mm-2439 === Subject: Re: Relative Cardinality > In that case your definition of exist is incomplete. According to that > a number exists if it can compared with all other numbers (and I would > presume you meant all other numbers in existence). 1) A number exists, if a fundamental set or an n-adic representation are available. 2) A number does not (yet) exist but could exist in principle, if existence as fundamental set or n-adic representation is possible. 3) A number cannot exist, if this is impossible. It is not easy do distinguish between the first two cases, in particular because existence is a relative notion. Some number may exist for someone at a certain time but not for another one at another time. The simple case is (3). > But whatever, as > soon as that rational number above is brought in existence, it can be > compared with sqrt(2). No, it cannot be compared with sqrt(2) but only with a rational which consists of the first 10^20 (or even some more) digits of sqrt(2). > So there is not yet a showing that sqrt(2) does > not exist. We know already from Floor(sqrt(2) * 10^10^100) and the same number with exchanged last digit, that they are incomparable by any means. We can prove this by induction for any due number with more digits. Your sophisticated arguing, however, is worthy of a set theorist and Cantor-follower. The same arguing is deluding the antidiagonal as different from every line number. 1) We find the antidiagonal different from every line number *which can be tested *. 2) We believe hat only such line numbers are of interest. 3) The antidiagonal is different from every line number (of interest). Concerning an earlier discussion I append a statement of Cantor's which I just read. His opinion on axioms (here he calls them Hypotheses) in a Letter to Veronese, 17.11.1890. Von Hypothesen ist in meinen arithmetischen Untersuchungen .9fber das Endliche und Transfinite .9fberall gar keine Rede, sondern nur von der Begr.9fndung des Realen in der Natur Vorhandenen. Sie hingegen glauben nach der Metageometer Riemann, Helmholtz und Genossen auch in der Arithmetik Hypothesen aufstellen zu k.9annen, was ganz unm.9aglich ist.... So wenig sich in der Arithmetik der endlichen Anzahlen andere Grundgesetze aufstellen lassen, als die seit Alters her an den Zahlen 1,2,3,... erkannten, ebensowenig ist eine Abweichung von den arithmetischen Grundwahrheiten im Gebiete des Transfiniten m.9aglich. Hypothesen welche gegen diese Grundwahrheiten versto¤en, sind ebenso falsch und widersprechend, wie etwa der Satz 2 + 2 = 5 oder ein viereckiger Kreis. Es gen.9fgt f.9fr mich, derartige Hypothesen an die Spitze irgend einer Untersuchung gestellt zu sehen, um von vorn herein zu wissen, da¤ diese Untersuchung falsch sein muss. Und der Erfolg hat es ja bei ihnen gezeigt, da Sie durch Ihre beklagenswerthen Hypothesen zu dem widersprechenden Begriffe actual unendlich kleiner linearer Gr.9a¤en gef.9fhrt worden sind! === Subject: Re: Relative Cardinality > In that case your definition of exist is incomplete. According to that > a number exists if it can compared with all other numbers (and I would > presume you meant all other numbers in existence). 1) A number exists, if a fundamental set or an n-adic representation > are available. A number exists if the axiom system in which one is working allows it to exist. > 2) A number does not (yet) exist but could exist in principle, if > existence as fundamental set or n-adic representation is possible. Representations, unless built into the axiom system, are irrelevant to the existence of numbers within that system. > 3) A number cannot exist, if this is impossible. WM has no idea of what is mathematically possible or impossible. In another post, he has insisted that there exist distinct real numbers with no rationals between them. This is only one of many evidences of just how little WM knows about mathematics. > But whatever, as > soon as that rational number above is brought in existence, it can be > compared with sqrt(2). No, it cannot be compared with sqrt(2) but only with a rational which > consists of the first 10^20 (or even some more) digits of sqrt(2). Does WM claim that if x < y and y < z one cannot deduce that x < z? > So there is not yet a showing that sqrt(2) does > not exist. > We know already from Floor(sqrt(2) * 10^10^100) and the same number > with exchanged last digit, that they are incomparable by any means. We > can prove this by induction for > any due number with more digits. > The same arguing is deluding the antidiagonal as > different from every line number. Which line number is it the same as? > 1) We find the antidiagonal different from every line number *which > can be tested *. But since one test serves *all* *simulteneously*, that is no restriction. If WM chooses to produce German quotes here, he must either produce also translations inot english (by other than himself) or have them ignored. This is about the current stae of mathematics, and is not limited to what was the case over a century ago. In some senses, the majority of the mathematics currently in existence has come into existence since then. That WM may be stuck in that past does not mean the rest of the world must cater to his faults. === Subject: Re: Relative Cardinality > > In that case your definition of exist is incomplete. According to that > a number exists if it can compared with all other numbers (and I would > presume you meant all other numbers in existence). > > 1) A number exists, if a fundamental set or an n-adic representation > are available. I have yet to see a definition of fundamental set. On the other hand, for all natural numbers an n-adic representation is available, although some of them can not be realised. So either all natural numbers exist, or your definition is circular. > 2) A number does not (yet) exist but could exist in principle, if > existence as fundamental set or n-adic representation is possible. > 3) A number cannot exist, if this is impossible. This is the umpteenth definition of exist I see. Earlier you gave as a definition that it should be comparable... > But whatever, as > soon as that rational number above is brought in existence, it can be > compared with sqrt(2). > > No, it cannot be compared with sqrt(2) but only with a rational which > consists of the first 10^20 (or even some more) digits of sqrt(2). So that number can only be compared to itself? Oh, well, how shocking. > So there is not yet a showing that sqrt(2) does > not exist. > > We know already from Floor(sqrt(2) * 10^10^100) and the same number > with exchanged last digit, that they are incomparable by any means. We > can prove this by induction for > any due number with more digits. I do not want to compare with numbers that do not exist. You try again the same trick as earlier. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Relative Cardinality > > 1) A number exists, if a fundamental set or an n-adic representation > > are available. > > I have yet to see a definition of fundamental set. On the other hand, > for all natural numbers an n-adic representation is available, although > some of them can not be realised. So either all natural numbers exist, > or your definition is circular. Cantor in a letter to Peano, 21.9.1895 Die Definitionen der endlichen Cardinalzahlen sind also diese: (The definition of the finte cadinal numbers is then this) 1 = (a), 2 = (a,b), 3 = (a,b,c), etc. There are always two overlines on the sets, which is Cantor's mark for cardinality. A fundamental set contains at least such a set as given by Cantor in order to explain this point. A fundamental set for the natural number n is a set which contains at least one set which contains exactly n elements. > > > 2) A number does not (yet) exist but could exist in principle, if > > existence as fundamental set or n-adic representation is possible. > > 3) A number cannot exist, if this is impossible. > > This is the umpteenth definition of exist I see. Earlier you gave > as a definition that it should be comparable... Comparability by magitude is the necessary and sufficient condition for a real number to exist. It is impossible to satisfy however, if there is not either a fundamental set or an n-adic representation. > > > But whatever, as > > soon as that rational number above is brought in existence, it can be > > compared with sqrt(2). > > > > No, it cannot be compared with sqrt(2) but only with a rational which > > consists of the first 10^20 (or even some more) digits of sqrt(2). > > So that number can only be compared to itself? Oh, well, how shocking. It can be compared with other existing numbers as well. But it cannot be compared with sqrt(2) because that one is not existing yet if only its first digits exist. > > > So there is not yet a showing that sqrt(2) does > > not exist. > > > > We know already from Floor(sqrt(2) * 10^10^100) and the same number > > with exchanged last digit, that they are incomparable by any means. We > > can prove this by induction for > > any due number with more digits. > > I do not want to compare with numbers that do not exist. You try again > the same trick as earlier. I believed that for you all numbers did exist. === Subject: Re: Relative Cardinality > > 1) A number exists, if a fundamental set or an n-adic representation > > are available. > > I have yet to see a definition of fundamental set. On the other hand, > for all natural numbers an n-adic representation is available, although > some of them can not be realised. So either all natural numbers exist, > or your definition is circular. Cantor in a letter to Peano, 21.9.1895 > Die Definitionen der endlichen Cardinalzahlen sind also diese: > (The definition of the finte cadinal numbers is then this) > 1 = (a), 2 = (a,b), 3 = (a,b,c), etc. > There are always two overlines on the sets, which is Cantor's mark for > cardinality. A fundamental set contains at least such a set as given by Cantor in > order to explain this point. A fundamental set for the natural number n > is a set which contains at least one set which contains exactly n > elements. Then WM's fundamental set is an equivalence class for Cantor cardinality, i.e., a Cantor cardinal. > Comparability by magitude is the necessary and sufficient condition for > a real number to exist. In whose axiom system? These may be WMnumbers, but are not anything else. It is impossible to satisfy however, if there > is not either a fundamental set or an n-adic representation. As every Cantor cardinality has an equivalence class (fundamental set), they all exist. === Subject: Re: Relative Cardinality Nntp-Posting-Host: hera.cwi.nl > > 1) A number exists, if a fundamental set or an n-adic representation > > are available. ... > A fundamental set contains at least such a set as given by Cantor in > order to explain this point. A fundamental set for the natural number n > is a set which contains at least one set which contains exactly n > elements. Ok. Take the set {1,2,3,4,5,...,n}, than {{1,2,3,4,5,...,n}} is a fundamental set for n? In that case n exists. (And this is irrespective of n.) > This is the umpteenth definition of exist I see. Earlier you gave > as a definition that it should be comparable... > > Comparability by magitude is the necessary and sufficient condition for > a real number to exist. It is impossible to satisfy however, if there > is not either a fundamental set or an n-adic representation. Can you prove that? > > But whatever, as > > soon as that rational number above is brought in existence, it can be > > compared with sqrt(2). > > > > No, it cannot be compared with sqrt(2) but only with a rational which > > consists of the first 10^20 (or even some more) digits of sqrt(2). > > So that number can only be compared to itself? Oh, well, how shocking. > > It can be compared with other existing numbers as well. But it cannot > be compared with sqrt(2) because that one is not existing yet if only > its first digits exist. Sorry, you said comparability by magnitude is necessary and sufficient. I claim that if that number exists, I can compare sqrt(2) with it. Rather, I claim that I can compare sqrt(2) with all existing numbers (using existing in your sense). > > So there is not yet a showing that sqrt(2) does > > not exist. > > > > We know already from Floor(sqrt(2) * 10^10^100) and the same number > > with exchanged last digit, that they are incomparable by any means. We > > can prove this by induction for > > any due number with more digits. > > I do not want to compare with numbers that do not exist. You try again > the same trick as earlier. > > I believed that for you all numbers did exist. That is irrelevant. I am trying to use *your* definition of exist, and can not find it is very consistent. I would think that with *your* definition and theorem (comparability is necessary and sufficient) sqrt(2) would exist, because I can compare it with every existing number. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Relative Cardinality > > A fundamental set contains at least such a set as given by Cantor in > > order to explain this point. A fundamental set for the natural number n > > is a set which contains at least one set which contains exactly n > > elements. > > Ok. Take the set {1,2,3,4,5,...,n}, than {{1,2,3,4,5,...,n}} is a > fundamental set for n? In that case n exists. (And this is irrespective > of n.) Three points are the beginning of all abuse of mathematics. No, n may mean a number but it is not a number. > > > This is the umpteenth definition of exist I see. Earlier you gave > > as a definition that it should be comparable... > > > > Comparability by magitude is the necessary and sufficient condition for > > a real number to exist. It is impossible to satisfy however, if there > > is not either a fundamental set or an n-adic representation. > > Can you prove that? I take it for granted as long as nobody is able to compare numbers in another way. > > > > But whatever, as > > > soon as that rational number above is brought in existence, it can be > > > compared with sqrt(2). > > > > > > No, it cannot be compared with sqrt(2) but only with a rational which > > > consists of the first 10^20 (or even some more) digits of sqrt(2). > > > > So that number can only be compared to itself? Oh, well, how shocking. > > > > It can be compared with other existing numbers as well. But it cannot > > be compared with sqrt(2) because that one is not existing yet if only > > its first digits exist. > > Sorry, you said comparability by magnitude is necessary and sufficient. > I claim that if that number exists, I can compare sqrt(2) with it. > Rather, I claim that I can compare sqrt(2) with all existing numbers > (using existing in your sense). You cannot compare sqrt(2) with all existing numbers (in my sense) because sqrt(2) does not exist in my sense. That is a consequence of the fact that it has not a periodic sequence of digits. You always compare a finite seqence of digits with another finite sequence of digits. But sqrt(2) is not among them. > > > > So there is not yet a showing that sqrt(2) does > > > not exist. > > > > > > We know already from Floor(sqrt(2) * 10^10^100) and the same number > > > with exchanged last digit, that they are incomparable by any means. We > > > can prove this by induction for > > > any due number with more digits. > > > > I do not want to compare with numbers that do not exist. You try again > > the same trick as earlier. > > > > I believed that for you all numbers did exist. > > That is irrelevant. I am trying to use *your* definition of exist, and > can not find it is very consistent. I would think that with *your* > definition and theorem (comparability is necessary and sufficient) sqrt(2) > would exist, because I can compare it with every existing number. Again: We now that sqrt(2) has no fundamental set (because it is not a natural) and it has not a complete decimal expansion, because it is not a rational. But to shorten the discussion: You cannot compare sqrt(2) with a number which in order to exist requires all the memory space of the universe because in that case there is no possibility to store sqrt(2) simultaneously. You see, my definition of exist circumvents also your sophisticated attacks. === Subject: Re: Relative Cardinality > Three points are the beginning of all abuse of mathematics. No, n may > mean a number but it is not a number. It is WM who is abusing mathematics here. How can n mean a number without being one? > > Sorry, you said comparability by magnitude is necessary and sufficient. > I claim that if that number exists, I can compare sqrt(2) with it. > Rather, I claim that I can compare sqrt(2) with all existing numbers > (using existing in your sense). You cannot compare sqrt(2) with all existing numbers (in my sense) > because sqrt(2) does not exist in my sense. That is a consequence of > the fact that it has not a periodic sequence of digits. You always > compare a finite seqence of digits with another finite sequence of > digits. But sqrt(2) is not among them. One can compare the continued fraction representations of sqrt(2) with arbitrary rationals to determine their relative size, so that sqrt(2) exists as much as any rational. Again: We now that sqrt(2) has no fundamental set (because it is not a > natural) and it has not a complete decimal expansion, because it is not > a rational. But to shorten the discussion: You cannot compare sqrt(2) with a number > which in order to exist requires all the memory space of the universe > because in that case there is no possibility to store sqrt(2) > simultaneously. You see, my definition of exist circumvents also your sophisticated > attacks. One can compare the square root of any natural to any rational for size, so such square roots exist as much as any rational exists. Similarly for roots of arbitrary rationals and of arbitrary index. So that WM falls on his face again. === Subject: Re: Relative Cardinality !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > >> because sqrt(2) does not exist in my sense. That is a consequence >> of the fact that it has not a periodic sequence of digits. You >> always compare a finite seqence of digits with another finite >> sequence of digits. But sqrt(2) is not among them. > > One can compare the continued fraction representations of sqrt(2) > with arbitrary rationals to determine their relative size, Oh good grief. Just square the rational in question and compare the resulting enumerator with two times the resulting denominator. If it is larger, the fraction is larger than sqrt(2). If it is smaller, the fraction is smaller than sqrt(2). If both are equal, you have a problem. No need to get into continued fractions or similar complications. You make it far too easy for WM to further evade. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Relative Cardinality <85d5pchvjf.fsf@lola.goethe.zz> > > >> because sqrt(2) does not exist in my sense. That is a consequence >> of the fact that it has not a periodic sequence of digits. You >> always compare a finite seqence of digits with another finite >> sequence of digits. But sqrt(2) is not among them. > > One can compare the continued fraction representations of sqrt(2) > with arbitrary rationals to determine their relative size, > > Oh good grief. Just square the rational in question and compare the > resulting enumerator with two times the resulting denominator. If it > is larger, the fraction is larger than sqrt(2). If it is smaller, the > fraction is smaller than sqrt(2). If both are equal, you have a > problem. Not if you only have a finite amount of memory available. If every number, there's no way to calculate its square (since there will be, roughly, twice as many decimal places). WM doesn't define a rational number as a number of the form a/b, either; he insists on having decimal places until it repeats. This is why I've kept on asking WM how he can compute anything in this system; it seems to only allow representation of numbers, not manipulation of numbers. This makes it rather useless ... === Subject: Re: Relative Cardinality !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw [WM:] >> > Comparability by magitude is the necessary and sufficient condition for >> > a real number to exist. It is impossible to satisfy however, if there >> > is not either a fundamental set or an n-adic representation. >> >> Can you prove that? > > I take it for granted as long as nobody is able to compare numbers > in another way. Well, let's see: 99/70 ?<> sqrt(2) | square both sides 9801/4900 ?<> 2 | multiply both sides with 4900 9801 > 9800 So I have established that 99/70 > sqrt(2) without reverting to fundamental sets or n-adic representations. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Relative Cardinality !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > > Comparability by magitude is the necessary and sufficient > > condition for a real number to exist. It is impossible to satisfy > > however, if there is not either a fundamental set or an n-adic > > representation. > > Can you prove that? > > > > But whatever, as soon as that rational number above is > > > brought in existence, it can be compared with sqrt(2). > > > > > > No, it cannot be compared with sqrt(2) but only with a > > > rational which consists of the first 10^20 (or even some > > > more) digits of sqrt(2). > > > > So that number can only be compared to itself? Oh, well, how > > shocking. > > It can be compared with other existing numbers as well. But it > > cannot be compared with sqrt(2) because that one is not existing > > yet if only its first digits exist. > > Sorry, you said comparability by magnitude is necessary and > sufficient. I claim that if that number exists, I can compare > sqrt(2) with it. Rather, I claim that I can compare sqrt(2) with > all existing numbers (using existing in your sense). Well, but that probably would require that all existing numbers can be squared. But if all atoms in the universe are already taken up by the number, there is no room for the square (besides, we know that perfect squares don't exist in the real world, just like perfect circles and triangles). So you have to throw away the original number when squaring, and afterwards it does not exist, so claiming that it is larger or smaller than sqrt(2) would be silly. Something like that. Do I get the fool's cap now for a moment, or is it welded to WM's head? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Complex Random Variables - Help Hi All Q1. What is the joint density (jpdf) of two correlated complex Gaussian random variables A = x + i y B = h + i w Where x and y are iid. h and w are iid, but A and B are correlated ( Need to show the correlation coefficient in the jpdf) Q2. How to convert the resultant jpdf from rectangular to polar coordinates Appreciate your help === Subject: Re: Complex Random Variables - Help > Q1. What is the joint density (jpdf) of two correlated complex Gaussian > random variables A = x + i y > B = h + i w Where x and y are iid. h and w are iid, but A and B are correlated ( > Need to show the correlation coefficient in the jpdf) See my reply to your last posting. Ciao Karl === Subject: Re: Motivation for matrix algebra > Abstract coordinate free linear algebra is one > motivation to define matrices and their addition > and > multiplication in the way they are defined. The > following > facts give a guideline: > > 1. Every finitely generated vector space V (over > some field > K) possesses a finite basis and is thus isomorphic > to the > vector space K^n for some n (called the dimension > of V). > > I learned that finite vector spaces will have a > ve a finite basis. When > you refer to over some field K, do you mean a field > to be like > whether real numbers, integers are used for values of > the matrices? > Not exactly. What I mean is this: a vector space V over a field K is a set of things (called vectors) that one can add such that the usual rules of addition are valid. (Of course one can make this precise writing down the axioms for addition.) Moreover one can multiply vectors v with elements of the field K (called scalars) such that the following rules hold: (a+b)v=av+bv (ab)v=a(bv) a(v+w)=av+aw where a,b are elements of K, v,w are elements of V. > 3. The set Hom(V,W) of all linear maps f:V-->W > forms > a vector space itself if one defines addition of > linear > maps pointwise, and multiplication with a scalar as > well. > > Using 2 for fixed bases B,C the vector space > Hom(V,W) > is isomorphic to the space M(mxn,K) of mxn matrices > with entries in K, where matrix addition is defined > in > the usual way. > Indeed this fact can be considered as the reason > why > addition is defined in the way it is. > > 4. The set End(V)=Hom(V,W) is a ring: addition is > defined > as in 3 and multiplication is the composition of > linear > maps. > Using 2 for fixed basis B,C this ring is ismorphic > to the > ring of square matrices M(nxn,K) with entries in K, > where the multiplication is defined in the usual > way. > Again this fact can be considered as the reason to > define > matrix multiplication in the way it is defined. > > I'm unable to follow the above argument since > since I've been taught > linear algebra from the applied sciences perspective > and so do not know > what Hom(V,W), End(V), M(mxn, K), refer to. However, > I would guess that > M is the set of all mxn matrices over the field K, > and Hom(V,W) is the > set of all homorphisms from V to W (but I've not > actually been taught > homorphisims). > That's right. Anyway I defined these sets (in words) in my original post. > > Remark: it is quite common among students to treat > matrices as if they were linear maps. But this > point of > view easily leads to a lot of confusion and to > unnecessary work. Rather one should think of > matrices > as coordinate descriptions of linear maps. Coordinate descriptions of linear maps because the > the matrices depend > on which basis used? Yes. > I'd appreciate a further > elaboration because I may > be one of the students you refer to. > Here is an example that hopefully shows the point: Let V be the set of all polynomial of the form Ax+B, where A,B are the real coefficients, and x is the variable of the polynomial. So V consists of all polynomials in one variable of degree <=1. V is a vector space (over the reals), because you can add polynomials of degree <=1 and get a polynomial of the same type. Also you can multiply with a real number. Now recall your analysis course and consider differentiation: the first derivative (Ax+B)' of a polynomial in V equals the constant polynomial A, right? Consider the map: D: V-->V, p-->p' that maps a polynomial in V to its first derivative. This is a linear map as you can easily check. (However, no matrix appearing here.) What does a matrix description of the map D look like? Ok, choose some basis of V. The obvious one is (1,x). Use this basis >>on both sides of the map V-->V<<, that is B=C in the notation of my original post. The coordinate vector of p=Ax+B with respect to (1,x) is the vector (A,B). So in particular the coordinate vectors of 1 and x are (0,1) and (1,0) respectively. The coordinate vector of d(Ax+B)=A is (0,A). matrix D representing the linear map d with respect to the basis (1,x) is the 2x2 matrix 0 0 0 1 === Subject: Re: Steven Cullinane is a Crank <19293059.1121879793457.JavaMail.jakarta@nitrogen.mathforum.org> Crank. === Subject: Re: Steven Cullinane is a Crank <23040840.1121793461996.JavaMail.jakarta@nitrogen.mathforum.org> > Actually, I can do better than that. If I remember correctly, all > finite simple groups are known to be generated by at most *two* > elements. I think this is a theorem of Neilsen-Schrier (correct me if > I'm wrong). At any rate, I can find 2 elements that generate the group > G. Same way I can find 2 generators each for H,I,J,K since they are all > simple groups. Yes, check Gorenstein's book on the classification of finite simple groups. It follows from the classification that every finite simple group is generated by two elements. Not sure if this follows from the Nielsen -Schrier theorem. I think Nielsen proved that every finitely generated subgroup of a free group is free. Schrier proved that the finitely generated hypothesis is not necessary, thus proving that every subgroup of a free group is free. Perhaps this is used to prove that all finite simple groups are 2-generated, but I'm not sure. But, coming back to your assertion, can you come up with any relations between your generators a,b,c? This could be non trivial! The word problem is known to be pretty damn hard! > BTW, do you mind calling my theorem something else? RAT sounds bad. You could arrange your array in a circle, and we'll call it CAT. === Subject: Re: Steven Cullinane is a Crank <23040840.1121793461996.JavaMail.jakarta@nitrogen.mathforum.org> Some notes on trying to find relations between crankbuster's generators a,b,c: I should qualify my earlier remark about the word problem; It is known that there *exists* a finitely presented group with an unsolvable word problem. That does not mean that finding relations between a,b,c is necessarily hard. It may or may not be hard. If you can find 2 generators for G (and for H,I,J,K), that would certainly be pretty impressive. === Subject: Re: Steven Cullinane is a Crank > Some notes on trying to find relations > between crankbuster's generators a,b,c > .... > If you can find 2 generators for G > (and for H,I,J,K), that would > certainly be pretty impressive. They aren't crankbuster's generators, they're Hall's. I can't find a two-generator presentation for the large Mathieu group, but the University of Birmingham can. See http://web.mat.bham.ac.uk/atlas/v2.0/spor/M24/. === Subject: Re: Steven Cullinane is a Crank <14156272.1121954428272.JavaMail.jakarta@nitrogen.mathforum.org> > They aren't crankbuster's generators, > they're Hall's. You idiot. These are Mathieu's original generators from his thesis in 1860. Doesn't your little cheat book from which you plagiarize tell you that? > I can't find a two-generator presentation > for the large Mathieu group, No, you are not competant to do any real math. > but > the University of Birmingham can. See > http://web.mat.bham.ac.uk/atlas/v2.0/spor/M24/. Doh. The classification was completed 20 years ago. === Subject: Re: Steven Cullinane is a Crank > You idiot. These are Mathieu's original > generators from his thesis in 1860. > Doesn't your little cheat book from > which you plagiarize tell you that? Congratulations. This is the first sign you have given of being anything other than an ignorant brat. Hall's Theory of Groups suggests a way to derive the generators, but does not say they are Mathieu's... I did consider remarking that those generators may not have originated with Hall, but that would have given you another opening to accuse me of crankery. You are correct in suggesting that my knowledge of the Mathieu groups is limited. My own discoveries concerned the geometry of the 4x4 array (and some more general results) and I did not know they were related to the Mathieu group and the 4x6 array of Curtis until he in March 1979. I do not claim to be competent to do real math now... I am too old, and my education falls short of the doctoral level. But it is not true to say I have never done ANY real math. === Subject: Re: Steven Cullinane is a Crank <14156272.1121954428272.JavaMail.jakarta@nitrogen.mathforum.org> Steven Cullinane is a Crank by Steven Cullinane. === Subject: Re: Steven Cullinane is a Crank > The Rectangular Array Theorem > by your humble superhero, Crankbuster > .... > Define (on my whim and fancy) the permutations > a = (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16, > 17,18,19,20,21,22) > b = (2,16,9,6,8)(4,3,12,13,18)(10,11,22,7,17) > (20,15,14,19,21) > c = (0,23)(1,22)(2,11)(3,15),(4,17)(5,9)(6,19) > (7,13)(8,20)(10,16)(12,21)(18,14) > > Let G be the group generated by > the permutations {a,b,c}... crankbuster's theorem: > But, coming back to your assertion, > can you come up with any relations > between your generators a,b,c? > This could be non trivial! I don't know about relations between these generators, but I do know plagiarism when I see it. The permutations defined on crankbuster's whim and fancy may be found on page 81 of The Theory of Groups, by Marshall Hall, Jr., The Macmillan Company, 1959, 11th printing, 1970. === Subject: Re: Steven Cullinane is a Crank <20566172.1121943858153.JavaMail.jakarta@nitrogen.mathforum.org> > [blah blah] You dirty low down crank, you are jealous of my theorem! Why don't we tell everybody where you plagiarize from. Everbody look at this beautiful site from a british artist http://www.mlyon.com/index.htm?personal/Tiling/tiling.htm~mainFrame Bet Steven Crank-in-Cheif Cullinane never took the artist's permission before blatantly stealing his ideas. Worse, making the artist's ideas look bad by adding his own idiotic nonsense about affine geometry. === Subject: Re: Steven Cullinane is a Crank <20566172.1121943858153.JavaMail.jakarta@nitrogen.mathforum.org> > Everbody look at this > beautiful site from a british artist > > http://www.mlyon.com/index.htm?personal/Tiling/tiling.htm~mainFrame > > Bet Steven Crank-in-Cheif Cullinane never took the artist's > permission before blatantly stealing his ideas. Worse, making the > artist's ideas look bad by adding his own idiotic nonsense about affine > geometry. tilings is really interesting. === Subject: Re: Steven Cullinane is a Crank >> Everbody look at this >> beautiful site from a british artist.... > Neither crankbuster nor Stewart indicate that the link is taken from Diamond Theory, http://m759.freeservers.com. Mike Lyon's tiling section lists work beginning in 1993. Diamond Theory dates from 1975-1976. === Subject: Re: To Google: > >> >>> Finding myself contact the customer support of a company which I did not >>> sign a sevice agreement with, violates consumer rights. I have another >>> preferred newsgroup reader which I pay for as part of my internet >>> service provider package. Google intrudes on my consumer rights and >>> represents illegal and even anticompetitive business aptitudes. >>> >> >> That's hyper-modern ultra-commercial greedy greasy globalization. >> I've had my account on ebay terminated for lack of response >> regarding intrusions into my account that I never established >> and often posts from Pay-Pal about my non-existent account. > I've never had an account on ebay, but I get a huge number of emails > telling me that they are going to close my account unless I connect to > some web site (which usually has weird numbers like 10.4.5.6 in them) > and cough up all my personal details. Actually I get quite a number of > these kinds of emails from various other banks as well. One time I even > got an email like this from a bank where I have an account. I have to > admit that I didn't follow it up, but fortunately for me my bank didn't > seem to notice, and my account is still working fine. Oh, and while I am at it, I also get quite a lot of emails from Nigeria > offering me huge sums of money. But then again, I guess I shouldn't be > telling you about this, because they ask me to keep their emails > confidential. Other than that, I do get the odd email offering me drugs at cut rates, > and quite a number written in what I think is Chinese or Japanese. even better are the ones offering to help me get a larger penis. I those come to has my name attached to it, which is quite obviously a female name. Luckily, my (primary) .edu acct is (so far) unfound by k wallace Oh, and once in a while I sometimes get an email from a friend or a > colleague. === Subject: Re: Estimation of extinction for thinned Poisson process > > >> The thinned process is Poisson, with modified rate equal to r = >> lambda*p. > I know this. Read my posting. > > Therefore, to >> estimate p (presumably through observations?) it is enough to estimate >> a Poisson rate r through observation, then divide by the known value of >> lambda. > I know this. Read my posting. The details (i.e. distribution of the > estimator, confidence intervals) I am interested in. > > As for the rest of your question: how do we know what you mean >> by not too good in maths? How much maths do you (or your client) >> know? > To repeat, I am looking for a simple explanation of the detailed > estimation process for somebody not too good in maths, i.e. almost no > knowledge of the involved maths and stats. > > Have you actually looked at some of the hits obtained in Google? > > >> would bet that some of them are pretty basic lecture notes at an >> elementary level. > > Fine, give me the link and I will be very grateful to you. > Elsewhere I would be grateful to you to shut up giving me useless advice. Well you can't hope for professional service on a voluntary newgroup. The problem is, there is hardly any maths to explain here. Measure the observed intensity by counting events over some time inverval. This count will have an error standard deviation of sqrt(n) where n is how many events you have counted. This is the best that can be done statistically. End of maths. Adjust the count by dividing by the observation time then divide by the known source intensity. This gives r to within some error. The standard deviation of the estimate of r will be r/ sqrt(n). === Subject: Re: Transcendental Dimensions Why not plainly showing the case you've got in mind? Looking forwards, Alain. === Subject: Re: Transcendental Dimensions > Hi While playing with fractals, I noticed that I am usually able to > create an equation, such that the dimension of the fractal is a root of > that equation... I am therefore skeptical regarding any object having transcendental > dimensions. I mean how would we go about proving that the dimension of > an object is not the root of any integer polynomial... maybe I am wrong, & if I am I would like to know some examples of > objects with transcendental dimensions. Gsax > Try the standard Cantor middle-thirds set. Its Hausdorff dimension is log(2)/log(3), which is transcendental, see this page: http://numbers.computation.free.fr/ Constants/Miscellaneous/classification.html#Hardy (the two lines need to be reattached to make a real URL). The author ascribes the proof that log(3)/log(2) is transcendental to Hardy and Wright. Dale. === Subject: The Proper Way to put a long URL in a NG post > http://numbers.computation.free.fr/ > Constants/Miscellaneous/classification.html#Hardy > > (the two lines need to be reattached to make a real URL). > Most of us use newsreaders that automatically wrap words, phrases, and sentences at or near a certain column and URLs are particularly prone to getting clobbered this way. BUT ... at least for Outlook Express, if the URL is enclosed in angle-brackets i.e. < ... > Like so then it doesn't matter how many lines the URL gets split across, it will show up as a single link in the reader. Norm === Subject: Re: Transcendental Dimensions > > > Try the standard Cantor middle-thirds set. Its Hausdorff dimension > is log(2)/log(3), which is transcendental, see this page: > > http://numbers.computation.free.fr/ > Constants/Miscellaneous/classification.html#Hardy > > (the two lines need to be reattached to make a real URL). > > The author ascribes the proof that log(3)/log(2) is transcendental > to Hardy and Wright. > > Dale. Hi everyone, any other objects with transcendental dimensions.. Also I read somewhere that most of the numbers are transcendental,... it is funny then that it takes so much trouble to produce their examples.. Gsax === Subject: Re: Transcendental Dimensions > any other objects with transcendental dimensions.. In some sense, most objects with non-integer dimension can be expected to have transcendental dimension. Dimensions of fractals often have the form (log p / log q) for some integers p and q. If the ratio isn't obviously a rational then it's irrational and hence transcendental. If log p / log q is a rational then p^n = q^m with m > 1, which tends to be pretty obvious. That's a fairly uncommon relationship, requiring that p and q be different powers of a common base. If that relationship doesn't hold, then log p / log q is transcendental. > Also I read somewhere that most of the numbers are > transcendental,... it is funny then that it takes so much trouble > to produce their examples.. It's easy to produce unlimited examples of transcendental numbers. It's also easy to come up with unlimited examples of numbers of unknown transcendentality. - Tim === Subject: Re: Transcendental Dimensions > Dimensions of fractals often have the form (log p / log q) for some > integers p and q. If the ratio isn't obviously a rational then it's > irrational and hence transcendental. If log p / log q is a rational > then p^n = q^m with m > 1, which tends to be pretty obvious. That's a > fairly uncommon relationship, requiring that p and q be different > powers of a common base. Irrational is not the same as transcendental. (A number can be irrational without being transcendental.) -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Transcendental Dimensions > > Dimensions of fractals often have the form (log p / log q) for some > integers p and q. If the ratio isn't obviously a rational then it's > irrational and hence transcendental. If log p / log q is a rational > then p^n = q^m with m > 1, which tends to be pretty obvious. That's a > fairly uncommon relationship, requiring that p and q be different > powers of a common base. > > Irrational is not the same as transcendental. > (A number can be irrational without being transcendental.) Yes -- but this is not a general case. Look up Gelfond-Schneider Theorem. If p and q are integers (and greater than 1), then log(p)/log(q) is either an integer or it is transcendental. Alternatively, if log(p)/log(q) is algebraic, then it must be an integer. Since all integers are rational, if log(p)/log(q) is not rational (i.e. if it is irrational), then it must indeed be transcendental. Michel. === Subject: Re: Transcendental Dimensions > any other objects with transcendental dimensions.. I would have thought it was fairly straightforward to construct a subset of [0,1] with any Hausdorff dimension between 0 and 1. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Transcendental Dimensions > Also I read somewhere that most of the numbers are transcendental,... Algebraic numbers are countable and therefore transcendental numbers have the same cardinal as the real numbers. > it is funny then that it takes so much trouble to produce their > examples.. It's very easy to create transcendental numbers. What is hard is to prove that a specific number is transcendental. Jose Carlos Santos === Subject: Re: Transcendental Dimensions >> Also I read somewhere that most of the numbers are transcendental,... > Algebraic numbers are countable and therefore transcendental numbers > have the same cardinal as the real numbers. >> it is funny then that it takes so much trouble to produce their >> examples.. > It's very easy to create transcendental numbers. What is hard is to > prove that a specific number is transcendental. The numbers that can be proved transcendental are countable. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: proved transcendental numbers (was: Transcendental Dimensions The numbers that can be proved transcendental are countable. Aren't there uncountably many Liouville numbers? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> The numbers that can be proved transcendental are countable. > >Aren't there uncountably many Liouville numbers? Of course there are, and they're all transcendental. Which in a sense contradicts what he said. Or maybe not: A proof is (or can be written as) a finite sequence of English words, and so there are only countably many proofs, hence only countably many provably transcendental numbers. Most Liouville numbers x are 'indescribable'; there exist only countably many Liouville numbers x such that there exists a proof of the fact that x is uncountable. ************************ === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > >>>> The numbers that can be proved transcendental are countable. >> >>Aren't there uncountably many Liouville numbers? > >Of course there are, and they're all transcendental. >Which in a sense contradicts what he said. Or maybe >not: > >A proof is (or can be written as) a finite sequence of >English words, and so there are only countably many >proofs, hence only countably many provably transcendental >numbers. Most Liouville numbers x are 'indescribable'; >there exist only countably many Liouville numbers x >such that there exists a proof of the fact that x is >uncountable. I am not contradicting any of what you say, but it is very easy to describe precisely an uncountable collection of transcendentals. Take any enumeration of the algebraic numbers in [0,1] as decimal expansions, and perform the Cantor diagonal construction, changing every 1 or 2 to either a 3 or a 4, and every other digit to either a 1 or a 2. Derek Holt. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions <210720050911387510%edgar@math.ohio-state.edu.invalid> > I am not contradicting any of what you say, but it > is very easy to describe precisely an uncountable > collection of transcendentals. Take any enumeration > of the algebraic numbers in [0,1] as decimal expansions, > and perform the Cantor diagonal construction, changing > every 1 or 2 to either a 3 or a 4, and every other digit > to either a 1 or a 2. Perhaps more relevant to the topics in this thread, there exists a subset of the reals with a transcendental Hausdorff dimension that contains only transcendental numbers. In fact, if you take any nowhere dense subset of the reals, in particular any nowhere dense subset that has a transcendental Hausdorff dimension, and if you randomly translate it on the real line, then with probability 1 the resulting set will contain only transcendental numbers (see the URL below). In other words, once you construct a Cantor set with transcendental Hausdorff dimension, all you have to do is move it at random on the number line and there'll be a 100 per cent chance of getting a transcendental dimension of transcendental numbers. Indeed, there's a 100 percent chance that you moved it a transcendental distance when you did this. Dave L. Renfro === Subject: Re: Transcendental Dimensions > http://numbers.computation.free.fr/Constants/Miscellaneous/classification.ht m l#Hardy > The author ascribes the proof that log(3)/log(2) is transcendental > to Hardy and Wright. Not exactly, the author merely says that *a* proof can be found in the cited 1979 publication by Hardy & Wright. The result itself is a trivial corollary of a theorem published independently in 1934 by Gelfond and Schneider: a^b is transcendental if a is algebraic and not 0 or 1, and b is an algebraic irrational. Let a = 2, b = log(3) / log(2). - Tim === Subject: Re: Transcendental Dimensions >> http://numbers.computation.free.fr/Constants/Miscellaneous/classification.ht m l#Hardy >> The author ascribes the proof that log(3)/log(2) is transcendental >> to Hardy and Wright. > >Not exactly, the author merely says that *a* proof can be found in the >cited 1979 publication by Hardy & Wright. There were no 1979 publications by Hardy & Wright. Lee Rudolph === Subject: Re: Transcendental Dimensions >>Not exactly, the author merely says that *a* proof can be found in the >>cited 1979 publication by Hardy & Wright. > There were no 1979 publications by Hardy & Wright. 1979 is the year in which the fifth edition of their book was published. Jose Carlos Santos === Subject: Re: Transcendental Dimensions > >>>Not exactly, the author merely says that *a* proof can be found in the >>>cited 1979 publication by Hardy & Wright. >> There were no 1979 publications by Hardy & Wright. > >1979 is the year in which the fifth edition of their book was published. ...which was in no reasonable sense a publication by Hardy & Wright (in the usual sense of &). Here is part of Apostol's Mathematical Reviews note on that edition. After Hardy's death in 1947, Professor Wright refrained from more thorough revisions for fear of disturbing Hardy's unique style. The most important changes in subsequent editions were the addition of an elementary proof of the prime number theorem in the third (1954) edition and an index of names in the fourth (1960) edition. Each new edition has also seen minor changes such as simplification of some proofs and updating of the Notes. A new Theorem 272 in the fourth edition evaluates Ramanujan's sum in terms of the Moebius function and the Euler totient. The main changes in this latest edition are in the Notes and in the index of names. A short two-page appendix at the end of the book mentions some recent advances concerning prime numbers, such as the work of Davis, Matijaseviv c, Putnam and Robinson on constructing a polynomial $R(x_1,cdots,x_k)$ whose positive values are primes for nonnegative integer values of $x_1,cdots,x_k$, and recent progress toward the solution of Goldbach's conjecture. An interesting observation is that none of the unsolved problems on primes mentioned in the first edition has been completely settled in the intervening 40 years. I see no mention of the Gelfond-Schneider theorem (or its special case previously mentioned) there, and deduce that the correct scholarly citation for the Hardy & Wright proof of the G-S theorem (or its special case) would have been to the first, 1938, edition. Sheesh. If a person can't be a pedant in sci.math, where *can* a person be a pedant? Lee Rudolph === Subject: Re: Transcendental Dimensions > I am therefore skeptical regarding any object having transcendental > dimensions. I mean how would we go about proving that the dimension > of an object is not the root of any integer polynomial... You mean like (ln 3 / ln 2)? It's known to be transcendental by the Gelfond-Schneider theorem. - Tim === Subject: prime.... hello...doctor~ for all ab in Z, if p|ab, p|a or p|b. => integer p>1 is prime. ---------------------------------------------- i think......... i can show that if p is composite number, there is some ab such that p|ab => p not divide a and p not divide b. namely, p=4, a=2, b=6, ab=12 so, 4|12 => 4 not divide 2 and 4 not divide 6 is this a no problem ?? so, i need your advice. thank you very much. === Subject: Re: prime.... >hello...doctor~ > >for all ab in Z, >if p|ab, p|a or p|b. >=> integer p>1 is prime. > >---------------------------------------------- >i think......... > >i can show that >if p is composite number, >there is some ab such that p|ab => p not divide a and p not divide b. >namely, p=4, a=2, b=6, ab=12 >so, 4|12 => 4 not divide 2 and 4 not divide 6 > >is this a no problem ?? So you are hoping to prove the contrapositive. That's fine. But examples aren't really good enough. What is your definition of prime? Presumably something like: For a positive integer p, p is a prime is and only if p>1 and p = nm -> n=+/- 1 or m=+/- 1. If this is the case, then if p>1 is not a prime, then there exists a factorization p = ab with a,b>1. Then p|ab, but since 11 is not a prime, then there exists a,b in Z such that p|ab and p does not divide a and p does not divide b). -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: prime.... >>hello...doctor~ >> >>for all ab in Z, >>if p|ab, p|a or p|b. >>=> integer p>1 is prime. >> >>---------------------------------------------- >>i think......... >> >>i can show that >>if p is composite number, >>there is some ab such that p|ab => p not divide a and p not divide b. >>namely, p=4, a=2, b=6, ab=12 >>so, 4|12 => 4 not divide 2 and 4 not divide 6 >> >>is this a no problem ?? > >So you are hoping to prove the contrapositive. That's fine. But >examples aren't really good enough. > >What is your definition of prime? Presumably something like: > > For a positive integer p, p is a prime is and only if p>1 and > p = nm -> n=+/- 1 or m=+/- 1. > >If this is the case, then if p>1 is not a prime, then there exists a >factorization p = ab with a,b>1. Then p|ab, but since 1cannot be that p|a or that p|b. This proves the contrapositive (if p>1 >is not a prime, then there exists a,b in Z such that p|ab and p does >not divide a and p does not divide b). Yes, Arturo's reply nails it. To emphasize the first point, proving the contrapositive is a fine way to do this proof, but when you write out the statement of the contrapositive, it involves a variable p, assumed composite, and you are trying to prove something about every such p, so verifying it for one example, p=4 is not enough. If a statement has an infinite number of case, verifying it for one case does not prove it. On the other hand, to disprove it, all you need to do is to find one case that fails (a single counterexample). quasi === Subject: Re: prime.... > hello...doctor~ > > for all ab in Z, > if p|ab, p|a or p|b. > => integer p>1 is prime. Let p = xy. Then consider xy|xy. If xy|x or xy|y then xy must be x or y and the other 1. Therefore no composites satisify the property. Let p be prime. If p|ab, it follows from unique factorizations of a and b, that p|a or p|b. Therefore all primes satisify the property. Combine the two and the property implies p is prime. === Subject: Re: prime.... > for all ab in Z, > if p|ab, p|a or p|b. => integer p>1 is prime. > That's often the definition of prime. p in N1, for all a,b in Z, (p | ab ==> p|a or p|b) ==> p irreduciable. If p = rs, then p | rs. p|r or p|s; r /= 0 /= s case p|r: some k with r = pk = rsk; sk = 1; s is unit case p|s: similar. Put it all together now, p is irredicable. === Subject: Double limits at the infinity. Given function f(x,y) I would need to know under which conditions over f, the limit of f as x,y go to +infinity does not depend on the way to go, for instance lim_{y-->inf} lim_{x-->inf) f(x,y) = lim_{x-->inf) lim_{y-->inf} f(x,y) = lim_{x-->inf} f(x,x) = I guess that the above relations are true if the limit exists, f is continuous and df(x,y)/dx and df/dy always go to zero as x and y go to +inf (in any way). Is this correct? Jes.9cs. === Subject: Re: Double limits at the infinity. >Given function f(x,y) I would need to know under which conditions over >f, the limit of f as x,y go to +infinity does not depend on the way to >go, for instance > >lim_{y-->inf} lim_{x-->inf) f(x,y) = >lim_{x-->inf) lim_{y-->inf} f(x,y) = >lim_{x-->inf} f(x,x) = A sufficient condition is that L = lim_{(x,y) -> (infty,infty)} f(x,y) exists, i.e. for every epsilon > 0 there is N such that whenever x > N and y > N, |f(x,y) - L| < epsilon. >I guess that the above relations are true if the limit exists, f is >continuous and df(x,y)/dx and df/dy always go to zero as x and y go to >+inf (in any way). >Is this correct? No (if by the limit you mean the three limits you mentioned). Consider, say, f(x,y) = arctan((x-y)/sqrt(x+y)). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Double limits at the infinity. >Given function f(x,y) I would need to know under which conditions over >f, the limit of f as x,y go to +infinity does not depend on the way to >go, for instance > >lim_{y-->inf} lim_{x-->inf) f(x,y) = >lim_{x-->inf) lim_{y-->inf} f(x,y) = >lim_{x-->inf} f(x,x) = > > >I guess that the above relations are true if the limit exists, f is >continuous and df(x,y)/dx and df/dy always go to zero as x and y go to >+inf (in any way). > >Is this correct? I'm not sure what you mean by the limit exists. Of course if you mean if lim_{(x,y) -> (inf,inf)} f(x,y) exists then the answer is yes, all the above are equal. But that's probably not what you mean, since that's pretty obvious, and also has nothing to do with the conditions on the derivatives. If you mean >I guess that the above relations are true if lim_{y-->inf} lim_{x-->inf) f(x,y) exists, f is continuous and df(x,y)/dx and df/dy always go to zero as x and y go to +inf (in any way) then the answer is no. >Jes.9cs. ************************ === Subject: Re: Double limits at the infinity. I'm not sure if this is related to what you asked, but there's a condition that implies the existence of the double limit. Suppose that for every x there exists Ly(x) = lim (y -> oo) (f(x,y) and that there exists L = lim (x -> oo) Ly(x). In addition, suppose the function of functions f(x,y) converges uniformly to Ly, in the sense that, for every eps>0, there exists a real K, depending only on eps, such that, if y > K, then |f(x,y) - Ly(x)| < eps for every real x. This implies the double limit exists and equals L, that is, for every eps >0, there exists A such that x > A and y> A implies |f(x,y - L| < eps. Im not quite sure, but I think this also implies the existence of the function Lx(y) = lim ( x-> oo) f(x,y) and that lim (y -> oo) Lx(y) = L. Amanda escreveu: > >Given function f(x,y) I would need to know under which conditions over >f, the limit of f as x,y go to +infinity does not depend on the way to >go, for instance > >lim {y-->inf} lim {x-->inf) f(x,y) = >lim {x-->inf) lim {y-->inf} f(x,y) = >lim {x-->inf} f(x,x) = > > >I guess that the above relations are true if the limit exists, f is >continuous and df(x,y)/dx and df/dy always go to zero as x and y go to >+inf (in any way). > >Is this correct? > > I'm not sure what you mean by the limit exists. > > Of course if you mean if lim {(x,y) -> (inf,inf)} f(x,y) > exists then the answer is yes, all the above are equal. > But that's probably not what you mean, since that's > pretty obvious, and also has nothing to do with the > conditions on the derivatives. > > If you mean > >I guess that the above relations are true if > lim {y-->inf} lim {x-->inf) f(x,y) exists, f is continuous > and df(x,y)/dx and df/dy always go to zero as x and y go to > +inf (in any way) > > then the answer is no. >Jes.9cs. > ************************ === Subject: Re: Teaching Special Relativity - What Would You Like To See Next? <%uDDe.1905$fx4.861@newssvr17.news.prodigy.com> > Spacetime is not endowed with coordinates. > Coordinates mean absolutely _nothing_ > without physical phenomena to define them. As Albert Einstein said: Any fool can make things bigger and more complex. It takes a touch of genius--and a lot of courage--to move in the opposite direction. My approach to special relativity isn't meant for religious relativists and chief editors of prestigious physics journals who are confused by special relativity in 1+1 dimensions. It's for little children who are capable of conceptualizing time with sliding rulers. http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Teaching Special Relativity - What Would You Like To See Next? Eugene Shubert: >> Spacetime is not endowed with coordinates. >> Coordinates mean absolutely _nothing_ >> without physical phenomena to define them. > >As Albert Einstein said: Any fool can make things bigger >and more complex. It takes a touch of genius--and a lot >of courage--to move in the opposite direction. > >My approach to special relativity isn't meant for religious Your ``approach to special relativity'' isn't meant for anyone who wants to understand the subject. >relativists and chief editors of prestigious physics >journals who are confused by special relativity in 1+1 >dimensions. It's for little children who are capable of >conceptualizing time with sliding rulers. Since you are confused by special relativity, you should refrain from confusing others. === Subject: Movement of a circle in contact with a rotating ellipse. Hello everyone. I am having some problems with this geometry problem I have been handed. A simple diagram will help my subsequent description: o o Two circle fixed in position 0 Ellipse o Circle The top two circles are fixed. The bottom circle can only move up and down. The ellipse is always in contact with all the circles. The problem is to describe the motion of the bottom circle as the ellipse is turned. Any help would be very much appreciated. Lee Russell === Subject: Re: Movement of a circle in contact with a rotating ellipse. >Hello everyone. > >I am having some problems with this geometry problem I have been >handed. > >A simple diagram will help my subsequent description: > >o o Two circle fixed in position > 0 Ellipse > o Circle > >The top two circles are fixed. The bottom circle can only move up and >down. The ellipse is always in contact with all the circles. The >problem is to describe the motion of the bottom circle as the ellipse >is turned. > If I understand your question correctly, your ellipse is not just turning (rotating), but its center is also moving to maintain tangency to the two circles, right? --Lynn === Subject: Re: How she works <99mdne_1JJKvvkTfRVn-og@whidbeytel.com> > Essentially: nature, or the universe operates in cycles; where there is > continuous aggregation, and disaggregation... > > The word 'essentially' is always almost followed with half-baked bull. > > - where George Lewis leSage's dispersed radiation gravitates... > > I have corrected you several times before... the name was 'Georges-Louis > LeSage' ... no 'Lewis' in the name! > > relative to each other, and then when they become large enough and > acquire enough spin, and angular momentum, they burst apart; > disaggregating suddenly like an overspun flywheel; to begin another > cycle. > > What you describe here is not an idea that LeSage is known for. I think you > are making it up. He came up with the idea of the kinetic theory of gravity, > of empirical evidence or experiment. You are a mindless freak Don... That's your worthless opinion(;^) Don === Subject: Re: compact operators, convergence pointwise / w.r.t. operator norm > >> Because the identity is not compact. > >Yes, and there exist compact injective T, e.g. T(x_n) := (x_n/n) >on ell_2. For injective T we have T^r to identity pointwise. >But not necessarily in norm, because else the identity would be >compact. > > (T^r _is_ a norm-continuous >> function of r for r > 0, right? Just a guess based on the >> heuristic...) > >Because the continuous functional calculus is, well, continuous. Heh-heh. >(Is it called continuous functional calculus because it is >continuous, or because it is defined on the space of continuous >functions defined on the spectrum of T?) I believe it's the latter, although the former would be more fun, as above. >Markus ************************ === Subject: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? Does String Theory have any postulates or laws? If you think it does, please share & please post them at: http://physicsmathforums.com Where we're handing out the physicsmathforums.com Nobel prize in physics to the first person who can name a postulate or law of string theory. Moving Dimensions Theory & On The Advancement Of Physics Physics has been furthered far more often by a rugged individual acknowledging the simple and obvious in a pursuit of the truth than book-keepers-in-training playing games in the abstruse in pursuit of tenure. The advancement of physics has ever depended far more on logic, reason, and Truth than government grants, tenure, group think, peer-reviewed journals, and aging bureaucracies. That is the way things are because that is the way things are, has lead to far more physics than the contemporary, things can't be that way because the math dictates that we live in thirty-three dimensions and four are curled up, and that is what NSF is funding. When experiments showed that light existed only in quantized packets, Einstein proclaimed that light only existed in quantized packets, and he won the Nobel Prize. When spectra from atoms showed discreet energies, Niels Bohr proclaimed that electrons orbits were quantized, and he received a Nobel Prize. When Maxwell's Equations had a recurring constant, Maxwell used c to denote it, and Einstein proclaimed that the speed of light must be constant for all observers-and so Special Relativity was born. When Einstein juxtaposed objects falling towards the earth getting closer together with the fact that two people starting at the equator, walking on originally parallel lines of longitude towards the North Pole, would come together because they were walking on a curve surface, Einstein proclaimed that the space-time around a massive object must also be curved. This along with Einstein's realization that the force of gravity would be rendered null in free-fall, lead to General Relativity. And so it is that in the above paragraph you have the roots of the greatest achievements of physics in the past 100+ years, dwarfing String Theory, Loop Quantum Gravity, and thousands of their variatons, which deal in the abstruse, complicated, muddled, and mythological worlds which are safe from physics simple rigor. Moving Dimensions Theory returns us to simpler times. It starts with the simple and keeps it simple. Light travels with a maximum velocity of c, because the fourth dimension is expanding at a rate relative to the three spatial dimensions at the velocity of c. A photon expands through space in a spherically symmetric manner. This is because the fourth dimension expands through the three spatial dimensions in a spherically symmetric manner. Energy and mass are equivalent, expressed by E=mc^2, because energy is nothing more than mass rotated into the expanding fourth dimension. The Einstein-Podolsky-Rosen effect (EPR) effect, which calls instantaneous action at a distance spooky, can be accounted for by the expanding dimension-as a point expands, it is yet a single locale in that dimension, and hence though separated the time dimension, and hence connected. The null vector of the photon, which remains 0 no matter how far the photon travels in space-time, may be accounted for by the fact that the fourth dimension is moving, and thus the only way to stay still in the four dimensions is to move with along with the expanding dimension. In Lorentzian Transformations, there is no way for an object to be rotated into the time dimension without it moving-this can be explained by the fact that the time result of the universe's existence upon a reality that has three stationary spatial dimensions and one expanding time dimension-when matter exists in the stationary dimensions, it is seen as mass, or a or a photon, or energy. Depending how we choose to observer matter are quantized bundles of energy that propagate at the velocity of c-this is because the fourth dimension is expanding relative to the three spatial dimensions in a quantized manner, in units of Planck's length at the rate of c. The Second Law of Thermodyamics, or the law of Entropy, states that the universe tends towards disorder. This is because the fourth dimension is expanding in a spherically symmetric from one another-thus a drop of food coloring in a pool will be carried outward and evenly distributed. In 1949 Godel published a paper showing that within the theory of relativity, time as we understand it, does not exist. Einstein recognized Godel's paper as an important contribution to the general theory of relativity, and since then physicists have not been able to find any logical shortcomings in Godel's work, and nobody has been able to account for the existence of time. But the Theory of Moving Dimensions accounts for time as we know it by showing that it is an emergent property of the underlying dimension's intrinsic relative movement. Relativity becomes increasingly exact at long-length scales but fails at short ones because space-time itself is quantized, as the time dimension is expanding in units of the Planck length. The concept of general relativity's smooth geometry, at large scales, disappears on short-distance scales-this has been a problem to string theorists, but only because they were never bold enough to recognize that's the way it is because that's the way it is. Realizing this might have lead one of them to see that the fourth dimension is expanding at a rate of c relative to the three spatial dimensions. So it is seen that Moving Dimensions Theory offers a simple model upon which all known phenomena of Relativity and Quantum Mechanics may rest. And because the underlying architecture of the universe is quantized-because the fourth dimension expands at the rate of c in units of the Planck length relative to the three spatial dimensions, quantum mechanics works for the small, while general relativity works for the large. That is the way it is because that is the way it is-this was the realization that lead to the postulate of MDT: the fourth dimension is expanding relative to the three spatial dimensions. http://physicsmathforums.com === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? greatbooksclassics@yahoo.com: >Does String Theory have any postulates or laws? If you think it does, >please share & please post them at: > >http://physicsmathforums.com > >Where we're handing out the physicsmathforums.com Nobel prize in >physics to the first person who can name a postulate or law of string >theory. > >Moving Dimensions Theory & On The Advancement Of Physics > >Physics has been furthered far more often by a rugged individual >acknowledging the simple and obvious in a pursuit of the truth than >book-keepers-in-training playing games in the abstruse in pursuit of >tenure. The advancement of physics has ever depended far more on logic, >reason, and Truth than government grants, tenure, group think, >peer-reviewed journals, and aging bureaucracies. That is the way >things are because that is the way things are, has lead to far more >physics than the contemporary, things can't be that way because >the math dictates that we live in thirty-three dimensions and four are >curled up, and that is what NSF is funding. > >When experiments showed that light existed only in quantized packets, >Einstein proclaimed that light only existed in quantized packets, and >he won the Nobel Prize. When spectra from atoms showed discreet >energies, Niels Bohr proclaimed that electrons orbits were quantized, >and he received a Nobel Prize. When Maxwell's Equations had a >recurring constant, Maxwell used c to denote it, and Einstein >proclaimed that the speed of light must be constant for all >observers-and so Special Relativity was born. When Einstein >juxtaposed objects falling towards the earth getting closer together >with the fact that two people starting at the equator, walking on >originally parallel lines of longitude towards the North Pole, would >come together because they were walking on a curve surface, Einstein >proclaimed that the space-time around a massive object must also be >curved. This along with Einstein's realization that the force of >gravity would be rendered null in free-fall, lead to General >Relativity. > > > >And so it is that in the above paragraph you have the roots of the >greatest achievements of physics in the past 100+ years, dwarfing >String Theory, Loop Quantum Gravity, and thousands of their variatons, >which deal in the abstruse, complicated, muddled, and mythological >worlds which are safe from physics simple rigor. > >Moving Dimensions Theory returns us to simpler times. It starts with >the simple and keeps it simple. Light travels with a maximum velocity >of c, because the fourth dimension is expanding at a rate relative to >the three spatial dimensions at the velocity of c. A photon expands >through space in a spherically symmetric manner. This is because the >fourth dimension expands through the three spatial dimensions in a >spherically symmetric manner. Energy and mass are equivalent, expressed >by E=mc^2, because energy is nothing more than mass rotated into the >expanding fourth dimension. The Einstein-Podolsky-Rosen effect (EPR) >effect, which calls instantaneous action at a distance spooky, >can be accounted for by the expanding dimension-as a point expands, >it is yet a single locale in that dimension, and hence though separated >the time dimension, and hence connected. The null vector of the photon, >which remains 0 no matter how far the photon travels in space-time, may >be accounted for by the fact that the fourth dimension is moving, and >thus the only way to stay still in the four dimensions is to move with >along with the expanding dimension. In Lorentzian