (e.g. the state space for the Hydrogen atom) in others it is not. And in some cases it is simply *wrong*, in the specific sense that looking at the action of an operator (= a measurable quantity in QM) on finite dimensional spaces does *not* give an arbitrarily accurate approximation. But all this, it just says that infinite dimensional state spaces are unavoidable. >The fact that Quantum Mechanical vector spaces are all separable, means >that each vector in the infinite space is expressible in the elements of >the basis. Which is the same as for finite vector spaces. You would have >an argument if you could show me a _useful_ infinite space with QM where >the vectors are _not_ expressible in any basis. Because then the latter >fact would be clear evidence that there exist _infinite_ vector spaces >which can _not_ be regarded as a limiting case of finite vector spaces. >With other words: this would prove the applicability of actual infinity >for vector spaces. I'm pretty sure that NO such evidence can be found. > Huh? *Every* Hilbert space has an orthonormal basis (choice needed here). Separability just means that the orthonormal basis is at most *countably infinite*. And yes, you need the *full countably infinite* set of orthonormal vectors to express every vector. So what you are saying doesn't make a lick of a sense. Perhaps, you are confusing the notions of Hamel basis and orthonormal basis? In finite dimensional spaces there are no big differences between the two concepts, but definitely *not so* in infinite dimensions. So there it is for your evidence. To add to the above: - First, you have no compelling *physical* reason (besides your own prejudice, that is) to rule out infinite dimensional state spaces. Even in the cases where you can argue that you can introduce a finite dimensional cut-off like the Hydrogen atom, you need the infinite dimensional space to prove certain things that you can compare with the lab experiments. - There are many situations where infinite dimensional state spaces are simply unavoidable, e.g. quantizing classical systems with infinitely many degrees of freedom like Q.E.D., or simply necessary to thermodynamic limits, etc.). - You cannot simply wave your hands and say oh, you can approximate by finite dimensional spaces, that is a cop-out answer that tells us nothing. You have to to say exactly what do you mean by that. In the above cases, even if it is possible to do so (a big if), it is also a highly *nontrivial* task to do it. If you can do it, better be prepared to win the Nobel prize or something. > We could multiply the examples infinitely (pun intended) drawing from > all sources from Classical Mechanics, Special and General Relativity, > etc. The fact is, is that all the successful theories that model our > universe use highly infinitary mathematical tools. They use moderately infinitary mathematical tools. The infinitary herein >can always be looked upon as very large but I don't know and don't care >how large but nevertheless finitary. See the example above. > So your qualm is the difference between the words moderatly and highly? To repeat myself: no, not all cases of infinity in physics are of the very large but I don't know and don't care how large type. This is simply wrong. Second, even if it were, it was incumbent upon *you* to prove that, since that is the way present-day theoretical physicists work: they just work directly on infinite dimensional spaces and have no qualms about it. Third, even if you proved it, why would theoretical physicists ditch what they already know? G. Rodrigues === Subject: Re: A Scientific Theory for Creation As the title for this thread suggests, I meant to explain my >scientific theory for creation and I tried to do so above the cries >of many critics and detractors. See >for a more direct unfolding of the physics. >Note to those who wish to post to sci.physics.foundations, it is a >moderated newsgroup. >Yeah, and I'm seeing the same reference over and over in your postings: >http://www.everythingimportant.org/relativity/special.pdf >Han de Bruijn > In the thread to which you refer at sci.physics.foundations: > I posted the following links: http://www.everythingimportant.org/relativity/special.pdf > http://arxiv.org/abs/physics/9811050 > http://en.wikipedia.org/wiki/Law_of_large_numbers > http://en.wikipedia.org/wiki/Measure_%28mathematics%29 > http://en.wikipedia.org/wiki/Probability_space > http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVN-4HM7Y9M-4&_us e r=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_ u rlVersion=0&_userid=10&md5=986e14bd53824cbfa6a12209871a3b31 The first link, in the complete course of the discussion, was only > posted two times. And you think that's over and over? Does the word > twice mean over and over to you? Try Google with Shubee special.pdf. And find: 1.050 references. Does that _not_ mean over and over to you? Han de Bruijn === Subject: Re: A Scientific Theory for Creation <4771e$478b3b09$82a1e228$7276@news1.tudelft.nl> posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > Try Google with Shubee special.pdf. And find: 1.050 references. > Does that _not_ mean over and over to you? Han de Bruijn The number of times that Shubee special.pdf appears doesn't strike me as being unusual. As David Hilbert has said, We must strive for it to become [a mathematical science]. We must extend the range of pure mathematics further and further, not only in our own mathematical interest but also for the sake of science as such. What I find unnatural, in fact, supernatural, is the number of mathematically incompetent posters that swarm to newsgroups, driven by the fantasy that they are qualified to comment on purely mathematical questions. Note that Stephen Montgomery-Smith is qualified to answer my mathematical questions. http://www.math.missouri.edu/~stephen/ . He has simply elected not to answer http://www.everythingimportant.org/viewtopic.php?t=1386 Shubee === Subject: Re: -- the final solution >Quasi, new Bishop of Angouleame. I'm just waiting for someone to do some numerology on the string -- to prove that it evaluates to the Number of the Beast (616, that is), proving that this is nothing less than the Apocalypse. -- Angus Rodgers Contains mild peril === Subject: Re: -- the final solution > Quasi, new Bishop of Angouleame. I'm just waiting for someone to do some numerology on the string > -- to prove that it evaluates to the Number of the Beast (616, > that is), proving that this is nothing less than the Apocalypse. Excellent. Somebody else spotted it as well. I thought of pointing it out, but I was curious if anyone else would see it first ;o) -- I.N. Galidakis === There are new job listings at http://jobs.phds.org -------------------------------------------------------------------- Title: Entry Level Equities Quant Employer: Tier 1 Investment Bank Location: London, England, United Kingdom World renowned investment house with one of the most prestigious equity derivatives quant teams in the market is seeking to strengthen their platform with a top class junior quant analyst. You will be responsible for the delivery of all equity... 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Full details: -------------------------------------------------------------------- Title: Structural Biology Opportunities at D. E. Shaw Research, LLC Employer: The D. E. Shaw group Location: New York, NY, United States Extraordinarily gifted structural biologists are sought to join a rapidly growing New York.89´.8bbased research group that is pursuing an ambitious, long-term strategy aimed in part at fundamentally transforming the process of drug discovery. Candidates... Full details: -------------------------------------------------------------------- Title: Post Doctoral Fellow .89´.8b Catchment Modelling- Water Quality, Hamilton Employer: National Institute of Water and Atmospheric Research Ltd Location: Hamilton, New Zealand NIWA is a leading environmental research institute and a key provider of atmospheric, freshwater and marine research in New Zealand. This is an exciting opportunity to undertake research on a two year Post Doctoral grant on the modelling of diffuse... Full details: -------------------------------------------------------------------- Title: Director's Postdoctoral Fellowship Employer: Argonne National Laboratory Location: Argonne, IL, United States Applications are being sought for the Director's Postdoctoral Fellowships. Candidates are selected based on their research and academic accomplishments, and the strength of their research proposal. They will collaborate with ANL scientists and... Full details: -------------------------------------------------------------------- Title: Senior Researcher/Manager - Surveys & Public Opinion Polling Employer: IFES Location: Washington, DC, United States Division: Applied Research Center Location: Washington, DC Responsibilities: * Management of research functions, with a particular focus on opinion research projects. * Developing relationships with programmatic staff to design resource efficient... 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Full details: -------------------------------------------------------------------- Title: Univeristy of Pennsylvania Biomedical Postdoctoral Teaching Fellowship Employer: University of Pennsylvania Location: Philadelphia, PA, United States PENN-PORT IRACDA Biomedical Postdoctoral Programs at the University of Pennsylvania School is pleased to announce the award of an NIH IRACDA University of Pennsylvania Postdoctoral Opportunities in Research and Teaching (PENN-PORT). The PENN-PORT... Full details: -------------------------------------------------------------------- Title: Exceptional Software Engineer Employer: Financial Institution Location: New York, NY, United States Portfolio Analytics team of the major Investment Bank is seeking experienced and detail oriented C++ software engineers to research, design, develop, and test Portfolio analytics software applications and components. Applicant will be immediately... 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Full details: -------------------------------------------------------------------- Title: Quant research analyst Employer: Hedge fund - originally dallas but role is in ny Location: New York, NY, United States Need two types, a risk management person and an equity derivatives person. Both C++ and SQL. Requires at least 3 years experience. Full details: -------------------------------------------------------------------- Title: Junior Desk Quant/FX Strategist Employer: Leading London based investment bank Location: London, England, United Kingdom A leading London based investment bank is looking for new PhDs with excellent quantitative skills to join its FX Strategy team This is an exciting opportunity for new Quants looking to get into the market Responsibilities of the role will include: *... Full details: -------------------------------------------------------------------- Title: Mid Level IR/FX Hybrid Quant Role Employer: Strong House need to fill this seat. Location: London, England, United Kingdom My client has a seat to fill as one of his Quants has moved over into the trading team within this bank. The role: Develop mathematical models for pricing, both for well established products to improve pricing and risk management, and for new... Full details: -------------------------------------------------------------------- Title: Reinsurance Actuarial Analyst (Bermuda) Employer: The D. E. Shaw group Location: Bermuda, Bermuda The D. E. Shaw group seeks a talented individual to join its reinsurance business in Bermuda. The Reinsurance Actuarial Analyst will play an integral role in establishing and growing the reinsurance strategy and will be involved in quantitative data... Full details: -------------------------------------------------------------------- Post your job (free!): http://jobs.phds.org/jobs/post PhDs.org: Science, Math, and Engineering Career Resources --------------------------------------------------------- * Job Listings: http://jobs.phds.org/ - Job board with hundreds of listings for Ph.D.s - Reach tens of thousands of Ph.D.s each month * Graduate School Rankings: http://graduate-school.phds.org/ - Comprehensive, customizable rankings of graduate programs * Career Resources: http://www.phds.org/ - Pointers to the best resources on the web for: + getting into graduate school + writing your dissertation + jobs for Ph.D.s in academia and industry * Engineering Science Weblog: http://blog.phds.org/ - Building better scientists and engineers === Subject: Correlation and Regression Analysis Software posting-account=ajkdngoAAAB2X3SxyQtiBYkcRETphklh 1.1.4322),gzip(gfe),gzip(gfe) Our latest online software allows computing Coefficient of Correlation and Equation of Regression Analysis, it displays the results along with scatter diagram and line graph For the Correlation program Visit http://www.thinkanddone.com/ge/Corr.aspx For the Regression program Visit http://www.thinkanddone.com/ge/Reg.aspx Asad === Subject: Re: the need for relevance Is mathematics an activity of economic value? >Is it rewarded? If yes, then feel the responsability >of producing something else than fantasy. >Hey, we're artists, not engineers. >Mathematics can be regarded as the art of thinking >abstractly. >Sure it's been applied, and sure it uses the >scientific method, but at heart, it's art, not science. I'm not happy with my above response to Han de Bruijn. There are elements of truth in what I said, but on the > whole, it's not what's really going on. I said Hey, we're artists, not engineers, but I take that back. We're more like architects. In that sense we're artists _and_ engineers. And as architects, we want the end product to be both structurally > sound and relevant, not just aesthetically satisfying. Good! An I suppose you want to build your structure on solid ground in the first place? > The mathematics community is very much aware of the need for > relevance. But we stretch it ... If theory A yields some insight into theory B, then at least in > principle, A applies to B. Of course, they might both apply to each > other to some degree, in which case we would have arrows from A to B > and also from B to A. If we have a path A to B to C, then indirectly, > A applies to C. Of course, this simple concept does not take into > account the nature and strength of the applicability. The core theories relate to concrete things -- for example, numbers, > geometric objects. At the foundation, there is logic, which allows us > to define and analyze methods of deduction for all the other theories. Mathematical theories are, by design, abstractions. Even the core > theories (numbers, geometry) are abstract, but less so. For example, > I'm sure Euclid was well aware that the concept of a straight line, > extending forever in both directions, is an abstraction, not > necessarily realizable in the physical universe. Even more so for the > concept of a point -- an object with a location but no size. True. But these abstractions can be materialized into things in the real world by ading a tiny bit of uncertainity to them. Once you add a little bit of thickness to an abstract (read: ideal, hence invisible) straight line, it readily becomes a line you can actually draw on say: a piece of paper or a computer screen. This is an important feature to remember. Uncertainity is the gateway from abstraction (idealization) to reality. > The point I'm trying to make is that Math does care about reality, but > readily allows further abstraction in order to get a handle on it. > Still, for any new theory, there is a sense of obligation to reveal, > however indirectly, something new about existing theories. In other > words, if some theory A sheds some new light on existing theories, and > if those existing theories lead, in some chain, down to the core > theories, then that provides some justification for A. This has clearly happened to e.g. Set Theory. The _ZFC_ system contanins many axioms which are nearly trivial (perhaps better: may be considered instead as _theorems_) for _finite_ sets. What has happened next is that infinite sets have be endowed with finite set like properties, in order to make them look a great deal like the already well known finite sets. So _theorems_ for finite sets have been adopted as _axioms_ for infinite sets, for the simple reason that, otherwise, we would have no starting point, at all, for the latter. But these choices seem rather arbitrary. And consequently, there is more than _one_ set theory for infinite sets. But the problem could have been solved otherwise, by simply not allowing other than finite sets, together with a limit concept of some sort, for the purpose of approaching infinite sets in the calculus way. > However, based on past experience, the mathematics community is very > tolerant of, and in fact encourages, free exploration, with little or > no requirement to demonstrate relevance at the start, especially if > the structure of the new theory seems intuitively right. Play with it, > see where it leads. If it leads nowhere, it may die a natural death, > or if the structure still feels just right, it may survive on its > own. That liberal do whatever you want credo is based on the > expectation that somewhere down the line (but not necessarily in the > current lifetime), there will be some tie back to existing theories. > But it's not a blank check. What I'm trying to say is that mathematics combines art (math as a > beautiful form of reasoning) with science (attempts to model and > explain reality). Math is not oblivious to the need for relevance, but > allows a lot of freedom in that regard. Eventual, potential relevance > is usually sufficient. I have a page which is called Snippets of Pure Applicable Mathematics or alternatively Purified Applied Mathematics. What you say is right: _potential_ relevance is sufficient. I would like to add: is NECESSARY. Mathematics must not be _applied_ per se, but it should be _applicable_, i.e. it should NOT BE IMPOSSIBLE TO FIND AN APPLICATION for it, somehow. Mathematics must be pure. And if it is applied, it should be _purified_ first, in order to get rid of the non-mathematical details and terms. It is strange practice that mathematics should be rigourous in its logic but at the same time should enjoy complete and unrestricted freedom with the choice of its axioms. This chain is as weak as its weakest link, and quite vulnerable to the nonsensical: from first principles. Han de Bruijn === Subject: Re: the need for relevance > Mathematics must not be _applied_ per se, but it should be _applicable_, > i.e. it should NOT BE IMPOSSIBLE TO FIND AN APPLICATION for it, somehow. One great difficulty with that thesis is that if an application must exist before the math is developed, as HdB would have it, then much of applicable mathematics would never have been developed, or at least would have been much delayed in its development. For example, the parts of number theory which now allow electronic banking and commerce were long thought to have no possible applications at all. === Subject: Re: the need for relevance fed this fish to the penguins: > The mathematics community is very much aware of the need for > relevance. But we stretch it ... If theory A yields some insight into theory B, then at least in > principle, A applies to B. Of course, they might both apply to each > other to some degree, in which case we would have arrows from A to B > and also from B to A. If we have a path A to B to C, then indirectly, > A applies to C. Of course, this simple concept does not take into > account the nature and strength of the applicability. The core theories relate to concrete things -- for example, numbers, > geometric objects. At the foundation, there is logic, which allows us > to define and analyze methods of deduction for all the other theories. Mathematical theories are, by design, abstractions. Even the core > theories (numbers, geometry) are abstract, but less so. For example, > I'm sure Euclid was well aware that the concept of a straight line, > extending forever in both directions, is an abstraction, not > necessarily realizable in the physical universe. Even more so for the > concept of a point -- an object with a location but no size. True. But these abstractions can be materialized into things in the real >world by ading a tiny bit of uncertainity to them. Once you add a little >bit of thickness to an abstract (read: ideal, hence invisible) straight >line, it readily becomes a line you can actually draw on say: a piece of >paper or a computer screen. This is an important feature to remember. Uncertainity is the gateway from abstraction (idealization) to reality. > Huh? What does it mean to add a little bit of thickness or uncertainty to say, the Hilbert spaces of quantum mechanics, the SU(2) group for the spin of an electron, the curvature tensor for GR? What do you mean by materializing into things in the real world? This should be fun to read. > The point I'm trying to make is that Math does care about reality, but > readily allows further abstraction in order to get a handle on it. > Still, for any new theory, there is a sense of obligation to reveal, > however indirectly, something new about existing theories. In other > words, if some theory A sheds some new light on existing theories, and > if those existing theories lead, in some chain, down to the core > theories, then that provides some justification for A. This has clearly happened to e.g. Set Theory. The _ZFC_ system contanins >many axioms which are nearly trivial (perhaps better: may be considered >instead as _theorems_) for _finite_ sets. What has happened next is that >infinite sets have be endowed with finite set like properties, in order >to make them look a great deal like the already well known finite sets. So _theorems_ for finite sets have been adopted as _axioms_ for infinite >sets, for the simple reason that, otherwise, we would have no starting >point, at all, for the latter. But these choices seem rather arbitrary. >And consequently, there is more than _one_ set theory for infinite sets. But the problem could have been solved otherwise, by simply not allowing >other than finite sets, together with a limit concept of some sort, for >the purpose of approaching infinite sets in the calculus way. > I see that you acknowledge that the problem *was solved*. Could be solved in any other way? And doesn't the very question tell you anything? I mean, besides wishful thinking on your part, what sort of evidence do you have that the problem *can* be solved by simply not allowing other than finite sets, together with a limit concept of some sort, for the purpose of approaching infinite sets in the calculus way? Hint: half-ass half-baked set-theory like axiom systems with funny names like Implementable Set Theory or something like it, do not count as solutions. And even if we assume you could do this, why would theoretical physicists, that already have to spend a large amount of time absorbing the current mathematical tools, would embark on *your* ship? Just because Mr. de Bruijn is prejudiced against infinite sets? And about real analysis, what exactly do you have in mind? Usual real analysis uses mostly countable choice. As far as I know, non-standard models use ultrafilters to build them (= full AC) and since they contain the usual reals, they are as infinitary as them. You could also go the axiomatic route, but somehow I do not think that that is what you have in mind. This should be fun to read. > However, based on past experience, the mathematics community is very > tolerant of, and in fact encourages, free exploration, with little or > no requirement to demonstrate relevance at the start, especially if > the structure of the new theory seems intuitively right. Play with it, > see where it leads. If it leads nowhere, it may die a natural death, > or if the structure still feels just right, it may survive on its > own. That liberal do whatever you want credo is based on the > expectation that somewhere down the line (but not necessarily in the > current lifetime), there will be some tie back to existing theories. > But it's not a blank check. What I'm trying to say is that mathematics combines art (math as a > beautiful form of reasoning) with science (attempts to model and > explain reality). Math is not oblivious to the need for relevance, but > allows a lot of freedom in that regard. Eventual, potential relevance > is usually sufficient. I have a page which is called Snippets of Pure Applicable Mathematics >or alternatively Purified Applied Mathematics. What you say is right: >_potential_ relevance is sufficient. I would like to add: is NECESSARY. Mathematics must not be _applied_ per se, but it should be _applicable_, >i.e. it should NOT BE IMPOSSIBLE TO FIND AN APPLICATION for it, somehow. > And oh enlightened Mr. de Bruijn, why exactly is that a necessary criterion for mathematics? I've said this innumerous times, but I will say it again: Mathematics is an *autonomous* discipline with a conceptual framework of its own. It enters in relations with other disciplines (e.g. theoretical physics) but is not bound by them -- that would be denying its autonomy. It seems even absurd to say this, when mathematicians *do work* on that assumption. So unless Mr. de Bruijn is ready to act as a censor and order mathematicians to do otherwise, they will continue to work autonomously. Note: and please do not tell me that Mathematics was born from trying to explain the real world (one of those expressions that without is meaningless). Astronomy was born from astrology; you are not suggesting that astronomers should be drawing horoscopes are you? And assuming that that is a necessary criterion, how can you hope to apply it? Given some mathematical theory how can you decide that it is *impossible* to find an application for it? This should be fun to read. >Mathematics must be pure. And if it is applied, it should be _purified_ >first, in order to get rid of the non-mathematical details and terms. > Huh? What exactly are the non-mathematical details and terms? >It is strange practice that mathematics should be rigourous in its logic >but at the same time should enjoy complete and unrestricted freedom with >the choice of its axioms. This chain is as weak as its weakest link, and >quite vulnerable to the nonsensical: from first principles. > Unrestricted freedom in the choice of its axioms? Since the vast majority of mathematicians in practice works within ZF(C), that hardly counts as unrestricted freedom. And, please, do not harp about the nonsensical in mathematics, since you have been unable to show a *single* example of contradiction. G. Rodrigues === Subject: need for solution posting-account=CdahlwoAAABUhXQiebVn5TkpF9bEPc5c Media Center PC 3.0; .NET CLR 1.0.3705),gzip(gfe),gzip(gfe) In the name of GOD, the Merciful I'm looking for solution of CMOS Analog Circuit Design 2nd Ed. by P.E. Allen - can you help me? please send it to my email ( rgolmohammadi64@gmail.com ) if it is possible. === Subject: Information on the sequences Could you please explain what are those sequences and how to find the values in place of ? in the following three questions? 1. What wil be the value of ? in the sequence 1, 6, 19, 45, ?, 161, 266 2. What will be the next number in the sequence 1, 3, 7, 19, 51, ? 3. What will be the next number in the sequence 1, 6, 21, 56, 120, 216, 336, 456, 546, ? === Subject: Re: Information on the sequences Do Lagrange interpolation. Ask your teacher justify why he thinks his method is any better. -- rhhardin@mindspring.com On the internet, nobody knows you're a jerk. === Subject: Re: Information on the sequences > Could you please explain what are those sequences and how to find the > values in place of ? in the following three questions? > 1. What wil be the value of ? in the sequence 1, 6, 19, 45, ?, 161, 266 Let a_n be the coefficient of x^4 in (1 + x + x^2)^n. Then, when _n_ goes from 1 to 7, a_n is 1, 6, 19, 45, 90, 161, 266 > 2. What will be the next number in the sequence 1, 3, 7, 19, 51, ? Let b_n be the coefficient of x^n in (1 + x + x^2)^n. Then, when _n_ goes from 1 to 6, b_n is 1, 3, 7, 19, 51, 141. > 3. What will be the next number in the sequence 1, 6, 21, 56, 120, 216, > 336, 456, 546, ? Let c_n be the coefficient of x^n in (1 + x + x^2 + x^3)^6. Then, when _n_ goes from 1 to 10, c_n is 1, 6, 21, 56, 120, 216, 336, 456, 546, 580. Jose Carlos Santos === Subject: Re: Information on the sequences Is there any easy method to expand those higher order polynomials? === Subject: Re: Information on the sequences > do you expand the polynomial like (1+x+x^2+x^3)^n? > Is there any easy method to expand those higher order polynomials? See what Euler had to say about this subject: http://arxiv.org/PS_cache/math/pdf/0505/0505425v1.pdf I am not Euler and I use Mathematica for this. Jose Carlos Santos === Subject: Re: Information on the sequences <5v14jjF1i3b2oU1@mid.individual.net> posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) Could you please explain what are those sequences and how to find the > values in place of ? in the following three questions? > 1. What wil be the value of ? in the sequence 1, 6, 19, 45, ?, 161, 266 Let a n be the coefficient of x^4 in (1 + x + x^2)^n. Then, when n > goes from 1 to 7, a n is 1, 6, 19, 45, 90, 161, 266 2. What will be the next number in the sequence 1, 3, 7, 19, 51, ? Let b n be the coefficient of x^n in (1 + x + x^2)^n. Then, when n > goes from 1 to 6, b n is 1, 3, 7, 19, 51, 141. 3. What will be the next number in the sequence 1, 6, 21, 56, 120, 216, > 336, 456, 546, ? Let c n be the coefficient of x^n in (1 + x + x^2 + x^3)^6. Then, when > n goes from 1 to 10, c n is 1, 6, 21, 56, 120, 216, 336, 456, 546, > 580. > Jose Carlos Santos Just curious: did you use Maple to find these generating functions? Mate === Subject: Re: Information on the sequences posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) >Could you please explain what are those sequences and how to find the >values in place of ? in the following three questions? >1. What wil be the value of ? in the sequence 1, 6, 19, 45, ?, 161, 266 Let a n be the coefficient of x^4 in (1 + x + x^2)^n. Then, when n > goes from 1 to 7, a n is 1, 6, 19, 45, 90, 161, 266 2. What will be the next number in the sequence 1, 3, 7, 19, 51, ? Let b n be the coefficient of x^n in (1 + x + x^2)^n. Then, when n > goes from 1 to 6, b n is 1, 3, 7, 19, 51, 141. 3. What will be the next number in the sequence 1, 6, 21, 56, 120, 216, >336, 456, 546, ? Let c n be the coefficient of x^n in (1 + x + x^2 + x^3)^6. Then, when > n goes from 1 to 10, c n is 1, 6, 21, 56, 120, 216, 336, 456, 546, > 580. > Jose Carlos Santos Just curious: did you use Maple to find these generating functions? Mate Never mind; you used http://www.research.att.com/~njas/sequences === Subject: Re: Information on the sequences > Could you please explain what are those sequences and how to find the > values in place of ? in the following three questions? > 1. What wil be the value of ? in the sequence 1, 6, 19, 45, ?, 161, 266 > Let a_n be the coefficient of x^4 in (1 + x + x^2)^n. Then, when _n_ > goes from 1 to 7, a_n is 1, 6, 19, 45, 90, 161, 266 > 2. What will be the next number in the sequence 1, 3, 7, 19, 51, ? > Let b_n be the coefficient of x^n in (1 + x + x^2)^n. Then, when _n_ > goes from 1 to 6, b_n is 1, 3, 7, 19, 51, 141. > 3. What will be the next number in the sequence 1, 6, 21, 56, 120, 216, > 336, 456, 546, ? > Let c_n be the coefficient of x^n in (1 + x + x^2 + x^3)^6. Then, when > _n_ goes from 1 to 10, c_n is 1, 6, 21, 56, 120, 216, 336, 456, 546, > 580. > Jose Carlos Santos > Just curious: did you use Maple to find these generating functions? > Mate Never mind; you used > http://www.research.att.com/~njas/sequences Yes, together with Mathematica to check the results. Jose Carlos Santos === Subject: Re: measure theory f_n goes to f in measure iff f_n goes to f in metric d, where metric d defines like this; d(f,g)=integral( [f-g] / ([f-g] + 1) ). On a measure space of finite measure, yes. ************************ David C. Ullrich === Subject: Re: divergent series haha DIVERGENT SO NO COUNTER ! <32243742.1200238998443.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Denis Feldmann David C. Ullrich a .8ecrit : > In the threath by Denis Feldmann named > divergent series > once again David C. Ullrich thinks he can > correct me. The word is thread, by the way. > but it is true !!! > keywords : Riemann's Series Theorem , Cauchy > product , Hilbert's Hotel. > this shows that any series with real terms that > containes oo negative and > oo positive terms can in fact be summed to any > desired real !!! > I DARE YOU DAVID C ULLRICH TO GIVE ME A You really enjoy making a fool of yourself, > right? >I gave a counterexample already. Let a n = > (-1)^n/n^2. A shorter one to type is (-1/2)^n > tommy1729 ************************ David C. Ullrich And, trivially, any absolutely convergent alternating > series, of which > there are many, will do as a counterexample. But THE TOPIC WAS : DIVERGENT SERIES !!! > try again , David and others. tommy1729- ***************************************************** You are sometimes a crank of the size of JSH, with the difference that you seem to enjoy being an ass and he, apparently, doesn't. You want a series with infinite number of positive elements and infinite number of negative elements that DIVERGES and, nevertheless, you can't re-arrange to make it sum to whatever you want? It's boringly easy: besides the numerous examples that others already gave you, what about the series with general element a n = [(-1)^n] * n ? Try to rearrange the element of this series and make it sum up to 35.5, say. I hope, in a rather hopelessly way (go figure!), that you'll be able to realize that the above indeed is a divergent series (or as you write it: DIVERGENT SERIES). You, once again, are confusing pretty elementary stuff: Riemann Series Theorem talks of a convergent series such that it is NOT absolutely convergent, meaning: the series of a n converges, but the series of | a n| diverges. The theorem says that then you can rearrange the elements of such a series so as to make the new arrangement sum up to ANY real number or even to diverge to +oo or -oo, a rather ashtonishing theorem. Too bad you're not humble enough as to stop writing heavy nonsenses even after you're proven to be an ass...tsk,tsk,tsk. Tonio === Subject: Re: divergent series haha DIVERGENT SO NO COUNTER ! > Denis Feldmann David C. Ullrich a .8ecrit : > In the threath by Denis Feldmann named > divergent series > once again David C. Ullrich thinks he can > correct me. >The word is thread, by the way. > but it is true !!! > keywords : Riemann's Series Theorem , Cauchy > product , Hilbert's Hotel. > this shows that any series with real terms > that > containes oo negative and > oo positive terms can in fact be summed to > any > desired real !!! > I DARE YOU DAVID C ULLRICH TO GIVE ME A >You really enjoy making a fool of yourself, > right? >I gave a counterexample already. Let a_n = > (-1)^n/n^2. > A shorter one to type is (-1/2)^n > tommy1729 >************************ >David C. Ullrich And, trivially, any absolutely convergent > alternating > series, of which > there are many, will do as a counterexample. But THE TOPIC WAS : DIVERGENT SERIES !!! > try again , David and others. > tommy1729 Ok, here's an example of a divergent series that is a > counterexample to > your claim. Let a_n = 1/n, for n odd > = -1/2^n, for n even. Then a_n is an alternating series, but no matter how > you arrange it, it > always diverges to +oo. > nope : hilbert's Hotel -> order a_n such that all the a_(n^2) come first. There is no such rearrangement, _in_ the sense relevant to the theorem you're talking about. The theorem is this: If sum a_n is convergent and sum |a_n| is divergent then for any a in R there exists a bijection s : N -> N such that sum a_{s(n)} = a. There is no bijection of N that puts all the n^2 before all the other natural numbers. Your understanding of the point to Hilbert's Hotel is as muddled as your understanding of almost everything that you SHOUT about. >now as a suborder a_(n^2) comes before a_((n+1)^2) this limit is clearly finite. game over , you lose. tommy1729 -- > Dave Seaman > Oral Arguments in Mumia Abu-Jamal Case heard May 17 > U.S. Court of Appeals, Third Circuit > ************************ David C. Ullrich === Subject: Re: divergent series haha DIVERGENT SO NO COUNTER ! > Denis Feldmann In the threath by Denis Feldmann named > divergent series > once again David C. Ullrich thinks he can > correct me. >The word is thread, by the way. > but it is true !!! > keywords : Riemann's Series Theorem , Cauchy > product , Hilbert's Hotel. > this shows that any series with real terms that > containes oo negative and > oo positive terms can in fact be summed to any > desired real !!! > I DARE YOU DAVID C ULLRICH TO GIVE ME A >You really enjoy making a fool of yourself, > right? >I gave a counterexample already. Let a_n = > (-1)^n/n^2. > A shorter one to type is (-1/2)^n tommy1729 >************************ >David C. Ullrich And, trivially, any absolutely convergent alternating > series, of which > there are many, will do as a counterexample. But THE TOPIC WAS : DIVERGENT SERIES !!! Actually the topic was _summability methods_ for divergent series, which really doesn't have all that much to do with the theorem of Riemann you're spewing nonsense about, which has to do with rearrangements. In any case, it's very easy to give a divergent series which is a counterexample to your version of the theorem. I'd do so except I see that Seaman already has. try again , David and others. >tommy1729 ************************ David C. Ullrich === Subject: Re: divergent series haha DIVERGENT SO NO COUNTER ! > Denis Feldmann > David C. Ullrich a .8ecrit : > In the threath by Denis Feldmann named > divergent series > once again David C. Ullrich thinks he can > correct me. >The word is thread, by the way. > but it is true !!! > keywords : Riemann's Series Theorem , > Cauchy > product , Hilbert's Hotel. > this shows that any series with real terms > that > containes oo negative and > oo positive terms can in fact be summed to > any > desired real !!! > I DARE YOU DAVID C ULLRICH TO GIVE ME A >You really enjoy making a fool of yourself, > right? >I gave a counterexample already. Let a_n = > (-1)^n/n^2. > A shorter one to type is (-1/2)^n > tommy1729 >************************ >David C. Ullrich And, trivially, any absolutely convergent > alternating > series, of which > there are many, will do as a counterexample. But THE TOPIC WAS : DIVERGENT SERIES !!! > try again , David and others. > tommy1729 Ok, here's an example of a divergent series that > is a > counterexample to > your claim. Let a_n = 1/n, for n odd > = -1/2^n, for n even. Then a_n is an alternating series, but no matter > how > you arrange it, it > always diverges to +oo. nope : hilbert's Hotel - > order a_n such that all the a_(n^2) come first. If you mean the order type is something other than > omega, then it's not an > infinite series any more. You lose. there are an infinite amount of squares ! so i dont lose. now as a suborder a_(n^2) comes before a_((n+1)^2) this limit is clearly finite. You haven't added all the terms. You lose. game over , you lose. Your claim was that you could get the series to sum > to any real number. > You can't make the sum -100. You lose. -- > Dave Seaman > Oral Arguments in Mumia Abu-Jamal Case heard May 17 > U.S. Court of Appeals, Third Circuit > === Subject: Re: divergent series haha DIVERGENT SO NO COUNTER ! > Ok, here's an example of a divergent series that > is a > counterexample to > your claim. Let a_n = 1/n, for n odd > = -1/2^n, for n even. Then a_n is an alternating series, but no matter > how > you arrange it, it > always diverges to +oo. > nope : hilbert's Hotel - > order a_n such that all the a_(n^2) come first. If you mean the order type is something other than > omega, then it's not an > infinite series any more. You lose. > there are an infinite amount of squares ! > so i dont lose. You didn't add any of the nonsquare terms. You also didn't read far enough. You failed to notice that there is another reason that you lose (see below). > Your claim was that you could get the series to sum > to any real number. > You can't make the sum -100. You lose. -- Dave Seaman Oral Arguments in Mumia Abu-Jamal Case heard May 17 U.S. Court of Appeals, Third Circuit === Subject: Re: Kuratowski Ordered Pair > æ æ æ æ The first thing you need to understand, it seems to me, is that the > characteristic property is vacuous. When we write: > æ æ æ æ (a, b) = (p, q) > æ æ æ æ æ => æ æ æ æ a=p and p=q > we are already treating (a, b) and (p, q) as ordered pairs, otherwise we > wouldn't know which element to equate with which in the entailed. >No, we prove the equalities in the consequent of the above formula. >If your point is that we READ the SYMBOLS in a certain order, then >that applies to any ordinary mathematical notation. > æ æ æ æ It isn't that. æWhen I say that the above entailment is vacuous, it > is of course vacuously true. It is a kind of repetition of the notion of an > ordered pair, that we have THIS (and not something else) here, and THAT > (and not something different from that) next. I don't see how you're using the term 'vacuously true' in its usual >sense. =

-> (x=p & y=q) is not vacusously true, since the antecedent is not always false. And I just can't make sense of what you mean the repetition you claim. It's a formula. It's proven from the axioms. I really don't know what >else you want. And I don't know what you want that is relevent to any >mathematical concern I might have. > æ æ æ æ We seem to be making a contrast with the unordered pair, where: > not æ æ [ (a, b) = (p, q) > æ æ æ æ æ => æ æ æ æ a=p and b=q.] >Actually, we need universal quantifiers on a, b, p, and q there, since >there are instances of {a b}={p q} -> (a=p & b=q). Since there are such instances is exactly the reason you need a >universal quantifier over a,b,p,q after 'not'. Oh, but it just occurred to me that I may have misunderstood your >intent. I guess you were saying, Let's take an instance such >that.... In that case, yes, nevermind what I said about quantifiers. > æ æ æ æ I'm not sure what you mean here. That {a, b}={p, q} is satisfied by > the equalities a=p and b=q (as it is by the equalities a=q and b=p) doesn't > mean it is sometimes an entailment, does it? In any case, the quantifiers > are a distraction. > æ æ æ æ But these two entailments are not comparable. Whatever is the case > in the second one, the pairs in the first are already ordered pairs, they > are already and . The difference is that with { } we can't universally generalize over >the conditional but with < > we can can universally generalize over >the conditional. >I don't know what you mean by already are that has any bearing. >The math is simple enough > = {{a} (a b}} >And we prove that =