Transformations, >there is no way for an object to be rotated into the time dimension >without it moving-this can be explained by the fact that the time >result of the universe's existence upon a reality that has three >stationary spatial dimensions and one expanding time dimension-when >matter exists in the stationary dimensions, it is seen as mass, or a >or a photon, or energy. Depending how we choose to observer matter >are quantized bundles of energy that propagate at the velocity of >c-this is because the fourth dimension is expanding relative to the >three spatial dimensions in a quantized manner, in units of Planck's >length at the rate of c. The Second Law of Thermodyamics, or the law of >Entropy, states that the universe tends towards disorder. This is >because the fourth dimension is expanding in a spherically symmetric >from one another-thus a drop of food coloring in a pool will be >carried outward and evenly distributed. In 1949 Godel published a paper >showing that within the theory of relativity, time as we understand it, >does not exist. Einstein recognized Godel's paper as an important >contribution to the general theory of relativity, and since then >physicists have not been able to find any logical shortcomings in >Godel's work, and nobody has been able to account for the existence >of time. But the Theory of Moving Dimensions accounts for time as we >know it by showing that it is an emergent property of the underlying >dimension's intrinsic relative movement. Relativity becomes >increasingly exact at long-length scales but fails at short ones >because space-time itself is quantized, as the time dimension is >expanding in units of the Planck length. The concept of general >relativity's smooth geometry, at large scales, disappears on >short-distance scales-this has been a problem to string theorists, >but only because they were never bold enough to recognize that's the >way it is because that's the way it is. Realizing this might have >lead one of them to see that the fourth dimension is expanding at a >rate of c relative to the three spatial dimensions. > >So it is seen that Moving Dimensions Theory offers a simple model upon >which all known phenomena of Relativity and Quantum Mechanics may rest. >And because the underlying architecture of the universe is >quantized-because the fourth dimension expands at the rate of c in >units of the Planck length relative to the three spatial dimensions, >quantum mechanics works for the small, while general relativity works >for the large. That is the way it is because that is the way it >is-this was the realization that lead to the postulate of MDT: the >fourth dimension is expanding relative to the three spatial dimensions. > >http://physicsmathforums.com > === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? > Does String Theory have any postulates or laws?... strings resolves the incompatibility between quantum mechanics and general relativity (which, as currently formulated, cannot both be right). properties-that is, the different masses and other properties of both four forces of nature (the strong and weak nuclear forces, electromagnetism, and gravity)-are a reflection of the various ways in which a string can vibrate. Just as the strings on a violin or on a piano have resonant frequencies at which they prefer to vibrate-patterns that our ears sense as various musical notes and their higher harmonics-the same holds true for the loops of string theory. But rather than producing musical notes, each of the preferred mass and force charges are determined by the string's oscillatory pattern. The electron is a string vibrating one way, the up-quark is a string vibrating another way, and so on. http://www.pbs.org/wgbh/nova/elegant/everything.html > ...If you think it does, > please share & please post them at: > > http://physicsmathforums.com > > Where we're handing out the physicsmathforums.com Nobel prize in > physics to the first person who can name a postulate or law of string > theory. > > Moving Dimensions Theory & On The Advancement Of Physics > > Physics has been furthered far more often by a rugged individual > acknowledging the simple and obvious in a pursuit of the truth than > book-keepers-in-training playing games in the abstruse in pursuit of > tenure. The advancement of physics has ever depended far more on logic, > reason, and Truth than government grants, tenure, group think, > peer-reviewed journals, and aging bureaucracies. That is the way > things are because that is the way things are, has lead to far more > physics than the contemporary, things can't be that way because > the math dictates that we live in thirty-three dimensions and four are > curled up, and that is what NSF is funding. > > When experiments showed that light existed only in quantized packets, > Einstein proclaimed that light only existed in quantized packets, and > he won the Nobel Prize. When spectra from atoms showed discreet > energies, Niels Bohr proclaimed that electrons orbits were quantized, > and he received a Nobel Prize. When Maxwell's Equations had a > recurring constant, Maxwell used c to denote it, and Einstein > proclaimed that the speed of light must be constant for all > observers-and so Special Relativity was born. When Einstein > juxtaposed objects falling towards the earth getting closer together > with the fact that two people starting at the equator, walking on > originally parallel lines of longitude towards the North Pole, would > come together because they were walking on a curve surface, Einstein > proclaimed that the space-time around a massive object must also be > curved. This along with Einstein's realization that the force of > gravity would be rendered null in free-fall, lead to General > Relativity. > > > > And so it is that in the above paragraph you have the roots of the > greatest achievements of physics in the past 100+ years, dwarfing > String Theory, Loop Quantum Gravity, and thousands of their variatons, > which deal in the abstruse, complicated, muddled, and mythological > worlds which are safe from physics simple rigor. > > Moving Dimensions Theory returns us to simpler times. It starts with > the simple and keeps it simple. Light travels with a maximum velocity > of c, because the fourth dimension is expanding at a rate relative to > the three spatial dimensions at the velocity of c. A photon expands > through space in a spherically symmetric manner. This is because the > fourth dimension expands through the three spatial dimensions in a > spherically symmetric manner. Energy and mass are equivalent, expressed > by E=mc^2, because energy is nothing more than mass rotated into the > expanding fourth dimension. The Einstein-Podolsky-Rosen effect (EPR) > effect, which calls instantaneous action at a distance spooky, > can be accounted for by the expanding dimension-as a point expands, > it is yet a single locale in that dimension, and hence though separated > the time dimension, and hence connected. The null vector of the photon, > which remains 0 no matter how far the photon travels in space-time, may > be accounted for by the fact that the fourth dimension is moving, and > thus the only way to stay still in the four dimensions is to move with > along with the expanding dimension. In Lorentzian Transformations, > there is no way for an object to be rotated into the time dimension > without it moving-this can be explained by the fact that the time > result of the universe's existence upon a reality that has three > stationary spatial dimensions and one expanding time dimension-when > matter exists in the stationary dimensions, it is seen as mass, or a > or a photon, or energy. Depending how we choose to observer matter > are quantized bundles of energy that propagate at the velocity of > c-this is because the fourth dimension is expanding relative to the > three spatial dimensions in a quantized manner, in units of Planck's > length at the rate of c. The Second Law of Thermodyamics, or the law of > Entropy, states that the universe tends towards disorder. This is > because the fourth dimension is expanding in a spherically symmetric > from one another-thus a drop of food coloring in a pool will be > carried outward and evenly distributed. In 1949 Godel published a paper > showing that within the theory of relativity, time as we understand it, > does not exist. Einstein recognized Godel's paper as an important > contribution to the general theory of relativity, and since then > physicists have not been able to find any logical shortcomings in > Godel's work, and nobody has been able to account for the existence > of time. But the Theory of Moving Dimensions accounts for time as we > know it by showing that it is an emergent property of the underlying > dimension's intrinsic relative movement. Relativity becomes > increasingly exact at long-length scales but fails at short ones > because space-time itself is quantized, as the time dimension is > expanding in units of the Planck length. The concept of general > relativity's smooth geometry, at large scales, disappears on > short-distance scales-this has been a problem to string theorists, > but only because they were never bold enough to recognize that's the > way it is because that's the way it is. Realizing this might have > lead one of them to see that the fourth dimension is expanding at a > rate of c relative to the three spatial dimensions. > > So it is seen that Moving Dimensions Theory offers a simple model upon > which all known phenomena of Relativity and Quantum Mechanics may rest. > And because the underlying architecture of the universe is > quantized-because the fourth dimension expands at the rate of c in > units of the Planck length relative to the three spatial dimensions, > quantum mechanics works for the small, while general relativity works > for the large. That is the way it is because that is the way it > is-this was the realization that lead to the postulate of MDT: the > fourth dimension is expanding relative to the three spatial dimensions. http://physicsmathforums.com === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? But what are the postulates? What are the laws? We can all name Newton's Laws and Einstein's Postulates, which lie at the base of classical and relativistic physics. But what are String Theory's postulates and laws? Perhaps there are none? Moving Dimensions Thoery has a postulate: The General Postulate of Moving Dimensions Theory: The fourth dimension is expanding relative to the three spatial dimensions. The Specific Postulate of Moving Dimensions Theory: The fourth dimension is expanding relative to the three spatial dimensions at the rate of c in quantized units of the Planck length. Classical physics, quantum mechanics, and relativity descend from this simple postulate. Light, and thus all energy, is quantized as the dimension which transports it expands in a quantized manner. Light travels at a constant velocity in all frames because velocity is measured relative to time which is measured relative to the light that is transported by the fourth expanding dimension. Thus both fundamental constants h and c emerge from the fundamental nature of the expansion of the fourth dimension relative to the three spatial dimensions. And thus MDT provides a simple, unifying postulate accounting for the classical, relativistic, and quantum mechanical properties of this universe. http://physicsmathforums.com === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? > But what are the postulates? Are you dense or something (sorry your posts have already answered that question) - he explained some of it postulates and laws. Bill > > What are the laws? > > We can all name Newton's Laws and Einstein's Postulates, which lie at > the base of classical and relativistic physics. > > But what are String Theory's postulates and laws? > > Perhaps there are none? > > Moving Dimensions Thoery has a postulate: > > The General Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions. > > The Specific Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions at the rate of c in quantized units of the Planck length. > > Classical physics, quantum mechanics, and relativity descend from this > simple postulate. Light, and thus all energy, is quantized as the > dimension which transports it expands in a quantized manner. Light > travels at a constant velocity in all frames because velocity is > measured relative to time which is measured relative to the light that > is transported by the fourth expanding dimension. Thus both > fundamental constants h and c emerge from the fundamental nature of the > expansion of the fourth dimension relative to the three spatial > dimensions. And thus MDT provides a simple, unifying postulate > accounting for the classical, relativistic, and quantum mechanical > properties of this universe. > > http://physicsmathforums.com > === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? jollyrogership@yahoo.com: >But what are the postulates? > >What are the laws? > >We can all name Newton's Laws and Einstein's Postulates, which lie at >the base of classical and relativistic physics. > >But what are String Theory's postulates and laws? Purcahse the two volume set: ``Quantum Fields and Strings,'' then get back to us once you've understood the material. > >Perhaps there are none? > >Moving Dimensions Thoery has a postulate: > >The General Postulate of Moving Dimensions Theory: >The fourth dimension is expanding relative to the three spatial >dimensions. > >The Specific Postulate of Moving Dimensions Theory: >The fourth dimension is expanding relative to the three spatial >dimensions at the rate of c in quantized units of the Planck length. > >Classical physics, quantum mechanics, and relativity descend from this >simple postulate. Light, and thus all energy, is quantized as the >dimension which transports it expands in a quantized manner. Light >travels at a constant velocity in all frames because velocity is >measured relative to time which is measured relative to the light that >is transported by the fourth expanding dimension. Thus both >fundamental constants h and c emerge from the fundamental nature of the >expansion of the fourth dimension relative to the three spatial >dimensions. And thus MDT provides a simple, unifying postulate >accounting for the classical, relativistic, and quantum mechanical >properties of this universe. > >http://physicsmathforums.com > === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? Go here : http://library.thinkquest.org/27930/ Click on Theoretical Cosmology and see the String Theory Series. -- Best, Frederick Martin McNeill Poway, California, United States of America mmcneill@fuzzysys.com http://www.fuzzysys.com http://members.cox.net/fmmcneill/ ************************* Phrase of the week : The reason why we are on a higher imaginative level is not because we have finer imagination, but because we have better instruments. -- Alfred North Whitehead (1861-1947) :-))))Snort!) ************************* === Subject: Re: Fractional derivatives >Hi > > I read someplace that fractional derivatives can be computed by >tinkering in the Fourier domain of that function. > >My question is, that not all functions have a Fourier transform ( >Dilet conditions)... >so how do we define their fractional derivatives? Not all functions have derivatives; why should every function have a fractional derivative? >Gsax ************************ === Subject: Re: Fractional derivatives > >>Hi >> >> I read someplace that fractional derivatives can be computed by >>tinkering in the Fourier domain of that function. >> >>My question is, that not all functions have a Fourier transform ( >>Dilet conditions)... >>so how do we define their fractional derivatives? > >Not all functions have derivatives; why should every function >have a fractional derivative? Does every C-infinity function have a fractional derivative? If not, what can be said about the set of values of c such that every/some/all-in-some-interesting-subset C-infinity function has a derivative of order c? Presumably these questions can actually be answered, maybe in an interesting way, if instead of C-infinity one puts in real analytic on all of R or maybe even defined by a single power series on all of R. Lee Rudolph === Subject: Re: Fractional derivatives > >> >>>Hi >>> >>> I read someplace that fractional derivatives can be computed by >>>tinkering in the Fourier domain of that function. >>> >>>My question is, that not all functions have a Fourier transform ( >>>Dilet conditions)... >>>so how do we define their fractional derivatives? >> >>Not all functions have derivatives; why should every function >>have a fractional derivative? > >Does every C-infinity function have a fractional derivative? I don't know. The question should really be whether it's possible to define a notion of fractional derivative that's applicable to every smooth function - there are various definitions of fractional derivative in existence, not all equivalent. I don't recall seeing any such definition that doesn't need some sort of global growth condition. And the definitions are not local, as has been pointed out. This would all tend to point towards a no. On the other hand, it's possible to define an ordinary derivative in terms of the Fourier transform; when you do that it's not at all obvious that the derivative is local, although it is. I doubt that there's going to be a reasonable notion of fractional derivative that will apply to every smooth function, but something like the following might be possible and/or well known: Say f is smooth. Let g = f phi, where phi is smooth with compact support and phi = 1 near a. Then g has a fractional derivative. If the value of this fractional derivative at a were independent of phi then that would be the fractional derivative of f at a. But I'm pretty sure that the value at a is not independent of phi. On the other hand it might be, for example, that the value at a was well-defined _modulo_ smooth functions, or some such. >If not, what can be said about the set of values of c such >that every/some/all-in-some-interesting-subset C-infinity >function has a derivative of order c? > >Presumably these questions can actually be answered, maybe >in an interesting way, if instead of C-infinity one >puts in real analytic on all of R or maybe even >defined by a single power series on all of R. > >Lee Rudolph ************************ === Subject: Re: Fractional derivatives > I don't recall seeing any such definition that doesn't need > some sort of global growth condition. And the definitions > are not local, as has been pointed out. This would all tend > to point towards a no. http://en.wikipedia.org/wiki/Differintegral http://mathworld.wolfram.com/FractionalDerivative.html === Subject: Re: Fractional derivatives > I read someplace that fractional derivatives can be computed by > tinkering in the Fourier domain of that function. Derivation goes over to multiplication by x in the Fourier domain; as I understand it fractional derivation corresponds to multiplication by x^c. > My question is, that not all functions have a Fourier transform ( > Dilet conditions)... > so how do we define their fractional derivatives? I very much doubt if you could define it in this case. Note that fractional derivatives are not usually local, like normal derivatives. They depend on the behaviour of f(t) for all t. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Model Aircraft Design <3k85fqFt3c12U1@individual.net> > I'm having trouble visualizing this. > > Where are the rings in relation to each other -- concentric or something > else? > > They are concentric rings > > Do the 36 vanes all meet in the center of the rings, like helicopter rotor > blades? > > No they don't. they are placed between the two concentric rings to forma > sort of fan with no hub. So, each vane will be 2 inches wide at the top and > bottom (for a 10 inch and 12 inch ring). For interest I was going to try to work out the error in approximating the ellipse arcs as circles - to see if it really was as small as I guess it will be - but reading through this stuff again I think maybe I don't understand the layout at all after all. Why is each vane 2 inches wide at the top and bottom? If the ring diameters are 10 and 12 inches as you state, then as far as I can see the lengths of the straight top and bottom edges of the vanes will be approximately one inch (the difference between the radii) - but not exactly one inch... they will in fact be slightly longer as these edges are not exactly radial. Where does the 2 inches come from? === Subject: Re: Model Aircraft Design >> the equation of a cilinder with radius R and the Z-axis as centerline >> x^2 + y^2 = R^2 >> a 45Á rotated plane containing the X-axis through (0,0,0) >> y = z >> the intersection of both is an ellips in the y = z plane >> if you transform it in the z = 0 plane, rotate over 45Á >> x' = x >> y' = sqrt(y^2 + z^2) >> the ellisp is >> x^2 + (y/2)^2 = R^2 > > Doesn't this ellipse have a major to minor axis ratio of 2:1? Don't we > need the ratio to be sqrt(2):1? That would make it x^2 + y^2/2 = R^2. yes, i did a miscalculation but if you see how i did it, you can see where i made the error x^2 + y^2/2 = R^2 is the right answer !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > As an undecided myself, I have three: > > 1. If a number is indescribeable and unrepresentable, it is irrelevant. > The set of relevant numbers has cardinality equal to the naturals. Problem is that _single_ numbers are irrelevant. If we talk about measures, then the quality of relevant and irrelevant numbers is the same. If we are talking about series in general terms, making a distinction between relevant and irrelevant numbers is completely artificial. So while you will never encounter isolated irrelevant numbers in practice, excluding them when looking at the general class of real numbers causes more trouble than it is worth. Much more trouble. > 2. No continuum has been discovered in physics -- everything seems > to change in finite units called quanta. That's the real world. Mathematics is not interested in the real world. Apart from that, the _probabilities_ of quantum changes appear to be quite well modeled with continuous distributions. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Wanless' ?Fifth? Conjecture > The number of denary digits, in the number which is the maximum number > of trials of base, a, required, using Wanless' Extended Factorization > algorithm, to prime-factorize _any_ (sic) (sic) (sic) number, is 54. ...There's two caveats to this, which I was hoping someone/sometwo would point out...? === Subject: Re: Wanless' ?Fifth? Conjecture > The number of denary digits, in the number which is the maximum >> number of trials of base, a, required, using Wanless' Extended >> Factorization algorithm, to prime-factorize _any_ (sic) (sic) (sic) >> number, is 54. ...There's two caveats to this, which I was hoping someone/sometwo would > point out...? One (fairly) serious, one somewhat less so J === Subject: Re: Wanless' ?Fifth? Conjecture > >>> The number of denary digits, in the number which is the maximum >>> number of trials of base, a, required, using Wanless' Extended >>> Factorization algorithm, to prime-factorize _any_ (sic) (sic) (sic) >>> number, is 54. >> >> >> ...There's two caveats to this, which I was hoping someone/sometwo >> would point out...? One (fairly) serious, one somewhat less so > J ...revolves about what exactly I mean by prime-factorize === Subject: Fields as universal algebras? Are fields an example of a pathological thing in algebra that doesn't really qualify as a special case of a universal algebra? The problem is that the unary multiplicative inverse isn't defined on 0. How is this usually handled? Or do universal algebraists just avoid fields? Confused, Snis Pilbor === Subject: Re: Fields as universal algebras? days. My association with the Department is that of an alumnus. > Are fields an example of a pathological thing in algebra that >doesn't really qualify as a special case of a universal algebra? The >problem is that the unary multiplicative inverse isn't defined on 0. >How is this usually handled? Or do universal algebraists just avoid >fields? Fields are not in general universal algebras if all you allow are full operations. However, if you allow partial operations (which are functions whose domains are subsets of powers of the underlying set) then you can define fields to be such objects; they are usually called partial algebras. You have to be very careful with partial algebras, however, because many of the theorems we are used to may fail (e.g., the image of a sub-partial algebra under a partial algebra homomorphism need not be a partial algebra, because the domain of the partial operation in the image properly contains the image of the domain of the corresponding partial operation). -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Fields as universal algebras? Are fields an example of a pathological thing in algebra that > doesn't really qualify as a special case of a universal algebra? The > problem is that the unary multiplicative inverse isn't defined on 0. > How is this usually handled? Or do universal algebraists just avoid > fields? It is perfectly good in universal algebra. But it is not equationally definable in the technical sense. If it were, then the cartesian product with coordinatewise operations would still be a field (since that operation preserves equations). -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Fields as universal algebras? > > Are fields an example of a pathological thing in algebra that > doesn't really qualify as a special case of a universal algebra? The > problem is that the unary multiplicative inverse isn't defined on 0. > How is this usually handled? Or do universal algebraists just avoid > fields? > > Confused, > Snis Pilbor In Unversal Algebra, it is standard to view a field as a special case of a ring, with inverse *not* considered a full-fledged operation. The point is that UA is concerned with the consequences of equations (universally quantified formulas) and the axiom for inverse isn't universally quantifed. It is interesting to note that the super-set of Universal Algebra, Model Theory, which doesn't limit itself to that which is expressible in equational logic, has a lot to say about the theory of fields. Note that in traditional abstract algebra the theory of fields is situated in the theory of rings in a stronger way than the way in which the theory of rings is situated inside in the theory of abelian groups. In field theory, you often consider subrings which are not subfields, but in ring theory you seldom consider additive subgroups which are not subrings. The fact that fields don't constitute a variety in the sense of UA can be regarded as an explanation of this state of affairs. As Lee Rudolph suggests, you can certainly generalize UA in a way that allows for partial operations. The theory isn't as nice as the classic UA theory, but you can still come up with things like generalized versions of the HSP theorem. See Partial Algebras - An Introductory Survey by Peter Burmeister for more details (from the conference proceedings Algebras and Orders, edited by Rosenberg and Sabidussi). Hope this helps -John Coleman === Subject: Re: Fields as universal algebras? > In field theory, you often consider subrings which are not subfields, > but in ring theory you seldom consider additive subgroups which are not > subrings. What about ideals? === Subject: Re: Fields as universal algebras? days. My association with the Department is that of an alumnus. > > > >> In field theory, you often consider subrings which are not subfields, >> but in ring theory you seldom consider additive subgroups which are not >> subrings. > >What about ideals? If you do not require your rings to have a 1, then ideals are subrings. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Fields as universal algebras? > > Are fields an example of a pathological thing in algebra that >doesn't really qualify as a special case of a universal algebra? The >problem is that the unary multiplicative inverse isn't defined on 0. >How is this usually handled? Or do universal algebraists just avoid >fields? Until Arthur Magidin or someone else who actually thinks of himself as a universal-algebrist (contrasted with universal algebraist), maybe like Michael Barr, speaks up, I will opine that there's no reason not to do universal algebra with partially-defined operations. Obviously (?) you will expect to get weaker theorems; you can still talk about laws, but I bet that the theory of varieties falls fairly flat. Well, we shall see. Lee Rudolph === Subject: Re: brachistochrone variations Brachistochrone Problem http://mathworld.wolfram.com/BrachistochroneProblem.html === Subject: entire functions Find all entire functions f such that |f(z)|=1 on the unit circle |z|=1 === Subject: Re: entire functions - The functions z -> a.z^n with nonnegative integral n and with |a|=1 are the only entire functions with absolute value 1 on the unit circle. This theorem is an exercise in the book Classical Complex Analysis by Liang-Shin Hahn and Bernard Epstein. (Jones and Bartlet, 1996, ISBN 0-86720-494-X). Chapter 6, Exercise 3c, p.203 I am afraid that this only solves the smaller part of your problem. Is it homework? Is it a term assignment? Happy studies: Johan E. Mebius >Find all entire functions f such that |f(z)|=1 on the unit circle |z|=1 > > > === Subject: Re: entire functions >Find all entire functions f such that |f(z)|=1 on the unit circle |z|=1 You know about those automorphisms of the disk often denoted phi_a, right? phi_a is an automorphism of the disk that sends a to 0. There is a product P of powers of phi_a's that has the same zeroes inside the disk as f, so that f/P has no zero in the disk. The maximum modulus theorem then shows that f/P must be constant in the disk, so that f = c P in the disk. Now you need to look and see when it happens that c P is actually entire and you're done. ************************ === Subject: Re: entire functions > Find all entire functions f such that |f(z)|=1 on the unit circle |z|=1 > Schwarz's Lemma might help you (http://planetmath.org/encyclopedia/SchwarzLemma.html). J. === Subject: Re: square mod > what is the more efficient algorithm for doing x^2 mod (n)? That depends on many things such as the size of n, the type of class n falls into and what system you are on. It also depends (as you point out) if this is a one-off calculation. For n much above 8K bits, I would expect a convolution/.FFT approach to be fastest. While it is well known that one can do multiplication by FFT, it is somewhat less known that one can do division by using Newton-Raphson with FFT multiplies. See Aho, Hopcrof, Ullman's book on Algorithms. The cutoff between classical division and FFT division will be problem, implementation, and hardware dependent. === Subject: Re: square mod > what is the more efficient algorithm for doing x^2 mod (n)? > > That depends on many things such as the size of n, the type of class n > falls into and what system you are on. > > It also depends (as you point out) if this is a one-off calculation. > > For n much above 8K bits, I would expect a convolution/.FFT > approach to be fastest. While it is well known that one can do > multiplication by FFT, it is somewhat less known that one can do > division by using Newton-Raphson with FFT multiplies. > See Aho, Hopcrof, Ullman's book on Algorithms. > > The cutoff between classical division and FFT division will be problem, > implementation, and hardware dependent. You can implement Montgomery reduction with interpolation multiplication techniques. At no point does x^2 mod n require division for odd n [and with precompute for even n] Tom === Subject: Re: square mod >(what is this twittering thing - looks like a robot fetching stuff >from Usenet and posting it here...) > > i'm a 'robot' but who are you? :)) The remark was about 'Twittering One' - the something that fills this thread with garbage... The Handbook of Applied Cryptography (HAC) is a very decent reference, Chapter 14 of it gives you lots of info on the subject. Squaring is described in 14.2.4. Karatsuba is a Russian mathematician, the author of the Karatsuba-Ofman algorithm for efficient multiplication. HAC mentions the algoritm but does not get into details. Anyway, you hardly need it for you 200 bits numbers. === Subject: discriminating SO3 invariance from general O3 invariance Given the space S of symmetric real 3x3 matrices, we operate on it with O3 via conjugation. I wish to discriminate SO3 conjugation classess from O3 conjugation classess. that is - i'm looking for a function of a matrix E which is invariant under rotations, but not under reflections (in general, at least). I believe i'm able to prove, with the aid of some earlier advice in this group, that such an invariant function cannot be a polynomial in the entries of E, of any degree. Still, I hope, the search is not lost. is anyone familiar with such invariants? any thoughts or just directions would be appreciated too. Ofek === Subject: Re: Euclidean postulates <7973055.1121924244120.JavaMail.jakarta@nitrogen.mathforum.org> > Any four points in space in a stationary positions > or in a random motion lie > exactly at a surface of a sphere at any moment of > time > > And,this is very simple to prove. > > My question ? > > Is this an euclidean geometry or euclidean > postulates. Yes. This is fact of Euclidean geometry. It is possible to find four noncoplanar points in hyperbolic space that do not lie on the surface of a sphere. Such a set of points might lie on the surface of a horosphere (sphere with 'infinite' radius) or a hypersphere (set of all points equidistant from a given plane). === Subject: Re: An interesting problem >>Problem: Let f(x) = 1 + x + (x^2)/2 + (x^3)/6 + ... + (x^2n)/2n. >> >> It doesn't make sense to me as written. >Taylor series expansion of e^x. It is the truncated verions that stops >at >the term (x^2n)/(2n)!. In response to another comment from another >poster, >I was not giving the general term of the expression, but rather the >final term >of the polynomial which also includes terms of odd degree. OK, got it. You should probably write the function as f(x,n) and of course include the factorial symbol at the end. It would avoid confusing people like me, anyway. :^) --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: An interesting problem Am 20.07.05 23:52 schrieb achava@hotmail.com: > Very nice solution! Now why didn't I think of that? > Maybe, you did not go to the desperation not being able to determine a general expression for the zeros of the function, as I did (so the considering of a minimum was just an escape)? :-) Gottfried Helms === Subject: Questions about co-np! I have a question, I have a undirected graph G and a set F of combinations of vertices in G. A minial vertex cover of G is a vertex cover whose size is not reducable which means removal of any vertex from the vertex cover will make it no longer a vertex cover. It's different with smallest vertex cover. eg, F={{v1,v2,v3},{v3,v4.v5}} Now I have a language L={|F is set of all minimal vertex covers of G} Assume, I know {|G has a vertex cover of size k} is in NP. How can I prove it's in coNP? Maybe I can prove the complement of L is in NP, then L is in coNP. But I have no idea how to do it in poly-time. Can anyone give me some idea on it? === Subject: Re: Questions about co-np! > I have a question, I have a undirected graph G and a set F of > combinations of vertices in G. > A minial vertex cover of G is a vertex cover whose size is not > reducable which means removal of any vertex from the vertex cover > will make it no longer a vertex cover. It's different with smallest > vertex cover. > eg, F={{v1,v2,v3},{v3,v4,v5}} Are you talking about removing sets (in F) or elements of the sets in F? Is the size of the cover the number of sets in it? > Now