-> (a=p & b=q). >And we prove >~Aabpq {a b}={p q} -> (a=p & b=q) > æ æ æ æ But no contrast is being made here. That a house is a dwelling and > that it is not the case that a crocodile is a dwelling does not mean that > the difference between a house and crocodile is that one is a dwelling and > the other is not. There is no prior notion of pair such that if we impose > this condition (the characteristic property entailment) then it becomes an > ordered pair, and such that our not imposing that condition leaves us with > an unordered pair. The idea that anything is being explained is entirely > illusory. Who proposed that anything is explained? Certain properties are chosen as relevant to some mathematics we want >to do. We find an encoding (not an explaination) and prove that the >encoding preserves those properties. And it works. And again, STILL, >you've not shown a certain a single mathematical result that can't be >derived because we've happened to choose the Kuratowski definition. > We need to find some other way of > contrasting ordered and unordered pairs. > æ æ æ æ The illusion that the Kuratowski set could be any kind of a > definition of the ordered pair begins, I think, with the background notion > that an unordered pair can be represented as a simple set: > æ æ æ æ (a, b) æ= æ{a, b} > But this doesn't work, because if a=b then the unordered pair (a, a) > becomes {a}, which while no doubt a set, is no kind of pair. Unordered > pairs need to be pairs, even in the special case that a=b. >They are. > = {{a} {a a}}, which is a pair. > æ æ æ æ It's not even clear what your notion of an unordered pair is. Sure it is. It's defined in any textbook on set theory. > Are > you saying that its {a, b}, provided ~(a=b), No. Rather, just consult the standard definition in any textbook of >set theory. > but if a=b then it's the > Kuratowski set with a=b, that is {{a}}? No. I have no idea where you came up with that. > I just find that bizarre. It is (at least as far as ordinary set theory). Where did you come up >with it? I am simply trying to make sense of what you said. It is there, above, unsnipped. When I said an unordered pair, (a, b), still needs to be a pair even if a=b, you said: It is: = {{a}, {a, a}}. The definitions you are talking about are definitions of {x, y}, not definitions of an unordered pair. I do not accept that they are the same thing, for in the case that x=y, or if these are understood as distinct variables, in the case that x=a and y=a, the pair has vanished. A multiset would work, but I don't think that is what you want. > æ æ æ æ What we need, to describe pairs, both ordered and unordered, is the > notion of content at a position. >You say we need this, but I don't know of any mathematics that can't >be expressed with the definition we have. > What we require, in either case, is that > there are distinct positions or slots. Call them S and S' (S not= S'). We > cannot identify the slots with their content æi.e. S is the slot which has > a, for in the event that a=b, we should again come down to a single slot. > Slots are not reducible to their contents. > æ æ æ æ A pair, (a, b), is: (a at S, b at S'). >As far as I can tell what it is you want, we can express it also in >set theory: >{{0 a} {1 b}}. > æ æ æ æ But this in not even a pair in the case where a=1 and b=0. This is > precisely the problem. You cannot skate over the distinction between value > and position. If a=1 and b=0, then 1 is the first and 0 is the second. Sorry, my thinking jumped tracks. The problem appears with triples and higher tuples. If a=1 and b=0, of course we still have a pair, but the triple would become: {{1,0}, {2, c}}. I think the bone of contention is that you regard this characteristic property entailment: = --> a=p and b=q as a property of an ordered pair. I don't believe it is, unless an ordered pair being an ordered pair is a property of an ordered pair. The ordered pair just means already THIS-a THAT-b, ie. a, not p or anything else (for if p=a we wouldn't have added anything), and likewise b, and not something other than b. You want to turn around and claim this is part of the meaning of an ordered pair. But it isn't. It's just an identity property on a and b. But once we have encodings of ordered pairs -- we could write this as E -- then it does become material to ask about the entailment: E = E --> a=p and b=q. For example if we encoded the ordered pair as: a + b/2, we find that E<1, 3> = E<2. 1> = 5/2, and the 'characteristic propery' fails. The characteristic propety belongs at the level of encodings of ordered pairs, not at the level of the ordered pairs themselves. We already have to be in the business of encoding to make sense of the Kuratowski definition. It does not by itself do anything to recommend the encoding project. Noel === Subject: Re: Kuratowski Ordered Pair >If we're not in a theory in the language of set theory then, yes, the >Kuratowski definition may be quite irrelevent. No one disputes that. >But so what? > æ æ æ æ You're twisting my words. I don't intend to. And you don't say in what way you think I have. It is held that set theory axiomatizes virtually all of ordinary >mathematics. That, for some people, is a part of the reason for >studying set theory. So, in that sense, if we wern't using set theory, >then I agree that we'd have no (or virtually no, or quite little) use >for the Kuratowski definition. > It is no good > explaining that it satisfies the 'characteristic property'. What happens in > contexts in which sets like the Kuratowski are required to be both ordered > pairs and simply that particular kind of set? >WHAT particular kind of set? > æ æ æ æ I simply mean a set of the form {{a}. {a, b}} > æ æ æ æ The Kuratowski set is an encoding of the ordered pair. (This wasn't > my idea, but I'll use it.) æYes, it is an effective encoding because it > satisfies the charateristic property (so-called). By 'effective' here we > mean that each distinct ordered pair will have a unique encoding. Another > immediate 'property' of the ordered pair is that (a, b) ~= (b, a) unless > a=b. One could base an encoding on this property, but it would not be > enough to ensure unique encodings. Effective here does not mean that it > captures any properties of the ordered pair. Indeed the meaning of the > ordered pair is irrelevant. It is effective in the same way that encoding > the letters of the English alphabet by the numbers 1 to 26 would be > effective. One can translate an English language statement into numbers > (with appropriate punctuation) and decode it back again. The only problem > that might arise is if we have an English language statement that combines > words and numbers (numerals). We could use number words, but suppose we > want to be able to use number-numerals. My point is that having shown that > the ordered pair can be encoded as a set, it isn't a case of > done-that-forget-it. Every time, in any context, a Kuratowski set appears, > we will have to know or decide there and then whether it functions as an > ordered pair or whether it functions simply as the native set. I've never needed to make such a decision, as well as I don't know >what native set means. It's hard not to draw the conclusion that you are being perverse. If I encode 'The Mark of the Beast' as 666, thereafter when I use 666 I may be referring simply to the native number, eg. the mumber obtained by multiplying 333 by 2, or I may be referring to the Mark of the Beast. > Sure, every > time an ordered pair appears we can reduce it, replace it by the Kuratowski > set, but everytime a Kuratowski set appears we have to decide which it is, > native set or encoding for ordered pair. I've never needed to make such a decision. Everytime a Kuratowski set, i.e. a set having that form, occurs, you are making that decision. > Have we really got rid ot the > notion of an ordered pair by this means? Who said anything about getting rid of any notions? > Sloppy language on my part. Presumably you think of the Kuratowski definition as some sort of reduction, a capturing of (the essential features of) the notion of an ordered pair in terms of sets. But it seems an odd sort of reduction which is sustained by the choice to interpret a Kuratowsk (-like) set as an ordered pair, or not, at every turn. Noel === Subject: Re: Kuratowski Ordered Pair >On the other hand, try to base set theory on the 01 strings of >computer programs, it might not work either, or it will be very, very >clumsy. I didn't hear about any attempt to this, not even from >constructivists. >http://hdebruijn.soo.dto.tudelft.nl/jaar2007/set_theory.pdf Fascinated. I read in Your paper: > The most important undefined notion in set theory is being a member > of a set. > If a is a member of A then we shall write, according to Halmos: > a (element sign) A > Fortunately, this notion can be mapped upon our TObject arrays in a > well > defined manner. Calling sequence defined by: > function is_in(e : TObject; X : Objects) : boolean;.... This is, what i make from it: > A computer seems to cover only finite set theory. Let A be a bunch of > memory cells, to be more precise, A is the adress of an array of > memory cells. When we move a number a into on of these cells, we cant > express this with set theory, but the state, when it is in the memory > cell, we can express in set theory: a is element of A. > As the number a can be the address of another memory cell, and as an > address is a number too, so we have > a number is the element of another number, a number A is pointing to > another number a. > a EURO A is the same as a----> A .There can be a direction in between > sets. > And when a certain a is element of A it can not be that > simultaneously, at the same time, A is element of a. > The memory cell with address number A can have stored the same number > (it's own address), but when it is stored, it is a, not A. > a =|= A., even when their value is equal. > The placing of a number a into a cell with address A is expressed by > ( 0 , a ), the first component denoting the content of cell A at a > certain time, the second at a later time, after the placement of a > into A. > ( .....,A ( t1 ),......., A ( t2),..... ) with t1 in time before t2. > Did i get it right? Yes. > Here in this computer approach, the element of relation of set theory > is an application of a function in simplest form, of Hamilton's > ordered couples (..pure time..). Now let's express the element of-relation with a Kuratowski set: > a EURO A , a element of A as { { a } , { a, A } }. This does not > express anything about the intrinsic properties of a and A which is > expressed in: a element of A. This is indeed my latest development. I'm basically in the process of re-inventing a wheel which was invented originally by Peter Aczel. For the main idea, read the first few pages of _this_ main reference: http://standish.stanford.edu/bin/detail?fileID=1817023219 , giving: http://sul-derivatives.stanford.edu/derivative?CSNID=00000056&mediaType=appl ication/pdf Han de Bruijn === Subject: Re: Kuratowski Ordered Pair > This is indeed my latest development. I'm basically in the process of > re-inventing a wheel which was invented originally by Peter Aczel. > For the main idea, read the first few pages of _this_ main reference: http://standish.stanford.edu/bin/detail?fileID=1817023219 , giving: > http://sul-derivatives.stanford.edu/derivative?CSNID=00000056&mediaType=appl i cation/pdf Peter Aczel's way of thinking gives rise to weird consequences. Such as the following. Consider the Kuratowski pair: (a,b) = {{a},{a,b}} . If the latter set is pictured as a membership graph, we find the edges: (a,{a}), ({a},{{a},{a,b}}), (a,{a,b}), (b,{a,b}), ({a,b},{{a},{a,b}}) Thus we need five Kuratowski ordered pairs in order to characterize one Kuratowski ordered pair. Plonk! Han de Bruijn === Subject: Re: -- Kuratowski Ordered Pair ( is the Kuratowski set, an asymmetrical set - here a simple result based on Hamiltons ordered couples or temporal quantities or directed quantities) posting-account=n4TzyQkAAADLWxrRHqyiUZ-1SZdOB4vv Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) An after-math The euro-sign EURO did not turn out. Written in the window, where You insert Your message with Google- groups, th group. (Here a try with notepad: EURO ) 0 times 0 = 0, so it is an involution in a space with just on element, where 0 = 1. Otherwize it's never an involution, as 0 * 0 =|= 1. Black magician is refering to cheating magic, an expression from people, who were made frightened with darkness at nights. Wessel's introduction of direction into Euclid's geometry is introducing a temporal concept into a static geometry. There was dynamic geometry already before the Elements: Hippias of Elis' Qudratrix, Archytas 'fingertip'. Sticking to a static geometry we can advance it with the concept of fixed alignment in space. An invariant of rotations in spatial space is an axis, like in Foucault's pendulum or in a gyro. Just the axis, without the orientation of the rotation, gives us a bi-direction, an alignment, fix in space, pointing to and from a fix-star ( distance / speed = very long duration). For every diameter of a sphere in Euclid's geometry there is one corresponding diameter in any other sphere, a parallel one, so all straight lines fall into equivalence classes, in a modern expression. Each equivalence class can be thought of as one alignment. Two straight lines through a common point are sharing two pairs of opposite angles. In this way an angle can be thought of as a difference of alignment. Two halflines, sharing a point as their common ends, display two angles. Without further information, the smaller one is taken as the angle. In a concav quadrangle, with the form of an arrowhead, one takes the bigger angle when talking about the inner angle of the concave corner, as here it is referenced. Three rotational axes, three points can reference any aligment in space, with an origin at Your position. Four points, four axes gives redundant information for error correction, best arranged as a tetrahedron. Every pair of opposite edges, taken together with their inner line of shortest distance gives perpendicularity. Four laser gyros, four equal angled triangles can be arranged as a tetra as well, giving the optimal platform of reference. Now back to Kuratowski. With friendly greetings Hero === Subject: -- a Dougster-like FLT-related conjecture Here is an FLT-related conjecture, adapted from Dougster's recent conjectures ... Conjecture: There do not exist coprime integers x,y,z > 1 such that the integers phi(x), phi(y), phi(z), (x + y + z - 2*max(x,y,z)) have an odd common prime factor. Remarks: If the above conjecture is true, a proof of it would yield an instant, alternative proof of FLT. However the other direction is not so clear. In other words, while the above conjecture would prove FLT, it's not clear how FLT can be used to prove the above conjecture. quasi === Subject: Re: -- a Dougster-like FLT-related conjecture Ok, so here's the revised version of Dougster's conjecture. This one should hold up. Dougster's conjecture: There do not exist positive integers x,y,z such that (1) x < y < z < x+y (2) x,y,z, are pairwise coprime (3) For some prime p > 2, x | z^p - y^p y | z^p - x^p z | x^p + y^p quasi === Subject: Re: -- a Dougster-like FLT-related conjecture >Ok, so here's the revised version of Dougster's conjecture. This one >should hold up. Dougster's conjecture: There do not exist positive integers x,y,z such that (1) x < y < z < x+y (2) x,y,z, are pairwise coprime (3) For some prime p > 2, x | z^p - y^p > y | z^p - x^p > z | x^p + y^p Whoops -- I forgot the new condition (which was the whole point of the revision). In addition to the conditions (1), (2), (3), above, we also require the following: (4) p | (x + y - z) Now it's correct (and probably not so easy to defeat with a counterexample). quasi === Subject: Re: -- a Dougster-like FLT-related conjecture >Here is an FLT-related conjecture, adapted from Dougster's recent >conjectures ... Conjecture: There do not exist coprime integers x,y,z > 1 such that the integers phi(x), phi(y), phi(z), (x + y + z - 2*max(x,y,z)) have an odd common prime factor. Remarks: If the above conjecture is true, a proof of it would yield an instant, >alternative proof of FLT. However the other direction is not so clear. >In other words, while the above conjecture would prove FLT, it's not >clear how FLT can be used to prove the above conjecture. Whoops -- l left out a few conditions. As stated above, there are immediate counterexamples, such as (x,y,z) = (7,9,13) I'll have to take this conjecture in for repair. quasi === Subject: Re: -- a Dougster-like FLT-related conjecture >Here is an FLT-related conjecture, adapted from Dougster's recent >conjectures ... >Conjecture: >There do not exist coprime integers x,y,z > 1 such that the integers > phi(x), phi(y), phi(z), (x + y + z - 2*max(x,y,z)) >have an odd common prime factor. >Remarks: >If the above conjecture is true, a proof of it would yield an instant, >alternative proof of FLT. However the other direction is not so clear. >In other words, while the above conjecture would prove FLT, it's not >clear how FLT can be used to prove the above conjecture. Whoops -- l left out a few conditions. As stated above, there are immediate counterexamples, such as (x,y,z) = (7,9,13) I'll have to take this conjecture in for repair. It's not fixable. I don't know what I was thinking when I proposed it. Instead, in a separate reply, I'll give a repaired version of Dougster's actual conjecture, incorporating his latest observation that p must divide (x + y - z). quasi === Subject: Physicists uncover new solution for cosmic collisions posting-account=X_LPLAoAAAC2yF3X3jJQ8mmNkEil0eAy http://www.theanalystmagazine.com/pr/6782628.htm It turns out that our math teachers were right: being able to solve problems without a calculator does come in handy in the real world. Two theoretical physicists at Rensselaer Polytechnic Institute have used what they call pen-and-paper math to describe the motion of interstellar shock waves -- violent events associated with the birth of stars and planets. === Subject: Student publishing project - Problem book editors required posting-account=Q-YppgoAAABMVjNhJN0GP6I7eJtbwwln CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) A press company that specializes in applied and scholar mathematics initiated the publication of a series of math problem books for medium through advanced level, with a new different structure. This editorial project is addressed to undergraduate and graduate students of mathematics institutions, as well as instructors, who can become authors/co-authors of problem books by collecting and editing the problems, according to publisher's guidelines. They also offer benefits, royalties and the option for the author to manage his own distribution locally or on the web. I have registered with them and have chosen the sub-domain Real Analysis - Sequences, Series (RAS). I am looking for other two guys to share the authorship of a problem book in this sub-domain, for a quick publication. See the details at www.infarom.com/problem_books.html . === Subject: Re: Student publishing project - Problem book editors required posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > I have registered with them and have chosen the sub-domain Real > Analysis - Sequences, Series (RAS). Given the recent publication of the following two books, I would advise picking a different topic. Wieslawa J. Kaczor and Maria T. Nowak, Problems in Mathematical Analysis I: Real Numbers, Sequences and Series, Student Mathematical Library #4, American Mathematical Society, 2000. Daniel D. Bonar and Michael J. Khoury, Real Infinite Series, Classroom Resource Materials, The Mathematical Association of America, 2006. By the way, besides the obvious choices of looking through the MAA journals (Math. Monthly, Math. Magazine, College Math. J.) and others (Math. Gazette), there are a number of journals which have ceased publication that I'd advise looking through for material. As just one example of probably a dozen I could name, there is Mathesis Recueil Mathematique (1881-1915), a journal I recently spent the past couple of months carefully going through every issue of and extracting (i.e. photocopying) everything of interest to me. Dave L. Renfro === Subject: Re: Student publishing project - Problem book editors required posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > As just one example of probably a dozen I could come up with, > there is Mathesis Recueil Mathematique (1881-1915), a journal > I recently spent the past couple of months carefully going > through every issue and extracting (i.e. photocopying) everything > of interest to me. Here's an example of a neat problem from their Questions D'Examen column, in Series 4, Volume 3, 1913, p. 54: Solve for x, if (a^2)(x^4) + ax^3 + bx - b^2 = 0. Here's one way of going about this: Begin by solving for b, using the quadratic formula. After cleaning things up a bit, you'll get b = [x +/- x(2ax + 1)] / 2, which leads to the following two equations: ax^2 + x - b = 0 ax^2 + b = 0 Now use the quadratic formula to solve for x in each of these equations. Dave L. Renfro === Subject: Re: Student publishing project - Problem book editors required > As just one example of probably a dozen I could come up with, > there is Mathesis Recueil Mathematique (1881-1915), a journal > I recently spent the past couple of months carefully going > through every issue and extracting (i.e. photocopying) everything > of interest to me. Here's an example of a neat problem from their Questions >D'Examen column, in Series 4, Volume 3, 1913, p. 54: Solve for x, if (a^2)(x^4) + ax^3 + bx - b^2 = 0. Here's one way of going about this: Begin by solving for b, using the quadratic formula. After cleaning things up a bit, you'll get b = [x +/- x(2ax + 1)] / 2, which leads to the following two equations: ax^2 + x - b = 0 ax^2 + b = 0 Now use the quadratic formula to solve for x in >each of these equations. Hmmm ... But the original polynomial factors easily by grouping: (a^2)(x^4) + ax^3 + bx - b^2 = ( (a^2)(x^4) - b^2 ) + (ax^3 + bx) = (a*x^2 + b)*(a*x^2 - b) + x*(ax^2 + b) = (a*x^2 + b)*(a*x^2 - b + x) quasi === Subject: Re: Student publishing project - Problem book editors required posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > Here's an example of a neat problem from their Questions > D'Examen column, in Series 4, Volume 3, 1913, p. 54: > Solve for x, if > (a^2)(x^4) + ax^3 + bx - b^2 = 0. > Here's one way of going about this: > Begin by solving for b, using the quadratic formula. > After cleaning things up a bit, you'll get > b = [x +/- x(2ax + 1)] / 2, > which leads to the following two equations: > ax^2 + x - b = 0 > ax^2 + b = 0 > Now use the quadratic formula to solve for x in > each of these equations. > Hmmm ... But the original polynomial factors easily by grouping: (a^2)(x^4) + ax^3 + bx - b^2 æ æ= ( (a^2)(x^4) - b^2 ) + (ax^3 + bx) æ æ= (a*x^2 + b)*(a*x^2 - b) + x*(ax^2 + b) æ æ= (a*x^2 + b)*(a*x^2 - b + x) Darn! I had not tried to work it by another way. I just assumed the method they outlined was easier than any other obvious approach would be. Still, the idea they gave is interesting. When I have time (and the desire), I think I'll see if I can come up with an example for which their method works and which is resistent to other approaches. Or, if anyone in sci.math has the time and interest to take up this challenge now, feel free to post an example if you can come up with one. Dave L. Renfro === Subject: Re: Student publishing project - Problem book editors required posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > I recently spent the past couple of months carefully going [...] Did I really write this? How about I spent the past couple of .... Dave L. Renfro === Subject: New Combination problem posting-account=-lJ1vQoAAACyGEAQJv-MyxCYxtv-f4x6 Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) hi.. anybody has tried this kind of problem. n={1,2,3,0} r=2 so permutation is n^r=16. the problem is to find out the possible no: of distinct sum like. (1,2)=3 and (1,3)=4 are distinct sum while (1,2)=3 and (0,3)=3 are not distinct sum === Subject: Re: New Combination problem posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) > hi.. anybody has tried this kind of problem. > n={1,2,3,0} > r=2 > so permutation is n^r=16. > the problem is to