I have a language L={|F is set of all minimal vertex covers > of G} > Assume, I know {|G has a vertex cover of size k} is in NP. > How can I prove it's in coNP? > Maybe I can prove the complement of L is in NP, then L is in coNP. > But I have no idea how to do it in poly-time. > > Can anyone give me some idea on it? What does the statement F is not a vertex cover mean? Can you use the information which makes F not a vertex cover, and verify it in polynomial time? === Subject: Re: set of a set etc. >>>>> >>>>>> The description is what I would call formal, not conceptual. My >>>>>> cat and the set of my cat {My cat} are different conceptually. >>>>>> My cat likes milk. The set of my cat does not, yet the two >>>>>> denotations are closely related. What is the conceptual >>>>>> relationship between the two? >>>>> Your cat is a member of the set of your cat. The set of your cat >>>>> is not a member of your cat. >>>>> >>>>> Sets are collections. A collection is distinct from the objects >>>>> therein (usually). Put a ring in a box. The box contains the >>>>> ring; the box and the ring are not the same thing. >>>> >>>> Just to confuse matters, W.V.O. Quine in Set Theory and Its Logic >>>> defines the law of extensionality and notes that a consequence of >>>> it is that there is only one memberless object. That is, since >>>> extensionality says that two things are identical if they have the >>>> same members, and indivduals do not have members, all indivduals >>>> are identical to the empty set and to each other. To avoid this, >>>> he could treat an individual as a different sort of object than a >>>> set, but instead he defines x in y as meaning x = y when y is >>>> an individual. A consequence of this is that individuals are >>>> identical to their unit sets, that is, x = {x} but ONLY when x is >>>> an individual. Of course, he retains x =/= {x} when x is a set. >>>> He takes some pains to show why this is harmless, but it does seem >>>> rather odd. >>>> >>>> --Mark >>> >>> make of it? >> >> Well, Quine discusses other possible solutions. He mentions using >> two different styles of variables, one for individuals and one for >> sets, which is I think the most natural approach. This is probably >> what many of your respondents have in mind when they point out that >> a cat is not the same as the set of a cat, etc. (Otherwise, they'd >> be forced to conclude that a cat is the same as the null set, since >> neither has any members [potential jokes about tom cats at this >> point notwithstanding].) Quine also mentions the possibility of >> adding a predicate that asserts individuality, or conversely, >> classitude, which would let us distinguish individuals from sets. >> But he prefers his solution as more elegant, since it doesn't >> require an extra predicate or separate variable styles. I quote >> from Quine: >> >> We are interested in x in y to begin with only for classes y; >> such are the only cases of x in y that are subject to >> preconceptions worth respecting. If for the sake of smooth >> systematization we see fit to assign meaning to further cases, let >> us assign a meaning that maximizes the smoothness.... Let us rule x >> in y true or false according as x = y or x =/= y, when y is an >> individual.... But what if y is an individual and z is the unit >> class of y? On our new interpretation ... x in y then becomes >> true if and only if x is the individual y; so (Ax)(x in y iff x in >> z) and therefore y = z. This result is prima facie unacceptable, >> since y is an individual and z is a class. But actually it is a >> harmless result; none of the utility of class theory is impaired by >> counting an individual, its unit class, the unit class of that unit >> class, and so on, as one and the same thing. True, we are well >> advised now to adjust our terminology to the extent of ceasing to >> explain individual as nonclass; let us take to saying that what >> constitutes them individuals is not inclassitude, but identity with >> their unit classes.... Everything comes to count as a class; still, >> individuals remain marked off from other classes in being their own >> sole members. ---End of quote >> >> The last point is a key one I think -- by this route, everything is >> a class to Quine, which simplifies some things. It's important to >> keep in mind that this is just the way Quine's axioms work. Many >> (probably most) other axiomatic systems don't consider x = {x} to be >> true, even if x is an individual. So there's no one true answer to >> the question about what this means, it depends on the axiom system >> you're using. But there's no argument when talking about >> multi-element sets: for Quine as for everyone else, the set {x,y} is >> different from the set {{x,y}}, since the first has two elements, >> and the second has one element. >> >> --Mark > > Before I say anything could you please explain the construction x in > y, especially the backslash. Sorry, I was just using in to mean the epsilon operator for set membership. > Also what distinction do you make > between set and class? I've seen them used interchangably but there > seems to be some distinction in Quine's useage. Quine does distinguish between set and class. He usually prefers the word class and reserves set for classes which are capable of being members of other classes. (Not all classes are sets in his theory, in order to avoid Russell's paradox.) > Also, as you have emphasized, the issue doesn't seem to have any > application to sets with more than one member. There is another > situation with multiple member sets however in that it seems like they > can somtimes be treated as if they were single member entities. I can > say something like I drove my car to work. or I could say I drove > the set of mechanical parts that I call my car to work.Here it seems > that the set {the mechanical parts that I call my car} is being > identified with the single object referred to with the expression my > car. (I don't think that the self referential aspect of the expresion > is a concern here, I'm just using to to avoid a complete list of what > the car parts actually are.) Do you have any thoughts or experience > with the possible use of sets of components of things as being > identified with the things that they comprise? Sets are conceptual objects, not physical ones. The fact that a car is physically composed of components has no bearing on what members it has, unless you decide that it does by calling it a set of components rather than a single entity. In other words, the set containing car is different from the set containing the components of the car, even though a physical box containing one car may be identical to a physical box containing the components of the car. Sets have properties based on how they're defined -- investigation into their physical properties can't discover properties that aren't derivable from their definitions. > (Or of using a set of > properties directly as a definition of a single entity?--As opposed to > For all x such that .. followed by the defining properties) I'm not sure I follow this, but sets are usually defined by their properties. E.g. the set of numbers divisible by 7 would normally specified like that, the set of numbers which have the property that they're divisible by 7. --Mark === Subject: Re: set of a set etc. The last point is a key one I think -- by this route, everything is a class > to Quine, which simplifies some things. It's important to keep in mind that > this is just the way Quine's axioms work. Many (probably most) other > axiomatic systems don't consider x = {x} to be true, even if x is an > individual. So there's no one true answer to the question about what this > means, it depends on the axiom system you're using. But there's no argument > when talking about multi-element sets: for Quine as for everyone else, the > set {x,y} is different from the set {{x,y}}, since the first has two > elements, and the second has one element. --Mark just learning about sets, so if this is a dumb question please pardon me. In your last sentence, is {x,y} different from {{x,y}} because the first has two elements (x and y) and the second has one element, the set {x,y}? So, would you say that the second has one element, and that one element contains two elements? Or would that not be valid? k wallace === Subject: Re: set of a set etc. <42DE957A.50802@netscape.net> > In your last sentence, is {x,y} different from {{x,y}} because the first > has two elements (x and y) and the second has one element, the set > {x,y}? Yes. > So, would you say that the second has one element, and that one element > Yes, that one element is a set, the set { x,y } which of course has two elements x and y, unless x = y. Then { x,y } = { x } = { y } has one element. nulset is a member of { nulset }, thus { nulset } has one element and just that one, viz nulset. Now nulset has no elements, so since nulset and { nulset } have different number of elements, they are different. Now {{ nulset }} and { nulset } are different because {{ nulset }} is the set that has only the element { nulset } and { nulset } is the set that only the element nulset. But as before, { nulset } /= nulset. Thus because the contents of {{ nulset }} and { nulset } are different, {{ nulset }} /= { nulset } even tho both have the same number of elements. === Subject: Re: Fixed-point algorithm for exponential function fitting for an embedded application Hello Vladim, samples of CORDIC algorithm for most common functions such sin, cos, atan2, ln...? Eric. > >>> We are seeking a light algorithm performing exponential function >>> (y=A*exp(-k*x)) fitting. >> >> >> First, take logarithms of the data points, so your fit will be >> >> ln y = ln A - k * x >> >> The logarithm function is a PITA to compute well, the MacLaurin >> series converges painfully slowly, so a lookup table with >> interpolation is in order. > CORDIC is fast and slim. It uses lookup table and is easily > implementable for fixed-point numbers. Vadim -- Eric Meurville +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ EPFL Tel.: +41 21 693 59 28 ( E. Meurville ) STi-IPR-LPM +41 21 693 38 17 ( Secretariat ) BM 2 143 Fax : +41 21 693 38 91 Station 17 CH-1015 Lausanne Switzerland E-mail : mailto:eric.meurville@epfl.ch Web : http://www.epfl.ch/ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ === Subject: Re: Fixed-point algorithm for exponential function fitting for an embedded application >samples of CORDIC algorithm for most common functions such sin, cos, >atan2, ln...? While you wait for information on CORDIC, you may also want to look at these: http://www.analog.com/UploadedFiles/Associated_Docs/727652249Chapter_4.pdf http://www.analog.com/UploadedFiles/Associated_Docs/32637427311745Chapter_2. pdf Jon -- { I'm keeping the cross-posted newsgroups intact, but I only read comp.arch.embedded. } === Subject: Re: Fixed-point algorithm for exponential function fitting for an embedded application > { I'm keeping the cross-posted newsgroups intact, > but I only read comp.arch.embedded. } >> >>Yes; dsPIC is the processor I am using. It is indeed equipped with a >>barrel shifter. > You aren't logging your data for later analysis, as one might normally > do if this were a research project at an educational institution. (I > had already looked up the epfl site.) This is something being done in > real time, which suggests that you are making an instrument for sale > or internal, practical use. Yes? Jon We are an educational institute very industry oriented. We are presently preparing a demonstrator which is actually a small embedded system and the processing must be performed in real time. This research project is presently financed by the Government and hopefully will become a product in few years. -- Eric Meurville === Subject: Re: Fixed-point algorithm for exponential function fitting for an embedded application >> { I'm keeping the cross-posted newsgroups intact, >> but I only read comp.arch.embedded. } >>> >>>Yes; dsPIC is the processor I am using. It is indeed equipped with a >>>barrel shifter. >> You aren't logging your data for later analysis, as one might normally >> do if this were a research project at an educational institution. (I >> had already looked up the epfl site.) This is something being done in >> real time, which suggests that you are making an instrument for sale >> or internal, practical use. Yes? >> Jon > >We are an educational institute very industry oriented. We are presently >preparing a demonstrator which is actually a small embedded system and >the processing must be performed in real time. This research project is >presently financed by the Government and hopefully will become a product >in few years. Understood. Optimizing the sensitivity and repeatable precision of measurement of exponential decay rates is something I've specialized in for over 15 years -- from mathematical development and refinement, through stochastics arguments and monte carlo simulation in helping to make optimal choices for measurement periods, sampling rates, etc., through to the software routines to minimize undesirable artifacts from their calculations in small microcontrollers and integer DSPs, at measurement rates (not sampling rates, but full measurement of the slope) spanning from 10k/sec to 1/sec. This means highly optimized and highly precise routines for very fast execution on processors without access to specialized hardware. I guess that explains why I was drawn to your original question. Jon === Subject: Re: Cantor and the binary tree > WM, claiming to be able to well-order anything, should be equally able > to give is, specifically and in all detail, his rule for well-ordering > the reals. > > WM > You did really study mathematics? There is a slight difference in > well-ordering countable and uncountable sets. You should advocate the > position that well-ordering of the reals was possible. i do not. But > the well-ordered rationals can be ordered by magnitude if Cantor lists > have complete antidiagonals. But didn't you say you proved that there exist no irrationals and that the reals were countable? I thought that's what this thread was all about. So even you don't believe your bogus arguments. Whew, that should restore a little bit of faith in humanity. Jiri === Subject: Re: Spherical Trigonometry text <42dde32f$0$49198$edfadb0f@dread12.news.tele.dk> Michael, I'm not sure why you would learn anything, but on the off chance that it helps, I'll tell you a little about the specific problem. Without going into to much detail, but briefly. My colleague had a plate held at a particular angle (both pitch angle and yaw angle, or basically rotated by certain angles around two orthogonal axes in sequence). He had a ray coming into the plate from a known direction. The question was how to find out what angle this ray makes with the normal to the plate. He had an engineering reference from decades earlier which solved the problem using a Pythagorean formula on tangents of the angles, but he suspected that this formula was wrong. The reference said that if x was one angle of rotation for the plate (pitch), and y was the other angle of rotation about an orthogonal axis (yaw), then the angle z he was seeking could be found by (tan z)^2 = (tan x)^2 + (tan y)^2. He doubted this formula and came to me to verify or disprove it. It took us a few hours (spread out over a week, since we both had other things to work on as well) to show that this is not correct. The true answer, from spherical trigonometry, was cos(z) = cos(x) * cos(y) using the spherical trig Law of Cosines and the fact that we were working with a right spherical triangle, so we could assume that A = pi / 2 and thus cos(A) = 0. We wondered why the published reference used a different formula. However, we wondered more why our engineering R&D center doesn't have a spherical trig reference in their tech library. We want something that discusses the Law of Cosines, the Law of Sines, and other key results in spherical trig, and should not be restricted only to right spherical triangles. It should be able to give an idea of both the derivation and the application of these rules, and it should (as I said before) be something that a working engineer could use on his/her own, even if not previously familiar with the subject. === Subject: Re: Spherical Trigonometry text <42dde32f$0$49198$edfadb0f@dread12.news.tele.dk> It appears, from searching www.loc.gov on the subject category spherical trigonometry, that the most recent books on the subject were published in Russian. David Ames === Subject: Re: Name for base-30 numbering system? >I am dealing with a software file format that represents >floating-point numbers in base 30 (!). Does anyone have a name >for this base? I found a webpage that said base 13 is called >tredecimal, but I couldn't find any with a name for base 30. Trigesimal. -- 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: Name for base-30 numbering system? I am dealing with a software file format that represents > floating-point numbers in base 30 (!). Does anyone have a name > for this base? I found a webpage that said base 13 is called > tredecimal, but I couldn't find any with a name for base 30. Suggestions? uvwxyzarefree? :-) some useful expressions in that: evaluate 1l0O 0O0O != O0O0 x = O * l; octal for the octagerians. -- Chuck F (cbfalconer@yahoo.com) (cbfalconer@worldnet.att.net) Available for consulting/temporary embedded and systems. USE worldnet address! === Subject: how to make money fast...... firs of all sup all? you have no idea how lucky you are for bump into this massage, so be for you close its, satisfied 5 min of curiosity ( like I did) and believe me, most of you wont regret .83 it worth that. Already as a beginning if you read this from a forum, you should pass what writing to the notepad or to the word and save, that you could read it carefully (in cash of the forumsÍ managers decide to delete the massage. And to bed, cause if so it them lost.) Just read it, crazy how genius it is. My first thought was that it is a fraud, I said to myself, ńyea right..î but as most of us I was curious and I keep reading, Was writing there that if I sand 1$ to each of the 6 address, that was written below, I could make lot of money in little time. Usually I ignore from this kind of letters, but the love and the attention that cause me this later (and the big consequences), show me that this one different, and this why I decided to take part in this. If this later continued as he suppose to, if we will put our suspects aside, surly, every one of us will make a profit. How absurd it to think that we been cheating us, how much energy we waste on the doubt?! Instead of taking this energy and spend it at the opportunity that been given us. The question I asked myself before I agreed to take part in this thing that changed my life, does it true? Yes it is! ItÍs true and itÍs work! The person who was above me (in list) telling: ńwithin 7 days I started get money in mail. I was shocked! I thought it will be finish soon and I didnÍt thought about it too much. But the money keeps coming! In the first week I get 25$ in the end of the second week 1000$, in the end of the third week I had more then 10000$!!! And it still grow! Now this the fourth week, I got sum of more then 42000$ and it keep coming!!! It currently was worth the 6$ + 6 stamps!!! For your intention! Follow after the simple instruction (that will be explain down) and you will see how the money coming!!! It easy, it legal and your invest it jest 6$ + 6 stamps!!! (lets say that 1$ for week + one stamp- just show you how much the invest close to nothing)! If all the next instructions will be done in attention you will achieve a big profit. This plan stay successful for the honesty and cooperation of people which read it. Please, keep the plane successful and help each one of us to help each other. Here the four steps to success: ! Step 1: Take 6 different piece of papers, and write the flowing sentence on each piece of paper: ńplease put me in your listî, write your names and the exact address on it. Now, you need to have 6 close envelope, each one have note with the above mentioned sentence, your name and address and 1$ in each of them. What you going to do is to create serves. You ask legitimate serves which is to be added to the list and you pay for it. 1. Gal Dor- P.O. box: 8216 polg Natanya industry area, Israel 2. Efrat Yakir - sîa 53 Tel Aviv, Israel 3. Shai Mida- brinshtein 3/3 Holon, Israel 4. Zevi Zesler- hadekel 2/2 Carmiel 21892, Israel 5. Naor Barouch- avenuesÍ Rotshild 27/7 Ashdod, Israel 6. Liran Rosenbaum- Ben Eliezer 8/16 Ashdod, Israel Those are the 6 address that you need to send the envelope to [CapitalEth] one each! Step 2: Delete the first name from the list ( number 1), move all the other names one up ( that who was number 6 becoming 5 and 5 becoming 4.83) and add your name and full address as number 6 in list. Step 3: Step 4: Also send all your friend this fix letter to their e-mail. Mark the title of the subject (this how all will see when they get in to specific group). Click on send message and you finished with the first group!!! Please remember! This plan stay successful because the honesty and the integration of the persons taking part, and by careful and intention of following instruction. Look like this: if you people of integration this plan will continue, and the money that so much people get will come to you too. So all letter get attention, and is instruction be done carefully- 6 people will get money (1$) each, for you to take part in this. Your name will move up in the list in geometric way, that until your name coming to the number 1 place, you already get thousands of dollars!!! That it, congratulations! All you need to do, it to pass from one group to other and send again. When you will get used to this it will take you like 30 seconds! Remember! As much as you send the message more you get. But you need to send minimum 300 messages. That it! Start getting cash from all over the word in days! Remember, when you adversities in out of country forums you need to translate the letter to the selected language. Now we will discuss the question why to take part? Now 343 people will send minimum 300 messages with my name as number 3 and each one will get just 7 responses, I will make 2410$ !!! o.k., now the fun part, each one from the 2401 people send minimum 300 letters with my name as number 2 and each one of them get just 7 answers. This will give me 16, 807$. Now each one from the 16,807 people will send this message to 300 groups with my name as number 1 and if still 7 will answers from all the groups I will get 117,649$ with a basically spend of 6$ and 6 stamps!!!!!!!! Remember ńhonesty it the best wayî, you donÍt need to cheat. Be fear and you will get the huge profit which is ńa lot of moneyî. So if all will get just 7 answers as the above mentioned description, you will get some of 117,649$ !!! would you be ready to miss this kind of opportunity for ńriskî of 6$ + 6 stamps ?! I didnÍt, and I hope you will join me that we will help each other. Notes: - maybe you will want to hire mailbox because the amount of mail that you will get. If you prefer to stay anonymous, you can use invented name. - If there is person in the list that you already known still send him the 1$, they part of this genius idea too! Would you prepare to take responsible and not to miss that simple wonderful business possible, for ńriskî of 6$ + 6 stamps ?? I hope that you will join me that we will be possible each other! - could be that you already saw this massage before because this business works! And the most important.83! If you try to cheat people by send the massages with your name in the list, without send the money to the 6 people that already was in the list [CapitalEth] you almost donÍt get a thing! Also donÍt try to change the other names. Already it a easy money and think what would happen if every one would cheat.83 exactly! Non of that would happen and you wouldnÍt have a thing!! So lets go no one try to be clever, going according the rules and the money in the way to you!!!! Good luck!!!! === Subject: easy and fast way to make money and a lot... first of all sup all? you have no idea how lucky you are for bump into this massage, so be for you close its, satisfied 5 min of curiosity ( like I did) and believe me, most of you wont regret .83 it worth that. Already as a beginning if you read this from a forum, you should pass what writing to the notepad or to the word and save, that you could read it carefully (in cash of the forumsÍ managers decide to delete the massage. And to bed, cause if so it them lost.) Just read it, crazy how genius it is. My first thought was that it is a fraud, I said to myself, ńyea right..î but as most of us I was curious and I keep reading, Was writing there that if I sand 1$ to each of the 6 address, that was written below, I could make lot of money in little time. Usually I ignore from this kind of letters, but the love and the attention that cause me this later (and the big consequences), show me that this one different, and this why I decided to take part in this. If this later continued as he suppose to, if we will put our suspects aside, surly, every one of us will make a profit. How absurd it to think that we been cheating us, how much energy we waste on the doubt?! Instead of taking this energy and spend it at the opportunity that been given us. The question I asked myself before I agreed to take part in this thing that changed my life, does it true? Yes it is! ItÍs true and itÍs work! The person who was above me (in list) telling: ńwithin 7 days I started get money in mail. I was shocked! I thought it will be finish soon and I didnÍt thought about it too much. But the money keeps coming! In the first week I get 25$ in the end of the second week 1000$, in the end of the third week I had more then 10000$!!! And it still grow! Now this the fourth week, I got sum of more then 42000$ and it keep coming!!! It currently was worth the 6$ + 6 stamps!!! For your intention! Follow after the simple instruction (that will be explain down) and you will see how the money coming!!! It easy, it legal and your invest it jest 6$ + 6 stamps!!! (lets say that 1$ for week + one stamp- just show you how much the invest close to nothing)! If all the next instructions will be done in attention you will achieve a big profit. This plan stay successful for the honesty and cooperation of people which read it. Please, keep the plane successful and help each one of us to help each other. Here the four steps to success: ! Step 1: Take 6 different piece of papers, and write the flowing sentence on each piece of paper: ńplease put me in your listî, write your names and the exact address on it. Now, you need to have 6 close envelope, each one have note with the above mentioned sentence, your name and address and 1$ in each of them. What you going to do is to create serves. You ask legitimate serves which is to be added to the list and you pay for it. 1. Gal Dor- P.O. box: 8216 polg Natanya industry area, Israel 2. Efrat Yakir - sîa 53 Tel Aviv, Israel 3. Shai Mida- brinshtein 3/3 Holon, Israel 4. Zevi Zesler- hadekel 2/2 Carmiel 21892, Israel 5. Naor Barouch- avenuesÍ Rotshild 27/7 Ashdod, Israel 6. Liran Rosenbaum- Ben Eliezer 8/16 Ashdod, Israel Those are the 6 address that you need to send the envelope to [CapitalEth] one each! Step 2: Delete the first name from the list ( number 1), move all the other names one up ( that who was number 6 becoming 5 and 5 becoming 4.83) and add your name and full address as number 6 in list. Step 3: Step 4: Also send all your friend this fix letter to their e-mail. Mark the title of the subject (this how all will see when they get in to specific group). Click on send message and you finished with the first group!!! Please remember! This plan stay successful because the honesty and the integration of the persons taking part, and by careful and intention of following instruction. Look like this: if you people of integration this plan will continue, and the money that so much people get will come to you too. So all letter get attention, and is instruction be done carefully- 6 people will get money (1$) each, for you to take part in this. Your name will move up in the list in geometric way, that until your name coming to the number 1 place, you already get thousands of dollars!!! That it, congratulations! All you need to do, it to pass from one group to other and send again. When you will get used to this it will take you like 30 seconds! Remember! As much as you send the message more you get. But you need to send minimum 300 messages. That it! Start getting cash from all over the word in days! Remember, when you adversities in out of country forums you need to translate the letter to the selected language. Now we will discuss the question why to take part? Now 343 people will send minimum 300 messages with my name as number 3 and each one will get just 7 responses, I will make 2410$ !!! o.k., now the fun part, each one from the 2401 people send minimum 300 letters with my name as number 2 and each one of them get just 7 answers. This will give me 16, 807$. Now each one from the 16,807 people will send this message to 300 groups with my name as number 1 and if still 7 will answers from all the groups I will get 117,649$ with a basically spend of 6$ and 6 stamps!!!!!!!! Remember ńhonesty it the best wayî, you donÍt need to cheat. Be fear and you will get the huge profit which is ńa lot of moneyî. So if all will get just 7 answers as the above mentioned description, you will get some of 117,649$ !!! would you be ready to miss this kind of opportunity for ńriskî of 6$ + 6 stamps ?! I didnÍt, and I hope you will join me that we will help each other. Notes: - maybe you will want to hire mailbox because the amount of mail that you will get. If you prefer to stay anonymous, you can use invented name. - If there is person in the list that you already known still send him the 1$, they part of this genius idea too! Would you prepare to take responsible and not to miss that simple wonderful business possible, for ńriskî of 6$ + 6 stamps ?? I hope that you will join me that we will be possible each other! - could be that you already saw this massage before because this business works! And the most important.83! If you try to cheat people by send the massages with your name in the list, without send the money to the 6 people that already was in the list [CapitalEth] you almost donÍt get a thing! Also donÍt try to change the other names. Already it a easy money and think what would happen if every one would cheat.83 exactly! Non of that would happen and you wouldnÍt have a thing!! So lets go no one try to be clever, going according the rules and the money in the way to you!!!! Good luck!!!! === Subject: Two distribution theory questions I'm reading up on distribution theory this summer and I became interested in the following question (it is not a home work question, I'm just interested): If Dk is the space of test functions f : R -> C with support in [-k,k], is Dk separable? What would be a countably dense subset? I'm thinking that maybe if {Fn} are the standard bump functions and Pmn is a polynomial with complex rational coefficients cut off in such a way that fn * Pmn lies in Dk then these will do? James === Subject: problem about polynomial distribution of binomial distributions Assume (S1,S2,S3,S4,S5) be a polynomial distribution of orden 5 with number of paramenter n and probability paramameters {p1,p2,p3,p4,p5} Show that (S1+S2,S3+S4,S5) is a polynomial distribution of orden 3 with number of parameters n and probability paramenters {p1+p2,p3+p4,p5} === Subject: Re: problem about polynomial distribution of binomial distributions >Assume (S1,S2,S3,S4,S5) be a polynomial distribution of orden 5 with number of paramenter n and probability paramameters {p1,p2,p3,p4,p5} > >Show that (S1+S2,S3+S4,S5) is a polynomial distribution of orden 3 with >number of parameters n and probability paramenters {p1+p2,p3+p4,p5} > What does (S1,S2,S3,S4,S5) a polynomial distribution of order 5 with number of parameter n and probability parameters {p1,p2,p3,p4,p5} mean? The terminology is obscure. Is this homework? -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: problem about polynomial distribution of binomial distributions polynomial distribution = multinomial distribution > >>Assume (S1,S2,S3,S4,S5) be a polynomial distribution of orden 5 with >>number of paramenter n and probability paramameters {p1,p2,p3,p4,p5} >> >>Show that (S1+S2,S3+S4,S5) is a polynomial distribution of orden 3 with >>number of parameters n and probability paramenters {p1+p2,p3+p4,p5} >> > > What does (S1,S2,S3,S4,S5) a polynomial distribution of order 5 with > number of parameter n and probability parameters {p1,p2,p3,p4,p5} mean? > The terminology is obscure. > > Is this homework? no === Subject: Re: problem about polynomial distribution of binomial distributions >polynomial distribution = multinomial distribution > > Top posting is very bad form. > > >> >> >> >>>Assume (S1,S2,S3,S4,S5) be a polynomial distribution of orden 5 with >>>number of paramenter n and probability paramameters {p1,p2,p3,p4,p5} >>> >>>Show that (S1+S2,S3+S4,S5) is a polynomial distribution of orden 3 with number of parameters n and probability paramenters {p1+p2,p3+p4,p5} >>> >>> >>> >> <>What does (S1,S2,S3,S4,S5) a polynomial distribution of order 5 with >> number of parameter n and probability parameters {p1,p2,p3,p4,p5} mean? >> the terminology is obscure. > So I suppose you mean to say that S1, S2, S3, S4, and S5 are *random variables* with multinomial distribution with parameters (n; p1, p2 , p3, p4, p5). This describes the number of items in each of five categories when n items are independently designated to be in these categories according to the distribution (p1, p2, p3, p4, p5). Define new categories: The new 1 is old 1 or old 2; the new 2 is old 3 or old 4; and the new 3 is the old 5. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: problem about polynomial distribution of binomial distributions with defining the new categories do you mean S1new = S1old or S2old? > So I suppose you mean to say that S1, S2, S3, S4, and S5 are *random > variables* with multinomial distribution with parameters (n; p1, p2 , p3, > p4, p5). This describes the number of items in each of five categories > when n items are independently designated to be in these categories > according to the distribution (p1, p2, p3, p4, p5). Define new > categories: The new 1 is old 1 or old 2; the new 2 is old 3 or old 4; > and the new 3 is the old 5. > > -- > Stephen J. Herschkorn sjherschko@netscape.net > Math Tutor in Central New Jersey and Manhattan > === Subject: Re: problem about polynomial distribution of binomial distributions with defining the new categories do you mean S1new = S1old or S2old? > So I suppose you mean to say that S1, S2, S3, S4, and S5 are *random > variables* with multinomial distribution with parameters (n; p1, p2 , p3, > p4, p5). This describes the number of items in each of five categories > when n items are independently designated to be in these categories > according to the distribution (p1, p2, p3, p4, p5). Define new > categories: The new 1 is old 1 or old 2; the new 2 is old 3 or old 4; > and the new 3 is the old 5. > > -- > Stephen J. Herschkorn sjherschko@netscape.net > Math Tutor in Central New Jersey and Manhattan > === Subject: Re: Finding the expected length to a pattern? > Hi all, > > I am interested in the following problem: > > Suppose I have a coin with probability of having a head H p, and tail T with > probability 1-p. > > What is the expected waited tosses before I get a pattern > > HTTHTTH, > HTHTHTHT, > These kinds of problems can be solved very efficiently with martingales. You can look up a discussion in Ross's Stochastic Processes or in David Williams' Probability with Martingales (look for the ABRACADABRA problem). Mike === Subject: Re: Finding the expected length to a pattern? > I am interested in the following problem: Suppose I have a coin with probability of having a head H p, and tail T with > probability 1-p. What is the expected waited tosses before I get a pattern HTTHTTH, > HTHTHTHT, etc. This is not a HW problem. I am interested in the fastest approach. Because I > already know how to do it using Markov chain modeling, but that takes a long > time. I am trying to understand Ross' approach(but his approach is hard to > understand...) so I want to see if anybody on the Internet has a > better/faster/easier-understanding approach. My apologies. I just posted a solution method using Markov chains, since I responded to a response which didn't include this paragraph. Are you interested in the fastest solution using pencil and paper? Scott -- Scott Hemphill hemphill@alumni.caltech.edu This isn't flying. This is falling, with style. -- Buzz Lightyear === Subject: Re: Finding the expected length to a pattern? > >> I am interested in the following problem: >> >> Suppose I have a coin with probability of having a head H p, and tail T >> with >> probability 1-p. >> >> What is the expected waited tosses before I get a pattern >> >> HTTHTTH, >> HTHTHTHT, >> >> etc. >> >> This is not a HW problem. I am interested in the fastest approach. >> Because I >> already know how to do it using Markov chain modeling, but that takes a >> long >> time. I am trying to understand Ross' approach(but his approach is hard >> to >> understand...) so I want to see if anybody on the Internet has a >> better/faster/easier-understanding approach. > > My apologies. I just posted a solution method using Markov chains, > since I responded to a response which didn't include this paragraph. > Are you interested in the fastest solution using pencil and paper? > > Scott > -- > Scott Hemphill hemphill@alumni.caltech.edu > This isn't flying. This is falling, with style. -- Buzz Lightyear Sure. I know there is a solution that only needs a few steps of pencil and paper... It is in Ross 8th Edition of Introduction to Probability Models... But I don't understand it so I don't know how to generalize that approach. It needs to first decompose the string HTHTHTT to maximal substrings, etc... Does anybody know how to do it? === Subject: Re: Finding the expected length to a pattern? > >> I am interested in the following problem: >> >> Suppose I have a coin with probability of having a head H p, and tail T >> with >> probability 1-p. >> >> What is the expected waited tosses before I get a pattern >> >> HTTHTTH, >> HTHTHTHT, >> >> etc. >> >> This is not a HW problem. I am interested in the fastest approach. >> Because I >> already know how to do it using Markov chain modeling, but that takes a >> long >> time. I am trying to understand Ross' approach(but his approach is hard >> to >> understand...) so I want to see if anybody on the Internet has a >> better/faster/easier-understanding approach. > > My apologies. I just posted a solution method using Markov chains, > since I responded to a response which didn't include this paragraph. > Are you interested in the fastest solution using pencil and paper? > > Scott > -- > Scott Hemphill hemphill@alumni.caltech.edu > This isn't flying. This is falling, with style. -- Buzz Lightyear > Sure. I know there is a solution that only needs a few steps of pencil and > paper... It is in Ross 8th Edition of Introduction to Probability Models... But I don't understand it so I don't know how to generalize that approach. It needs to first decompose the string HTHTHTT to maximal substrings, > etc... Does anybody know how to do it? Here are the mechanics. (See your reference for a proof as to why this works.) I'll use your HTTHTTH as an example. For the first toss, bettor number one joins the game with a total wealth of 1 unit. He places a bet on H. If he loses, he leaves the game with zero wealth. If he wins he receives a payout of 1/p units. The game is fair, because he has a 1-p chance of zero and a p chance of 1/p units, for an expected value of 1 unit. If he loses, he retires from the game. If we wins, he bets all his wealth on T for the second toss. Again, if he loses, he leaves with nothing. If he wins, his total wealth will now be 1/p * 1/(1-p) to make it fair. He will continue in this manner, betting successively on the the remaining letters ..THTTH to finish the pattern in the successive tosses. If he wins, he leaves the game with 1/p^3 * 1/(1-p)^4 units, reflecting 3 bets on heads and 4 bets on tails. Also, for the second toss, a second bettor joins the game, bringing with him his total wealth of 1 unit. He places his bet on the first letter of the pattern, H. He follows the same procedure that the first bettor does, just one toss later. Each new toss brings one new bettor to the game. When a bettor finally wins, the total wealth in the game will be owned by three bettors: the one who just won, having bet on HTTHTTH, with a total wealth of 1/p^3 * 1/(1-p)^4. the one who came in three tosses later, having been successful so far with HTTH, and having a total wealth of 1/p^2 * 1/(1-p)^2. the one who came in six tosses later, having just bet on the first H in the pattern, and having a total wealth of 1/p. So the total wealth (no matter how many tosses have occurred) is 1/p^3 * 1/(1-p)^4 + 1/p^2 * 1/(1-p)^2 + 1/p But since each new bettor increases the expected total wealth of the whole group by 1 unit when he joins the game, the expected number of betters (which is the same as the expected number of tosses) is the above number: 1/p^3 * 1/(1-p)^4 + 1/p^2 * 1/(1-p)^2 + 1/p Scott -- Scott Hemphill hemphill@alumni.caltech.edu This isn't flying. This is falling, with style. -- Buzz Lightyear === Subject: Re: Finding the expected length to a pattern? > Hi Kiki > Hi all, > > I am interested in the following problem: > > Suppose I have a coin with probability of having a head H p, and tail T with > probability 1-p. > > What is the expected waited tosses before I get a pattern > > HTTHTTH, > HTHTHTHT, > Could you please elucidiate what you mean by a pattern? HHTHHTT is as much a pattern as HTHTHTHT as is HHHTTHHH as is HTHHHTHTHHH I expect Kiki means the specific string HTTHTTH as one problem, and the specific string HTHTHTHT as another. Here's how to solve such a problem. I'll work on finding the expected number of tosses of another string. You can use this technique on the strings you are interested in. I'll choose HTHTT. First I define states that represent the partial progress towards the desired goal. State I is the initial state. Tossing H takes you to state H. Tossing T takes you back to state I. State H. Tossing T takes you to state HT. Tossing H takes you to state H. (you don't have to start completely over since you have already tossed an H. State HT. Tossing H takes you to state HTH. Tossing T takes you to state I. State HTH. Tossing T takes you to state HTHT. Tossing H takes you to state H. State HTHT. Tossing T takes you to state HTHTT. Tossing H takes you to state HTH. State HTHTT. This is the goal. There are no further states. Now I construct a transition matrix, which defines the probabilities of moving from one state to another. T = 1-p p 0 0 0 0 0 p 1-p 0 0 0 1-p 0 0 p 0 0 0 p 0 0 1-p 0 0 0 0 p 0 1-p 0 0 0 0 0 0 Now I define a state vector x_k, which contains the probabilities of being in each of these states. The initial state vector x_0 is {1,0,0,0,0,0}, i.e. the probability of being in state I is one. After one toss, the state vector x_1 = x_0 . T = {1-p,p,0,0,0,0}. After two tosses, the state vector is x_2 = x_1 . T = x_0 . T^2 = {1-2p+p^2,p,p-p^2,0,0,0}. In general, after k tosses, x_k = x_0 . T^k. We are interested in the probability that the goal is reached on the k'th toss, which is the last position in the state vector. We can select this position by multiplying by the column vector (0) (0) s = (0) (0) (0) (1) So the probability of achieving the goal after exactly k tosses is x_0 . T^k . s The expected number of tosses is then: Sum(k=0,Infinity; k x_0 . T^k . s) This sum can be rewritten as: x_0 . Sum(k=0,Infinity; k T^k) . s If t is a scalar, it isn't too hard to show that Sum(k=0,Infinity; k t^k) is t/(1-t)^2 (with some restrictions on t, so that the sum converges). There is a similar result for a matrix T: Sum(k=0,Infinity; k T^k) = T . (I-T)^-2 (again, with some restrictions on T, which are satisifed in this case, so that the sum converges). I-T = p -p 0 0 0 0 0 1-p p-1 0 0 0 p-1 0 1 -p 0 0 0 -p 0 1 p-1 0 0 0 0 -p 1 p-1 0 0 0 0 0 1 The inverse of this is, well, a little messy. Here it's good to have a math package: This is Mathematica's output: 1 -1 + p - p -2 1 -2 1 {{-------, -----------, (-1 + p) + -, (-1 + p) , -----, 1}, 2 3 3 p 1 - p p - p (-1 + p) p 2 1 -2 -1 + p - p -2 1 -2 1 > {----- + p , -----------, (-1 + p) + -, (-1 + p) , -----, 1}, 1 - p 3 p 1 - p (-1 + p) p 1 -2 -3 1 -2 1 -2 1 > {----- + p , -(-1 + p) + -, (-1 + p) + -, (-1 + p) , -----, 1}, 1 - p p p 1 - p 1 -3 -2 -2 1 > {------, -(-1 + p) , (-1 + p) , (-1 + p) , -----, 1}, 2 1 - p p - p 1 p p p 1 > {-----, -(---------), ---------, ---------, -----, 1}, 1 - p 3 2 2 1 - p (-1 + p) (-1 + p) (-1 + p) > {0, 0, 0, 0, 0, 1}} So now all you do is square this, multiply by T, multiply on the left by x0 and on the right by s, and you get: 1 ----------- p^2 (1-p)^3 Scott -- Scott Hemphill hemphill@alumni.caltech.edu This isn't flying. This is falling, with style. -- Buzz Lightyear === Subject: Re: Finding the expected length to a pattern? > >> Hi Kiki >> >> >> Hi all, >> >> I am interested in the following problem: >> >> Suppose I have a coin with probability of having a head H p, and tail T >> with >> probability 1-p. >> >> What is the expected waited tosses before I get a pattern >> >> HTTHTTH, >> HTHTHTHT, >> >> >> Could you please elucidiate what you mean by a pattern? >> >> HHTHHTT >> >> is as much a pattern as >> >> HTHTHTHT >> >> as is >> >> HHHTTHHH >> >> as is >> >> HTHHHTHTHHH > > I expect Kiki means the specific string HTTHTTH as one problem, and > the specific string HTHTHTHT as another. Here's how to solve such a > problem. > > I'll work on finding the expected number of tosses of another string. You > can use this technique on the strings you are interested in. I'll choose > HTHTT. > > First I define states that represent the partial progress towards the > desired goal. > > State I is the initial state. > Tossing H takes you to state H. > Tossing T takes you back to state I. > > State H. > Tossing T takes you to state HT. > Tossing H takes you to state H. (you don't have to start completely over > since you have already tossed an H. > > State HT. > Tossing H takes you to state HTH. > Tossing T takes you to state I. > > State HTH. > Tossing T takes you to state HTHT. > Tossing H takes you to state H. > > State HTHT. > Tossing T takes you to state HTHTT. > Tossing H takes you to state HTH. > > State HTHTT. > This is the goal. There are no further states. > > Now I construct a transition matrix, which defines the probabilities of > moving from one state to another. > > T = > 1-p p 0 0 0 0 > 0 p 1-p 0 0 0 > 1-p 0 0 p 0 0 > 0 p 0 0 1-p 0 > 0 0 0 p 0 1-p > 0 0 0 0 0 0 > > Now I define a state vector x_k, which contains the probabilities of being > in > each of these states. The initial state vector x_0 is {1,0,0,0,0,0}, i.e. > the probability of being in state I is one. After one toss, the state > vector > x_1 = x_0 . T = {1-p,p,0,0,0,0}. After two tosses, the state vector is > x_2 = x_1 . T = x_0 . T^2 = {1-2p+p^2,p,p-p^2,0,0,0}. In general, after > k tosses, x_k = x_0 . T^k. > > We are interested in the probability that the goal is reached on the k'th > toss, which is the last position in the state vector. We can select this > position by multiplying by the column vector > > (0) > (0) > s = (0) > (0) > (0) > (1) > > So the probability of achieving the goal after exactly k tosses is > x_0 . T^k . s > > The expected number of tosses is then: > > Sum(k=0,Infinity; k x_0 . T^k . s) > > This sum can be rewritten as: > > x_0 . Sum(k=0,Infinity; k T^k) . s > > If t is a scalar, it isn't too hard to show that Sum(k=0,Infinity; k t^k) > is > t/(1-t)^2 (with some restrictions on t, so that the sum converges). > > There is a similar result for a matrix T: > > Sum(k=0,Infinity; k T^k) = T . (I-T)^-2 (again, with some restrictions on > T, > which are satisifed in this case, so that the sum converges). > > I-T = > p -p 0 0 0 0 > 0 1-p p-1 0 0 0 > p-1 0 1 -p 0 0 > 0 -p 0 1 p-1 0 > 0 0 0 -p 1 p-1 > 0 0 0 0 0 1 > > The inverse of this is, well, a little messy. Here it's good to have a > math package: This is Mathematica's output: > > 1 -1 + p - p -2 1 -2 1 > {{-------, -----------, (-1 + p) + -, (-1 + p) , -----, 1}, > 2 3 3 p 1 - p > p - p (-1 + p) p > > 2 > 1 -2 -1 + p - p -2 1 -2 1 >> {----- + p , -----------, (-1 + p) + -, (-1 + p) , -----, 1}, > 1 - p 3 p 1 - p > (-1 + p) p > > 1 -2 -3 1 -2 1 -2 1 >> {----- + p , -(-1 + p) + -, (-1 + p) + -, (-1 + p) , -----, 1}, > 1 - p p p 1 - p > > 1 -3 -2 -2 1 >> {------, -(-1 + p) , (-1 + p) , (-1 + p) , -----, 1}, > 2 1 - p > p - p > > 1 p p p 1 >> {-----, -(---------), ---------, ---------, -----, 1}, > 1 - p 3 2 2 1 - p > (-1 + p) (-1 + p) (-1 + p) > >> {0, 0, 0, 0, 0, 1}} > > So now all you do is square this, multiply by T, multiply on the left by > x0 and on the right by s, and you get: > > 1 > ----------- > p^2 (1-p)^3 > > Scott > -- > Scott Hemphill hemphill@alumni.caltech.edu > This isn't flying. This is falling, with style. -- Buzz Lightyear 1. How do you verify your result is correct? 2. Suppose you are given only 5 minutes with paper&pencil, can you do it? === Subject: Re: Finding the expected length to a pattern? > Hi all, > > I am interested in the following problem: > > Suppose I have a coin with probability of having a head H p, and tail T with > probability 1-p. > > What is the expected waited tosses before I get a pattern > > HTTHTTH, > HTHTHTHT, I don't know if this is exactly what you want, but the expected number of trials (t) to achieve m successes is t = m/p where p is the probability of 1 success. For example, to get 4 sixes when rolling a single die, you would expect to have to roll it t = 4/(1/6) = 24 times. But that's not the same as getting 4 sixes in a row. With a coin, the probability of getting HHHH is 1/16. The pattern of consecutive H (or T) forms a negaitve binomial distribution. That means if you flip until the HHHH pattern comes up, you will have twice as many HHH paterns as HHHH, twice as many HH as HHH and twice as many H as HH. So if we stop at the first occurence of HHHH, there are 1 HHHH 2 HHH 4 HH 8 H The total H that occured is then 1 * 4 = 4 2 * 3 = 6 4 * 2 = 8 8 * 1 = 8 --------- 26 Now since H only accounts for half the flips, there were 26 T also. So it takes 52 flips to get HHHH. > > etc. > > This is not a HW problem. I am interested in the fastest approach. Because I > already know how to do it using Markov chain modeling, but that takes a long > time. I am trying to understand Ross' approach(but his approach is hard to > understand...) so I want to see if anybody on the Internet has a > better/faster/easier-understanding approach. > === Subject: Re: Finding the expected length to a pattern? > Hi all, > > I am interested in the following problem: > > Suppose I have a coin with probability of having a head H p, and tail T with > probability 1-p. > > What is the expected waited tosses before I get a pattern > > HTTHTTH, > HTHTHTHT, > > I don't know if this is exactly what you want, but the expected > number of trials (t) to achieve m successes is > > t = m/p > > where p is the probability of 1 success. > > For example, to get 4 sixes when rolling a single die, you would > expect to have to roll it > > t = 4/(1/6) = 24 > > times. But that's not the same as getting 4 sixes in a row. > > With a coin, the probability of getting HHHH is 1/16. The pattern > of consecutive H (or T) forms a negaitve binomial distribution. > That means if you flip until the HHHH pattern comes up, you will > have twice as many HHH paterns as HHHH, twice as many HH as HHH > and twice as many H as HH. So if we stop at the first occurence of > HHHH, there are > > 1 HHHH > 2 HHH > 4 HH > 8 H > > The total H that occured is then > > 1 * 4 = 4 > 2 * 3 = 6 > 4 * 2 = 8 > 8 * 1 = 8 > --------- > 26 > > Now since H only accounts for half the flips, there were 26 T also. > So it takes 52 flips to get HHHH. > > This is not correct. The right answer is that it takes 30 flips to get HHHH (with a fair coin). One way to see this is as follows: Let X be the expected number of flips to get HHHH. A. If we start T, it takes X more flips on average to get HHHH. B. If we start HT, it takes X more flips on average to get HHHH. C. If we start HHT, it takes X more flips on average to get HHHH. D. If we start HHHT, it takes X more flips on average to get HHHH. E. IF we start HHHH, it takes 4 flips on average to get HHHH (duh). Combining the five cases, we get the equation X = (1/2)*(X+1) + (1/4)*(X+2) + (1/8)*(X+3)+ (1/16)*(X+4) + (1/6)*4 Solving for X gives X = 30. Mike Mike === Subject: Re: Finding the expected length to a pattern? >> Hi all, >> I am interested in the following problem: >> Suppose I have a coin with probability of having a head H p, and tail T with >> probability 1-p. >> What is the expected waited tosses before I get a pattern >> HTTHTTH, >> HTHTHTHT, >> I don't know if this is exactly what you want, but the expected >> number of trials (t) to achieve m successes is >> t = m/p >> where p is the probability of 1 success. >> For example, to get 4 sixes when rolling a single die, you would >> expect to have to roll it >> t = 4/(1/6) = 24 >> times. But that's not the same as getting 4 sixes in a row. >> With a coin, the probability of getting HHHH is 1/16. The pattern >> of consecutive H (or T) forms a negaitve binomial distribution. >> That means if you flip until the HHHH pattern comes up, you will >> have twice as many HHH paterns as HHHH, twice as many HH as HHH >> and twice as many H as HH. So if we stop at the first occurence of >> HHHH, there are >> 1 HHHH >> 2 HHH >> 4 HH >> 8 H >> The total H that occured is then >> 1 * 4 = 4 >> 2 * 3 = 6 >> 4 * 2 = 8 >> 8 * 1 = 8 >> --------- >> 26 >> Now since H only accounts for half the flips, there were 26 T also. >> So it takes 52 flips to get HHHH. >This is not correct. The right answer is that it takes 30 flips to get >HHHH (with a fair coin). One way to see this is as follows: >Let X be the expected number of flips to get HHHH. >A. If we start T, it takes X more flips on average to get HHHH. >B. If we start HT, it takes X more flips on average to get HHHH. >C. If we start HHT, it takes X more flips on average to get HHHH. >D. If we start HHHT, it takes X more flips on average to get HHHH. >E. IF we start HHHH, it takes 4 flips on average to get HHHH (duh). >Combining the five cases, we get the equation >X = (1/2)*(X+1) + (1/4)*(X+2) + (1/8)*(X+3)+ (1/16)*(X+4) + (1/6)*4 >Solving for X gives X = 30. >Mike This is essentially the same as the Markov Chain approach. There are four states, plus the absorbing state, and one wants to find the expected time to the absorbing state. A state in this simple problem is the number of consecutive head which have been observed since the last tail. The transition probabilities are 0 -> 0 1-p 0 -> 1 p 1 -> 0 1-p 1 -> 2 p 2 -> 0 1-p 2 -> 3 p 3 -> 0 1-p 3 -> end p -- 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: Finding the expected length to a pattern? On 21 Jul 2005 10:04:19 -0700, Mike Hochster >> ... >> With a coin, the probability of getting HHHH is 1/16. The pattern >> of consecutive H (or T) forms a negaitve binomial distribution. >> That means if you flip until the HHHH pattern comes up, ... >> ... >> So it takes 52 flips to get HHHH. > >This is not correct. The right answer is that it takes 30 flips to get >HHHH (with a fair coin). One way to see this is as follows: > >Let X be the expected number of flips to get HHHH. > >A. If we start T, it takes X more flips on average to get HHHH. >B. If we start HT, it takes X more flips on average to get HHHH. >C. If we start HHT, it takes X more flips on average to get HHHH. >D. If we start HHHT, it takes X more flips on average to get HHHH. >E. IF we start HHHH, it takes 4 flips on average to get HHHH (duh). > >Combining the five cases, we get the equation >X = (1/2)*(X+1) + (1/4)*(X+2) + (1/8)*(X+3)+ (1/16)*(X+4) + (1/6)*4 >Solving for X gives X = 30. > >Mike > Great solution, so clear and simple. quasi === Subject: Re: Finding the expected length to a pattern? > Hi all, > > I am interested in the following problem: > > Suppose I have a coin with probability of having a head H p, and tail T with > probability 1-p. > > What is the expected waited tosses before I get a pattern > > HTTHTTH, > HTHTHTHT, > > I don't know if this is exactly what you want, but the expected > number of trials (t) to achieve m successes is > > t = m/p > > where p is the probability of 1 success. > > For example, to get 4 sixes when rolling a single die, you would > expect to have to roll it > > t = 4/(1/6) = 24 > > times. But that's not the same as getting 4 sixes in a row. > > With a coin, the probability of getting HHHH is 1/16. The pattern > of consecutive H (or T) forms a negaitve binomial distribution. > That means if you flip until the HHHH pattern comes up, you will > have twice as many HHH paterns as HHHH, twice as many HH as HHH > and twice as many H as HH. So if we stop at the first occurence of > HHHH, there are > > 1 HHHH > 2 HHH > 4 HH > 8 H > > The total H that occured is then > > 1 * 4 = 4 > 2 * 3 = 6 > 4 * 2 = 8 > 8 * 1 = 8 > --------- > 26 > > Now since H only accounts for half the flips, there were 26 T also. > So it takes 52 flips to get HHHH. > > > This is not correct. The right answer is that it takes 30 flips to get > HHHH (with a fair coin). One way to see this is as follows: > > Let X be the expected number of flips to get HHHH. > > A. If we start T, it takes X more flips on average to get HHHH. > B. If we start HT, it takes X more flips on average to get HHHH. > C. If we start HHT, it takes X more flips on average to get HHHH. > D. If we start HHHT, it takes X more flips on average to get HHHH. > E. IF we start HHHH, it takes 4 flips on average to get HHHH (duh). > > Combining the five cases, we get the equation > X = (1/2)*(X+1) + (1/4)*(X+2) + (1/8)*(X+3)+ (1/16)*(X+4) + (1/6)*4 > Solving for X gives X = 30. > Yeah, but my program shows that it takes...30. Oops. Mike === Subject: Re: A question about planar graphs (graph theory) >I have a question. Any graph-theory book has something like this: > >Let G be a connected planar simple graph with v vertices, where v >= 3 and e >edges. Then e <= 3v - 6. > >I have the impression that this corollary holds for general graphs (i.e., those >with closed simple curves and loops with two nodes). Am I right? You're proposing modifying the theory, substituting connected planar simple graph with ____? If you throw out simple, the number of edges is unbounded, because any edge can be duplicated without crossing another. If you throw out planar, e <= v(v-1)/2. If you throw out connected, nothing changes. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: A missing factor to solve a functional equation . Here is the idea , let us look at -1Á) (1-1/x)*f(x) = f(3*x)*(1-3/x) , f real, strictly monotonous , x > 0 pairing factors from both sides this way f(3*x) = 1-1/x f(x) = 1-3/x we've got compatible equations so we obtain a solution f(x)=1-3/x to generalize ... -2Á) (1-4*x^2)*f(x) = 2*f(x)*(1-x^2) pairing factors dosn't work , just have multiplying both sides by x (1-4*x^2)*x*f(x) = 2*x*f(x)*(1-x^2) then x*f(x) = (1-x^2) (1-4*x^2) = 2*x*f(x) and we obtain a solution f(x)=1/x - x to generalize ... IS IT OK? Alain. === Subject: conditional probability If the only information I know is that P(A), P(B), P(C), P(A and B), P(A and C) and P(B and C), (A, B and C are not independent) is there a way to compute P(A and B and C)? Ryan === Subject: Re: conditional probability > If the only information I know is that P(A), P(B), P(C), P(A and B), P(A and > C) and P(B and C), (A, B and C are not independent) is there a way to > compute P(A and B and C? P(A and B and C) = P(A and B) + P(A and C) + P(B and C) - P(A) - P(B) - P(C) + P(A or B or C). So you can compute P(A and B and C) if you also know P(A or B or C); otherwise you can get upper and lower bounds on P(A and B and C) from 0 <= P(A or B or C) <= 1. === Subject: Re: conditional probability >If the only information I know is that P(A), P(B), P(C), P(A and B), P(A and >C) and P(B and C), (A, B and C are not independent) is there a way to >compute P(A and B and C)? > >Ryan > No, not unless you also know P(A or B or C). quasi === Subject: Re: conditional probability >If the only information I know is that P(A), P(B), P(C), P(A and B), P(A and C) and P(B and C), (A, B and C are not independent) is there a way to >compute P(A and B and C)? > No. You need seven parameters to specify the probabilities of all the atoms (i.e., there are seven degrees of freedom), yet you give only six parameters above. You should be able to determine bounds on P(ABC). Why is your subject, conditional probability? -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: conditional probability how do you know seven paramenters are needed ? > >>If the only information I know is that P(A), P(B), P(C), P(A and B), P(A >>and C) and P(B and C), (A, B and C are not independent) is there a way to >>compute P(A and B and C)? >> > > No. You need seven parameters to specify the probabilities of all the > atoms (i.e., there are seven degrees of freedom), yet you give only six > parameters above. You should be able to determine bounds on P(ABC). > > Why is your subject, conditional probability? > > -- > Stephen J. Herschkorn sjherschko@netscape.net > Math Tutor in Central New Jersey and Manhattan > === Subject: Re: conditional probability >how do you know seven paramenters are needed ? > > Top posting is really bad form. > > > >> >> >> >>>If the only information I know is that P(A), P(B), P(C), P(A and B), P(A and C) and P(B and C), (A, B and C are not independent) is there a way to compute P(A and B and C)? >>> >>No. You need seven parameters to specify the probabilities of all the >>atoms (i.e., there are seven degrees of freedom), yet you give only six >>parameters above. You should be able to determine bounds on P(ABC). >> There are eight atoms whose probabilities must be determined. Letting ' denote complement, these are ABC, ABC', AB'C,..., and A'B'C'. Since the probabilities sum to 1, there are seven degrees of freedom. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: conditional probability > >If the only information I know is that P(A), P(B), P(C), P(A and B), P(A and C) and P(B and C), (A, B and C are not independent) is there a way to >compute P(A and B and C)? > > > No. You need seven parameters to specify the probabilities of all the > atoms (i.e., there are seven degrees of freedom), yet you give only six > parameters above. You should be able to determine bounds on P(ABC). > > Why is your subject, conditional probability? > > -- > Stephen J. Herschkorn sjherschko@netscape.net > Math Tutor in Central New Jersey and Manhattan > Well I had been going down the path P( A and B and C ) = P(A) P(B|A) P(C|(A and B)) and so I guess conditional probability was on my brain, but I agree it was not the best choice. Some pointers on how to generate an upper bound would also be useful. === Subject: Re: conditional probability > > >> >> >> >>> <>If the only information I know is that P(A), P(B), P(C), P(A and >>> B), P(A and C) and P(B and C), (A, B and C are not independent) is >>> there a way tocompute P(A and B and C)? >> >>> >>> >>No. You need seven parameters to specify the probabilities of all the >>atoms (i.e., there are seven degrees of freedom), yet you give only six >>parameters above. You should be able to determine bounds on P(ABC). >> >> >> >Some pointers on how to generate an upper bound would be useful. > With binary indices, let p_ijk denote the probabilities of the atoms. I.e., letting ' denote complement, p111 = P(ABC), p110 = P(ABC'), p101 = P(AB'C), etc. You have seven linear equations in eight variables; here are three of them: sum(j,k; p_1jk) = PA sum(k; p_11k) = P(AB) sum(i,j,k; p_ijk) = 1. Solve for all the other p_ijk's in terms of p111. Each of these values must be between 0 and 1, inclusive. This gives you seven inequalities that p111 must satisfy. Numerically, you can also do this by linear programming. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: conditional probability > >> >> >>> >>> >>> >>>> <>If the only information I know is that P(A), P(B), P(C), P(A and >>>> B), P(A and C) and P(B and C), (A, B and C are not independent) is >>>> there a way tocompute P(A and B and C)? >>> >>> >>>> >>> >>> No. You need seven parameters to specify the probabilities of all the >>> atoms (i.e., there are seven degrees of freedom), yet you give only six >>> parameters above. You should be able to determine bounds on P(ABC). >>> >>> >> >> Some pointers on how to generate an upper bound would be useful. >> > > With binary indices, let p_ijk denote the probabilities of the > atoms. I.e., letting ' denote complement, p111 = P(ABC), p110 = > P(ABC'), p101 = P(AB'C), etc. You have seven linear equations in > eight variables; here are three of them: > > sum(j,k; p_1jk) = PA > sum(k; p_11k) = P(AB) > sum(i,j,k; p_ijk) = 1. > > Solve for all the other p_ijk's in terms of p111. Each of these > values must be between 0 and 1, inclusive. This gives you seven > inequalities that p111 must satisfy. Correction : fourteen inequalities -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan > Every bit of Cantorianism has been well enough defined for the > understanding of thousands upon thousands of people. That TO fails where > so many have succeeded says more about TO than about the adequacy of > Cantorianism's explanations. The fact that a faith has millions of adherants doesn't say anything > about its validity. It says something about the society wherein it is > accepted, though. Han de Bruijn If TO or WM or anyone else wants to object to the faith of Cantorianism the way to do it is not to complain about the consequences of the axioms on which they are based but to replace that axiom system with something that is at least not self-contradictory. But that is not what the typical anti-Cantrian does. Discussion, linux) >> There is no mention of one historical or living figure who is >> anti-Cantorian, what their objections were > > Hmm, not quite. I did mention Kronecker. Did you miss that? > Since you are intent on expressing common anti-Cantorian ideas, I suppose that you should mention that many anti-Cantorians believe that there are infinite natural numbers. We have seen that claim come up repeatedly in this group (most recently with Tony Orlow). Will you include this viewpoint in your survey of anti-Cantorians? Also, when *will* you define what you mean by Cantor's theory? Do you mean ZF? If so, why call it Cantor's theory? If not, what *is* Cantor's theory? -- Jesse F. Hughes That's what's brutal about mathematics! When you're wrong, you can have spent years, and lots of effort, and come out at the end with nothing. -- James S. Harris on the path of self-discovery (?) <87sly8axyo.fsf@phiwumbda.org> Discussion, linux) > [...] > Also, when *will* you define what you mean by Cantor's theory? Do you > mean ZF? If so, why call it Cantor's theory? If not, what *is* > Cantor's theory? Actually, I see that he made motions towards this in the other thread Updated:.... But it is still utterly opaque how one is supposed to react to his criticisms. Petry doesn't say that ZF is bad exactly, but that some reasoning about infinite sets should be disallowed. Which reasoning? As far as I can tell: that which yields results he finds absurd. There is no obvious principle to be derived from criticisms like: Certainly infinite sets and power sets exist as absractions. But, abstractions don't necessarily obey exactly that same laws of logic as directly observable objects. Infinite sets? They're okay. Power set too. But reasoning about those is iffy, according to Petry. They might not obey the same laws of logic as observable objects (like what? 2? Pi? The set of primes less than 10^10^10?). Might not? Uh oh. -- Clouds are always white and the sky is always blue, And houses it doesn't matter what color they are, And ours is made of brick. -- A new song by Quincy P. Hughes >> infinitely large would be a property associated with a single >> member. arbitrarily large is a property associated with a >> collection of members that form an infinite subset. > > Seems harsh to niggle with what is a pretty clear exposition of the > basic problem, but you have to be ultra-careful with words hereabouts. > The arbitrarily large is a property not of the *collection of > elements*, but rather a property of the individual elements, under > some quantification - I guess I don't see how arbitrarily large could be a property of any individual number, and I don't understand what under some quantification adds -- though that might well be simply a failure of imagination on my part. At the same time, it doesn't seem entirely natural to say that being arbitrarily large is a property of a *set* of numbers either. Natural language (English, anyway) suggests its a property of the elements, collectively, like the property of, say, lifting a piano, applied collectively to four men: The members of S are arbitrarily large iff, for each n in S, there is an m in S that is larger than n. Granted, if you think plurals like the men or the members of S refer to sets, then the property in question is a property of sets, but that's at the least a questionable view of the semantics of plurals. (The *set* of four men, for example, didn't lift the piano; the men did.) Chris Menzel > objections to Cantor's Theory, which I plan to contribute > to the Wikipedia. I would be interested in having mathematicians think this etc... > While the pure mathematicians almost unanimously accept > Cantor's Theory (with the exception of a small group of > constructivists), there are lots of intelligent people who > believe it to be an absurdity. Typically, these people > are non-experts in pure mathematics, but they are people > who have who have found mathematics to be of great practical > value in science and technology, and who like to view > mathematics itself as a science. Many mathematicians, including pure mathematicians, view mathematics as a science (hence for example: MSRI = Mathematical SCIENCES Research Institute). Further vast majority of applied mathematicians accept Cantor's theory. In fact I would argue that more applied mathematicians probably do then pure mathematicians. This is because of the multitude of useful results that it has brought which are used all over in physics, engineering, statistics and other sciences in general. > These anti-Cantorians see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. They claim that that paradigm includes all > of the mathematics which has the potential to be applied to > the task of understanding phenomena in the real world (e.g. > in science and engineering). Tell that to the CS masters student at a local university here that got the unfortunate, purely practical, task of writing a program to detect infinite loops in pascal programs (the idea was to test freshmen programs before running them so that they wouldn't hog the processors). Only after a year of effort did he find out that such a task was in fact impossible. By a theory very similar to Cantor (actually results for computability were derived from Cantor's argument). Most definately not a fantasy world assignment. > Cantor's Theory, if taken seriously, would lead us to believe > that while the collection of all objects in the world of > computation is a countable