find out the possible no: of distinct sum like. Peermutations don't have distinct sums, i.e., (0,1) has same sum as (1,0). So why not use Combinations instead? > (1,2)=3 and (1,3)=4 are distinct sum while > (1,2)=3 and (0,3)=3 are not distinct sum Hmm...tricky. Set of numbers 0123 Combinations without replacement ['01', '02', '03', '12', '13', '23'] Distinct sums set([1, 2, 3, 4, 5]) Set of numbers 01234 Combinations without replacement ['01', '02', '03', '04', '12', '13', '14', '23', '24', '34'] Distinct sums set([1, 2, 3, 4, 5, 6, 7]) Set of numbers 012345 Combinations without replacement ['01', '02', '03', '04', '05', '12', '13', '14', '15', '23', '24', '25', '34', '35', '45'] Distinct sums set([1, 2, 3, 4, 5, 6, 7, 8, 9]) So far, so good. But the starting set is sequential numbers. What if they're not? Set of numbers 012389 Combinations without replacement ['01', '02', '03', '08', '09', '12', '13', '18', '19', '23', '28', '29', '38', '39', '89'] Distinct sums set([1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 17]) For the same size starting set, same number of combinations, yet different number of distinct sums. === Subject: easy asymptotics posting-account=nxIY5QoAAABDCgBR1GsKew18eMJbX3b6 Gecko/20071127 Firefox/2.0.0.11 eMusic DLM/4.0_1.0.0.1,gzip(gfe),gzip(gfe) Hi everybody, I am working on the following. //////////////////////////////////////////////////////////////////////////// /////////////////////////// Replace each of the following quantities by a simpler one which consists of only one of the terms and is asymptotically equivalent: (i) log n +1/2*n; (ii) log n + log (log n); (iii) n2 + en. bn is asymptotically equivalent with an (bn ~ an) when lim bn/an=1 when n goes to infty //////////////////////////////////////////////////////////////////////////// /////////////////////////// I am thinking (i) log n +1/2*n~1/2*n (ii) log n + log (log n)~log n (iii) n^2 + exp(n)~exp(n) I greatly appreciate your help. === Subject: Re: easy asymptotics > Hi everybody, I am working on the following. //////////////////////////////////////////////////////////////////////////// //// /////////////////////// > Replace each of the following quantities by a simpler one which > consists > of only one of the terms and is asymptotically equivalent: > (i) log n +1/2*n; > (ii) log n + log (log n); > (iii) n2 + en. > bn is asymptotically equivalent with an (bn ~ an) > when lim bn/an=1 when n goes to infty > //////////////////////////////////////////////////////////////////////////// //// /////////////////////// I am thinking > (i) log n +1/2*n~1/2*n > (ii) log n + log (log n)~log n > (iii) n^2 + exp(n)~exp(n) I greatly appreciate your help. Correct. -- I.N. Galidakis === Subject: Long-term behaviour of ODEs posting-account=KMurQQkAAACkDDGELZpG-7yQAg7fSfzi Gecko/20061201 Firefox/2.0.0.6 (Ubuntu-feisty),gzip(gfe),gzip(gfe) If I have an autonomous locally Lipschitz system of n coupled (non- linear) ODEs, for which all (unique) solutions are bounded to remain in some compact subset of real^n (and thus all solutions exist for all t >= t_0), am I guaranteed that as t -> infty, solutions will converge to some equilibrium solution? If so, where can I find this result? I have a feeling I read such a result (or something similar) a while back but can't remember where. If not, is there usual way to prove that all solutions to an ODE system do exhibit this behaviour? That they always do is clear numerically but I've no idea how to go about proving it (if the above result does not hold). An example might be: da/dt = -a / (a + b) + c db/dt = -b / (a + b) + a / (a + b) dc/dt = -c + b / (a + b) with initial conditions a_0, b_0, c_0 > 0. The class of ODEs in question model a fixed population, i.e. satisfy x'_1(t) + ... + x'_n(t) = 0 for all times t >= t_0 in case this helps. Similar ODEs I presume arise in biological/chemical context, but I can't find any useful results. Richard. === Subject: Re: Long-term behaviour of ODEs its strange that you ask this question, decades after chaotic solutions have been detected. Also periodic and quasiperiodic solutions are good candidates. Alois === Subject: Re: -- random points on the unit n-sphere .................. >If x_1,...,x_n are independent, identically distributed random >variables such that x/|x| is uniformly distributed on the unit >n-sphere (where x = (x_1,...,x_n)), must the variables x_1,...,x_n be >normally distributed? >Actually, I guess the answer has to be yes, otherwise the algorithm >for generating points on the n-sphere using normally distributed >random variables wouldn't work. >No, that's not correct. >After all, if we take the actual distribution for a single coordinate >on the unit sphere, it's bounded, so can't be normal. >Hmmm ... >quasi We are diving x by |x|; boundedness is no problem. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: -- random points on the unit n-sphere >What is an efficient algorithm to find random points on the unit >n-sphere? .................. >As there are avaiable fast methods to compute normal >random variables with mean 0 and variance 1 from a uniform >supply, it is unlikely that one can do better than dividing >a vector of n such normal random variables by its length. >Yes, the idea is so fundamentally simple -- essentially jjust >directional symmetry. >I wonder if a kind of converse holds. >In other words, what about the following question ... >If x_1,...,x_n are independent, identically distributed random >variables such that x/|x| is uniformly distributed on the unit >n-sphere (where x = (x_1,...,x_n)), must the variables x_1,...,x_n be >normally distributed? >Actually, I guess the answer has to be yes, otherwise the algorithm >for generating points on the n-sphere using normally distributed >random variables wouldn't work. >quasi I believe the answer must be yes, but this is not a proof. If one assumes the distribution of x must be symmetric, it is true for n >= 2; Gauss proved this assuming a sufficiently differentiable density, but this can be proved. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Solution Manual List (2008) upgraded posting-account=EMSFowoAAAAWVX65y-y79hF-VVqOrx2m SV1),gzip(gfe),gzip(gfe) If you need a Solution manual immediately . Please Email me at MANUALSELLER (at) GMAIL(dot)COM . don't give up Email Me. All the manual solutions author names are capitalize for you to find easily . ** don't post your message here ** I won't be here again . ** email me at manualseller at gmail (dot)com ** Mechanical Vibration 3th Edition by RAO !!! including project ( 95% same as 4th ed ) Fluid Mechanics 6th edition WHITE !! 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Engineering Mechanics Dynamics 11th Edition by HIBBELER !!! MicroComputer Engineering Third Edition by GENE.H.MILLER 68HC11 Semiconductor devices--physics and technology [SZE, s m ] !! Control Systems Engineering -4th edition NISE !! Introduction To Fluid Mechanics 5th edition FOX ! ! ! Adaptive Control 2nd Edition by KARL.J.ASTROM !!! Advanced Modern Engineering Mathematics 3rd Edition GLYN Antenna for all application 3rd Edition by JOHN . D . KRAUS !!! Applied Numerical Analysis 7Ed CURTIS F.GERALD PATRICK O WHEATLEY ! Communication Systems Engineering 2nd Edition by PROAKIS J !! Design of Analog CMOS Integrated Circuits solutions McGraw RAZAVI !! Digital Communications 4th Edition Solution manual by PROAKIS !! Digital Image Processing 2nd Edition by GONZALEZ !!! Digital Integrated Circuits by 2nd Edition by RABAEY ! ! ! Digital Signal Processing 2nd Edition by MITRA ! ! ! Digital Signal Processing by THOMAS J CAVICCHI ! ! 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Still if you could not find out the solution above list . please email me at **MANUALSELLER AT gmail (dot) com ** OR **check this out ** www.manualseller.com** click on solution manuals for updated solutions !! === Subject: Re: JSH: Seems correct -------------------------------------------------------------- > That means 50% of the time, you can't solve for z modulo a given prime > p... > My way always gives z modulo a given prime p that factors > nontrivially, but its always mixed in with (p-1)/2 or (p+1)/2 other > z's mod p that do not factor nontrivially, which is bad. >Yeah, you keep talking about your way, but you post as if you're >evaluating mine, when you're not. > I expect your 50% success rate to drop as p rises above 7. > Enrico >You're just looking for excuses to talk about your own stuff, not >evaluating mine at all, and then just guessing, like your high level >argument against my entire approach, claiming it would give (p-1)/2 >solutions--except it doesn't. simply, the JSH method fails as the summation SUM (1/2 + 1/3 + 1/5 +1/7 +1/11 + 1/13 + ...1/prime....) It fails to work at all above about 30 or 40. That is why plain old random guessing is orders of magnatude faster than the JSH method. In fact the JSH method makes RSA even more secure by orders of magnatude. === == Subject: asymptotics posting-account=nxIY5QoAAABDCgBR1GsKew18eMJbX3b6 Gecko/20071127 Firefox/2.0.0.11 eMusic DLM/4.0_1.0.0.1,gzip(gfe),gzip(gfe) Hello everybody. I am working on the following: ///////////////////////////////////////////////////////////// If bn Á.9c 0, and an and bn are both of the same order as cn, what can you say about the relation of an and bn? Note: bn is of the same order as cn means |bn/cn| is bounded away from both 0 and Áfi, i.e., there exist constants 0 Hello everybody. I am working on the following: > ///////////////////////////////////////////////////////////// > If bn -> 0, and an and bn are both of the same order as cn, what can > you say about the relation of an and bn? Note: > bn is of the same order as cn means |bn/cn| is bounded away from both > 0 and oo, i.e., there exist constants 0 that m < |bn/cn| n0. > ///////////////////////////////////////////////////////////// I am thinking > |bn/cn| is bounded away from both 0 and oo > |an/cn| is bounded away from both 0 and oo so that > |bn/an| is bounded away from both 0 and oo for some m*, M* and n0* > meaning that bn is of the same order as an. I think that's true, since |bn/an| = |bn/cn| * |cn/an| and you could argue that |cn/an| is bounded away from 0 and oo. now, since bn->0 , I think that an also has to go to 0 asymptotically. Yes. You could make a sandwich argument. Am I right? I appreciate your help and/or pointers. I think you're on the right track, you just have to make these arguments rigorous. - Randy === Subject: Re: asymptotics > Hello everybody. I am working on the following: > ///////////////////////////////////////////////////////////// > If bn '.9c 0, and an and bn are both of the same order as cn, what can > you say about the relation of an and bn? Note: > bn is of the same order as cn means |bn/cn| is bounded away from both > 0 and 'fi, i.e., there exist constants 0 that m < |bn/cn| n0. > ///////////////////////////////////////////////////////////// I am thinking > |bn/cn| is bounded away from both 0 and 'fi > |an/cn| is bounded away from both 0 and 'fi so that > |bn/an| is bounded away from both 0 and 'fi for some m*, M* and n0* > meaning that bn is of the same order as an. now, since bn'.9c 0, I think that an also has to go to 0 asymptotically. Am I right? I appreciate your help and/or pointers. You should post using only ASCII ... to me there are lots of strange characters in your message. === Subject: Re: asymptotics > Hello everybody. I am working on the following: > ///////////////////////////////////////////////////////////// > If bn '.9c 0, and an and bn are both of the same order as cn, what can > you say about the relation of an and bn? Note: > bn is of the same order as cn means |bn/cn| is bounded away from both > 0 and 'fi, i.e., there exist constants 0 that m < |bn/cn| n0. > ///////////////////////////////////////////////////////////// I am thinking > |bn/cn| is bounded away from both 0 and 'fi > |an/cn| is bounded away from both 0 and 'fi so that > |bn/an| is bounded away from both 0 and 'fi for some m*, M* and n0* > meaning that bn is of the same order as an. now, since bn'.9c 0, I think that an also has to go to 0 asymptotically. Am I right? I appreciate your help and/or pointers. Try rewriting this using only standard text; sci.math is an ASCII newsgroup. W^3 -------- sci.math, write -- at the beginning of your subject line. Set up a === Subject: Solutions Manual posting-account=zZKe-AoAAABe4u9R_oyyNrC-dHe3wHpx CLR 1.0.3705),gzip(gfe),gzip(gfe) I would like to purchase the following solutions manual. Foundations of Geometry - Gerard Venema (5th edition) (ISBN: 0131437003) === Subject: Solution Manuals - Accounting, Finance, Taxation posting-account=Oef_qQoAAAC_nhbXm5pHPNx-3IJ2uJp1 1.1.4322; .NET CLR 2.0.50727; InfoPath.1; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) I have the COMPREHENSIVE SOLUTION MANUALS in PDF and WORD format for the following textbooks: 1. Intermediate Accounting 12th edition by Kieso, Weygandt, Warfield; ISBN-10: 0471749559; ISBN-13: 978-0471749554 2. Advanced Accounting 9th edition by Fischer, Taylor, Cheng; ISBN-10: 0324304013; ISBN-13: 978-0324304015 For this one the TEST BANK is available too. 3. Contemporary Financial Management 10th edition by Moyer McGuigan, Kretlow; ISBN: 0-324-28908-1 4. West Federal Taxation 2007, 30th edition by Willis, Hoffman, Maloney, Raabe ISBN-10: 0324313489 ISBN-13: 978-0324313482 If you are interested send me an email to prodigy...@gmail.com to order. alt.law-enforcement === Subject: Re: Rotating 1,1,1,...,1 to 0,0,...,0,1 <1edf4ab4-ab1e-49bb-8630-200d03c32d5b@p69g2000hsa.googl egroups.com>, > I'm looking for a rotation matrix R that rotates vector 1,1,1,..,1 to > align with vector 0,0,...,0,1 in n dimensions. Since there may be many > such matrices, I'm looking for a one with the nicest symbolic > expression You cannot have such a rotation matrix because the pre-image and the image have different lengths. Look up Householder's transformation. Suppose v = (v_1, v_2, ..., v_n). Set e_1 = (1, 0, 0, ..., 0) Define H_u = I - u u^T / r_u u = e_1 + v/|v| r_u = 1 + v_1/|v| H_u is an orthogonal transformation. H_u(v) = (|v|, 0, 0, ..., 0). In the case v = (1, 1, ..., 1) we define a = 1/sqrt(n), r_u = 1 + a, and [ -1+aa/r_u -1+aa/r_u ... -1+aa/r_u a ] R = [ -1+aa/r_u -1+aa/r_u ... -1+aa/r_u a ] [ ... ] [ a a ... a a ] then R(v) = (0, 0, ..., sqrt(n)). -- Michael Press === Subject: Re: Rotating 1,1,1,...,1 to 0,0,...,0,1 , > <1edf4ab4-ab1e-49bb-8630-200d03c32d5b@p69g2000hsa.googl > egroups.com>, I'm looking for a rotation matrix R that rotates vector 1,1,1,..,1 to > align with vector 0,0,...,0,1 in n dimensions. Since there may be many > such matrices, I'm looking for a one with the nicest symbolic > expression You cannot have such a rotation matrix because > the pre-image and the image have different lengths. Look up Householder's transformation. Suppose v = (v_1, v_2, ..., v_n). > Set e_1 = (1, 0, 0, ..., 0) > Define > H_u = I - u u^T / r_u > u = e_1 + v/|v| > r_u = 1 + v_1/|v| H_u is an orthogonal transformation. > H_u(v) = (|v|, 0, 0, ..., 0). In the case v = (1, 1, ..., 1) > we define a = 1/sqrt(n), r_u = 1 + a, and [ -1+aa/r_u -1+aa/r_u ... -1+aa/r_u a ] > R = [ -1+aa/r_u -1+aa/r_u ... -1+aa/r_u a ] > [ ... ] > [ a a ... a a ] Misprint here. Should be [ -1+aa/r_u aa/r_u ... aa/r_u a ] R = [ aa/r_u -1+aa/r_u ... aa/r_u a ] [ ... ] [ a a ... a a ] then R(v) = (0, 0, ..., sqrt(n)). -- Michael Press === Subject: more asymptotics posting-account=nxIY5QoAAABDCgBR1GsKew18eMJbX3b6 Gecko/20071127 Firefox/2.0.0.11 eMusic DLM/4.0_1.0.0.1,gzip(gfe),gzip(gfe) Hello everybody. Apologies for reposting this. I am posting the ASCII version this time. I am working on the following: ///////////////////////////////////////////////////////////// If bn goes to 0 as n goes to infty, and an and bn are both of the same order as cn, what can you say about the relation of an and bn? Note: bn is of the same order as cn means |bn/cn| is bounded away from both 0 and infty, i.e., there exist constants 0 n0. ///////////////////////////////////////////////////////////// I am thinking |bn/cn| is bounded away from both 0 and infty |an/cn| is bounded away from both 0 and infty so that |bn/an| is bounded away from both 0 and infty for some m*, M* and n0* meaning that bn is of the same order as an. now, since bn goes to 0 as n goes to infty, I think that an also has to go to 0 asymptotically. Am I right? I appreciate your help and/or pointers. === Subject: Re: Leo Wapner's problem, trigonometric products > Almost a year ago, Leo Wapner presented the product: > a(n) = 2sin(1) 2sin(2) 2sin(3) ... 2sin(n). > The natural questions to ask are whether |a(n)| is bounded or not > as n --> oo, whether it converges, diverges to +oo, and so on. > Cf.: > In the message > Robert Israel showed that > (a) product_{j=1}^{n-1} sin(j pi/n) = n/2^(n-1) [ for any n>1] > and > (b) product_{j=1}^{n-1} cos(j pi/n) = (-1)^((n-1)/2)/2^(n-1) > [ n >1 and odd] > ---- > ``Hyperbolic geometry: The first 150 years, > This paper is available at no charge from Project Euclid: > of polyhedra in hyperbolic 3-space, for example from the measures > of some of their dihedral angles. > It's convenient to use a Cyrillic 'letter l' to denote a function > the angle theta: > ell(theta) := - int_{0 ... theta} log(2|sin u|) du , for any real u > using improper integrals for points where sin u is zero. > Lemma 1 states that ell is and odd periodic function of period pi > and satisfies the identity: > ell(n*theta) = sum_{ complete residue set of k's mod n} > ell(theta + k*pi/n) , for any n. > By the way, Lemma 2 gives a way of computing the volume of a > hyperbolic 3-simplex with the four vertices at infinity from > the values of the the dihedral angles for three edges meeting at > some vertex; it involves the ell-function at the three angles. > Milnor starts his proof of lemma 1 with the identity: > 2 sin (n*theta) = product_{k=0 ... n-1} 2sin(theta + k*pi/n). > He starts by writing the polynomial z^n-1 in C[X] as a product of > n linear factors, and concludes using analytic continuation; then, > he goes on with the proof of lemma 1. > ---- > Let n>1 be odd, and let theta = pi/2. Then, 2sin(n*theta) > = 2 sin(n*pi/2) = 2* (-1)^((n-1)/2). > Also, product_{k=0 ... n-1} 2sin(theta + k*pi/n) > = product_{k=0 ... n-1} 2 sin( pi/2 + k*pi/n) > = product_{k=0 ... n-1} 2 cos(k*pi/n). > So in this case, Milnor's identity in his Lemma 1 proof implies: > product_{k=0 ... n-1} 2 cos(k*pi/n) = 2* (-1)^((n-1)/2), or > product_{k=0 ... n-1} cos(k*pi/n) = (-1)^((n-1)/2)/(2^(n-1)), or > product_{k=1 ... n-1} cos(k*pi/n) = (-1)^((n-1)/2)/(2^(n-1)). > This is the same as Robert Israel's identity (b). > ---- > [...] There was a paper by Erdos and Szekeres in 1959 entitled > ``On the Product Pi_{k=1 ... n} (1- z^(a_k) ) , which is available > on the Web from: < http://www.renyi.hu/~p_erdos/1959-17.pdf > . It's a given that a_1 <= a_2 <= ... a_n are positive integers. They define M(a_1, ... a_n) = max_{|z| = 1} | product_{k=1 ... n} (1- z^(a_k) ) | . To consider already a non-trivial case, > suppose a_k = k. Erdos and Szekeres define > f(n):= min_{allowable a_k} M(a_1, ... a_n). They write: ``The determination of f(n) seems to be a > very difficult question [...]. They prove: > (1) lim_{n -->oo} ( f(n)^(1/n) ) = 1 > and > (2) f(n) >= sqrt(2n). Before proving these, they ask various questions and give results that > hold almost everywhere. They use the notation < k*alpha>, for integer k, and real alpha, > as defined by: > < k*alpha > := | 1 - exp(2pi*i*alpha) |. ^^^ | 1 - exp(2pi*i*k*alpha) | > According to a routine calculation I did, > |1 - exp(2pi*i*alpha)| = 2|sin(pi*alpha)|. One of these is the result (3), which says that: limsup_{n -> oo} product_{k=1 ... n} < k*alpha > = oo > holds for almost all real numbers alpha. > limsup_{n -> oo} product_{k=1, ... n} 2|sin(pi*alpha)| = oo rather: (3') limsup_{n -> oo} product_{k=1, ... n} 2|sin(pi*k*alpha)| = oo (holds for almost all real numbers alpha.) > holds for almost all real numbers alpha. Also, their result (1) can be rewritten as: (1') > liminf_{n->oo} product_{k=1, ... n} 2|sin(pi*k*alpha)| = 0, [ k inserted above ...] > for almost all real numbers alpha. For (1) and (3) to hold for a given alpha, it is sufficient that > there exists infinite sequences p_n and q_n such that: > |alpha - (p_n)/(q_n) | = o(1/( (q_n)^2 * log(q_n) ) ). (***) [note: presumably the 'o' has to do with letting n go to infinity] The proofs of (1) and (3) are only outlined (see: > by a simple computation). I don't know how Erdos and Szekeres proved (1) and (3); I've looked at the rest of their paper and related papers such as Halberstam and Freiman (``On a product of sines, Acta Arith. 49 (1987), 377-385). If we re-copy the known identity (a), we have: product_{j=1}^{n-1} sin(j pi/n) = n/(2^(n-1)). One idea is to choose n=355 [to approach (3') above ], and other numerators from the good rational approximations to pi. What can be done is to divide 2sin(1)... * 2sin(354) by 2sin(113*pi/355) ... 2sin(113*pi/355 *354): (11:55) gp > prod(X=1,354,sin(X)/sin(X*113*Pi/355)) %56 = 0.994792655184620570391980159185 (11:57) gp > prod(X=1,354,2*sin(X*113*Pi/355)) %57 = 355.000000000000000000000000000 (11:58) gp > %56*355 %58 = 353.151392590540302489152956511 (11:58) gp > prod(X=1,354,2*sin(X)) %59 = 353.151392590540302489152956511 355/113 is an unusually good approximation to pi. 333/106 is more typical. (13:03) gp > sin(311) %105 = 0.0176717854673708710252783801529 // 333 - 22 = 311 // ~= (106 - 7)*pi ~= 99*pi (13:07) gp > sin(311*106*Pi/333) %106 = 0.00943407222589675496378893516980 So sin(311) ~= 1.873 * sin(311*106*Pi/333) Similarly, but where instead sin(k)/sin(k*106*Pi/333) <1 , (13:33) gp > sin(22)/sin(22*106*Pi/333) %129 = 0.938227848850554612016287290497 (13:34) gp > sin(15*22)/sin(15*22*106*Pi/333) %130 = 0.938601666608783727877443039562 (13:34) gp > 106.0/113.0 %131 = 0.938053097345132743362831858407 (13:34) gp > 107.0/114.0 %132 = 0.938596491228070175438596491228 Finally, (13:38) gp > prod(X=1,332,sin(X)/sin(X*106*Pi/333)) %134 = 2.59932035523657157448716381009 and also for products of quotients up to 22*6, 22*7, ... 22*11 terms: (13:59) gp > prod(X=1,132,sin(X)/sin(X*106*Pi/333)) %152 = 0.711525309458178970157742414511 (14:00) gp > prod(X=1,154,sin(X)/sin(X*106*Pi/333)) %153 = 0.691907842742196740499042657140 (14:00) gp > prod(X=1,176,sin(X)/sin(X*106*Pi/333)) %154 = 0.683295568770184822358901156406 // minimum (14:00) gp > prod(X=1,198,sin(X)/sin(X*106*Pi/333)) %155 = 0.688218275999137931379939611480 (14:00) gp > prod(X=1,220,sin(X)/sin(X*106*Pi/333)) %156 = 0.711109166102398539427824541101 (14:01) gp > prod(X=1,242,sin(X)/sin(X*106*Pi/333)) %157 = 0.760241928504197705114723299379 Maybe we can use: 22-7*Pi ~= 0.008851424871447 David Bernier > The result for almost all alpha, sequences p_n , q_n exist so that > (***) holds, is a special case of a well-known result of > Khintchine. (essentially recopying Erdos & Szekeres verbatim). It's a pity they don't give references. ---- In > < http://crd.lbl.gov/~dhbailey/dhbpapers/khinchine.pdf > D.H. Bailey, J.M. Borwein and R.E. Crandall write: ``It is remarkable that, even though a random fractionÍs > limiting geometric mean exists and furthermore equals > the Khintchine constant with probability one, not a > single explicit real number (e.g., a real number cast > in terms of fundamental constants) has been demonstrated > to have elements whose geometric mean equals K_0. With respect to so-called Khintchine means for continued > fraction expansions, D.H Bailey et al. give the geometric > mean and the harmonic mean for the first 17,001,303 continued > fraction elements of pi. The GM is: 2.686393 > and the harmonic mean is: 1.745882. The constant K_0 is known to a relatively high precision (hundreds, > even thousands of decimal places). According to Plouffe, Xavier Gourdon has computed 110,000 digits > of K_0; > cf: > < http://pi.lacim.uqam.ca/piDATA/khintchine.txt > K_0 = 2.68545200106530644530 ... > and the Khintchine geometric mean for ~ 17,000,000 c.f. elements of pi: > 2.686393 ---- Erdos and Szekeres don't state which special case of > Khintchine's result implies (***) for almost all alpha. D.H. Bailey et al. give Khintchine's 1964 book > Continued fractions as their reference [13], so > maybe that reference would mention the special case needed > for (***). --- It's possible that some people, known or unknown to me, > any omission, I apologize. David Bernier === Subject: integration problem curvature and cartesian coordinates posting-account=HdOodgoAAADCIgJBgUGTlkxHMm6OhIQB 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) I wondered if anyone could help me with this problem. If I have a curve which begins at (x,y)=(0,0) and I move a distance s along the curve, what are the x and y coordinates at this point? I know the curvature at all points on my curve (as it is part of a circle) I also know that (ds/dx)=(1+(dy/dx)^2)^(1/2) (ds/da)=R (R is radius of curvature, a is angle of tangent) tan(a)=(dy/dx) dx) also I think that means y=integral(tan(a)dx, but I've no idea how to solve this. Can anyone help? === Subject: Primary ideal and its associated Looking for a hint in commutative algebra, Suppose a given ideal I in a commutative ring ring A with 1, I need to show if there is only one prime ideal of the form r((I:x)), where x is in A, then I is primary. Little hint will be appreciated. Jose === Subject: #525 second proof that Peano Axioms are self contradictory and that the notion of set of all finite integers is nonsense <478A6CF4.2080607@hotmail.com> posting-account=fsC03QkAAAAwkSNcSEKmlcR-W_HNitEd Gecko/20021120 Netscape/7.01,gzip(gfe),gzip(gfe) Alright I gave a proof outline of using the concept even versus odd numbers to intertwine with the concepts of finite versus infinite integers in order to prove that the concept of finite integer is foggy and ill-defined. With AP-adics what is destroyed of the old mathematics as decrepit are these concepts: (1) transcendental is destroyed (2) rational and irrational are destroyed because of numbers such as 0888888....77777 (3) finite is destroyed because of numbers such as 08888....77777 What is saved? Surprisingly, concepts such as these are saved and invigorated: (1) primeness (2) odd versus even (3) countability for it is even revived in Reals where it existed all along What is new that AP-adics ushers in? (1) that numbers have a native and instrinsic geometry (2) that Positive AP-adics is Elliptic geometry (3) that Negative AP-adics is Hyperbolic geometry (4) that the union of Elliptic + Hyperbolic is Euclidean geometry (5) that we have a North Pole and South Pole point as pi and (2pi) as imaginary AP-adics (6) that we have a hemisphere of imaginary numbers in AP-adics (7) that we have multiplication describing angular momentum in AP- adics That is only a partial list of what is new in mathematics by the AP- adics. But let me offer a second proof that the concept of *finite integer* in mathematics is old, stale, decrepit and destroyed. I should say that the concept of finite integers is dead, for the moment that a scientist declares death to a old concept of science, even though the entire rest of the mathematics community is not yet on board, makes little to no difference. We say Quantum Mechanics was discovered in the very early 1900s even though only a few persons believed in QM and that it took 50 years later for the physics community to come on board. So here is a second proof that the concept of finite integer or the entire concept of Natural Numbers as finite integers is all dead, done and buried. PROOF, NATURAL NUMBERS as finite entities is a dead concept: I am going to use the exact same Euclidean method of proving the Primes are infinite. If old mathematics of their finite-integer concept is correct then the part of the proof where we multiply the lot and add 1 and call it W+1 is going to be examined. Examined not for whether it is prime or not prime but examined as to whether it is finite-integer or infinite-integer. And the crux of the proof is that if it is a finite- integer as the old math community would applaud and cheer for is going to have contradictory consequences for the entire rest of mathematics. And the reason for this is because if W+1 is a finite integer means there are an infinity of primes beyond W+1 for which they are unaccounted for. Primes such as these 08888.....77777, or 13333.....999991 or .....1413121110987654321 or ........99997999799797. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Can Events of Zero Probability Happen? posting-account=vI5-YAoAAACpb1I_2s__b0LrjNDZjNTS Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > Recently, I had a debate with a physicist on the physical meaning of > probability theory. My view is that mathematical models of toy > universes do exist that are logically consistent where measure zero > events can happen. His expressed position was that Zero probability means that an event > cannot happen. He also stated that If you change that, you are no > longer talking about probability but about something else, and > whatever you say becomes incomprehensible. I responded by saying: In finite probability spaces, you are correct. In general, all that > is needed for a 'Probability space' is a measure on a sigma-algebra of > events that assigns to each event a number between 0 and 1. If we are talking physics (QM) then it is not correct because there are negative probability (<0), zero probability (0), and positive probability (>0) === Subject: Re: Can Events of Zero Probability Happen? <961fo35uieus57unl9fg2n80663l2m0tlo@4ax.com> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) >Ask him this: Suppose that X is a real number between 0 and 1, >chosen at random. What is the probability that X = 1/2? >This seems a little like cheating, because when you instead ask >suppose X is an integer chosen at random, what is the probability >that it's 2?, the mathematician is just as stuck as the physicist. Puhleeze. In the first case the words with a uniform distribution >are understood (or added explicitly if it turns out they're not >understood). In the second case the mathematician is not stuck, >he simply points out that the distribution needs to be specified >since there is no natural default. I was assuming a uniform distribution in both cases. æIf you can > choose the distribution there's no problem in either case. -- Richard It is quite easy to show that for a countably infinite set of objects, > such as the integers, a uniform distributions cannot exist. At least with all probabilities belonging to the set of the standard > reals.- Hide quoted text - - Show quoted text - I agree completely with the above. But please explain - I will choose a number at random from the unit interval. My selection will be 0 < X < 1 with probability 1. Consider 2 subintervals, and my selection process gurantees that I will choose from one or the other subinterval with probability 1/2. Divide again, and I will select a number from one of 4 subintervals each with probability 1/4. Continue this process indefinitely. I have countably infinitely many subintervals, each with equal probability of containing my random number. Even though the number of subintervals increases exponentially it is still a coutable set. Am I wrong here ? === Subject: Re: Can Events of Zero Probability Happen? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >I will choose a number at random from the unit interval. >My selection will be 0 < X < 1 with probability 1. >Consider 2 subintervals, and my selection process gurantees that I >will choose from one or the other subinterval with probability 1/2. >Divide again, and I will select a number from one of 4 subintervals >each with probability 1/4. >Continue this process indefinitely. >I have countably infinitely many subintervals, each with equal >probability of containing my random number. Even though the number of subintervals increases exponentially it is >still a coutable set. There are contably many subintervals obtained by successively dividing [0,1] in half, since each of the corresponds to a finite sequence of choices. But to identify your real uniqely, you need an infinite sequence of such choices. -- Richard -- :wq === Subject: Re: Can Events of Zero Probability Happen? <961fo35uieus57unl9fg2n80663l2m0tlo@4ax.com> posting-account=-eQqtQoAAACZVM-kNEsOn3k7GSvoJoS4 MathPlayer 2.0; .NET CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >Ask him this: Suppose that X is a real number between 0 and 1, >chosen at random. What is the probability that X = 1/2? >This seems a little like cheating, because when you instead ask >suppose X is an integer chosen at random, what is the probability >that it's 2?, the mathematician is just as stuck as the physicist. Puhleeze. In the first case the words with a uniform distribution >are understood (or added explicitly if it turns out they're not >understood). In the second case the mathematician is not stuck, >he simply points out that the distribution needs to be specified >since there is no natural default. I was assuming a uniform distribution in both cases. æIf you can >choose the distribution there's no problem in either case. -- Richard It is quite easy to show that for a countably infinite set of objects, > such as the integers, a uniform distributions cannot exist. At least with all probabilities belonging to the set of the standard > reals.- Hide quoted text - - Show quoted text - I agree completely with the above. But please explain - I will choose a number at random from the unit interval. > My selection will be 0 < X < 1 with probability 1. > Consider 2 subintervals, and my selection process gurantees that I > will choose from one or the other subinterval with probability 1/2. > Divide again, and I will select a number from one of 4 subintervals > each with probability 1/4. > Continue this process indefinitely. > I have countably infinitely many subintervals, each with equal > probability of containing my random number. Even though the number of subintervals increases exponentially it is > still a coutable set. Am I wrong here ? You are wrong. [0, 1] is an uncountable set, and yet it can be represented by (countable) sequences of zeroes and ones. All this proves is that 2^oo = c, the cardinality of the continuum. (And to be perfectly clear about how the sequences correspond to your intervals: The first digit i1 of .i1i2i3i4... is 0 if you choose the left interval and is 1 if you choose the right interval, etc.) === Subject: Re: Can Events of Zero Probability Happen? <961fo35uieus57unl9fg2n80663l2m0tlo@4ax.com> posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) >Ask him this: Suppose that X is a real number between 0 and 1, >chosen at random. What is the probability that X = 1/2? >This seems a little like cheating, because when you instead ask >suppose X is an integer chosen at random, what is the probability >that it's 2?, the mathematician is just as stuck as the physicist. >Puhleeze. In the first case the words with a uniform distribution >are understood (or added explicitly if it turns out they're not >understood). In the second case the mathematician is not stuck, >he simply points out that the distribution needs to be specified >since there is no natural default. I was assuming a uniform distribution in both cases. æIf you can >choose the distribution there's no problem in either case. -- Richard It is quite easy to show that for a countably infinite set of objects, >such as the integers, a uniform distributions cannot exist. At least with all probabilities belonging to the set of the standard >reals.- Hide quoted text - - Show quoted text - I agree completely with the above. But please explain - I will choose a number at random from the unit interval. > My selection will be 0 < X < 1 with probability 1. > Consider 2 subintervals, and my selection process gurantees that I > will choose from one or the other subinterval with probability 1/2. > Divide again, and I will select a number from one of 4 subintervals > each with probability 1/4. > Continue this process indefinitely. > I have countably infinitely many subintervals, each with equal > probability of containing my random number. Even though the number of subintervals increases exponentially it is > still a coutable set. Am I wrong here ? You are wrong. [0, 1] is an uncountable set, and yet it can be > represented by (countable) sequences of zeroes and ones. æAll this > proves is that 2^oo = c, the cardinality of the continuum. (And to be > perfectly clear about how the sequences correspond to your intervals: > The first digit i1 of .i1i2i3i4... is 0 if you choose the left > interval and is 1 if you choose the right interval, etc.)- Hide quoted text - - Show quoted text - === Subject: Re: Can Events of Zero Probability Happen? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) _If_ you're interested in resolving the difference between > you and the physicist you'd ask what I suggested you > ask and see what he says. Because it's possible that > he'd have no problem with choose a real between 0 and > 1 at random, and in that case you could explain why > he's simply _wrong_ about the impossibility of events > of probability zero happening. Sure, on the face of it, it seems possible to reason with a physicist that believes that conceptualizing events that occur with zero probability is unfathomable. The problem is, he explicitly said that even an event of incredibly small probability can't happen. Shubee === Subject: Re: Can Events of Zero Probability Happen? > _If_ you're interested in resolving the difference between > you and the physicist you'd ask what I suggested you > ask and see what he says. Because it's possible that > he'd have no problem with choose a real between 0 and > 1 at random, and in that case you could explain why > he's simply _wrong_ about the impossibility of events > of probability zero happening. Sure, on the face of it, it seems possible to reason with a physicist >that believes that conceptualizing events that occur with zero >probability is unfathomable. The problem is, he explicitly said that >even an event of incredibly small probability can't happen. First, if he said that why didn't you say so? There's a big difference between that and saying that events of zero probability can't happen. Second, again you should simply ask him a question. First ask him for an epsilon > 0 such that an event of probability < epsilon can't happen. Second, calculate an N such that 2^(-N) < epsilon. Third, ask him to flip a coin N times and tell you what sequences of heads and tails resulted. Then point out that the probability of that sequence of heads and tails is < epsilon. Third, no he _didn't_ say that! He said Probabilities this low are generally taken to mean the event could not have happened. That's _true_. Yes it is. Suppose I tell you that I was watching a glass of water the other day, and with no outside energy applied it just happened that half of it froze solid while the other half boiled away. Would you believe me? Shubee ************************ David C. Ullrich === Subject: Re: Can Events of Zero Probability Happen? posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) Sure, on the face of it, it seems possible to reason with a physicist >that believes that conceptualizing events that occur with zero >probability is unfathomable. The problem is, he explicitly said that >even an event of incredibly small probability can't happen. First, if he said that why didn't you say so? If events of zero probability can happen, then events of fantastically small probability can happen. Didn't Oh No assert that Zero probability means that an event cannot happen? > There's a big difference between that and saying > that events of zero probability can't happen. That is correct. If I titled this thread, Can Events of Fantastically Small Probability Happen? I probably wouldn't get a serious mathematical answer and I might even be told that asking such a fantastically stupid question doesn't belong at sci.math. > Second, again you should simply ask him a question. First ask him for > an epsilon > 0 such that an event of probability < epsilon can't > happen. Second, calculate an N such that 2^(-N) < epsilon. > Third, ask him to flip a coin N times and tell you what sequences > of heads and tails resulted. Then point out that the probability > of that sequence of heads and tails is < epsilon. David, going that route presupposes that the physicist respects precise and elegantly stated mathematical reasoning. The way I look at it, I have already arrived at an apparently insolvable impasse. Didn't Oh No make it clear that his philosophical perspective supercedes all established mathematical understanding? > Third, no he _didn't_ say that! He said Probabilities this low > are generally taken to mean the event could not have happened. > That's _true_. More precisely, those are weasel words in the context of the discussion. Definition: A weasel word is used to avoid making a straightforward statement. Weasel words are also used to deceive, distract, or manipulate an audience. Weasel wording conceals the full picture. In this way, one may evade responsibility for what may be inferred. http://en.wikipedia.org/wiki/Weasel_words Please note the meaning of the physicist's whole paragraph in response to my question: >Quantum mechanically, is there a nonzero probability for the Red Sea >to split (Exodus 14:21) and for a man to be fully formed out of the >inanimate material of the earth in a single day? (Genesis 2:7). Although, as in qm, when events are governed by probability, it may be technically possible to find a non-zero probability for extremely unlikely events, there must be some doubt about the meaning of the mathematics. Probabilities this low are generally taken to mean the event could not have happened. I interpret that answer as Yes, but. The key line is there must be some doubt about the meaning of the mathematics. Do you really believe that an expert physicist can rationally justify having doubt about the meaning of the mathematics? Oh No's answer Yes, but, when you mod out all the weasel words in the whole paragraph, clearly affirms my claim that Oh No explicitly said that even an event of incredibly small probability can't happen. > Yes it is. Suppose I tell you that I was watching a glass > of water the other day, and with no outside energy applied > it just happened that half of it froze solid while the other > half boiled away. Would you believe me? > David, thank you for bringing up this very familiar illustration in quantum physics. You have proven my point. The accepted and widely acknowledged answer by the experts in quantum physics is that the event that you described can happen, although with fantastically small, non-zero probability. Now, please consider the meaning of this amusing curiosity. When mainstream physicists interpret quantum physics and assert that miraculous events can happen in a glass of water, the meaning of fantastically small probability is not disputed. When I ask about the quantum mechanical chances for the Red Sea to part (Exodus 14:21) and for a man to be fully formed out of the inanimate material of the earth in a single day (Genesis 2:7), then suddenly those events call into question the meaning of fantastically small probabilities. Shubee http://www.everythingimportant.org/creationism === Subject: Re: Can Events of Zero Probability Happen? P = .... > Whether or not something can really happen is not > a > mathematical question. agreed. but what should we call such a question then ? Ask him this: Suppose that X is a real number between > 0 and 1, > chosen at random. What is the probability that X = > 1/2? good example. if he answers : 0 , it cant happen. well than he's an idiot :) of course he might never bump into such questions perhaps. on the other hand physics prof are seldom ashamed to steal math even if they dont fully understand the concept. Where to go from there depends on how he replies. It > _could_ > be, for example, that the way he's using the term, > choose a real number between 0 and 1 at random is > simply > not something that can happen... considering that quantum mechanics is considered as a discrete consideration ... in discrete space probability 0 is really 0. *** as for P [ pick [0,1] = 1/2 ] the mathematical discussion somewhat continues though... ( your example in the beginning rewritten ) some would say P = 0 , others would say P = h , with h an infinitesimal small number. some others might argue h with h^2 = 0 (nilpotent h ) more rare ideas , but certainly sensible would also be P = t1 [ in venkat's post and mine , if you recall , t1 is the smallest possible number > 0 ( not h ! ) ] and i can continue : P = 1 / aleph_1 for example. whereas finitists might even say : P does not exist. i dont know if there is a real answer , maybe its just about what definitions you chose and what axioms you accept... Recently, I had a debate with a physicist on the > physical meaning of >probability theory. My view is that mathematical > models of toy >universes do exist that are logically consistent > where measure zero >events can happen. His expressed position was that Zero probability > means that an event >cannot happen. He also stated that If you change > that, you are no >longer talking about probability but about something > else, and >whatever you say becomes incomprehensible. I responded by saying: In finite probability spaces, you are correct. In > general, all that >is needed for a 'Probability space' is a measure on > a sigma-algebra of >events that assigns to each event a number between 0 > and 1. http://en.wikipedia.org/wiki/Measure_%28mathematics%2 > 9 >http://en.wikipedia.org/wiki/Probability_space I believe that my learned opponent is wrong for > three reasons, two of >which are purely mathematical. First, it seems that > he is unable to >detach himself from his own emotional feelings about > the universe to >even consider the meaning of zero probability from a > measure-theoretic >point of view. Second, he thinks that the occurrence > of a measure-zero >probability event is incomprehensible. > s/browse_frm/thread/c4de9ae9a364fc79 Who is right? Shubee Shubee , you are right ;-) > ************************ David C. Ullrich tommy1729 === Subject: Re: Can Events of Zero Probability Happen? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >Ask him this: Suppose that X is a real number between 0 and 1, >chosen at random. What is the probability that X = 1/2? This seems a little like cheating, because when you instead ask >suppose X is an integer chosen at random, what is the probability >that it's 2?, the mathematician is just as stuck as the physicist. Puhleeze. In the first case the words with a uniform distribution >are understood (or added explicitly if it turns out they're not >understood). In the second case the mathematician is not stuck, >he simply points out that the distribution needs to be specified >since there is no natural default. I was assuming a uniform distribution in both cases. If you can > choose the distribution there's no problem in either case. >It is quite easy to show that for a countably infinite set of objects, >such as the integers, a uniform distributions cannot exist. Right. The original post was about a physicist not being happy with zero-probability events that can happen. David Ullrich gave the example of a uniform distribution on the unit interval as a way to show the physicist that it must be possible. But a (uniformly distributed) random integer is not intuitively more impossible that a random real, so a theory that works for one but not the other is not much use for addressing the physicist's intuition. I'll try again. Suppose you said to the physicist: choose a random integer, what's the probability that it's 2. That would be just as convincing as the [0,1] case, but it would be wrong. That's why I said it was cheating: you're using a form of argument that only works because of the particular example you chose. -- Richard -- :wq === Subject: Re: Can Events of Zero Probability Happen? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) > Ask him this: Suppose that X is a real number between 0 and 1, > chosen at random. What is the probability that X = 1/2? This seems a little like cheating, because when you instead ask > suppose X is an integer chosen at random, what is the probability > that it's 2?, the mathematician is just as stuck as the physicist. >Puhleeze. In the first case the words with a uniform distribution > are understood (or added explicitly if it turns out they're not > understood). In the second case the mathematician is not stuck, > he simply points out that the distribution needs to be specified > since there is no natural default. I was assuming a uniform distribution in both cases. Try to read. There is *no* such thing aas uniform distribution on integers That was exactly the point: once we switch from the unit interval to the integers, a uniform distribution becomes as impossible for mathematicians as for physicists. is trying to convey something. Presumably you're not so rude in real life. -- Richard -- :wq === Subject: Re: Can Events of Zero Probability Happen? Richard Tobin a .8ecrit : Ask him this: Suppose that X is a real number between 0 and 1, > chosen at random. What is the probability that X = 1/2? > This seems a little like cheating, because when you instead ask > suppose X is an integer chosen at random, what is the probability > that it's 2?, the mathematician is just as stuck as the physicist. > Puhleeze. In the first case the words with a uniform distribution > are understood (or added explicitly if it turns out they're not > understood). In the second case the mathematician is not stuck, > he simply points out that the distribution needs to be specified > since there is no natural default. > I was assuming a uniform distribution in both cases. > Try to read. There is *no* such thing aas uniform distribution on integers That was exactly the point: once we switch from the unit interval to > the integers, a uniform distribution becomes as impossible for > mathematicians as for physicists. is trying to convey something. You were trying very badly, as your argument were starting by I was assuming A, *after* it hasd been said A was impossible. Anyway, as (from other threads) it becomes clear your argument had no substance (The general shape is How can we convince X that there exists possible events of probability 0 Answer : give him the example of uniform distribution on [0,1] and you conter But the example of random integer would not be convincing, so your example is bad...) Presumably you're not so rude in real > life. > I am with fools like you > -- Richard === Subject: Re: Can Events of Zero Probability Happen? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) > Presumably you're not so rude in real life. >I am with fools like you You really are a most unpleasant person. -- Richard -- :wq === Subject: Solutions to Vector Mechanics for Engineers: Stacis 8th posting-account=boPtewoAAACKJMPsRR3xcmlYiuov3Nrm Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) Does Anyone have the book and solutions to Vector Mechanics for Engineers: Stacis 8th edition? Please send either to my email at === Subject: Re: question <28893826.1200037758652.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=q2ruDwoAAACW4OvF4soI61szUIc6xJHw Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) I'd like to see the complete sol-n, if ever solved. Showing f(0)=0 is f(x^3) = x() ==> f(0) = f(0^3) = 0() = 0. === Subject: Re: question <28893826.1200037758652.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=q2ruDwoAAACW4OvF4soI61szUIc6xJHw Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) Original equation: f(x^3+y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2) To show: f(a*x)=a*f(x) for real a,x? Intermediate results: f(0)=0, f(1) in {0,1}, f(-1) in {-1,0} f(x^3) = x[f(x)^2] Interesting suggestions of previous poster. Assuming that f(1) = 1, I still had trouble showing that f(n) = n ==> f(n+1) = n+1. Taking z=n^(1/3) and assuming f(n)=n gives: n = f(n) = f(z^3) = z[f(z)^2] ==> f(z) = +/-(z). Assuming that f(z) = +(z) gives: f(n+1) = f(z^3 + 1) = (z+1)[z^2 - z + 1] = z^3 + 1 = n+1. However, f(z) = -(z) gives: f(n+1) = (z+1)[z^2 + z + 1] which is NOT = z^3 + 1. Specifically, if f(1)=1 and f(2)=2, can't f(2^[1/3]) = -[2^(1/3)]. === Subject: Re: question hi zugzwang, >I'd like to see the complete sol-n, if ever solved. Indeed, so would I; alas, a dense subset of R is not R. >I still had trouble showing that f(n) = n ==> f(n+1) = n+1. The original statement was: f(x^3 + y^3) = (x + y)(f(x)^2 - f(x)f(y) + f(y)^2). Thus f(n + 1) = (n^(1/3) + 1)(f(n^(1/3))^2 - f(n^(1/3))f(1) + f(1)^2). So f(n + 1) = (n^(1/3) + 1)(n^(2/3) - n^(1/3) + 1) = n + 1. The proof that f(n - 1) = n - 1 is very similar. Namaste, -- Charles === Subject: Re: question Charles, What about adjusting the question to show f(a*x)=a*f(x) for real x BUT integer a? Is this easier to work with? If we use the method of induction that you pointed out to earlier we have f(x)=x for all integer x and hence f(ax)=ax=f(x) for integer a and x (right?) but is there a way to have this for all real x but with a still being an integer? Just thinking out loud here. Best wishes. === Subject: Re: question posting-account=q2ruDwoAAACW4OvF4soI61szUIc6xJHw Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) This is my stumbling point. Under the inductive assumption that f(n)=n, I agree that f(x^3) = xf(x)^2 implies that [f(n^(1/3))]^2 = [n^(1/3)]^2. However, I see no way of eliminating the possibility that [f(n^(1/3))] = -[n^(1/3)]. === Subject: Re: question in alt.math.undergrad: > This is my stumbling point. Under the inductive assumption that > f(n)=n, I agree that f(x^3) = xf(x)^2 implies that > [f(n^(1/3))]^2 = [n^(1/3)]^2. > However, I see no way of eliminating the possibility that > [f(n^(1/3))] = -[n^(1/3)]. The only x in R satisfying f(x) = -x is x = 0. Suppose that f(x) = -x, and let y = x^(1/3). Then -y^3 = -x = f(x) = f(y^3) = y f(y)^2, so -y^2 = f(y)^2, which is possible if and only if y = 0 = f(y), in which case x = 0. In fact I can show that if S = {x in R : f(x) = x}, then S is closed under addition, cubing, taking cube roots, and multiplication by 2^(1/3), and -S = S. Unfortunately, our classes just started today, and I'm teaching a heavier load than usual, so I've not had time to think about the problem properly. Brian === Subject: I'm so old..... Hi all, This is my first post. It has been quite a while since I've been in math so I might have a few 'elementary' questions. Here is the first one: 3/x-1 is less than or equal to -2/x Ok, I know how to solve for the answer of 2/5, and I understand that the answer can be greater than equal to, or less than equal to that answer. The problem I am having is distinguishing when the solution works or not. It says the answer is less than 0, and 2/5 greater than or equal to x less than 1. I just don't understand how I am supposed to decipher this easily. I have understood a few convoluted run-throughs on the net, but how can I do this without taking half an hour to do it? Help! === Subject: Re: I'm so old..... <14261626.1200276290659.JavaMail.jakarta@nitrogen.mathforum.org>, > Hi all, > This is my first post. It has been quite a while since I've been in math so > I might have a few 'elementary' questions. Here is the first one: > 3/x-1 is less than or equal to -2/x > Ok, I know how to solve for the answer of 2/5, and I understand that the > answer can be greater than equal to, or less than equal to that answer. The > problem I am having is distinguishing when the solution works or not. It > says the answer is less than 0, and 2/5 greater than or equal to x less than > 1. > I just don't understand how I am supposed to decipher this easily. I have > understood a few convoluted run-throughs on the net, but how can I do this > without taking half an hour to do it? > Help! Here is a variation of Barb's reply. Suppose we had the equation 3/(x - 1) = -2/x. You can do a two step process: Multiply by x - 1 to get 3 = (-2x + 2)/x then multiply by x to get 3x = -2x + 2. Then finish up x = 2/5. With inequalities, we want to do the same thing but complications arise. To solve 3/( x - 1) <= -2/x we want to multiply by x - 1 just as in the equation but we need to worry about the sign of x - 1 so we do two cases. If x - 1 > 0 then x > 1. Draw a line mark 1 on it and draw an arrow starting at 1 and going off to the right to infinity - or until you get tried of drawing it whichever comes first. Anyway, now multiply to get 3 <= (-2x + 2)/x. We now want to multiply by x but the same problem arises so we take two subcases. Suppose x > 0. Mark 0 on your line that you drew to the left of 1. Draw an arrow starting at 0 and going to the right. Multiply by x to get 3x <= -2x + 2 or 5x <= 2 or x <= 2/5 [5 is bigger than zero]. Take your picture put 2/5 between 0 and 1 and draw an arrow to the _left_ starting at 2/5. Look at the over lap of all three arrows. There is none; x can't be both less than 2/5 and larger that 1. Now suppose x < 0 (and x - 1 is still bigger that 0). Take your line with the 0 and 1 on it but this time draw an arrow to the left starting at 0. The two arrows don't overlap so this case has no solution - x can't be both bigger than 1 and less than 0. That's the end of the subcases - we get no solutions with x - 1 > 0 Now, what if x - 1 < 0? That is x < 1. Draw a line with an arrow going left from 1. Do the x > 0 case, draw the arrows you'll get an overlap of your three arrows between 2/5 and 1 note 2/5 is included but 1 is excluded. Now do the x < 0 case, draw the arrows, check the overlap which turns out to be to be to the left of 0. So, x satisfies 3/(x - 1) <= -2/x if (and only if) 2/5 <= x < 1 _or_ x < 0. Just to see if I am right (always a good idea) check x = -1, 1/5, 3/5, 2. -- Paul Sperry Columbia, SC (USA) === Subject: Re: I'm so old..... <14261626.1200276290659.JavaMail.jakarta@nitrogen.mathforum.org>, > Hi all, > This is my first post. It has been quite a while since I've been in math so > I might have a few 'elementary' questions. Here is the first one: > 3/x-1 is less than or equal to -2/x > Ok, I know how to solve for the answer of 2/5, and I understand that the > answer can be greater than equal to, or less than equal to that answer. The > problem I am having is distinguishing when the solution works or not. It > says the answer is less than 0, and 2/5 greater than or equal to x less than > 1. > I just don't understand how I am supposed to decipher this easily. I have > understood a few convoluted run-throughs on the net, but how can I do this > without taking half an hour to do it? > Help! 3/(x-1) <= -2/x [the parentheses are important] We want to clear the denominators by multiplying by (x-1) and x, but the signs of these are important, since multiplying by a negative number reverses the direction of the <=. So there are 3 cases: (a) 1 0. (b) 0 0. (c) x<0, in which case both x-1 and x are < 0. We multiply both sides by (x-1) and x. In case (a) both are positive, and in (c) both are negative so their product is positive, so we don't reverse the <=: 3x <= -2(x-1) for 1= -2(x-1) for 0= 2 x >= 2/5 So, for case (b), 0 < x < 1 and 2/5 <= x, so 2/5 <= x < 1 is another solution. HTH. -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: List of solutions manual (thousands) posting-account=h2vHUgoAAABxsDdBcDh0Jgtjz-I5l8nS 1.1.4322),gzip(gfe),gzip(gfe) hi > i want the solution for æRadiative Heat Transfer (2nd Ed., Michael > Modest) > how much is it? question: > what is the format for the solution, well printed, handwrite or e- > file? > does the solutions included all the problem ( even and odd )? thank you > My List of Solutions Manual > > contact me to : æ newbergh...@yahoo.com > If your wanted solutions manual ins't on this list, also can ask me if > is available . These are some only. > This list (not links) is available from : >http://rapidshare.com/files/59002351/List of solutions manual.txt > > - Mechanics, Mechanical Engineering & Aerospace Engineering: > > Classical mechanics (2nd Ed., Goldstein) > Classical Mechanics (Douglas Gregory) + original Ebook > Advanced Dynamics (Greenwood) + original Ebook > Advanced Engineering Dynamics (2nd Ed., Jerry Ginsberg) + Ebook > Classical Dynamics (Jorge V. Jos.8e) + Ebook > Impact Mechanics (W.J. Stronge) > Introduction to Mechanical Engineering (Rizza) > Mechanical Engineering Principles (Bird & Ross) + original Ebook > Mechanics of Fluids (8th Ed., Massey) + original Ebook > Fluid Mechanics (5th Ed., White) + Ebook > Fluid Mechanics (6th Ed., White) > Viscous Fluid Flow (3rd Ed., White) + Ebook > Fundamentals of Thermal-Fluid Sciences (1st Ed., Cengel) + original > Ebook > Fundamentals of Thermal-Fluid Sciences (2nd Ed., Cengel) + original > Ebook > Fundamentals of Thermal-Fluid Sciences with Student Resource CD (3rd > Ed., Cengel & Turner) > Thermodynamics: An Engineering Approach (5th Ed., Cengel) + original > Ebook > Thermodynamics: An Engineering Approach (6th Ed., Cengel) + original > Ebook > Essentials of Fluid Mechanics: Fundamentals and Applications (1st Ed., > Cengel) + original Ebook > Fluid Mechanics (1st Ed., Cengel) + original Ebook > Heat Tranfer (2nd Ed., Cengel) + original Ebook > Heat and Mass Transfer: A Practical Approach (3rd. Ed., Cengel) + > original Ebook > Design and Simulation of Thermal Systems (Suryanarayana & Arici) > Introduction to Fluid Mechanics (6th Ed., Robert Fox, Alan McDonald & > Philip Pritchard) > Fluid Mechanics (5th Ed., Douglas) > Fluid Mechanics (3rd Ed., Kundu) > Fluid Mechanics with Engineering Applications (Finnemore) > Fundamentals of Fluid Mechanics, 4th Ed (Bruce R. Munson, Donald F. > Young, Theodore H. Okiishi) + original ebook > Fundamentals of Fluid Mechanics, 5th Ed (Bruce R. Munson, Donald F. > Young, Theodore H. Okiishi) > A Brief Introduction to Fluid Mechanics, 3rd Ed (Donald F. Young, > Bruce R. Munson, Theodore H. Okiishi) > A Brief Introduction to Fluid Mechanics, 4th Ed (Donald F. Young, > Bruce R. Munson, Theodore H. Okiishi, Wade W.) > Engineering Fluid Mechanics, 7th Ed (Clayton T. Crowe, Donald F. > Elger, John A. Roberson) > Engineering Fluid Mechanics, 8th Ed (Clayton T. Crowe, Donald F. > Elger, John A. Roberson) > Mechanics of Fluids (3rd Ed., Potter) > Mechanics of Fluids (4th Ed., Shames) > Extended Irreversible Thermodynamics (3rd Ed., D. Jou, J. Casas- > Vazquez & G. Lebon) > Thermodynamics: An Integrated Learning System (Schmidt, Ezekoye, > Howell & Baker) > Introduction to Thermal and Fluids Engineering (Kaminski & Jensen) > Heating, Ventilating and Air Conditioning Analysis and Design (6th > Ed., McQuiston) > An Introduction to Fluid Dynamics: Principles of Analysis and Design > (Middleman) > Introduction to Mass and Heat Transfer: Principles of Analysis and > Design (Middleman) > Heat Transfer (2nd Ed., Mills) > Convective Heat and Mass Transfer (4th Ed., Kays & Crawford) > Advanced Engineering Thermodynamics (3rd Ed., Bejan) > Convection Heat Transfer (2nd Ed., Bejan) > Convection Heat Transfer (3rd Ed., Bejan) > Thermal Design and Optimization (Bejan) > Shape and Structure, from Engineering to Nature (Bejan) > An Introduction to Combustion: Concepts and Applications (2nd Ed., > Turns) > Thermodynamics: Concepts and Applications (Stephen Turns) > Thermal-Fluid Sciences: An Integrated Approach (Stephen Turns) > Principles of Heat Transfer (Kaviany) > Heat Convection (Latif M. Jiji) + original Ebook > Heat Transfer (9th Ed., Holman) > Fundamentals of Momentum, Heat and Mass Transfer (4th Ed., Welty) > Momentum, Heat, and Mass Transfer Fundamentals (Kessler) + original > Ebook > Analytical Methods for Heat Transfer and Fluid Flow Problems (Bernhard > Weigand) > Heat Tranfer (Rao) > Heat Conduction (kakac) > Heat Exchanges (Kakac) > Convective Heat Transfer (kakac) > Heat Exchangers: Selection, Rating and Thermal Design (2nd Ed. Sadik > Kakac & Hongtan Liu) > Fundamentals of Engineering Thermodynamics, 5th Ed (Michael J. Moran, > Howard N. Shapiro) + original Ebook > Fundamentals of Engineering Thermodynamics, 6th Ed (Michael J. Moran, > Howard N. Shapiro) > Fundamentals of Heat and Mass Transfer (5th Ed., Incropera, DeWitt) > Fundamentals of Heat and Mass Transfer (6th Ed., Incropera, DeWitt) > Introduction to Heat Transfer (4th Ed., Incropera, DeWitt) > Introduction to Heat Transfer (5th Ed., Incropera, DeWitt) > Radiation Detection and Measurement (3rd Ed., Glenn Knoll) > Radiative Heat Transfer (2nd Ed., Michael Modest) > Engineering Heat Transfer (2nd Ed., Janna) > Engineering Thermodynamics: Work and Heat Transfer (4th Ed., G.F.C. > Rogers & Y.R. Mayhew) > Elements of Heat Transfer (Yildiz Bayazitoglu and M. Necati Ozisik) > Inverse Heat Transfer: Fundamentals and Applications (M.N. Ozisik & > Helcio R.B. Orlande) > Thermal Radiation Heat Transfer (4th Ed.,Robert Siegel & John R. > Howell) > Computational Heat Transfer (2nd Ed., Jaluria) > Principles of Combustion (2nd Ed., Kenneth Kuan-yun Kuo) > Incompressible Flow (3rd Ed., Panton) > Modern Compressible Flow: With Historical Perspective (3rd Ed., John > D. Anderson) > Non-Newtonian Flow : Fundamentals and Engineering Applications (R P > Chhabra & J F Richardson) + original Ebook > Computational Techniques for Fluid Dynamics (Srinivas, K., Fletcher, > C.A.J.) > Ebook > Theory of Applied Robotics: Kinematics, Dynamics and Control (Reza N. > Jazar) > Kinematic Chains and Machine Components Design (Dan B. Marghitu) + > original Ebook > Kinematics and Dynamics of Machinery (3rd Ed., Wilson & Sadler) > Kinematics, Dynamics, and Design of Machinery (2nd Ed., Waldron & > Kinzel) > Mechanism Design: Analysis and Synthesis-Volume 1 (4th Ed., Erdman & > Sandor) > Machines and Mechanisms: Applied Kinematic Analysis (3rd Ed., > Myszka) > Mechanical Design: A Components Approach (Peter Childs) > Mechanical Design of Machine Elements and Machines: A Failure > Prevention Perspective (Collins) > Fundamentals of Machine Component Design (3rd Ed., Juvinall) > Fundamentals of Machine Component Design (4th Ed., Juvinall) > Design of Machine Elements (8th Ed., Spotts) > Machine Design (Wentzell) > Solutions Manual to the text : Problems on the Design of Machine > Elements (Faires) > Machine Elements in Mechanical Design (4th Ed., Mott) > Mechanical Design: An Integrated Approach (1st Ed., Ugural) > Design of Machinery (3rd Ed., Norton) > Design of Machinery (4th Ed., Norton) > Machine Design (2nd Ed., Norton) > Machine Design : An Integrated Approach (3rd Ed., Norton) > Mechanical Engineering Design (6th Ed., Shigley) > Mechanical Engineering Design (7th Ed., Shigley) > Shigley's Mechanical Engineering Design (8th Ed., Budynas) > Fundamentals of Machine Elements (1st Ed., Hamrock) > Fundamentals of Machine Elements (2nd Ed., Hamrock) > Mechanics of Materials: A Modern Integration of Mechanics and > Materials in Structural Design (Christopher Jenkins & Sanjeev Khanna) > Mechanics of Materials (3th Ed., Beer) > Mechanics of Materials (5th Ed., Gere) > Mechanics of Materials (6th Ed., Gere) > Mechanics of Materials (Ugural) > Simplified Mechanics and Strength of Materials (6th Ed., James > Ambrose) > Engineering Mechanics, Statics, 2nd Ed (William F. Riley, Leroy D. > Sturges) > Engineering Mechanics, Dynamics, 2nd Ed (William F. Riley, Leroy D. > Sturges) > Engineering Mechanics - Statics, 5th Ed (J. L. Meriam, L. G. Kraige) + > Ebook > Engineering Mechanics - Statics, 6th Ed (J. L. Meriam, L. G. Kraige) > Engineering Mechanics - Dynamics, 5th Ed (J. L. Meriam, L. G. Kraige) > Engineering Mechanics - Dynamics, 6th Ed (J. L. Meriam, L. G. Kraige) > Vector Mechanics for Engineers: Statics (7th Ed., Ferdinand P. Beer) > Vector Mechanics for Engineers: Statics (8th Ed., Ferdinand P. Beer) > Vector Mechanics for Engineers: Dynamics (7th Ed., Ferdinand P. Beer) > Vector Mechanics for Engineers: Dynamics (8th Ed., Ferdinand P. Beer) > Statics: Analysis and Design of Systems in ... read more Hi there > I am Extremely interested in the solution manual for > Statics and Strengths of Materials Sixth Edition by H.W Morrow and > Robert P.Kokernak > Is there any possibility I might be able to receive that from you? > Francisco Hello I was wondering if you could send me the solution manual to An Introduction to Fluid Dynamics: Principles of Analysis and Design === Subject: Re: Solutions Manual for the 8th Edition of Electric Circuits by Nilsson an <20739117.1194493179732.