set, and while the collection of all > identifiable abstractions derived from the world of computation > is a countable set, there nevertheless exist uncountable sets, > implying (again, according to Cantor's logic) the existence > of a super-infinite fantasy world having no connection to the > underlying reality of mathematics. The anti-Cantorians see > such a belief as an absurdity (in the sense of being > disconnected from reality, rather than merely counter-intuitive). This paragraph seems to be biased towards anti-Cantorians rather then describing what they believe. Given all the results that modern analysis (which is heavily based on Cantor's set theory) has brought us, it seems that reality at least seems to work as in our models which do use Cantor's ideas, rather then not. What we know of the real world is based on models that were derived or proved from Cantor's theory. There is a difference in arguing that Cantor's theory has no basis in reality on intuitive means and in arguing that Cantor's theory is wrong. Perhaps it is that Cantor's theory is only useful in approximate models of the reality. So far no one has come up with a system of mathematics capable of solving real world problems that for example doesn't use the completeness of the real numbers which is the property that guarantees uncountability of the real numbers. In any case, most anti-Cantorians that post to sci.math seem to be not claiming that Cantor's theory is the wrong basis for modeling reality on, but they seem to be claiming that Cantor's theory is wrong mathematically, and that is simply rediculous. > In the contemporary mainstream mathematical literature, there > is almost no debate over the validity of Cantor's Theory. That is because mathematically it is a sound theory. You will NOT find any further debate on this in mainstream mathematical literature. Pure mathematicians are interested in showing something is or is not mathematically correct. Cantor's theory IS correct mathematically, and that is why there won't be any debate over it in mathematical journals. Maybe someone will find a more palatable basis for mathematics which allows the sort of analysis that is needed in other sciences. That won't invalidate Cantor's set theory, and even if such a new theory will be gotten there won't be any debate over Cantor's set theory being wrong in any mathematical journal. > It was the advent of the internet which revealed just how > prevalent the anti-Cantorian view still is; there seems to be a > never-ending heated debate about Cantor's Theory in the Usenet > newsgroups sci.math and sci.logic. Typically, the > anti-Cantorians accuse the pure mathematicians of living in a > dream world, and the mathematicians respond by accusing the > anti-Cantorians of being imbeciles, idiots and crackpots. All the posters I saw were trying to purport mathematical proofs that Cantor's set theory (mostly the diagonalization argument) is wrong. That is the same as people claiming to have squared the circle or claiming to have computed pi to be 3.125. That is why they are called imbeciles, idiots and crackpots. > It is plausible that in the future, mathematics will be split > into two disciplines - scientific mathematics (i.e. the science > of phenomena observable in the world of computation), and > philosophical mathematics, wherein Cantor's Theory is > merely one of the many possible theories of the infinite. This paragraph is total nonsense put in purely because you were lacking Mathematics already ranges from mathematical physics, all the way to pure mathematics. It is people on the right end of the spectrum (pure mathematics) that contain those that try to do mathematics without certain tools such as Cantor's set theory. The people on the left, who are more interested in modeling the real world are probably rarely interested if their model uses the complete reals or not, they use the best theory suited for the job, and that theory is based on Cantor's set theory. Those who try to find alternatives to Cantor's set theory are interested in the foundations of mathematics, which is about as pure mathematics as you can get. Jiri Robert Low > The least number principle does express that (N,<) is a well-ordering. > > > But in the weakened sense that it's only sets of naturals > which make some P(n) true that have to have a least element. For > example, the set of infinite elements of a non-standard model has > no least element, but that's OK because there's no way of expressing > 'n is infinite' in the language. Right. --- Jeff <42DC3F5F.30904@netscape.net> <85pste8vvp.fsf@lola.goethe.zz> <854qaq8u5i.fsf@lola.goethe.zz> <85mzoi5wgm.fsf@lola.goethe.zz> > and that I was trying to prove > things about sets, not numbers, which is also bull, since I was > proving a property regarding a set DEFINED by a natural number, which > is ultimately a property of that number. > > But the set N is not defined by any one natural number Under Tony's theory, the number representing N is defined by a string of an infinite number of ones. Yes, more than one Turing machine can produce this. karl m <42DC3F5F.30904@netscape.net> <85ll43dvil.fsf@lola.goethe.zz> <858y03c4do.fsf@lola.goethe.zz> <85zmsic1xg.fsf@lola.goethe.zz> <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> > There are four schemes of interest: > - the usual induction scheme [v]; > - the scheme of total induction [iv]; > - the least number principle [i]; > - induction up to z. > These are all equivalent (in first-order PA). More details in R. Kaye 1991, > _Models of PA_, pp. 43-6. But since first-order PA includes the induction scheme, this is misleading. After all, in first-order PA (and so assuming the induction scheme), (1) the usual induction scheme is equivalent to the well-ordering principle scheme is equivalent to just (2) the usual induction scheme implies the well-order principle scheme What would be interesting if you could show (1) in first-order PA minus the induction scheme. But apparently you cannot, since in first-order PA minus the induction scheme you cannot show that the well-order principle scheme implies the induction scheme. See or if this address is too big, search in Google sci.logic with Equivalence Pigeon Hole Principle Induction and the relevant thread will be the first choice. Helene.Boucher@wanadoo.fr > >> There are four schemes of interest: >> - the usual induction scheme [v]; >> - the scheme of total induction [iv]; >> - the least number principle [i]; >> - induction up to z. >> These are all equivalent (in first-order PA). More details in R. Kaye >> 1991, >> _Models of PA_, pp. 43-6. > > But since first-order PA includes the induction scheme, this is > misleading. It depends upon how you formulate PA. They are equivalent over the base theory PA-, which has (a) axioms stating that + and x are associative, commutative, and satisfy the distributive law, (b) axioms stating that 0 is an identity for + and a zero for x, and that 1 is an identity for x; (c) axioms saying that < is linear order. (d) axioms saying that + and x respect order; (e) if x < y, then (Ez)(x+z = y) (f) that 0 is the least number and that < is discrete. See Kaye 1991, p. 16 and pp 45-6. ard Kaye then defines PA as PA- plus the induction scheme (p. 43). Other authors define PA as the six axioms for successor, plus and times plus the induction scheme. > After all, in first-order PA (and so assuming the > induction scheme), > > (1) the usual induction scheme is equivalent to the well-ordering > principle scheme > > is equivalent to just > > (2) the usual induction scheme implies the well-order principle scheme > > What would be interesting if you could show (1) in first-order PA minus > the induction scheme. But apparently you cannot, since in first-order > PA minus the induction scheme you cannot show that the well-order > principle scheme implies the induction scheme. If by PA minus the induction scheme, you mean the six usual axioms for successor, + and x, I don't know. But they are equivalent when the above theory PA- is the base theory, as I noted above. --- Jeff <42DC3F5F.30904@netscape.net> <85ll43dvil.fsf@lola.goethe.zz> <858y03c4do.fsf@lola.goethe.zz> <85zmsic1xg.fsf@lola.goethe.zz> <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> > It depends upon how you formulate PA. > > They are equivalent over the base theory PA-, which has > (a) axioms stating that + and x are associative, commutative, and satisfy > the distributive law, > (b) axioms stating that 0 is an identity for + and a zero for x, and that 1 > is an identity for x; > (c) axioms saying that < is linear order. > (d) axioms saying that + and x respect order; > (e) if x < y, then (Ez)(x+z = y) > (f) that 0 is the least number and that < is discrete. > See Kaye 1991, p. 16 and pp 45-6. > ard Kaye then defines PA as PA- plus the induction scheme (p. 43). > > Other authors define PA as the six axioms for successor, plus and times plus > the induction scheme. > non-standard definition of PA. Indeed I don't think it's warranted to call this particular axiomatization PA unless some kind of warning is given ! > > If by PA minus the induction scheme, you mean the six usual axioms for > successor, + and x, Yes that's what I (and I think most other people) mean. Helene.Boucher@wanadoo.fr > > >> It depends upon how you formulate PA. >> >> They are equivalent over the base theory PA-, which has >> (a) axioms stating that + and x are associative, commutative, and satisfy >> the distributive law, >> (b) axioms stating that 0 is an identity for + and a zero for x, and that >> 1 >> is an identity for x; >> (c) axioms saying that < is linear order. >> (d) axioms saying that + and x respect order; >> (e) if x < y, then (Ez)(x+z = y) >> (f) that 0 is the least number and that < is discrete. >> See Kaye 1991, p. 16 and pp 45-6. >> ard Kaye then defines PA as PA- plus the induction scheme (p. 43). > >> >> Other authors define PA as the six axioms for successor, plus and times >> plus >> the induction scheme. >> > > non-standard definition of PA. Indeed I don't think it's warranted to > call this particular axiomatization PA unless some kind of warning is > given ! This subtheory is nice because it relates PA to the theory of the (non-negative part) of a discretely ordered ring. It includes all the obvious algebraic and order-theoretic properties of (N, 0, 1, +, x, <). (Of course, the notion of successor is not treated as primitive here: Sx may be defined as x+1 or as the smallest y > x.) > > >> >> If by PA minus the induction scheme, you mean the six usual axioms for >> successor, + and x, > > Yes that's what I (and I think most other people) mean. I agree. It's usually something like Q (with the definition of <), plus the induction scheme. So, your original questions are, I think, these: (a) Does Q + induction scheme imply the least-number principle? (b) Does Q + induction scheme imply the principle of total induction? The answer is yes. (See Hajek/Pudlak 1993, _Metamathematics of First-Order Arthmetic_, p. 35). Or did you mean to ask about the converse in (a), (c) Does Q + least number principle imply induction scheme? I'm guessing, since I haven't checked the proof, but I'd be surprised if that wasn't true as well. --- Jeff <85ll43dvil.fsf@lola.goethe.zz> <858y03c4do.fsf@lola.goethe.zz> <85zmsic1xg.fsf@lola.goethe.zz> <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> >> If by PA minus the induction scheme, you mean the six usual axioms for >> successor, + and x, > > Yes that's what I (and I think most other people) mean. > > I agree. It's usually something like Q (with the definition of <), plus the > induction scheme. > > So, your original questions are, I think, these: Actually it wasn't a question, more a comment... > (a) Does Q + induction scheme imply the least-number principle? > (b) Does Q + induction scheme imply the principle of total induction? > The answer is yes. (See Hajek/Pudlak 1993, _Metamathematics of First-Order > Arthmetic_, p. 35). I wasn't mentioning this direction... > > Or did you mean to ask about the converse in (a), > (c) Does Q + least number principle imply induction scheme? > I'm guessing, since I haven't checked the proof, but I'd be surprised if > that wasn't true as well. > This direction is the problem. See the link I provided (or Google as I suggested). Helene.Boucher@wanadoo.fr >> Or did you mean to ask about the converse in (a), >> (c) Does Q + least number principle imply induction scheme? >> I'm guessing, since I haven't checked the proof, but I'd be surprised if >> that wasn't true as well. >> > > This direction is the problem. See the link I provided (or Google as I > suggested). I looked at the link. But it appears to concern the equivalence of PHP and induction, a different topic (though the topic of these equivalences is discussed in Hajek/Pudlak). We have that PA- + least number principle implies induction, where PA- is the base theory Kaye describes. Also, the least number principle and order-induction are equivalent (in predicate logic). So: the question is: does Q + order-induction imply induction? This is how Hajek and Pudlak discuss the topic. They discuss this for classes of Sigma_n and Pi_n formula. Let L_{phi} be the least-number principle formula for phi. Let I_{phi} be the induction formula for phi. Let I'_{phi} be the order-induction formula for phi. The theory ISigma_n is Q U {I_{phi} : phi a Sigma_n formula}. The theory I'Sigma_n is Q' U {I'_{phi} : phi a Sigma_n formula}. where Q' is Q augmented with the axiom x < Sx. Lemma 2.12 (p. 64) of Hajek/Pudlak shows that ISigma_n is equivalent to I'Sigma_n, and IPi_n is equivalent to I'Pi_n. They remark (p. 63) that it is unknown whether IDelta_n and LDelta_n are equivalent (although LDelta_{n+1} implies IDelta_{n+1}.). Hajek and Pudlak are mainly concerned with fragments of PA defined by the complexity of formulas. I can't see if they show that Q + order-induction implies ordinary induction, but I shall think about it. Perhaps it's an open issue. Finally, the axioms of PA- are all theorems of ISigma_1 (possibly even IOpen). Therefore, we have ISigma_1 + order-induction implies ordinary induction. What the small amount of ordinary induction is used for is to show the general algebraic properties of + and x (commutativity, etc.) and the general order properties of < (i.e., discrete linear order with a least element). In short, order-induction (equiv., least number principle) implies ordinary induction modulo a weak base theory which includes these properties of +, x and <. --- Jeff <858y03c4do.fsf@lola.goethe.zz> <85zmsic1xg.fsf@lola.goethe.zz> <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> > >> If by PA minus the induction scheme, you mean the six usual axioms for >> successor, + and x, > > Yes that's what I (and I think most other people) mean. > > I agree. It's usually something like Q (with the definition of <), plus the Sorry, I didn't see you sneak in the definition of < (in terms of addition). This turns the problem into whether Well Ordering Principle for this particular ordering implies induction, instead of (what I was thinking) whether Well Ordering Principle for some generic < implies induction. In your case the proof appears trivial. In order to prove (Well Ordering Principle for < in terms of addition => Induction), suppose WOP; you need to prove that every number has a predecessor. If not then there is a least number without a predecessor, say L. L is not 1, since 1 has 0 as a predecessor. Consider the set {L,1}. This can be well-ordered, which implies that 1 < L. So K + 1 = L for some K, so L has a predecessor. <85zmsic1xg.fsf@lola.goethe.zz> <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> > suppose WOP; you need to prove that every number has a predecessor. Ugh. Please read, Every *non-zero* number has a predecessor. <85zmsic1xg.fsf@lola.goethe.zz> <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> > >> If by PA minus the induction scheme, you mean the six usual axioms for >> successor, + and x, > > Yes that's what I (and I think most other people) mean. > > I agree. It's usually something like Q (with the definition of <), plus the > > Sorry, I didn't see you sneak in the definition of < (in terms of > addition). This turns the problem into whether Well Ordering Principle > for this particular ordering implies induction, instead of (what I was > thinking) whether Well Ordering Principle for some generic < implies > induction. In your case the proof appears trivial. In order to prove > (Well Ordering Principle for < in terms of addition => Induction), > suppose WOP; you need to prove that every number has a predecessor. If > not then there is a least number without a predecessor, say L. L is > not 1, since 1 has 0 as a predecessor. Consider the set {L,1}. This > can be well-ordered, which implies that 1 < L. So K + 1 = L for some > K, so L has a predecessor. I should add the following for clarity's sake. Consider PA - induction + (Ex)Px -> (Ey)(Py & (Az)(Pz -> z>= y)) where one does not provide a definition for <. Then you can't prove induction, since the set of one-variable polynomials with coefficients in the set of natural numbers provides a counterexample. The standard ordering on such polynomials is a well-ordering, and the structure satisfies all the Peano axioms of successor, addition and multiplication, but it does not satisfy induction (and in particular, not every element has a predecessor). Helene.Boucher@wanadoo.fr >> It's usually something like Q (with the definition of <)... Andrew Boucher: > Sorry, I didn't see you sneak in the definition of < (in terms of > addition). This is a perfectly standard axiom of Q. > This turns the problem into whether Well Ordering Principle > for this particular ordering implies induction, instead of (what I was > thinking) whether Well Ordering Principle for some generic < implies > induction. In your case the proof appears trivial. In order to prove > (Well Ordering Principle for < in terms of addition => Induction), > suppose WOP; you need to prove that every number has a predecessor. If > not then there is a least number without a predecessor, say L. L is > not 1, since 1 has 0 as a predecessor. Consider the set {L,1}. This > can be well-ordered, which implies that 1 < L. So K + 1 = L for some > K, so L has a predecessor. I don't get this (second-order) proof at all. Here is the answer to the original question you raised (in first-order logic). For the system Q, see Hajek/Pudlak (1993), p. 28; Boolos, Burgess & Jeffrey (2002), p. 215, call it R. Q is defined by the eight axioms: Q1 Sx =/= 0 Q2 Sx = Sy -> x = y Q3 x =/= 0 -> Ey(x = Sy) Q4 x + 0 = x Q5 x + Sy = S(x + y) Q6 x * 0 = 0 Q7 x * Sy = x * y + x Q8 x < y <-> Ez(z=/=0 & y = x + z) PA may be axiomatized by adding the induction scheme to Q1-Q8. Defn: 1 = S0 Lemma: x+1 = x + S0 = S(x+0) = Sx (using Q4, Q5) Then consider the schemes: IND: [P0 & (Ax)(Px -> P(x+1))] -> (Ax)Px LNP: ExPx -> Ex(Px & (Ay P(x+1) and (Ex)~Px. Thus, by LNP, (Ex)(~Px & (Ay Py. Hence, Pd. But Px -> P(x+1). Thus, Pd -> P(d+1). Thus P(d+1). Thus, Pc. Contradiction. QED. In other words, using the axioms Q1, Q3, Q4, Q5 and Q8 as our base theory, we can show that the Well-Ordering (represented as a scheme: the least number principle) implies the usual Induction Scheme. --- Jeff Helene.Boucher@wanadoo.fr >> >>> If by PA minus the induction scheme, you mean the six usual >>> axioms for >>> successor, + and x, >> >> Yes that's what I (and I think most other people) mean. >> >> I agree. It's usually something like Q (with the definition of <), >> plus the >> >> Sorry, I didn't see you sneak in the definition of < (in terms of >> addition). This turns the problem into whether Well Ordering Principle >> for this particular ordering implies induction, instead of (what I was >> thinking) whether Well Ordering Principle for some generic < implies >> induction. Nobody is suggesting that Well-Ordering for a generic relation < implies induction. Actually, it doesn't even make sense. That (X, <) is well-ordered doesn't imply that it is isomorphic to (omega, <)! They are discussing the principle of total induction (or least number principle, or well-ordering) for the usual < on N, and this order < will be governed by some axioms of some sort (e.g., < is a discrete linear order with a least element) or given by the usual definition, that x < y iff (Ez)(z =/=0 & y = x + z). The technical question you raised is this. If we take Q as standard, including the definition of < (e.g., as in Hajek/Pudlak), then does Q + total induction imply ordinary induction. I suspect that the answer is yes, but the proof might be a bit fiddly. >>In your case the proof appears trivial. In order to prove >> (Well Ordering Principle for < in terms of addition => Induction), >> suppose WOP; you need to prove that every number has a predecessor. >>If >> not then there is a least number without a predecessor, say L. L is >> not 1, since 1 has 0 as a predecessor. Consider the set {L,1}. This >> can be well-ordered, which implies that 1 < L. So K + 1 = L for some >> K, so L has a predecessor. I cannot follow this. Every number has a predecessor? That isn't induction. Isn't even true. The axiom that every *non-zero* number has a predecessor is one of the axioms of Q, and has nothing to do with induction. Induction says: if X contains 0 and is closed under successor, then X = N. > I should add the following for clarity's sake. Consider PA - induction > + > (Ex)Px -> (Ey)(Py & (Az)(Pz -> z>= y)) > where one does not provide a definition for <. Then you can't prove > induction, since the set of one-variable polynomials with coefficients > in the set of natural numbers provides a counterexample. The standard > ordering on such polynomials is a well-ordering, and the structure > satisfies all the Peano axioms of successor, addition and > multiplication, but it does not satisfy induction (and in particular, > not every element has a predecessor). I don't follow this at all. Nobody is sneaking in a definition of <. All workers in this field either take the definition of < as standard, or include some basic axioms for <. You seem to have been thinking of just some generic order. This is not relevant. Unfortunately, people do take Q to be different things. In Boolos, Burgess and Jeffrey (2002, p. 208), Q is taken to consists of the six axioms for successor, plus and times, with the following axioms for < [i] ~(x<0), [ii] x < Sy <-> (x < y v x = y) [iii] x < y v x = y v y < x. On the other hand, Hajek and Pudlak (1993, p. 28) take Q to consist of what Boolos and Jeffrey (1989, p. 159) originally called Q, which is the axioms for successor, plus and times, and with the axiom x =/= 0 -> (Ey)(x = Sy), but with the definition [iv] x < y <-> (Ez)(z =/= 0 & y = x+z) Burgess, Boolos and Jeffrey (2002, p. 215) discuss a different theory R (which they call Robinson arithmetic), which is the same as Hajek/Pudlak's Q. We can formulate PA without < of course. But then it is usually extended by the usual definition of <, which is [iv] above. I think that by PA minus induction you are referring to just the six axioms for successor, plus and times. But everyone would---and does---include the standard definition of < in any work on this topic (e.g., Hajek/Pudlak). If fact, no one talks of PA minus induction. They talk of various base theories, such as the two versions of Q above, or Kaye's PA-, or various others. --- Jeff Nntp-Posting-Host: hera.cwi.nl > Dik T. Winter said: > ... > > The only reason to reject this bijection is > > if one clings to the idea that all natural numbers are finite, which is > > impossible. > > Back on your horse again. Tell me about the binary numbers (extended to the > left with 0's) where the leftmost 1 is in a finite position. Are all those > numbers finite? Are there only finitely many of them? > > yes and yes I think you should apply for the reward for solving Collatz' problem. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ >>>> I will have my web pages published before too long, so I am not >>>> getting into a mosh pit with you again right now. Just be aware that >>>> anti-Cantorians are sick of being called crackpots, and the day will >>>> soon come when the crankiest Cantorians will eat their words, and >>>> this rot will be extricated from mathematics. >> >> > I never claimed you would be cast from the ranks of mathematics, but that > you > will see the errors that you are currently ignoring, and that the rot of > Cantorian cardinality will be removed from mainstream thought and replaced > with > ideas that don't lead to absurdity like Banach-Tarski. I do see the > ramifications of this nonsense in many areas. Until you can demonstrate > that > the theory is really correct, Correct? What does that mean? Really, I have no idea what the word means in this context. Inconsistent perhaps? If so, how? >I am well within my rights to disagree with your > axioms and conclusions, How do you disagree with an axiom? How about Group Theory, for example? Do you disagree that there is element e such that g*e=g for every g? What does it mean that you disagree with an axiom? Is it that you don't think that there is a model which satisfies the axiom? If so, are you contending that N doesn't model PA? Could you explain exactly which axiom(s) you disagree with, what disagree with means, and why you disagree with them? We can then move on to conclusions, where I don't understand what disagree with them means either. > and if that right is challenged, I will continue to > defend it and challenge your theory. It takes two to tango, and if you end > up > with people vowing vengeance, well hell, you probably deserve it. Then > again, > maybe JSH is mentally unstable, but then so was Cantor, and so was Godel. > -- You might think that infinite ordinals and cardinals are absurd and nonsense. You are in good company historically. Lots of people thought that zero was absurd and nonsense, as were negative numbers (show me -3 cows), imaginary numbers, irrational numbers, positional notation ... of course, 100 years after each of these absurd and nonsensical concepts were introduced, it was only the cranks that continued to rail against them. Its like somebody saying 100 years after Copernicus that the geocentric model leads to absurdities and obvious nonsense like the earth is spherical, when it is obviously flat. This would be likely to lead to the same type of contempt from astronomers as you are getting from the mathematicians here. > >>>Nothing of any importance about mathematics would change if we >>>substituted different words for the basic concepts. In contrast, >>>your arguments are about nothing *but* terminology. To me, that >>>shows that there is no actual content to your arguments. An actual >>>mathematical argument does not depend on word choice. As a >>>challenge, see if you can express your claims about infinite sets, >>>or infinite naturals, or set size, or whatever, *without* using the >>>words infinite, larger, size, etc. >> >>This clearly represents the formalist (Hilbertian) view on >>mathematics as a senseless game with symbols. So, it represents the view that it does not make sense to talk about > different things using the same words. No. You're twisting my words and those of Daryl McCullough. (Though I feel much more comfortable with your interpretation than with his.) No trouble with the rest of your writeup [ deleted ]. Han de Bruijn > One issue here is that TO keeps ignoring standard mathematical > definitions, however often presented, and then declaring that that the > defined words and phrases must have other meanings than the ones > mathematicians have agreed on. Meanings ? Agreed on ? Look at Daryl McCullough's arguments, where he asks Tony to think about fluffy pink flying elephants. So what meanings have you mathematicians agreed on ? Han de Bruijn Han de Bruijn says... > > >> One issue here is that TO keeps ignoring standard mathematical >> definitions, however often presented, and then declaring that that the >> defined words and phrases must have other meanings than the ones >> mathematicians have agreed on. > >Meanings ? Agreed on ? > >Look at Daryl McCullough's arguments, where he asks Tony to think about >fluffy pink flying elephants. You've got that completely backwards. Tony is the one who wants to talk about undefined terms. I want to stick to basic mathematical terms: sets, naturals, membership, addition, multiplication, functions, etc. -- Daryl McCullough Ithaca, NY >> One issue here is that TO keeps ignoring standard mathematical >> definitions, however often presented, and then declaring that that the >> defined words and phrases must have other meanings than the ones >> mathematicians have agreed on. > Meanings ? Agreed on ? Yes. When communicating with others you have to use the agreed upon meanings of words. Is that really such a hard concept? Otherwise, banana coaster throat warbling yachtsman. > Look at Daryl McCullough's arguments, where he asks Tony to think about > fluffy pink flying elephants. So what meanings have you mathematicians > agreed on ? There is agreed upon definition for infinite set, countably infinite, bijection, cardinality, etc. They are all fairly simply definitions. Stephen > Guess what? That means you reject the continuum hypothesis! Isn't that > exciting? Don't you just want to go out and learn all about it rather > than just spouting vague, uninformed, and often silly nonsense? Really, do you mean that _accepting_ the Continuum Hypothesis somehow represents no-nonsense behaviour ? Han de Bruijn > These seems to be another common misconception among > the anti-Cantorians that words cannot have specific > meanings in specific contexts. Somehow they > think an all-encompassing definition of 'infinite' must > be provided before someone can say what an infinite set is. > I am not sure what they mental hangup is. I wonder > how any of them would ever learn a foreign language. Ha, ha, ha. _This_ anti-Cantorian has learned six languages: Dutch, German, French, English, Latin and Greek. We in the Netherlands are privileged with our knowledge of foreign languages. Yet I find that an all-encompassing definition of 'infinite' must be provided. For the simple reason that 'infinity' is not a concept that is limited to mathematics alone. It spreads out i.e. into physics, and gives rise there to singularities that exist but one can never perceive them, due to a Cosmic Censorship that prevents us to take a look into the inside Han de Bruijn >> These seems to be another common misconception among >> the anti-Cantorians that words cannot have specific >> meanings in specific contexts. Somehow they >> think an all-encompassing definition of 'infinite' must >> be provided before someone can say what an infinite set is. >> I am not sure what they mental hangup is. I wonder >> how any of them would ever learn a foreign language. > Ha, ha, ha. _This_ anti-Cantorian has learned six languages: Dutch, > German, French, English, Latin and Greek. We in the Netherlands are > privileged with our knowledge of foreign languages. Yet I find that > an all-encompassing definition of 'infinite' must be provided. That seems silly. > For the simple reason that 'infinity' is not a concept that is limited > to mathematics alone. The word 'infinity' is not limited to mathematics alone, but 'infinite set' is limited to mathematics. 'infinite set' has a precise definition, and the definition does not have to include all possible uses of the word 'infinity' or 'infinite' in other areas. > It spreads out i.e. into physics, and gives rise > there to singularities that exist but one can never perceive them, due > to a Cosmic Censorship that prevents us to take a look into the inside Why are you bringing physics into this? Whether black holes exist or not has nothing to do with set theory. Stephen > Why are you bringing physics into this? Whether black > holes exist or not has nothing to do with set theory. Theoretically: no. In practice: yes. Because some consequences of set theory have invaded into physics by the fact that mathematics becomes somewhat _applied_ there, huh ! Geez ... Han de Bruijn >> Why are you bringing physics into this? Whether black >> holes exist or not has nothing to do with set theory. > Theoretically: no. In practice: yes. Because some consequences of set > theory have invaded into physics by the fact that mathematics becomes > somewhat _applied_ there, huh ! Geez ... > Han de Bruijn Are you claiming that set theory is directly applied in General Relativity? Do you actually have evidence of that? Ignoring mathematical foundations, which physicists, and most everybody else, typically do, it seems that the limit of 1/sqrt(1-v^2/c^2) as v approaches c is infinite with or without set theory. Stephen > Are you claiming that set theory is directly applied in General > Relativity? Do you actually have evidence of that? I have seen a paper in which transfinite induction was used. And in the existence of maximal solutions to the initial value problem, appeal is (at least sometimes) made to Zorn's lemma. !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > >> Why are you bringing physics into this? Whether black >> holes exist or not has nothing to do with set theory. > > Theoretically: no. In practice: yes. Because some consequences of > set theory have invaded into physics by the fact that mathematics > becomes somewhat _applied_ there, huh ! Geez ... Physics can influence where mathematics is heading, but not what it is finding there. Its verdict on mathematics can't be true/false, but just interesting/irrelevant. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > >> It spreads out i.e. into physics, and gives rise >> there to singularities that exist but one can never perceive them, due >> to a Cosmic Censorship that prevents us to take a look into the inside > > Why are you bringing physics into this? Whether black > holes exist or not has nothing to do with set theory. Something that is so dense that everything you throw at it does not leave an impact apart from even more denseness? I am not sure this can be called irrelevant to this thread. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum > >> These seems to be another common misconception among >> the anti-Cantorians that words cannot have specific >> meanings in specific contexts. Somehow they >> think an all-encompassing definition of 'infinite' must >> be provided before someone can say what an infinite set is. >> I am not sure what they mental hangup is. I wonder >> how any of them would ever learn a foreign language. > >Ha, ha, ha. _This_ anti-Cantorian has learned six languages: Dutch, >German, French, English, Latin and Greek. We in the Netherlands are >privileged with our knowledge of foreign languages. Yet I find that >an all-encompassing definition of 'infinite' must be provided. Does a Christmas tree in certain engineering contexts actually have be a conifer? Martin >> >>an all-encompassing definition of 'infinite' must be provided. Does a Christmas tree in certain engineering contexts actually have > be a conifer? No. But it seems that you have deleted an essential add-on: >> For the simple reason that 'infinity' is not a concept that is limited >> to mathematics alone. It spreads out i.e. into physics, and gives rise >> there to singularities that exist but one can never perceive them, due >> to a Cosmic Censorship that prevents us to take a look into the inside Han de Bruijn > >>I asked for a definition of infinite, and no one could give me a >>definition of that word. The best I could get was that an infinite >>set can have a bijection with a proper subset, which is hardly a >>definition of the word infinite. On the contrary, that's a perfectly good definition of the concept > infinite set. It's the standard definition of the actual infinite, but it is not perfectly good. Worse. It's not good at all. Han de Bruijn Han de Bruijn says... > > >> >>>I asked for a definition of infinite, and no one could give me a >>>definition of that word. The best I could get was that an infinite >>>set can have a bijection with a proper subset, which is hardly a >>>definition of the word infinite. >> On the contrary, that's a perfectly good definition of the concept >> infinite set. > >It's the standard definition of the actual infinite, but it is not >perfectly good. Worse. It's not good at all. It's perfectly good in the sense that it allows for mathematical reasoning about the infinite. We can reason that certain sets must be infinite, and that other sets must be finite. In contrast, Tony's talk about infinite naturals doesn't allow any kind of reasoning to be made about them. -- Daryl McCullough Ithaca, NY On 21 Jul 2005 13:37:22 -0700, stevendaryl3016@yahoo.com (Daryl In contrast, Tony's talk about infinite naturals doesn't allow > any kind of reasoning to be made about them. > I read: Tony's talk ... doesn't allow [for] any kind of reasoning... Same for M.9fckenheim, btw. F. > No, give a definition that *doesn't* use the words > infinite, finite, limitless, limit, boundless, > etc. Give a *mathematical* definition. What he means is a *Hilbertian*, formalists definition. That's the way they suck you into their camp. Han de Bruijn > Well, that's about as close to a lie as one can get, eh? I asked for a > definition of infinite, and no one could give me a definition of that word. The > best I could get was that an infinite set can have a bijection with a proper > subset, which is hardly a definition of the word infinite. In fact I went to > the etymology, which literally means without end. Finite means with a known > end or bound, and infinite means without end. Of course, I got all sorts of > flack for my definition, from those that couldn't even suggest one outside of > the set theory they were regurgitating. let's try to be straight here, and no > more insulting than necessary, so it doesn't come back to bite us, why don't > we? The infinite they define as that an infinite set can have a bijection with a proper subset of itself is known as _actual_ infinite, which is rejected by most anti-Cantorians as sheer nonsense. Apparently, you are rather talking about _potential_ infinity: something finite that becomes larger and larger. The indisputably useful concept of a limit falls within the latter category. Everything that involves the infinite and cannot be handled with limits is rather suspect IMHO. Han de Bruijn !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > >>>Nah, nah. First comes the theorem. And then comes the proof. >> You cannot prove something without axioms. So if you >> start with your conclusions, you have to create some >> axioms before you can find a proof. Unfortunately they >> never seem to actually define any axioms. Look at Tony Orlow >> in this thread for a fine example of this sort of behavior. > > You cannot prove something without having a clue what to prove. So > the conjecture comes first. Then we have to find out, eventually, > which set of axioms fits our conjecture. Some people arrive at wrong > conclusions because they create the _wrong axioms_. That does not invalidate the conclusions as conclusions. It invalidates the model. The problem is that our cranks here don't bother replacing the axioms while protesting some of their conclusion as unintuitive, and wanting to keep others. And that's not going anywhere. The rules are: if you don't like some conclusions, you have to change the axioms, and then you lose all other conclusions (many of them might be easy to reacquire, but that process is not automatic). -- David Kastrup, Kriemhildstr. 15, 44793 Bochum > The rules are: if you don't like some conclusions, you have to change > the axioms, and then you lose all other conclusions (many of them > might be easy to reacquire, but that process is not automatic). But what if your method is not axiomatic ? I mean, in intuitionism, the emphasis is not on formal reasoning and axions, but constructiveness. Han de Bruijn !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > >> The rules are: if you don't like some conclusions, you have to >> change the axioms, and then you lose all other conclusions (many of >> them might be easy to reacquire, but that process is not >> automatic). > > But what if your method is not axiomatic ? Then neither is your reasoning, and there are no truths you can derive. Cherrypicking from the axiomatic mathematics is intellectually dishonest. You can't just put together the conclusions you like and call that a complete building. That's like telling an architect that you want to have a penthouse view from a twelve-story house, but to save costs, he should leave off the lower 10 stories. > I mean, in intuitionism, the emphasis is not on formal reasoning and > axions, but constructiveness. That's like saying that in the Minotaur's maze, the emphasis should be on finding the right direction and not on following some stupid thread. The line between truth and falseness spins into a hairline when you are exploring the far ranges of mathematics. And this thread always leads back to the axioms from which you started. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Square roots - decimal expansions -- >I don't understand where you want to go. You're the one who wanted to go somewhere. You asked whether 424 was an exception (to the rule that tells whether a number has a rational square root, and thus whether the square root of that decimal expansion has an ultimately periodic decimal expansion). The answer is no, it's not an exception. I have no idea why you would have thought it might be an exception. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Square roots - decimal expansions -- It is very bizarre what you say I would say I don't know what you would say but I would say it is very bizarre. === Subject: Re: Square roots - decimal expansions -- > Something like paper dolls ? (3.1111111111111111101111111111111111191111111111111*sqrt(2))^2 = 19.3580246913580246789135802469135803484691358024689974988641975308642421033 0 864197530828641975308641 Like this ? === Subject: Re: Square roots - decimal expansions -- > Something like paper dolls ? (3.1111111111111111101111111111111111191111111111111*s > qrt(2))^2 = > 19.358024691358024678913580246913580348469135802468997 > 49886419753086424210330864197530828641975308641 Like this ? sum(i^2, i, 1, 987654321)/sum(i, i, 1, 987654321) = 1975308643/3 = 658436214.33333333333333333... sum(i^2, i, 1, 12345678987654321)/sum(i, i, 1, 12345678987654321) = 24691357975308643/3 = 8230452658436214.3333333333... sum(i^2, i, 1, 123456789)/sum(i, i, 1, 123456789) = 246913579/3 = 82304526.333333333333333333... Or maybe those ? === Subject: Re: a question about multivariate function > Is there any multivariate function f which satisfies: if f(a1,a2,...,an)=f(a1',a2',...,an') > then a1=a1',a2=a2',...,an=an' > Yes, but if f maps to a space of lower dimension than n, then f cannot be continous. Like for instance f: R^n -> R a1= ...d1_3d1_2d1_1d1_0.d1_-1d1_-2... a2= ...d2_3d2_2d2_1d2_0.d2_-1d2_-2... . . . a_n=...dn_3dn_2dn_1dn_0.dn_-1dn_-2... f = ...d3_0d2_1d1_2d2_0d1_1d1_0.d1_-1d1_-2d_2-1d_1-3d_2-2d_3-1 in a kind of double diagonal ordering. This way each image is unique. -- Jos van Kan www.josvankan.tk === Subject: Questions wrt computable order of numbers. at: http://en.wikipedia.org/wiki/Computable_numbers it says that the order relations of computable numbers are not computable. We are talking about determining which of two distinct numbers is less, etc. This concerned me a bit because I like to have an order function that would eventually complete on whatever computable numbers I choose to implement in a program (it makes me feel better knowing, even if completion is after the big party at the end of the Universe). It is clear that there are some subsets of the computable numbers in which the order of any two numbers is easily computed with finite steps, for example, the set of rationals or the set of 99th roots of rationals. An approximation of the order relation within a specified margin of error did not seem meaningful. The order relation is not really a typical number in the first place to fit nicely into the definition of computable number versus uncomputable number. It is right or wrong, not approximate. The definition of computability of the order relation that seemed more useful to me was whether the order can be exactly determined within a finite number of steps (even if not a predictable number of steps). Any of my programs will only represent a countable subset of reals, because finite representations are countable. Let's say that I perform enough normalization on the representations to provide an equality function that just works by comparing representations. Are there convenient greater subsets of the computables (how about all algebraic numbers) for which it is always possible (assuming pre-detection of the equal case) to guarantee that the combined error of two approximations will eventually be less than the difference so it will be possible to establish the order after a finite number of steps? My definition of computability of the order relation may be different there an example of two non-equal computable numbers whose order cannot be established with certainty within any finite number of steps? It seems like it might depend upon the representation, so with suitable choice of the representations, would order computation within algebraic numbers ever be a problem? Where should I look for this sort of thing? Ray Whitmer === Subject: Re: Kaprekar's constant > We all know abt the Kaprekar constant. Certainly not all. >In the base 10 representation of numbers, if we lok at four-digit >numbers then the Kaprekar iterative process has a single fixed point >attractor with which we end up, regardless of the intial 4 digit number >we began with.. With 77 exceptions which end up at 0: see >Hence let us denote above process by K(10,4), i.e. the iterative >process uses base-10, 4-digit numbers. So K(10,4) has a single >attractor 6174. >let us denote the iteration function as f()..so f(6174)=6174.... In other words, for base b you look at the function f_b(x) = x' - x where x' has the base-b digits of x arranged in descending order and x has the digits arranged in ascending order. > When we look at K(10,j) for some j>4, we do not always find single >attractors,.... we may also find cycles ...i.e. >f(a)=f(b)=f(c)=...=f(a). Usually the length of the cycles is much >smaller than the total number of j-digit numbers. >Let's call a single fixed point as cycle of length 0... It's more logical to call this a cycle of length 1. >My question is, do there exist, i &j, such that K(i,j) has a single >cycle that runs through all the j-digit numbers in base i. No. Since f(x) = f(y) if x and y are rearrangements of the same digits, this can't happen. Also, there are always numbers x such that f(x) = 0, namely those where all the digits are the same. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Finding the inverse solution to a non-linear differential equation >This nonlinear equation describes the motion of a body subject to drag. >M*(d^2x/dt^2) + D*(dx/dt)^2 - F(t) = 0 >Where M is mass, D is a drag factor and F is a forcing function. [ and presumably you're in a case where dx/dt won't ever be negative ] >This is straightforward to solve numerically to find x and dx/dt given >t and some initial conditions x[0] and dx/dt[0], but how does one >approach the solution if x is given and t is the desired solution? Rewrite the DE with x as the independent variable and t the dependent variable: if v = dx/dt, we have dt/dx = 1/v dv/dx = (dv/dt)/(dx/dt) = (F(t) - D v^2)/(M v) d^2 t/dx^2 = - v^(-2) dv/dx = (- F(t) + D v^2)/(M v^3) = (-F(t) (dt/dx)^3 + D dt/dx)/M which can be solved numerically just as easily as the original starting with an initial condition t(x_0) = t_0, t'(x_0) = 1/v_0 (of course you'll want v_0 <> 0: if v_0 = 0 you'll want to first do a few steps with the original DE to obtain a suitable initial condition for this one). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Questions on log-normal distributions Are log-normal distributions (lnd) applicable for any distributions for whom the lower bound value is 0 (i.e. human height can't go below 0)? In theory, if something like human height is modeled as a normal distribution, one can calculate the probability of a person being less than 0 feet tall! I'm under the impression that lnd-models are the perfect models for quantifying any phenomena (i.e. life-span, height, rate-of-returns, heights of buildings, average speed, etc.) for whose lower limit is 0 to infinity. What are the differences/limitations of a normal distribution bounded by 0 to infinity VS a lnd? And when are bounded normal distributions valid but a lnd not valid? === Subject: Re: Questions on log-normal distributions >Are log-normal distributions (lnd) applicable for any distributions for >whom the lower bound value is 0 (i.e. human height can't go below 0)? What do you mean by applicable? >In theory, if something like human height is modeled as a normal >distribution, one can calculate the probability of a person being >less than 0 feet tall! One can also calculate the probability of being more than 10^6 feet tall. In either normal or log-normal this would be nonzero, but it really should be 0 (anything that tall would not be called a person). I don't think that in itself is a serious objection to using those distributions in a model, nor is the problem with negative heights a very serious objection to using a normal distribution. Of course it might depend on what you want to do with the distribution. >I'm under the impression that lnd-models are the perfect models for >quantifying any phenomena (i.e. life-span, height, rate-of-returns, >heights of buildings, average speed, etc.) for whose lower limit is 0 >to infinity. Any phenomena? Only those where the model is a good fit to reality, or arises in a natural way. There are lots of other distributions on the interval (0,infinity), e.g. exponential, gamma, Weibull, ... Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Update: Objections to Cantor's Theory On 20 Jul 2005 22:35:14 -0700, Proginoskes > >> [...] >> That reminds me of when I wanted to understand the concept of a >> non-deterministic machine. I asked someone, supposedly an expert, to >> explain it to me. He started out by saying a non-deterministic machine >> was a 7-tuple ... and I stopped him right there. Even though I really >> didn't know what they were, I knew that a formal definition as some >> kind of 7-tuple was not the way to understand the essence. > >There's a big gap between understanding something and being able to >explain it. This is why there are so many good researchers in colleges >who are such horrible teachers. > >Someone with more of the mind of a teacher would have probably said >something like: A non-deterministic machine is an algorithm that >reaches points where a decision has to be made. It will choose the >correct decision, provided that there is a success somewhere further >down the road, among the future choices. I hope so, anyway, because I > > Yes, I like that explanation. It's similar to the way I think about it, and it does capture the essence. Once the basic concept is communicated, then it's reasonable to ask the question Ok, now how can we formalize the concept?. After all, the definers must have had the essence in mind when they devised the definition, so it's unfair to just hit someone with an abstract, formal definition without first motivating it with the concept of what the definition is trying to model. quasi === Subject: Re: Update: Objections to Cantor's Theory On 20 Jul 2005 12:47:05 -0700, david petry When Cantor introduced his ideas, there was a heated debate > about whether they should be accepted as mathematics. For > reasons which are not entirely clear, the ideas were > accepted [...] > Actually, the reasons are rather clear. David Hilbert described Cantor's work as the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity. F. === Subject: Re: Update: Objections to Cantor's Theory >If mose of the debate on the internet is junk, then >it seems strange to use the internet as evidence >for the prevalence of the anti-Cantorian view. For many of the anti-Cantorians, it is most obvious that there's something absurd about Cantor's Theory, but they really haven't a clue about how to argue the point to the mathematicians. === Subject: Re: Update: Objections to Cantor's Theory >>If mose of the debate on the internet is junk, then >>it seems strange to use the internet as evidence >>for the prevalence of the anti-Cantorian view. > For many of the anti-Cantorians, it is most obvious that > there's something absurd about Cantor's Theory, but they > really haven't a clue about how to argue the point to the > mathematicians. What is absurd about there not existing a bijection between a set and its powerset? In any case, as you said, most of the anti-Cantorian arguments are nonsense. It seems strange to then conclude that there is a serious and prevalent anti-Cantorian view. Stephen === Subject: Re: Update: Objections to Cantor's Theory > >>If mose of the debate on the internet is junk, then >>it seems strange to use the internet as evidence >>for the prevalence of the anti-Cantorian view. > > For many of the anti-Cantorians, it is most obvious that > there's something absurd about Cantor's Theory, but they > really haven't a clue about how to argue the point to the > mathematicians. > > What is absurd about there not existing a bijection between > a set and its powerset? The view is that one can be constructed. karl m === Subject: Re: Update: Objections to Cantor's Theory On 21 Jul 2005 11:40:06 -0700, david petry For many of the anti-Cantorians, it is most obvious that > there's something absurd about Cantor's Theory, but they > really haven't a clue about how to argue the point to the > mathematicians. > That's because the lack the necessary mathematical abilities. Btw: If the h a d them, the wouldn't be anti-Cantorians. (It's r e a l l y that simple!) F. -- I am the very model of a modern non-Cantorian, With insights mathematical as good as any saurian. I rattle the Establishment foundations with prodigious ease, And populate the counting numbers with some new infinities. I've never studied axioms of sets all theoretical, But that's just ted'ous detail; whereas MY thoughts are heretical And cause the so-called experts rather quickly to exasperate, While I sit back and mentally continue just to .... (Barb Knox) === Subject: Re: Update: Objections to Cantor's Theory > Set theory is a disease from which mathematics > will one day recover (Poincare) > Poincare never said that. You will not be able to give a valid > reference for this. I can't, but Keith Ramsay already has. :-) === Subject: Re: Update: Objections to Cantor's Theory > |> tSet theory is a disease from which mathematics > |> twill one day recover (Poincare) > There was a discussion of this on the Historia Mathematica > issue in the Mathematical Intelligencer, vol. 13 (1991) > p. 19-22. Allegedly it concludes he didn't say it. > > On 7 Sept. 1998 Michael Detlefsen commented that the original > of the remark appears in Poincare's 1908 essay, The Future > of Mathematics (appearing as chapter 2 of _Science and > Method_). Here's how he quotes it (I word-wrapped it): > > #... it has come to pass that we have encountered certain > #paradoxes, certain apparent contradictions that would have > #delighted Zeno the Eleatic and the school of Megara. And > #then each must seek the remedy. For my part, I think, and > #I am not the only one, that the important thing is never to > #introduce entities not completely definable in a finite > #number of words. Whatever be the cure adopted, we may > #promise ourselves the joy of the doctor called in to follow > #a beautiful pathologic case. > > So it seems that rather than describing set theory as a > disease, he is describing set theory as a patient to be > cured of a disease, i.e. paradoxes. His idea of a proper > cure wasn't the same as Cantor's, nor was it the same as > yours. However, I don't think you are interpreting it correctly. Look at his proposed cure: For my part, I think, and I am not the only one, that the important thing is never to introduce entities not completely definable in a finite number of words. That is almost exactly the anti-Cantorian view. There don't exist more than a countable number of entities, given Poincare's cure. He is claiming that it is Cantor's ideas about set theory that is the disease. I don't think anyone is arguing that there is something wrong with sets per se. So I'm sticking with the quote. Keith Ramsay === Subject: Re: Update: Objections to Cantor's Theory > > > > [...] the mathematicians respond by accusing the > anti-Cantorians of being imbeciles, idiots and[/or] crackpots. > And that's indeed true in most (if not all) cases. > Could y o u, David, show me a single exception? which is an example of an anti-Cantorian who is far from being a Crackpot. ************ 5. dav2222 Jul 28 1999, 3:00 am show options === Subject: More Cantor Reply to Author | Forward | Print | Individual Message | Show original | Report Abuse Hello sci.math participants. My question relates to a topic that has created a lot of heated discussion in this group. I have done some searches ondejanews and the world wide web and have not found a convincing answer yet. Like many persons, I am troubled by Cantor's diagonalization proof.I feel that my math teachers gave me a little bait and switch. Here's what I mean: They taught me about whole numbers, then integers, then fractions.I learned that fractions are rational numbers. (by fraction I mean something like A/B where A and B are integers) Then I was taught that rationals are not enough. For example, thesquare root of two can't be represented as a fraction. Ditto for pi. So then they introduce real numbers -- so far so good. Then comes the diagonalization proof, which seems pretty cool, but a few weeks after learning about it, it occurred to me that thereal numbers in the proof are qualitatively different from numbers like pi - this is the bait and switch What I mean is, numbers like pi can be described in English. The Enlish description can be mapped to a whole number -- for example by letting each letter be a digit in a base 26 numbering system . (Maybe a higher base would be useful to get spaces, symbols, etc. in) By the way, I realize that this is not an original thought. Please bear with me and I will reach my question. Anyways, it seems like one could make a distinction between meaningful real numbers and meaningless or undescribable real numbers. I don't have the mathematical skill to formally define meaningful numbers, but informally, I would offer the following definition: A meaningful number is one that can be described unambiguously, to a person of reasonable intelligence, in a finite string of English letters , Arabic numerals, and commonly used symbols. Ellipses are only permissible when reasonable people would agree on the natural continuation of the preceding sequence or series of characters, numerals, and/or symbols. Thus THE SQUARE ROOT OF TWO is meaningful. Also, 3.1415926 . . . is meaningful. But, 0.97364380 . . . , a number that might be found in a diagonalization proof, is meaningless by my definition. In fact, I am troubled even calling 0.97364390 . . . a number -- arguably it is just an ambiguous string of numerals and symbols. THE SMALLEST NUMBER NOT DEFINABLE IN LESS THAN 19 SYLLABLES is possibly meaningless. At any rate, my instinct is that the question is not relevant the the question I ask below. So here's my question: What happens to mathematics if we get rid of meaningless numbers ? Note that I'm not trying to argue that the diagonalization proof is wrong - My understanding is that mathematics is no longer tied down by the real world and that many mathematicians consider themselves free to make whatever assumptions lead to interesting results. e.g., non-Euclidean geometries . Do we need meaninglessnumbers in the same way that we need sqrt (2)? What uses are there for meaningless numbers? (besides proofs about aleph-0, cardinality, and the like) I also believe that without meaningless numbers many of Cantor's results fall. Iam comfortable with this - I'm just wondering what else we lose. Can one construct a coherent mathematics without all that stuff? My feeling is yes, but I don't really know. By the way, I have given the above some thought and realize that arguably there is something more fundamental at stake than meaningless numbers. I would propose getting rid of meaningless sets. By analogy to the above definition, the following are examples of meaningful sets: {1, 2 , 9, 23} {2, 3, 5, 7, 11, 13, 17, 19, 23 . . . } {ALL EVEN INTEGERS} (You get the point). {ALL SETS THAT ARE NOT MEMBERS OF THEMSELVES} is possibly meaningless. {9, 97, 973, 9736, 97364, 973643, 9736438, 97364380 . . .} is meaningless. It seems to me that if you define sets in such a way -- i.e. get rid of meaningless sets -- then 's proof about power sets always being of greater cardinaliy does not work anymore. As above, my question is -- what else do you lose? Is mathematics still coherent? Do we need meaningless sets? What uses are there for them? would appreciate an answer. I apologise for my ignorance of mathematics - please don't flame me for getting something wrong. I am a lawyer and have not studied mathematics in many years. This is just something that has always troubled me. I have done many searches and have not come across a satisfactory answer. P.S. It would be great if someone would give me somelingo to describe this problem. e.g.: Aha - the problem you are describing is known as Kubo's conundrum === Subject: Re: Update: Objections to Cantor's Theory >> >> >> >>>> [...] the mathematicians respond by accusing the >>>>anti-Cantorians of being imbeciles, idiots and[/or] crackpots. >>> And that's indeed true in most (if not all) cases. >>Could y o u, David, show me a single exception? which is an example of an anti-Cantorian who is far from being a > Crackpot. He's not an anti-Cantorian. He is a person who is just learning the subject, and is asking some interesting questions. His ideas about meaningless sets and meaningless real numbers are early attempts to try to get to notions of recursively enumerable sets and computable real numbers. If he had opportunity to study these very proper subjects, he would come to find out that they don't really solve his problems, and indeed diagonal arguments can be used with those to find uncomputable or non-primitive recursive real numbers. He will come up against problems like Turing's halting theorem (again effectively Cantor's diagonal argument). magazine called Wireless world which debunked Einstein's theory of relativity. I got very interested, and started asking the same figured out that Einstein really had got it right. That deep thinking really helped me understand a lot of the issues. I think that this kid has potential to go in the same direction. Just give him time (and hopefully good advice). ************ 5. dav2222 Jul 28 1999, 3:00 am show options > === > Subject: More Cantor > Reply to Author | Forward | Print | Individual Message | Show original > | Report Abuse Hello sci.math participants. My question relates > to a topic that has created a lot of heated > discussion in this group. I have done some > searches ondejanews and the world wide web and > have not found a convincing answer yet. Like many persons, I am troubled by Cantor's > diagonalization proof.I feel that my math > teachers gave me a little bait and switch. > Here's what I mean: > They taught me about whole numbers, then > integers, then fractions.I learned that fractions > are rational numbers. (by fraction I mean > something like A/B where A and B are integers) Then I was taught that rationals are not > enough. For example, thesquare root of two can't > be represented as a fraction. Ditto for pi. > So then they introduce real numbers -- so far so > good. Then comes the diagonalization proof, which seems > pretty cool, but a few weeks after learning about > it, it occurred to me that thereal numbers in the > proof are qualitatively different from numbers > like pi - this is the bait and switch > What I mean is, numbers like pi can be described > in English. The Enlish description can be mapped > to a whole number -- for example by letting each > letter be a digit in a base 26 numbering > system > . (Maybe > a higher base would be useful to get spaces, > symbols, etc. in) By the way, I realize that this is not an > original thought. Please bear with me and I will > reach my question. Anyways, it seems like one could make a > distinction between meaningful real numbers and > meaningless or undescribable real numbers. > I > don't have the mathematical skill to formally > define meaningful numbers, but informally, I > would offer the following definition: > A meaningful number is one that can be described > unambiguously, to a person of reasonable > intelligence, in a finite string of English > letters > , Arabic numerals, and commonly used > symbols. Ellipses are only permissible when > reasonable people would agree on the natural > continuation of the preceding sequence or series > of characters, numerals, and/or symbols. Thus THE SQUARE ROOT OF TWO is meaningful. > Also, 3.1415926 . . . is meaningful. > But, 0.97364380 . . . , a number that might be > found in a diagonalization proof, is meaningless > by my definition. In fact, I am troubled even > calling 0.97364390 . . . a number -- > arguably it is just an ambiguous string of > numerals and symbols. > THE SMALLEST NUMBER NOT DEFINABLE IN LESS THAN > 19 SYLLABLES is possibly meaningless. At any > rate, my instinct is that the question is not > relevant the the question I ask below. So here's my question: What happens to > mathematics if we get rid of meaningless > numbers > ? Note that I'm not trying to argue that > the diagonalization proof is wrong - My > understanding is that mathematics is no longer > tied down by the real world and that many > mathematicians consider themselves free to make > whatever assumptions lead to interesting results. > e.g., non-Euclidean geometries > . Do we need > meaninglessnumbers in the same way that we need > sqrt (2)? What uses are there for meaningless > numbers? (besides proofs about aleph-0, > cardinality, and the like) I also believe that without meaningless numbers > many of Cantor's results fall. Iam comfortable > with this - I'm just wondering what else we > lose. Can one construct a coherent mathematics > without all that stuff? My feeling is yes, but I > don't really know. > By the way, I have given the above some thought > and realize that arguably there is something more > fundamental at stake than meaningless numbers. > I would propose getting rid of meaningless > sets. By analogy to the above definition, the > following are examples of meaningful sets: > {1, 2 , 9, 23} > {2, 3, 5, 7, 11, 13, 17, 19, 23 . . . } > {ALL EVEN INTEGERS} (You get the point). > {ALL SETS THAT ARE NOT MEMBERS OF THEMSELVES} is > possibly meaningless. > {9, 97, 973, 9736, 97364, 973643, 9736438, > 97364380 . . .