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=i_ucagoAAACYhkyVZLsYfkiQ8_eX4Q5V Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) yeah, here too. It would be very helpful! hey do you have solution manual of the electric circuits 8th edition? > if you have please forward it to me. please... Forward here also. lemmingsownyou@gmail.com === Subject: Re: Solutions Manual for the 8th Edition of Electric Circuits by Nilsson an <20739117.1194493179732.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=2z_lFwoAAAC4rmPCkloWoytKqTC2eSPF Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) If you have the solutions manual that would be very helpful can any of you forward it to me at tyler.jones@aggiemail.usu.edu === Subject: Re: solutions manual in PDF format GetSolution Team posting-account=2QudPgoAAAAKl90hCIq723K8ysSin4pl CLR 1.1.4322; InfoPath.2),gzip(gfe),gzip(gfe) Electric Circuits 8th edition Nilsson Riedel. [CapitalYAcute] need it. can you send it to me? === Subject: Re: Electric Circuits Nilsson 8th Edition posting-account=2z_lFwoAAAC4rmPCkloWoytKqTC2eSPF Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > If anyone has a copy of the solutions manual, could you forward it to has anyone found it? tyler.jones@aggiemail.usu.edu please forward to me === Subject: Correlation and Regression Analysis Software posting-account=ajkdngoAAAB2X3SxyQtiBYkcRETphklh 1.1.4322),gzip(gfe),gzip(gfe) Our latest online software allows computing Co-efficient of correlation, and Linear equation of regression anlysis. It displays the results, scatter diagram, and line graph For Correlation Visit http://www.thinkanddone.com/ge/Corr.aspx For Regression Visit http://www.thinkanddone.com/ge/Reg.aspx === Subject: Question about ripples posting-account=25ZpqAoAAADxgZ18ZCFxSYfhae8RiqQ3 Gecko/20071204 Ubuntu/7.10 (gutsy; Google-TR-3) Firefox/2.0.0.11,gzip(gfe),gzip(gfe) I was wondering if anyone could push me in the right direct of mathematical eqs for calculating ripples of water; not only a flat surface of water but the maybe a spherical shape? Keith === Subject: Looking for equations for ripples posting-account=25ZpqAoAAADxgZ18ZCFxSYfhae8RiqQ3 Gecko/20071204 Ubuntu/7.10 (gutsy; Google-TR-3) Firefox/2.0.0.11,gzip(gfe),gzip(gfe) I am looking for equations for ripples, like on water. I am looking for all types of equations you might have for this application, be it lake surface or maybe something like a spherical shape. With the spherical shape I am talking a long the lines of a bubble. You have an empty space with in the bubble, yet there is a surface- layer in which the ripple effects. Keith === Subject: Re: Tough function question >2) In the second case, why did you define g(x) as (f(x) - f(-x))/2 and why >does |g(x)|'s turning point being >=1/4 mean that the max of |f(x)|>=1/4? Hint: x^3+bx is an even function and ax^2+c is odd. Namaste, -- Charles === Subject: Re: Tough function question >2) In the second case, why did you define g(x) as (f(x) - f(-x))/2 and why >does |g(x)|'s turning point being >=1/4 mean that the max of |f(x)|>=1/4? Oops! Hint: x^3+bx is an ODD function and ax^2+c is EVEN. Namaste, -- Charles === Subject: Re: Tough function question > Suppose we have the function f whose value at x is defined as > f(x)=x^3+ax^2+bx+c where a,b,c are real coefficients. How could we show > that max[-1<= x <= 1] of abs( f(x) ) is > or = to 0.25. >You can't: you need the values of the coefficients. > That was my first reaction. But I'm not so sure. Note that > we're _given_ the coefficient of x^3. It's clear that there > is _some_ number delta such that the maximum of every > polynomial as above on [-1,1] is >= delta. > In fact, come to think of it: > If |1 + b| >= 1/4 then |f(1) - f(-1)| >= 1/2 so > max |f(x)| >= 1/4. Suppose that |1 + b| < 1/4. > Let g(x) = (f(x) - f(-x))/2 = x^3 + bx. Then > g'(x) = 3x^2 + b. Note that b is negative, so > this has the real root x_0 = sqrt(-b/3). > And |b| < 5/4 shows that |x_0| < 1. Now, > |g(x_0)| = |x_0(x_0^2 + b)| = sqrt(|b|/3)(2/3)|b| > = |b|^(3/2)*2/(3sqrt(3)). > Since |b| >= 3/4 this shows that > |g(x_0)| >= (3/4)^(3/2)*2/(3*sqrt(3)) = 1/4, > QED. David, well done and thank you very much indeed. If you don't mind, may I >ask you a few trivial questions about your proof: 1) May I ask if you chose to divide the proof into the cases of |1 + b| >= >1/4 and |1 + b| < 1/4 because that way you would have |f(1) - f(-1)|>= 1/2 >(and hence prove the first case)?? In other words, did the two cases come >directly from evaluating |f(1) - f(-1)| and realising that to have this >= >1/2 you needed |1 + b| >= 1/4? Yes, more or less the first thing I thought of was when could we show that |f| >= 1/4 at one of the endpoints, and then the second thing I thought of was that looking at f(1) - f(-1) would simplify things. >2) In the second case, why did you define g(x) as (f(x) - f(-x))/2 and why >does |g(x)|'s turning point being >=1/4 mean that the max of |f(x)|>=1/4? You really want me to do _everything_ for you, eh? I mean I gave you a solution to your problem, you're not willing to think about the details even a little bit? Good luck with the rest of your homework. >3) Not related to your proof, but would you have an idea how to show when >equality occurs in this example? > ************************ > David C. Ullrich > ************************ David C. Ullrich === Subject: Re: Fluid Mechanics (1st Ed., Cengel) + Ebook <20281248.1199973330425.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=l6ac3woAAABdpyrwEWLg5LHYGUvRPiVt 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; InfoPath.2),gzip(gfe),gzip(gfe) > hey i was looking for the solution manual for Fluid Mechanics fundemental and application Solutions 1st Ed by Cengel and Cimbala + ebook if u did find it, please help me my email is saif ud...@hotmail.com or mycolle...@hotmail.com > do you still looking for solutions manual for that one? i actually have that one... i will charge you $20 for fluid mechanics cengel 1e === Subject: Re: Any Diff Eq books written like Ash Calculus Tutoring Book? > Ash's Calculus Tutoring Book is famous in some circles for the > unique and direct approach. Terrific sequencing--gradually builds a > solid base for subsequent chapters. It's not a typical school textbook > in that it does not stress proofs. More emphasis on developing > intuition. That's what I'm after. If she or Banner had written an ODE > book, I'd be set. I was able to read an excerpt from Carol Ash's book on Amazon. It does look pretty good. It does not have the super-easy-but-with-2/3-of-the-info-missing feel of the books that I mentioned; IOW, it looks as though it would be a decent stand-alone text and not a mere supplement. One of my instructors once told me you need to be able to explain stuff in trashy language (i.e. in an intuitive fashion) as well as mathematical language. Otherwise, you memorize stuff like the definition of a fuction and talk about domain and range without understanding what a function is. Ash's first few pages seem to do just that, impart the understanding and not get bogged down in the vocabulary. As to the books that I mentioned, I do not have the Diff Eqns Demystified, but I do have the Schaum's Easy Outline. Normally, I get the regular Schaum's Outlines, but for Diff Eqns, I simply wanted a supplement for the class text, not a second textbook. My class starts next Tuesday and we are using Brannan & Boyce with supplemental info from Boyce & DiPrima. The first dozen pages of Brannan & Boyce were not badly written, but then they started going all over the place and I was struggling to figure out what they were saying. It looks to me as though this information is identical to the first chapter of Boyce & DiPrima, for whatever that info is worth. I think that it would be a hard book for self-study. Yes, with the instructor to clarify things, the book may be useable, but I was trying to get through the first two chapters prior to the start of the semester to be a bit ahead for a change as in calculus I was always a bit behind. At any rate, although the Schaum's Easy Outlines is a bit too easy to use as a self-instruction source, it looks as though it can help to smooth over a rough spot or two which is all that I need it for. The price is right as well. Also, I have found it somewhat useful to use the REA Problem Solver book for classes over the last few years. Generally, I do not need it for 90% of the stuff, but there is that 10% where seeing lots more worked-out examples was helpful. There is also this free, online wikibook: http://en.wikibooks.org/wiki/Differential_Equations It seems to be a nice supplement. At least it gives a different style than Boyce. Some other online textbooks here with 2 or 3 related to Diff Eqns (some are lecture notes, some are books) http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html > I have the Tenenbaum ODE book, which is very good, but I still get > hung on something once in a while. I have decided to buy that as well after taking a preview look at it. It does look very well written. > One of the normal Diff Eq school texts may take an approach that lends > toward self-teaching, but without reading quite a bit of each book > (Boyce-Diprima? Nagle-Saff? Zill?), I won't know which is better in > that respect. Anyone familiar with those? Although not any easier than a typical textbook, there are online videos of a Diff Eqns class taught at MIT. The lecturer, Arthur Mattuck, is quite good, IMO. Here is that link: http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.h tm === Subject: Re: Any Diff Eq books written like Ash Calculus Tutoring Book? > Ash's Calculus Tutoring Book is famous in some circles for the > unique and direct approach. Terrific sequencing--gradually builds a > solid base for subsequent chapters. It's not a typical school textbook > in that it does not stress proofs. More emphasis on developing > intuition. That's what I'm after. If she or Banner had written an ODE > book, I'd be set. I was able to read an excerpt from Carol Ash's book on Amazon. It does >look pretty good. It does not have the >super-easy-but-with-2/3-of-the-info-missing feel of the books that I >mentioned; IOW, it looks as though it would be a decent stand-alone text and >not a mere supplement. Yes it would--that's why it is well known in some circles. But most courses stress 'proofs' rather than intuition, so it seems unlikely that it would be used. Ideal for self-study or for anyone who is looking for simply how to use math (us engineering types). > One of my instructors once told me you need to be >able to explain stuff in trashy language (i.e. in an intuitive fashion) as >well as mathematical language. Otherwise, you memorize stuff like the >definition of a fuction and talk about domain and range without >understanding what a function is. Ash's first few pages seem to do just >that, impart the understanding and not get bogged down in the vocabulary. Exactly. Also true for Adrian Banner's Calculus Lifesaver. That book did start out as a lecture series. The book is almost conversational. Before you realize it, he has covered lots of ground and no one is asleep. but I do have the Schaum's Easy Outline. Normally, I get the regular >Schaum's Outlines, but for Diff Eqns, I simply wanted a supplement for the >class text, not a second textbook. That makes sense. I've bought the Easy Outline books for a couple other subjects (bio, chem, etc). They are good for refresher for terms, but I rarely bother to pick them up any more. The regular Schaum's books are a lot more linear. Even then, too terse for just reading. >My class starts next Tuesday and we are using Brannan & Boyce with >supplemental info from Boyce & DiPrima. The first dozen pages of Brannan & >Boyce were not badly written, but then they started going all over the place >and I was struggling to figure out what they were saying. It looks to me as >though this information is identical to the first chapter of Boyce & >DiPrima, for whatever that info is worth. I think that it would be a hard >book for self-study. I actually did work through several chapters of Boyce and Diprima by myself, but started hitting some rough road (many authors seem to get tired of being nice to their audience). That's why I was wondering about others. Maybe a combination of two books (though that would take longer). >.... >There is also this free, online wikibook: >http://en.wikibooks.org/wiki/Differential_Equations It seems to be a nice >supplement. At least it gives a different style than Boyce. >Some other online textbooks here with 2 or 3 related to Diff Eqns (some are >lecture notes, some are books) >http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html > I have the Tenenbaum ODE book, which is very good, but I still get > hung on something once in a while. I have decided to buy that as well after taking a preview look at it. It >does look very well written. It gets consistently good reviews. It is a Dover book, so it has that older 'black and white' look about it (tough to describe unless you own a bunch of Dover books). >Although not any easier than a typical textbook, there are online videos of >a Diff Eqns class taught at MIT. The lecturer, Arthur Mattuck, is quite >good, IMO. Here is that link: >http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index. htm Excellent! The man knows how to connect. I found videos for the lecture series from Adrian Banner's calculus course. Also very good, especially if you have his book. May be useful for anyone looking for help with calculus. http://press.princeton.edu/video/banner/8351.html There is also a link to a sample chapter on that page. Now if I could find the equivalent for Differential Equations... Here's a question for those familiar with -any- of the normal ODE texts. Of the following, which do you think are less formal and more 'intuitive' in approach: Boyce-Diprima Trench Zill Nagle-Saff others? Someone has to be familiar with a couple of those. It's difficult to tell anything from a brief glance at a bookstore (even if I could find all of them). === Subject: Re: Any Diff Eq books written like Ash Calculus Tutoring Book? reply-type=response > Ash's Calculus Tutoring Book is famous in some circles for the > unique and direct approach. Terrific sequencing--gradually builds a > solid base for subsequent chapters. It's not a typical school textbook > in that it does not stress proofs. More emphasis on developing > intuition. That's what I'm after. If she or Banner had written an ODE > book, I'd be set. I was able to read an excerpt from Carol Ash's book on Amazon. It does > look pretty good. It does not have the > super-easy-but-with-2/3-of-the-info-missing feel of the books that I > mentioned; IOW, it looks as though it would be a decent stand-alone text > and not a mere supplement. One of my instructors once told me you need to > be able to explain stuff in trashy language (i.e. in an intuitive > fashion) as well as mathematical language. Otherwise, you memorize stuff > like the definition of a fuction and talk about domain and range without > understanding what a function is. Ash's first few pages seem to do just > that, impart the understanding and not get bogged down in the vocabulary. As to the books that I mentioned, I do not have the Diff Eqns Demystified, > but I do have the Schaum's Easy Outline. Normally, I get the regular > Schaum's Outlines, but for Diff Eqns, I simply wanted a supplement for the > class text, not a second textbook. My class starts next Tuesday and we are using Brannan & Boyce with > supplemental info from Boyce & DiPrima. The first dozen pages of Brannan > & Boyce were not badly written, but then they started going all over the > place and I was struggling to figure out what they were saying. It looks > to me as though this information is identical to the first chapter of > Boyce & DiPrima, for whatever that info is worth. I think that it would be > a hard book for self-study. Yes, with the instructor to clarify things, > the book may be useable, but I was trying to get through the first two > chapters prior to the start of the semester to be a bit ahead for a change > as in calculus I was always a bit behind. At any rate, although the Schaum's Easy Outlines is a bit too easy to > use as a self-instruction source, it looks as though it can help to smooth > over a rough spot or two which is all that I need it for. The price is > right as well. Also, I have found it somewhat useful to use the REA > Problem Solver book for classes over the last few years. Generally, I do > not need it for 90% of the stuff, but there is that 10% where seeing lots > more worked-out examples was helpful. There is also this free, online wikibook: > http://en.wikibooks.org/wiki/Differential_Equations It seems to be a nice > supplement. At least it gives a different style than Boyce. > Some other online textbooks here with 2 or 3 related to Diff Eqns (some > are lecture notes, some are books) > http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html > I have the Tenenbaum ODE book, which is very good, but I still get > hung on something once in a while. I have decided to buy that as well after taking a preview look at it. It > does look very well written. > One of the normal Diff Eq school texts may take an approach that lends > toward self-teaching, but without reading quite a bit of each book > (Boyce-Diprima? Nagle-Saff? Zill?), I won't know which is better in > that respect. Anyone familiar with those? > Although not any easier than a typical textbook, there are online videos > of a Diff Eqns class taught at MIT. The lecturer, Arthur Mattuck, is > quite good, IMO. Here is that link: > http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.h t m p.s. - when I took Calculus III last semester, I found that I really liked Paul's Online Notes. I just noticed that he also has notes for a Diff Eqns class. His writing is quite clear and I found the Calc III stuff to be a valuable supplement to the Stewart text. http://tutorial.math.lamar.edu/Classes/DE/DE.aspx === Subject: Re: Latest advances with factoring congruences posting-account=W_v73AoAAADTM_uett2a7ZO49gKTBMKy Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > There has been a burst of research lately on an outgrowth of > mathematical research I've pioneered for about four years into the > question of whether or not factoring one number could be used to > factor another. That there has been a steady and persistent achievement of milestones > over a period of years may surprise many of you who just know about > the effort from cases when a flurry of newsgroup postings show up, > where a lot of people criticize a particular research effort. The latest and most important chapter though to cap years of basic > research is the remarkable discovery of just a few basic equations-- > factoring congruences: Given z^2 = y^2 + nT at least 50% of the time, you can solve for z modulo a given prime p > coprime to y, z and odd nT coprime to 3, with the following congruence > relations: z = (2a)^{-1}(1 + 2a^2)k mod p and k^2 = (a^2+1)^{-1}(nT) mod p where 'a' is chosen such that k exists i.e. (a^2+1)^{-1}(nT) is a > quadratic residue modulo p, and k is coprime to nT. Where also, y = (1+2a^2)^{-1} z mod p, or y = -(1+2a^2)^{-1} z mod p. And I've included some of the latest research found just yesterday > from the existence proof, which shows the relations will work for a > particular prime p at least 50% of the time, and from which I also > found the equations giving y. That y is given in that way versus my prior belief that you had to > solve for y, from y^2 = z^2 - nT removes the one area where speed might have been an issue. That is > because for any particular 'a' that you choose, 'n' can be chosen such > that (a^2+1)^{-1}(nT) is a quadratic residue modulo p, which means > you can force solutions easily. The one warning there is that n > cannot be divisible by 3, should not be even and should not contain > small primes e.g. primes under 100. The relations are easily derived, and I've made that derivation > available so that others can look it over to see if it contains > mistakes, and none have been noted. The existence proof was worked > out by me mostly yesterday so there is a possibility that there is > some error in it, but I doubt that at this time. Then the mathematics says that the factoring congruence relations are > valid, and that they will work with a given prime p when all the > conditions listed are met, about 50% of the time. There has been a push for me to develop a working algorithm from them, > and demonstrate that they work by factoring a large number, which may > seem like a reasonable request to many of you. But in the meantime while I work at making sure that my current > research is correct, and later work at writing a computer program, the > proof that they work is freely available, so it would be a remarkable > part of this story for a refusal of the mathematical community to > accept proof to be a large part of the delay in informing the world of > the situation. I have sent an early paper to a math journal, which might have been a > bit too early as it did not have the existence proof, some of the > conditions or the information that they work 50% of the time for a > given prime, but there is that to have some hope on. Also I have emailed some mathematician directly. If the factoring congruences are correct, and if they do factor about > 50% of the time when you use a p>sqrt(nT), then they could allow > factoring of what is generally called a public key, which is part of > the encryption system developed by RSA. If that is true, then they > would probably allow rapid factoring of it as well, which would mean a > system that was thought to be capable of protecting information for > decades, will have been proven defunct, almost literally overnight. Of course, there are those who would rather that be demonstrated > first, and not left as a claim based solely on mathematical proof, so > there is a call for me to factor a public key, and show that RSA > encryption is over. And that covers all the key items and should bring you up to speed on > the the latest. James Harris First! -- THE Troll Feeder Beating other troll feeders to the trowel since 2008. === Subject: ... The Gift of Power, Discovery of DIVINE Creation! posting-account=moli1AoAAADCMdYtxdzC78HhVQJXOHI0 rv:1.8.1) Gecko/20061010 Firefox/2.0,gzip(gfe),gzip(gfe) blessings Bobbie Ann, I KNEW you would understand!! Millennium post that lengthy soliloquy. I'm not sure what to make of most of it, but I did understand: The reason why institutional, 'politically- correct' and especially covert, physics has never been able to do this -- is because it has NEVER FOCUSED ON THE INCARNATION OF EXPERIENCE, OF PHENOMENA, OF PERSONA, OF SPIRIT, OF CONSCIOUSNESS ... THEY FOREVER DENIED THE POSSIBILITY OF THEIR SUCCESS BY DENYING TRUTH, DENYING SPIRITUALITY! BY DENYING EXPERIENCE, KNOWLEDGE, SENSATION, CONSCIOUSNESS, THE EMPIRICAL -- BY INSISTING ON A DEAD PHYSICS AND A DEAD COSMOS -- THEY DENIED THEMSELVES THE OPPORTUNITY OF EVER WITNESSING LIFE AND ITS UBIQUITOUS CREATION ... EVERYWHERE AND EVERY-WHEN! Bobbie Ann blessed Mother, rock my world! Millennium . .. the electric charge, first and only unit of matter, definer of space and times, distance, measure. the tickler, which moves our hearts to beat ... our voices to sing ... the tingle, moving down the uppermost leaves and branches of our cousin tree, down through its body to the deepest filamentary fibrous roots touching the Earth ... day's warmth, Grandmother Sun's caress, on our tear-filled faces, carried throughout our being, aye, even unto our happiest of toes, stretching ... these thoughts, emotions, in this very now ... that light in our child's eyes! ~~~ how does it all come to be? (this ephemeral 'certainty'? ) this experience divine, this knowing and growing, this sharing amongst family, this sacred creation? our life of wondrous sensation, immeasurable communication, our physical bodies and material world? how does electromagnetic spirit, consciousness, light, song become manifest as fire, electricity, plasma ... and thence atmosphere (breath) ... ocean ... earth ... garden ... life ... all our ancestral relations? (whale, dolphin, bear, deer, coyote, crow, lizard, frog, cricket ...) the two forms of matter, electron and proton (fire, electricity) from which all materiality is built and cooled! science and mathematics, language and number, culture and philosophy! as one of our many yahoo physics denizens noted, even Wolfgang Pauli couldn't put a finger on the origin of the Fine Structure Constant ... of 137, discovered by Arnold Sommerfeld in 1916 in the measurements of the frequencies of the spectrum of hydrogen; finally revealed and explained in the discoveries by Millennium Twain in 1994, in the standing-electromag netic-wave structures of the electron and (superluminal) proton, and neutron! Werner Heisenberg or Neils Bohr would have been 'shocked' to hear someone even ask the forbidden question ... and Max Planck would have erased the heretical dialog from the 'net' [But not Michael Faraday, Andre-Marie Ampere, or Nikola Tesla! not Leonardo DaVinci or Galileo Galilei, nor Kristian Birkeland of Norway, Jack Piddington of Australia ...] how did the covert bankers, governments, 'corp- a-rations' arrange that ALL of physical science knowledge would be suppressed, indeed hidden by a 'pseudo-German' physics guild, approved of and followed by the malleable French and manipulating English, and the gullible Americans? [The obedient Japanese and the whole enslaved world?] by pretending that physics, knowledge, the living 'Electro-verse' , spoke Deutsche? That creation was German?? They thought (correctly) that all the world's 'corpse-a-rations' , 'gov-a-ments' , media, publishers, schools -- would go along with that European scam, the dumbing-down of all peoples? yes. and they were 'right', for a hundred years. during that violent 20th century run by the churches, the mafias, the militaries. until now, the universal 'now' ... of all times, all cultures, all knowing, all our ancestors. all spirit ... .. . [to be added -- transcribe text from pages 8-9 of The Electron from 1993, proving how creation is accomplished, where the number 137 comes from, in the light from hydrogen ... and how the electro- magnetic waves become an electron, charge ... and also pages 10-11 showing how the frequencies and wavelengths of light (red, ultraviolet, gamma) make up the structure of the electron, and how they and other frequencies are and may be released in hydrogen reactions ... to warm & illumin our world, provide our power needs and transportation. ] http://groupkos.com/mtwain/TheElectron.pdf [add discussion from chapter 2 of Metric Relativity, The Proton, showing how the number 137, discovered in the light given off by hydrogen, an electron wrapped around a proton, is simply a harmonic, which comes from the 137-loop fundamental frequency and structure of the *superluminally- spinning* proton!] origin of number, math, song, being! http://groupkos.com/mtwain/TheProton.pdf NuclearStructure and DiosasAncianos2012 on the four 'dimensions' of experience, sensation, which I proved to be encapsulated in the totality of the helicoidal electromagnetic wave -- transverse electric and magnetic 'feelings', and centripetal/ centrifugal 'love', and longitudinal sound, song!] the WHOLE of experience, creation, power and knowing. gift of global consciousness, the blue star, the maitreyah, maadi, maschiak, messiah, adi shakti ... morphic grid, morphogenic field, goddess light, aboriginal grandmother' s wisdom ... our collective feminine returned. sung by all our sacred relations. 31 December 2007 .. . 1 January 2008 Re: The Gift ... Discovery Of DIVINE Creation!!! draft Transcription and explanation from pages 8-9 of The Electron from 1994. http://groupkos.com/mtwain/TheElectron.pdf . .. Historical interpretation of the observational, experimental and theoretical properties of an electron have yielded three basic 'scales' for its domains. Names associated with those scales are: 1) the 'Classical' electron radius, 2) the Compton scattering wavelength, and 3) the Bohr 'orbital' radius. If you used the first two scales to depict a toroid of small 'twist' diameter and large 'spin' ray (twist) radius of 2.81794 fermi. [A fermi is ten-to-the-minus- thirteen centimeters. ] The large (spin) radius is from Compton X-RAY scattering and is 386.19 fermi. To convert to wavelength multiply by 2 time Pi, to get 2426.3 fermi. The other radius, wavelength, frequency is the electronic P-Shell ULTRAVIOLET separation distance of the electron-wave from the proton nucleus of the hydrogen atom. These Bohr 'orbital' radii are on the order of 52,918 fermi. A very large number, compared to the 3/4 fermi radius of the proton, yet still a small enough wavelength, large enough frequency, to be in the UV or BLACKLIGHT range of emission spectra. The Conception of Charge Now it may not be clear from the above discussion [see pictures in TheElectron] just why closure of an electromagnetic wave into a circle or figure-8 or other closed Lissajous topology -- why that closure results in the property of 'charge'! It is quite simple actually. Recall your basic electricity and electromagnetism: 1) Electric potential [Voltage] is an electroSTATIC charge force [gradient] resulting from the physical separation of 'charge carriers', charges (i.e., electrons on one side of a capacitor, or ions separated in a chemical battery). 2) An electric current [DC = direct current] results when a conductor is place across the potential barrier, and current flows until charge is neutralized (work is done, voltage then goes to zero.) 3) STATIC electricity has no magnetic field associated with it. When direct current flows, it generates a static perpendicular magnetic field. [Similarly, the motion of a magnetic field generates current flow.] When an AC (alternating current) signal flows, it generates a dynamic magnetic field which rotates around the conductor varying 'sinusoidally' [actually helicoidally] from positive to negative and back to positive. It is then able to separate from conductor and propagate into 'space'. Viz, a radio wave for example. 4) An electromagnetic wave propagates at high velocity, with 300,000 km/sec as the average value of 'lightspeed' . As it passes through the media, an antenna for example, it induces oscill- ating magnetic fields and charge flow. In a radio receiver that will result in AC energy absorption providing an audio signal. No net CHARGE results from an AC (phase) signal. 5) Charge is by definition a separation of static 'carriers'. Electrostatic charge can ONLY build up because electrons have the property of assuming ANY VELOCITY, INCLUDING ZERO VELOCITY. They can be static! An electromagnetic wave, by definition and observation, must propagate at high velocity, on the order of 300,000 km/sec. It cannot be confined. [It may be bent, reflected, absorbed, re-admitted, etc. but it conveys NO charge to the surrounding media.'] It is merely a signal, NOT a charge. The Metric Relativity model of a standing lightwave electron -- just described above with the Classical and Compton spectra dimensions -- has transformed its EXTERNAL linear propagation velocity of an electromagnetic wave into an INTERNAL cyclical path and vibration. This creates the total absence of EXTERNAL motion! It, the electron, can therefore be trapped, localized, contained, 'cooled' (i.e., confined, grouped and separated), thereby creating the phenomenon of electric GROUP amplitude potential which we also know as Voltage! Charge, voltage, is the consequence of Creation!! Experience, sensation, history, encapsulated in matter!!! ~~~ You have just witnessed another proof, amongst many in The Undiscovered Physics, of the Discovery of the Creation of Charge, the Creation of Matter. The reason why institutional, 'politically- correct' and especially covert, physics has never been able to do this -- is because it has NEVER FOCUSED ON THE INCARNATION OF EXPERIENCE, OF PHENOMENA, OF PERSONA, OF SPIRIT, OF CONSCIOUSNESS ... THEY FOREVER DENIED THE POSSIBILITY OF THEIR SUCCESS BY DENYING TRUTH, DENYING SPIRITUALITY! BY DENYING EXPERIENCE, KNOWLEDGE, SENSATION, CONSCIOUSNESS, THE EMPIRICAL -- BY INSISTING ON A DEAD PHYSICS AND A DEAD COSMOS -- THEY DENIED THEMSELVES THE OPPORTUNITY OF EVER WITNESSING LIFE AND ITS UBIQUITOUS CREATION ... EVERYWHERE AND EVERY-WHEN! ELECTRICITY, MAGNETISM, LOVE, SONG ... KNOWN BY ALL OUR SACRED RELATIONS! .. . === Subject: Lies, Damn Lies and Statistics posting-account=ajkdngoAAAB2X3SxyQtiBYkcRETphklh 1.1.4322),gzip(gfe),gzip(gfe) Our new applet allows computing Range, Median, Arithmetic, Geometric, and Harmonic means, Mean Deviation, Standard Deviation, Variance, Coefficient of Variance, Moments and Moments ratios for ungrouped data. http://www.thinkanddone.com/ge/Stat1.aspx === Subject: Re: Skybuck's Racing Random Number Generator V3 (algorithm fixed, should be no more many duplicates) > So after I correct this, it produced some amazing result: > Entropy = 1.000000 bits per bit. > Optimum compression would reduce the size of this 640000 bit file by 0 > percent. > Chi square distribution for 640000 samples is 0.32, and randomly would > exceed this value 50.00 percent of the times. > Arithmetic mean value of data bits is 0.5004 (0.5 = random). > Monte Carlo value for Pi is 3.127278182 (error 0.46 percent). > Serial correlation coefficient is -0.000450 (totally uncorrelated = > 0.0). Ok, although the error on Pi seems a bit high. Should get better with more samples. 640000 bits isn't that much test data. === Subject: Re: Skybuck's Racing Random Number Generator V3 (algorithm fixed, should be no more many duplicates) > So after I correct this, it produced some amazing result: > Entropy = 1.000000 bits per bit. > Optimum compression would reduce the size of this 640000 bit file by 0 > percent. > Chi square distribution for 640000 samples is 0.32, and randomly would > exceed this value 50.00 percent of the times. > Arithmetic mean value of data bits is 0.5004 (0.5 = random). > Monte Carlo value for Pi is 3.127278182 (error 0.46 percent). > Serial correlation coefficient is -0.000450 (totally uncorrelated = > 0.0). > Ok, although the error on Pi seems a bit high. Should get better with more samples. 640000 bits isn't that much test data. and skybuck was trying to do this testing with only 100 random numbers and could not figure out what the problem was. with 100 numbers out of 2 billion. anything would look random. you need volumes to show random. Jim P. === Subject: Final call for papers posting-account=H_onOgoAAADscC8Mv_EiDpbOw2hhNnM7 .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) Final call for papers The 2008 MULTICONF (website: www.PromoteResearch.org ) will be held during July 7-10 2008 in Orlando, FL, USA. We invite draft paper submissions and the deadline for paper submission is very close. The event consists of the following conferences. * International Conference on Artificial Intelligence and Pattern Recognition (AIPR-08) * International Conference on Automation, Robotics and Control Systems (ARCS-08) * International Conference on Bioinformatics, Computational Biology, Genomics and Chemoinformatics (BCBGC-08) * International Conference on Enterprise Information Systems and Web Technologies (EISWT-08) * International Conference on High Performance Computing, Networking and Communication Systems (HPCNCS-08) * International Conference on Software Engineering Theory and Practice (SETP-08) * International Conference on Theoretical and Mathematical Foundations of Computer Science (TMFCS-08) The website contains more details. John Edward === Subject: Re: Cholesterol Scale of Measurement? > Jim Langston explained : > Hello guys > hope you can help me with this simple question: > would you be so kind to tell me what is the Scale of Measurement > of Cholesterol ? > is that an interval or ratio or whatever scale? > i found this question in a quiz about statistics. > http://en.wikipedia.org/wiki/Cholesterol > It seems to be Xmg/dL > yes but what i need to know is: > is that a ratio scale, an interval scale or what? > http://www.faqs.org/faqs/diabetes/faq/part1/section-9.html > milligarms/deciliter. Kinda like ppm Parts Per Million. > So a cholesterol reading of 200 would be 200 milligrams of > cholesterol per a deciliter of blood. > Those are called units. What the op means by scale is not clear to > me. > Ah I see what the OP is getting at, having high cholesterol myself I > took the opportunity to answer the question I was interested in. Not > the one asked. Isn't the technical term a concentration. Which is effectively a > ratio. The ratio of specific substance/total mixture. Well, the data provides the answer if the links are followed, whatever the OP meant by Scale. If > 200 mg/dL is good or bad is arbitrary. We say that over 200 is bad. I would say that ~100 mg/dL is good. So twice as much is bad *shrug* It is how much cholesterol is in the blood. It's not a percentage, but could be calculated to a percentage fairly easy. I'm quite sure what an interval scale is so don't know if it is or not. It's simply how much cholesterol (in mmols or mg) are in a certain amount of blood. -- Jim Langston tazmaster@rocketmail.com === Subject: Re: Question about pi > Not too long ago I heard that someone had come up with a method for > determining the Nth digit of pi, so that if I asked for the > 3,240,077,555th digit, there is a formula that could somehow > determine > what that digit would be without actually iterating pi. Is this > true? > If it is true, how is it possible? > Intuitively the answer must be no. > Sorry, but your intuition has failed. > There are a number of formulas, the most well known beinghte BBP > formula, > for determining a specific digit in pi. AFAIK all report a value in > some > power of two ---- BBP reports in jex. > Breifly, these ormulas compute pi as an infinite sum, and by choosing > which > term you want to start with yo ignore all previous digits. >However that term is meaningless without the terms before it ,you cannot >get the nth digit because it depends on the value of previous terms. >I don't fail. > The Bailey-Borwein-Plouffe formula and algorithm ... > And right at the end it admits it doesn't work even if you could > understand > the useless explaination which does even appear to produce > a forumula for the result in the end,or at least not clearly. > Perhaps you could supply it? When you do supply the formular I will prove it is wrong. > That probaly why they did not supply a formula!!! > They knew it could be shown to be garbage!! Its not garbage, and indeed has been used to determine (for example) the > trillionth value in the hexadecimal representation of pi - despite the > fact that we don't know the value of the 999,000,000,000 preceding > hexadecimal digits. I other words.. Despite the fact they cannot prove it is correct......... http://www.sciencenews.org/pages/sn_arc98/2_28_98/mathland.htm === Subject: Probability Quesion. I might answer this myself as I go along but anyway here goes. Say two people roll a dice, the highest number wins you would expect each person to win 50% of the time. But what if one person was luckier and won 55% of the time. You could say he was 5% luckier than average. How extend that to 3 people, you would expect each person to win 33% of the time. How say one persoon was luckier as in the first example, and also the same %age luckier. Would that be 5% luckier, ie 33+5=38%? I think this is too high, I think the answer is he would be 3.3% luckier for it to be the same amount of 'luckierness' than in the first example. Also I think I am into problems here with sample size possibly. Going back to the first example another way is to say 55% is 10% luckier than normal (5/50 times 100).Thats how I get the 3.3% value for the second example. So for four players it would be 2.5%, 5 players=2%, 10 players=10% etc... You see I want to compare how lucky a person is in poker 'showdowns' there would be no problem if it was just 2 players but sometimes there are 3 or maybe 4, the max possible (but rather unlilkely) being 10. Problem X ------------ But lets take 10 players as an example, say one player wins 20% of the time, which is twice as lucky as expect. Now if he got the same amount of luck in 2 player games what would the figures be? It can't be twice because twice 50% is 100% and that don't seem right. ----------------- I think I am into sample size problems here too, I never did statistics unfortunately. Well I pretty much know I am into sample size stuff because I know that over say a million game even winning 5% more than average would be impossible (at least statistically anyway). I guess I need to look at something like standard deviation (yuk) or something like that? OK back to problem X. I can't even guess the right answer so I am going to make it easier and say it is on a sample of 100 to see if that helps. So in the 10 player games normal is 10 in 100 but he wins 20 in 100 So in a 2 player game normal is 50 in 100 but he wins ?? in 100 with the same amount of good luck. What is the correct value of ?? One guess would be 60 but that seems to low, another guess is 100 but that seems way to high. 75 feels about right but I can't explain why. OK I will have a crack at Problem X. I think a better way of thinking is how often would he win 20 in 100 which would have a value 'x' which means it would happen once in every x number of trials so then I would need to work out the value which happened once in every 'x' trials for two players, and this would be the answer!?? Trouble is I am not sure how to do it (yet). I think I could do a simulation though. On my computer :O) I would just get the maximum value which occured in 'x' trials. I better still obtain an average value over repeated trials?? I expect there is a (well known?) equation for this, which I might be able to work out if I spent few years thinking about it!!! Any thoughts/answers? === Subject: Correlation and Regression Analysis Software posting-account=ajkdngoAAAB2X3SxyQtiBYkcRETphklh 1.1.4322),gzip(gfe),gzip(gfe) Our latest online software allows computing Coefficient of Correlation and Equation of Regression Analysis, it displays the results along with scatter diagram and line graph For the Correlation program Visit http://www.thinkanddone.com/ge/Corr.aspx For the Regression program Visit http://www.thinkanddone.com/ge/Reg.aspx Asad === Subject: Singapore math textbooks approved in California posting-account=NgMGSwkAAABJni6NIYF05Yc4jNzXwHf- 1.1.4322),gzip(gfe),gzip(gfe) http://www.marketwire.com/mw/rel_us_print.jsp?id=792848 === Subject: Education Schools: Helping or Hindering Potential Teachers? posting-account=NgMGSwkAAABJni6NIYF05Yc4jNzXwHf- 1.1.4322),gzip(gfe),gzip(gfe) At the web site http://concernedctparent.blogspot.com/ Education Schools: Helping or Hindering Potential Teachers? === Subject: Trig question (sin A)^2 = 2 sin 2A {-180from gives four results as follows -104.0, 0, 76.0, 180 degrees. Where does >the 0 and 180 degrees come from? > By inspection, A = 0 is a root of the equation, hence also any integer multiple of pi radians (180 degrees). Therefore in the range -180 < A <= 180, A = 0 and A = 180 degrees are solutions. To find this solution algebraically, the equation can be written as sin(A) [ sin(A) - 4 cos(A) ] = 0 Then sin(A) = 0 or sin(A) - 4 cos(A) = 0, and A = asin(0) = 0 is a solution, hence also A = 180 degrees. === Subject: Re: Trig question (sin A)^2 = 2 sin 2A {-180from gives four results as follows -104.0, 0, 76.0, 180 degrees. Where does >the 0 and 180 degrees come from? > By inspection, A = 0 is a root of the equation, hence also any integer multiple of pi radians (180 degrees). Therefore in the range -180 < A <= 180, A = 0 and A = 180 degrees are solutions. To find this solution algebraically, the equation can be written as sin(A) [ sin(A) - 4 cos(A) ] = 0 Then sin(A) = 0 or sin(A) - 4 cos(A) = 0, and A = asin(0) = 0 is a solution, hence also A = 180 degrees. Nic.