} is meaningless. > It seems to me that if you define sets in such a > way -- i.e. get rid of meaningless sets -- then > 's proof about power sets always being of > greater cardinaliy does not work anymore. > As above, my question is -- what else do you > lose? Is mathematics still coherent? Do we > need meaningless sets? What uses are there for > them? > would appreciate an answer. > I apologise for my ignorance of mathematics - > please don't flame me for getting something > wrong. I am a lawyer and have not studied > mathematics in many years. This is just > something that has always troubled me. I have > done many searches and have not come across a > satisfactory answer. > P.S. It would be great if someone would give me > somelingo to describe this problem. e.g.: Aha - > the problem you are describing is known > as Kubo's conundrum > === Subject: Re: Update: Objections to Cantor's Theory (...) > which is an example of an anti-Cantorian who is far from being a > Crackpot. > I think that this kid has potential to go in the same direction. Just > give him time (and hopefully good advice). > I apologise for my ignorance of mathematics - > please don't flame me for getting something > wrong. I am a lawyer and have not studied > mathematics in many years. This is just > something that has always troubled me. I have > done many searches and have not come across a > satisfactory answer. === Subject: Re: Update: Objections to Cantor's Theory > (...) >>which is an example of an anti-Cantorian who is far from being a >>>Crackpot. >>I think that this kid has potential to go in the same direction. Just >>give him time (and hopefully good advice). >>>I apologise for my ignorance of mathematics - >>>please don't flame me for getting something >>>wrong. I am a lawyer and have not studied >>>mathematics in many years. This is just >>>something that has always troubled me. I have >>>done many searches and have not come across a >>>satisfactory answer. Well, you got me. I guess that invalidates everything else I said. :-) === Subject: Re: Update: Objections to Cantor's Theory On 21 Jul 2005 10:36:43 -0700, david petry which is an example of an anti-Cantorian who is far from being a > Crackpot. > *lol* You really don't know what you are talking about, do you? Just a quote from that post: I don't have the mathematical skill to formally define meaningful numbers, but [bla and bla] It's always the same storry..., here's another one: I was asked that before, and never got around to fully analyzing the axioms for lack of time, but [bla and bla] (Tony Orlow) -------------------- I am the very model of a modern non-Cantorian, With insights mathematical as good as any saurian. I rattle the Establishment foundations with prodigious ease, And populate the counting numbers with some new infinities. I've never studied axioms of sets all theoretical, But that's just ted'ous detail; whereas MY thoughts are heretical And cause the so-called experts rather quickly to exasperate, While I sit back and mentally continue just to .... (Barb Knox) F. === Subject: Re: Update: Objections to Cantor's Theory On 21 Jul 2005 00:33:56 -0700, Keith Ramsay said: > |> > |> tSet theory is a disease from which mathematics > |> twill one day recover (Poincare) > |> > |> This quote has been questioned in the past. What is the > |> actual reference? > | > |For all I know, it's just folklore, but it has > |appeared in various reputable works. > > There was a discussion of this on the Historia Mathematica > issue in the Mathematical Intelligencer, vol. 13 (1991) > p. 19-22. Allegedly it concludes he didn't say it. > > On 7 Sept. 1998 Michael Detlefsen commented that the original > of the remark appears in Poincare's 1908 essay, The Future > of Mathematics (appearing as chapter 2 of _Science and > Method_). Here's how he quotes it (I word-wrapped it): > > #... it has come to pass that we have encountered certain > #paradoxes, certain apparent contradictions that would have > #delighted Zeno the Eleatic and the school of Megara. And > #then each must seek the remedy. For my part, I think, and > #I am not the only one, that the important thing is never to > #introduce entities not completely definable in a finite > #number of words. Whatever be the cure adopted, we may > #promise ourselves the joy of the doctor called in to follow > #a beautiful pathologic case. > > So it seems that rather than describing set theory as a > disease, he is describing set theory as a patient to be > cured of a disease, i.e. paradoxes. His idea of a proper > cure wasn't the same as Cantor's, nor was it the same as > yours. Excellent detective work. I've wondered for years about the legitimacy of that quote. === Subject: Re: Update: Objections to Cantor's Theory >>...l >>the essence of the what the real numbers _really are_. (And what they >>really are ain't sets!) >>... > > Haha. > > That reminds me of when I wanted to understand the concept of a > non-deterministic machine. I asked someone, supposedly an expert, to > explain it to me. He started out by saying a non-deterministic machine > was a 7-tuple ... and I stopped him right there. Even though I really > didn't know what they were, I knew that a formal definition as some > kind of 7-tuple was not the way to understand the essence. Well, it might not be the way to *introduce* the concept to a beginner -- one needs to start with an informal account of the underlying intuitions. But that doesn't mean that it isn't, at the end of the day, the right way to understand the concept's essence. === Subject: Re: Update: Objections to Cantor's Theory > Well, we know now that ZFC is consistent (assuming ZF is) I didn't know that ZFC is consistent. And why should I assume that ZF is consistent ? Isn't that just wishful thinking ? Han de Bruijn === Subject: Re: Update: Objections to Cantor's Theory > >> Well, we know now that ZFC is consistent (assuming ZF is) > >I didn't know that ZFC is consistent. I didn't say it was known to be consistent. Read my whole sentence again. Or read the work of Godel and of Cohen. >And why should I assume that ZF >is consistent ? Isn't that just wishful thinking ? Yes, it is just wishful thinking. But isn't everything _you_ espouse also wishful thinking? That is, your particular point of view seems to be to put greater faith in the conclusions which are drawn from _inductive_ reasoning; rather than conclusions drawn _deductively_ from axioms, you want to work with statements which are borne out in reality. This is an eminently reasonable way to do some things (physical science, for example), but of course it's just wishful thinking to suppose that there are no exceptions nor contradictions out there somewhere, simply because you haven't encountered any yet. Was Newtonian mechanics correct? Experience said yes and so with wishful thinking we believed it. Eventually we found it had to be tweaked. So too with ZF. We actually know we can't prove it consistent (unless it's inconsistent) but based on our experience we believe it to be consistent. The experiments testing its consistency consist of the zillions of proofs which have been constructed of propositions phrased, ultimately, on the ZF axioms of set theory. No contradictions have yet been found. Some day a contradiction will emerge, and we'll have to tweak things just as the physicists did. I wouldn't lose any sleep over it; the experimental evidence is really strong that our wishful thinking is in fact correct. dave === Subject: Re: Update: Objections to Cantor's Theory > I think that I would object to the use of the word anti-Cantorian and > I would prefer that you use the word intuitionist. Though I feel much sympathy for intuitionism, I would object to that. For the reason that intuitionism has already a well-defined meaning, which conflicts on many issues with David Petry's description of an anti-Cantorian. Han de Bruijn === Subject: Re: Update: Objections to Cantor's Theory > I think that I would object to the use of the word anti-Cantorian >> and I would prefer that you use the word intuitionist. > Though I feel much sympathy for intuitionism, I would object to that. For the reason that intuitionism has already a well-defined meaning, > which conflicts on many issues with David Petry's description of an > anti-Cantorian. Somebody else emailed me exactly that objection, and I agree with them and you. They suggested the word constructivist. === Subject: Re: Update: Objections to Cantor's Theory > I still await the inclusion of the answers to the following questions. To which axioms of ZF do anti-Cantorians object, exactly? The last five or so. :-) Han de Bruijn === Subject: Re: Update: Objections to Cantor's Theory > On 20 Jul 2005 12:47:05 -0700, david petry > Rationale: (1) It's definitely n o t plausible. You might f i r s t ask > professionals of mathematics before claiming such ridiculous things. (2) It obviously reflects *your own* opinion concerning this topic - (3) It's complete nonsense. >It is plausible that in the future, mathematics will be split >>into two disciplines - scientific mathematics (i.e. the science >>of phenomena observable in the world of computation), and >>philosophical mathematics, wherein Cantor's Theory is merely >>one of many possible formal theories of the infinite. As I've said in the other thread: > Worse. It _has_ already split into two disciplines: http://huizen.dto.tudelft.nl/deBruijn/nag.htm Han de Bruijn === Subject: Re: Update: Objections to Cantor's Theory > As I've said in the other thread: > Worse. It _has_ already split into two disciplines: >> >> http://huizen.dto.tudelft.nl/deBruijn/nag.htm > Han de Bruijn venture to say that it is describing social problems rather than philosophical problems. I also share with you a sense that mathematics has recently tended to be a little overboard on the abstract. However, I do believe that the trend is reversing. You don't want it to reverse too quickly otherwise you will get a French Revolution rather than an American War of Independence (I hope you get my meaning). Also, I think that some tension between the applied mathematicians and pure mathematicians is always going to be healthy. I don't really think that objections to the axioms of ZF are at the root of this problem. Well the emphasis on axiomization might have had a psychological effect upon mathematicians. But these effects are reversable. And in the meantime, people are trying to develop automatic proof techniques, which will certainly prove to be useful in the future. Anyway I have another post on this thread on this subject. I would be interested in your comments on it. Stephen === Subject: Re: Update: Objections to Cantor's Theory > > Worse. It _has_ already split into two disciplines: > > http://huizen.dto.tudelft.nl/deBruijn/nag.htm > Nonsense. === Subject: Re: Update: Objections to Cantor's Theory ... > The anti-Cantorians claim that while infinite sets and > power sets of those infinite sets are undeniably useful > abstractions, Not all anti-Cantorians claim that. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Update: Objections to Cantor's Theory >> The anti-Cantorians claim that while infinite sets and >> power sets of those infinite sets are undeniably useful >> abstractions, >Not all anti-Cantorians claim that. from the noise. === Subject: Re: Update: Objections to Cantor's Theory Discussion, linux) > >>> The anti-Cantorians claim that while infinite sets and >>> power sets of those infinite sets are undeniably useful >>> abstractions, > >>Not all anti-Cantorians claim that. > > from the noise. > By what principle? by you and supported (you say) by Han de Bruijn[1]. There are other self-proclaimed anti-Cantorians on this group and a number of them are committed to arguments involving infinite natural numbers as a way of refuting Cantor[2]. I'd wager I can find more people that support this tactic than that support the view that infinity is a useful abstraction but not bound by normal logic. So by what principle have you determined that your view is signal and this other view noise? What test have you applied? Is it all just a matter of whether you agree with the claims? Footnotes: [1] Sorry if I get the name wrong. Maybe you mentioned someone else. [2] Orlow is currently the loudest, but I can try to find other names if necessary. -- Now I realize that he got away with all of that because sci.math is not important, and the rest of the world doesn't pay attention. Like, no one is worried about football players reading sci.math postings! -- James S. Harris on jock reading habits === Subject: Re: Update: Objections to Cantor's Theory On 21 Jul 2005 11:25:43 -0700, david petry > > I'm trying to separate the signal from the noise. > There is no signal, only noise (produced by mathematical crackpots). F. === Subject: Re: Update: Objections to Cantor's Theory !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > >>> The anti-Cantorians claim that while infinite sets and >>> power sets of those infinite sets are undeniably useful >>> abstractions, > >>Not all anti-Cantorians claim that. > > from the noise. Where signal is defined as what you happen to believe in... -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Update: Objections to Cantor's Theory > It is plausible that in the future, mathematics will be split > into two disciplines - scientific mathematics (i.e. the science > of phenomena observable in the world of computation), and > philosophical mathematics, wherein Cantor's Theory is merely > one of many possible formal theories of the infinite. Call this one relegious mathematics or matheology. The Cantorians close their eyes in order to avoid obvious contradictions and to maintain their useless pet, simply insisting that logic is not valid in the infinite. One example: Take the set Q of all rationals q, in normal order. Multiply each one by pi. This gives an equinumerous (by bijection) set X of irrationals q*pi. The union of both sets consists of rationals and irrationals. By symmetry reasons there is never a pair of irrationals without a rational between them in this set. Now add one further irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot exist other than that two irrationals, namely sqrt(2) and one of the elements q*pi of X have no rational q of Q between them. This is a contradiction because in the whole set of real numbers in normal order, there are never wo irrationals without a rational between them, as is easy to prove. Alone the latter fact is proof enough to see that there are not more irrationals than rationals. Probable solution: There are no irrational numbers existing at all, as Kronecker already knew. === Subject: Re: Update: Objections to Cantor's Theory mueckenh@rz.fh-augsburg.de says... > >> It is plausible that in the future, mathematics will be split >> into two disciplines - scientific mathematics (i.e. the science >> of phenomena observable in the world of computation), and >> philosophical mathematics, wherein Cantor's Theory is merely >> one of many possible formal theories of the infinite. > >Call this one relegious mathematics or matheology. The Cantorians close >their eyes in order to avoid obvious contradictions and to maintain >their useless pet, simply insisting that logic is not valid in the >infinite. You've got that completely backwards. The one thing that (classical) mathematicians insist on is that logic works the same regardless of whether the domain is naturals, reals, infinite sets, or whatever. Cantorians would never claim that logic is not valid for infinite sets. -- Daryl McCullough Ithaca, NY === Subject: Re: Update: Objections to Cantor's Theory > It is plausible that in the future, mathematics will be split > into two disciplines - scientific mathematics (i.e. the science > of phenomena observable in the world of computation), and > philosophical mathematics, wherein Cantor's Theory is merely > one of many possible formal theories of the infinite. Call this one relegious mathematics or matheology. It is WM's beeleifs that are a matter of religion, as they require simulteneous belief in the mutually exclusive. > The Cantorians close their eyes in order to avoid obvious > contradictions and to maintain their useless pet, simply insisting > that logic is not valid in the infinite. Actually, it is the Cantorian insistence that logic IS valid in the infinite that the the anti-Cantorian rabble objects to so fervently. One example: Take the set Q of all rationals q, in normal order. > Multiply each one by pi. This gives an equinumerous (by bijection) set > X of irrationals q*pi. The union of both sets consists of rationals and > irrationals. By symmetry reasons there is never a pair of irrationals > without a rational between them in this set. Now add one further > irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot > exist other than that two irrationals, namely sqrt(2) and one of the > elements q*pi of X have no rational q of Q between them. This is a > contradiction because in the whole set of real numbers in normal order, > there are never wo irrationals without a rational between them, as is > easy to prove. The only contradiction here is produced by WM assuming, as usual, what is patently false. Between EVERY two reals there is a rational, in fact, between any two distinct reals there are infinitely many rationals. Alone the latter fact is proof enough to see that there are not more > irrationals than rationals. The latter fact referred to is a falsehood which only someone as mathematically incompetent as WM would be guilty of foisting on sci.math. Probable solution: There are no irrational numbers existing at all, as > Kronecker already knew. Solution: WM is too incompetent to discuss any mathematical issue, as sci.math already knew. > === Subject: Re: Update: Objections to Cantor's Theory > > It is plausible that in the future, mathematics will be split > into two disciplines - scientific mathematics (i.e. the science > of phenomena observable in the world of computation), and > philosophical mathematics, wherein Cantor's Theory is merely > one of many possible formal theories of the infinite. > > Call this one relegious mathematics or matheology. The Cantorians close > their eyes in order to avoid obvious contradictions and to maintain > their useless pet, simply insisting that logic is not valid in the > infinite. > CORRECTION: > One example: Take the set Q of all rationals q, in normal order. > ADD pi TO EACH ONE. This gives an equinumerous (by bijection) set > X of irrationals q+pi. The union of both sets consists of rationals and > irrationals. By symmetry reasons there is never a pair of irrationals > without a rational between them in this set. Now add one further > irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot > exist other than that two irrationals, namely sqrt(2) and one of the > elements q+pi of X have no rational q of Q between them. This is a > contradiction because in the whole set of real numbers in normal order, > there are never two irrationals without a rational between them, as is > easy to prove. > > Alone the latter fact is proof enough to see that there are not more > irrationals than rationals. > > Probable solution: There are no irrational numbers existing at all, as > Kronecker already knew. > === Subject: Re: Update: Objections to Cantor's Theory > > It is plausible that in the future, mathematics will be split > into two disciplines - scientific mathematics (i.e. the science > of phenomena observable in the world of computation), and > philosophical mathematics, wherein Cantor's Theory is merely > one of many possible formal theories of the infinite. > > Call this one relegious mathematics or matheology. The Cantorians close > their eyes in order to avoid obvious contradictions and to maintain > their useless pet, simply insisting that logic is not valid in the > infinite. > > CORRECTION: > One example: Take the set Q of all rationals q, in normal order. > ADD pi TO EACH ONE. This gives an equinumerous (by bijection) set > X of irrationals q+pi. The union of both sets consists of rationals and > irrationals. By symmetry reasons there is never a pair of irrationals > without a rational between them in this set. Now add one further > irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot > exist other than that two irrationals, namely sqrt(2) and one of the > elements q+pi of X have no rational q of Q between them. WM is deliberately trying to promulgate a falsehood here. Between ANY two distinct reals there are as many rationals as in Q itself. Since the union Q u X u {sqrt(2)} contians Q, there must be infinitely many rationals between any two distinct members of it, including those mentioned by WM. > This is a contradiction because in the whole set of real numbers in > normal order, there are never two irrationals without a rational > between them, as is easy to prove. The only contradiction here is that WM contradicting the truth. > > Alone the latter fact is proof enough to see that there are not more > irrationals than rationals. Alone, the latter fact is proof enough that WM is too mathematically incompetent to assess any mathematical statements beyond the level of 2+2=4. > > Probable solution: There are no irrational numbers existing at all, as > Kronecker already knew. Probable solution: WM, get thee to a shrink posthaste! === Subject: Re: Update: Objections to Cantor's Theory > > Set theory is a disease from which mathematics > will one day recover (Poincare) > > > > Poincare never said that. You will not be able to give a valid > reference for this. There is no actual infinity. The Catorians have forgotten that and have fallen into contradictons. [H. Poincar.8e, Les math.8ematiques et la logique III, Rev. m.8etaphys. morale 14, p. 316, (1906).] === Subject: Re: Update: Objections to Cantor's Theory > > Set theory is a disease from which mathematics > will one day recover (Poincare) > > > > Poincare never said that. You will not be able to give a valid > reference for this. There is no actual infinity. The Catorians have forgotten that and have > fallen into contradictons. [H. Poincar.8e, Les math.8ematiques et la > logique III, Rev. m.8etaphys. morale 14, p. 316, (1906).] Poincare never said that. === Subject: Re: why did the Apollo missions all land in the same area? Have you a photo of the hidden face of the moon ? Can you send me it ? === Subject: Re: why did the Apollo missions all land in the same area? > today, he saw a map > of where all the Apollo misisons landed. Why did all the missions seem > to land in the same > general patch of moon? Is there something special about that area? > You would think they would > spread out a bit more! And be sure to zoom in all the way. But the real question is why isn't it green? === Subject: Re: why did the Apollo missions all land in the same area? Hey, Mark did not know the moon is yellow color and so smooth up close. Mark saw a moon rock in a museum and it was not yellow. === Subject: Re: why did the Apollo missions all land in the same area? !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Hey, Mark did not know the moon is yellow color and so smooth up > close. Mark saw a moon rock in a museum and it was not yellow. That's to be expected if you store cheese for too long outside of a vacuum. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: why did the Apollo missions all land in the same area? <85u0ioggoy.fsf@lola.goethe.zz> Is it possible for a vacuum to contain cheese? David Ames === Subject: Re: why did the Apollo missions all land in the same area? > Is it possible for a vacuum to contain cheese? Yes, but it's a good idea to change the bag when it does. --Mark === Subject: Re: why did the Apollo missions all land in the same area? today, he saw a map > of where all the Apollo misisons landed. Why did all the missions seem > to land in the same > general patch of moon? Is there something special about that area? > You would think they would > spread out a bit more! They didn't. They were all over the place. Latitude is constrained by launch paramters. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: why did the Apollo missions all land in the same area? > today, he saw a map > of where all the Apollo misisons landed. Why did all the missions seem > to land in the same > general patch of moon? Is there something special about that area? > You would think they would > spread out a bit more! As others have said, naturally it would be easiest to work on the hemisphere that faces the earth all the time but in addition to that, that map seems to be a Mercator or similar projection of the hemisphere of the moon that faces earth. The distortions of such a projection makes things in the center appear smaller and closer together. -- rb === Subject: Re: why did the Apollo missions all land in the same area? > today, he saw a map > of where all the Apollo misisons landed. Why did all the missions > seem to land in the same > general patch of moon? So they could use the same studio set, of course! === Subject: Re: why did the Apollo missions all land in the same area? > today, he saw a map > of where all the Apollo misisons landed. Why did all the missions > seem to land in the same > general patch of moon? > > So they could use the same studio set, of course! LOL === Subject: Re: automorphisms of subspaces of the reals > > Actually, assuming the axiom of choice, there is a dense subset U of > the real line such that the only continuous mapping f:U-->U is the > identity. Nonsense! For one thing, there are the *constant* functions f:U-->U. Maybe I meant to say that the only continuous *injection* from U into U is the identity. Let me think this over. === Subject: Re: automorphisms of subspaces of the reals > >>Q. Does there exist an infinite subspace X of the reals (with the >>relative topology on X), such that the only automorphism of X is the >>identity. >> >>A. I don't believe so. Here is a proof using the Cantor--Bendixson >>analysis of subsets of the reals. >>... > >Very nice. > >quasi Well, nice try anyway. I was about to ask about the irrationals. quasi === Subject: Re: automorphisms of subspaces of the reals > >> >>>Q. Does there exist an infinite subspace X of the reals (with the >>>relative topology on X), such that the only automorphism of X is the >>>identity. >>> >>>A. I don't believe so. Here is a proof using the Cantor--Bendixson >>>analysis of subsets of the reals. >>>... >> >>Very nice. >> >>quasi > >Well, nice try anyway. > >I was about to ask about the irrationals. > >quasi It's clear that the irrationals are far from perfect, and removing a countable set won't help. Nevertheless, the irrationals do have infinitely many automorphisms, for example x -> ax+b where a,b are rational and a is nonzero. quasi === Subject: Re: automorphisms of subspaces of the reals On 20 Jul 2005 21:44:37 -0700, Butch Malahide > > >> Q. Does there exist an infinite subspace X of the reals (with the >> relative topology on X), such that the only automorphism of X is the >> identity. >> >> A. I don't believe so. Here is a proof using the Cantor--Bendixson >> analysis of subsets of the reals. >> >> Let U be an infinite subset of the reals. By Cantor--Bendixson, we can >> write U as the union of a perfect closed set and a countable set. > >The Cantor-Bendixson theorem applies to *closed* sets. How do you write >the set of all irrational numbers as the union of a perfect closed set >and a countable set? > >> [. . .] >> >> The proof actually shows that every infinite set of reals has >> infinitely many autohomeomorphisms. > >Actually, assuming the axiom of choice, there is a dense subset U of >the real line such that the only continuous mapping f:U-->U is the >identity. > >Hint: Use transfinite induction. To start with, you can assume that U >contains the set Q of all rational numbers. Then, any continuous >mapping f:U-->R will be determined by its restriction to Q. There are >just continuum many continuous (or otherwise) mappings from Q into R. >In constructing the set U, make sure that none of those functions >(except the identity) extends to a continuous mapping of U into U. Ah. I thought there should be a counterexample, but I couldn't see how to construct one explicitly. What you say here seems probably right - having thought about it for about a minute I'm stuck on one detail (maybe you can fill in the detail or say how the thing should be approached differently so as to avoid the problem): If f : Q -> R is continuous say B(f) is the set of all real x such that either f has no limit at x or f has a limit at x but this limit does not equal x. In general say _f(x) is the limit of f at x if there is such a limit, or 0 if the limit does not exist. It seems clear that if f : Q -> R is continuous but not the identity then B(f) has cardinality c; this is the detail I don't quite see how to prove, not that I've tried very hard. Assuming this: Say the continuous functions f : Q -> R other than the identity are enumerated as f_a for ordinals a < c. We construct two increasing families of sets U_a and T_a for ordinals a < c, such that U_a intersect T_a is empty for all a, as follows: Let U_0 = Q and T_0 = {}. For each a, choose x in B(f) such that x is not in U_b for any b < a, and such that _if_ f has a limit at x then _f(x) is not in T_b for any b < a. Let U_a be the union of the previous U_b's with x added, and let T_a be the union of the previous T_b's, with _f(x) added if f has a limit at x. The fact that B(f) has cardinality c says that it's always possible to find such an x. Now let U be the union of the U_a for a < c and let T be the union of the T_a. Now if f : U -> U is continuous and not the identity then there exists a such that the restriction of f to Q is f_a. Say x = x_a is the real that was added to U_a above. Now the fact that f is continuous and x is in U shows that f_a has a limit y at x, and that f(x) = y. But this is a contradiction, since the construction shows that y is in T, hence not in U. ************************ === Subject: Euclidean Postulates II What postulate is more general? a.- For a point external to a straight line can pass a straight line and only one straight line. Or b.- A triangle can be translated and/or rotated without changing its sides or its angles. Ludovicus === Subject: mathematica command I need a mathematica command that does the following: Assume X={1,2,3} and I would like to evaluate the funcion F[1,2,3] . I mean I need a command to remove the braces off my X so I can use them as paramenter to my function F. I could use for example F[X[[1]],X[[2]],X[[3]]] but I hope somebody knows a command so that I don't have to do this.. === Subject: Re: mathematica command > I need a mathematica command that does the following: > Assume X={1,2,3} > and I would like to evaluate the funcion F[1,2,3] . I mean I need a command > to remove the braces off my X so I can use them as paramenter to my function > F. I could use for example F[X[[1]],X[[2]],X[[3]]] but I hope somebody > knows a command so that I don't have to do this.. Apply[F,X] will do what you want. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: A little bit of History prime (adj.) 1399, from L. primus first, from pre-Italic *prismos, superl. of Old L. pri before, from PIE base *per- beyond, *pro- before (see pre-). To prime a pump (c.1840) meant to pour water down the tube, which saturated the sucking mechanism and made it draw up water more readily. Arithmetical sense (prime number) is from 1570; prime meridian is from 1878; prime minister is from 1646, applied to the First Minister of State of Great Britain since 1694. Priming first coat of paint is from 1609. Prime time originally (1503) meant spring time; broadcasting sense of peak tuning-in period is attested from 1964. pre- prefix meaning before, from O.Fr. pre- and M.L. pre-, both from L. pr.be (adv.) before, from PIE *prai- (cf. Oscan prai, Umbrian pre, Skt. pare thereupon, Gk. parai at, Gaul. are- at, before, Lith. pre at, O.C.S. pri at, Goth. faura, O.E. fore before), variant of base per- beyond. The L. word was active in forming compound verbs. === Subject: Re: A little bit of History >prime (adj.) >1399, from L. primus first, from pre-Italic *prismos, superl. of Old >L. pri before, from PIE base *per- beyond, *pro- before (see >pre-). To prime a pump (c.1840) meant to pour water down the tube, which >saturated the sucking mechanism and made it draw up water more readily. >Arithmetical sense (prime number) is from 1570; prime meridian is from >1878; prime minister is from 1646, applied to the First Minister of >State of Great Britain since 1694. Priming first coat of paint is from >1609. Prime time originally (1503) meant spring time; broadcasting >sense of peak tuning-in period is attested from 1964. > >pre- >prefix meaning before, from O.Fr. pre- and M.L. pre-, both from L. >príě (adv.) before, from PIE *prai- (cf. Oscan prai, Umbrian pre, Skt. >pare thereupon, Gk. parai at, Gaul. are- at, before, Lith. pre >at, O.C.S. pri at, Goth. faura, O.E. fore before), variant of base >per- beyond. The L. word was active in forming compound verbs. ... and your point is? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: A little bit of History >>prime (adj.) >>1399, from L. primus first, from pre-Italic *prismos, superl. of Old ... >... and your point is? Two is the only even prime. Primus inter pares has the Pope as its unique referent. The Pope and Bertrand Russell are two. Therefore... Lee Rudolph (damn, it's hard to keep sorites going in this weather) === Subject: does anybody know how to remove the braces of a list in mathematica so it can be sendt as parameters to a function? I need a mathematica command that does the following: Assume X={1,2,3} and I would like to evaluate the funcion F[1,2,3] . I mean I need a command to remove the braces off my X so I can use them as paramenter to my function F. I could use for example F[X[[1]],X[[2]],X[[3]]] but I hope somebody knows a command so that I don't have to do this.. === Subject: Re: does anybody know how to remove the braces of a list in mathematica so it can be sendt as parameters to a function? >I need a mathematica command that does the following: >Assume X={1,2,3} >and I would like to evaluate the funcion F[1,2,3] . I mean I need a command >to remove the braces off my X so I can use them as paramenter to my function >F. I could use for example F[X[[1]],X[[2]],X[[3]]] but I hope somebody >knows a command so that I don't have to do this.. > I believe the command is Apply. Look it up. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: Self-interest and other-interest What has this got to do with mathematics????????????????? === Subject: Cardinality of N and P(N) BACKGROUND Cantor defined the cardinality of N, the set of natural numbers, as aleph-0, which is the first transfinite number. Thus card(N) = aleph-0. The cardinality of the power set of N (i.e., the set of all possible subsets of N) is 2^aleph-0. Thus card(P(N)) = 2^aleph-0. This implies that N (the infinite set of natural numbers) cannot be put in one-to-one correspondence with the members of P(N). In other words, N is a countably infinite set, but P(N) is an uncountably infinite set. QUESTION My question is, what are the ramifications if it could be demonstrated that P(N) is, in fact, countable; i.e., what if the members of P(N) could be ordered in such a way as to be countable, and thus mappable to members of N? Would this impact Cantor's hierarchy of transfinites (aleph-1, aleph-2, etc.)? Would this impact the Continuum Hypothesis? -drt === Subject: Re: Cardinality of N and P(N) !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > BACKGROUND > > Cantor defined the cardinality of N, the set of natural numbers, as > aleph-0, which is the first transfinite number. Thus card(N) = > aleph-0. > > The cardinality of the power set of N (i.e., the set of all possible > subsets of N) is 2^aleph-0. Thus card(P(N)) = 2^aleph-0. > > This implies that N (the infinite set of natural numbers) cannot be put > in one-to-one correspondence with the members of P(N). In other words, > N is a countably infinite set, but P(N) is an uncountably infinite set. > > QUESTION > > My question is, what are the ramifications if it could be > demonstrated that P(N) is, in fact, countable; i.e., what if the > members of P(N) could be ordered in such a way as to be countable, > and thus mappable to members of N? Anyway, assume a mapping f(n) that maps n to a subset of N. The mapping will not cover {n|n not in f(n)}. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cardinality of N and P(N) > My question is, what are the ramifications if it could be demonstrated > that P(N) is, in fact, countable; i.e., what if the members of P(N) > could be ordered in such a way as to be countable, and thus mappable to > members of N? Would this impact Cantor's hierarchy of transfinites (aleph-1, aleph-2, > etc.)? Would this impact the Continuum Hypothesis? Which impact would have the discovery that 1 + 1 is actually equal to 3? Jose Carlos Santos === Subject: Re: Cardinality of N and P(N) On 21 Jul 2005 14:54:39 -0700, David R Tribble > > QUESTION My question is, what are the ramifications if it could be demonstrated > [say in ZFC] that P(N) is, in fact, countable [...] > This would mean that there is a contradiction in our standard set theory, ZFC. Bad thing... F. === Subject: Re: Cardinality of N and P(N) > BACKGROUND > > Cantor defined the cardinality of N, the set of natural numbers, as > aleph-0, which is the first transfinite number. Thus card(N) = > aleph-0. > > The cardinality of the power set of N (i.e., the set of all possible > subsets of N) is 2^aleph-0. Thus card(P(N)) = 2^aleph-0. > > This implies that N (the infinite set of natural numbers) cannot be > put in one-to-one correspondence with the members of P(N). In other > words, N is a countably infinite set, but P(N) is an uncountably > infinite set. > > QUESTION > > My question is, what are the ramifications if it could be demonstrated > that P(N) is, in fact, countable; i.e., what if the members of P(N) > could be ordered in such a way as to be countable, and thus mappable > to members of N? It would imply that the supposed demonstration is wrong. It cannot be demonstrated that P(N) is countable, at least using any standard set theory axioms, because has been proven that P(N) is uncountable. The proof is nearly trivial. One might just as well ask what the ramifications would be if it could be demonstrated that 1 + 1 = 3. --Mark === Subject: Re: Cardinality of N and P(N) > QUESTION My question is, what are the ramifications if it could be demonstrated > that P(N) is, in fact, countable; i.e., what if the members of P(N) > could be ordered in such a way as to be countable, and thus mappable to > members of N? Would this impact Cantor's hierarchy of transfinites (aleph-1, aleph-2, > etc.)? Would this impact the Continuum Hypothesis? Yes, in a very major way. In fact, every set theoretic question could be answered, both affirmitively and negatively at the same time. Stephen === Subject: Re: Cardinality of N and P(N) >> My question is, what are the ramifications if it could be demonstrated >> that P(N) is, in fact, countable; i.e., what if the members of P(N) >> could be ordered in such a way as to be countable, and thus mappable to >> members of N? >> >> Would this impact Cantor's hierarchy of transfinites (aleph-1, aleph-2, >> etc.)? >> >> Would this impact the Continuum Hypothesis? > Yes, in a very major way. > In fact, every set theoretic question could be answered, both > affirmitively and negatively at the same time. Wouldn't it just prove that the notion of counting infinite sets could be incorrect, at least for power sets? Assuming that it only applied to the countability of power sets, and could not be extended to other non-countable sets, would this really affect the rest of Cantor's (et al) theorems? -drt === Subject: Re: Cardinality of N and P(N) >>My question is, what are the ramifications if it could be demonstrated >>>that P(N) is, in fact, countable; i.e., what if the members of P(N) >>>could be ordered in such a way as to be countable, and thus mappable to >>>members of N? >>> >>>Would this impact Cantor's hierarchy of transfinites (aleph-1, aleph-2, >>>etc.)? >>> >>>Would this impact the Continuum Hypothesis? >Yes, in a very major way. >>In fact, every set theoretic question could be answered, both >>affirmitively and negatively at the same time. Wouldn't it just prove that the notion of counting infinite > sets could be incorrect, at least for power sets? Assuming that > it only applied to the countability of power sets, and could not > be extended to other non-countable sets, would this really affect > the rest of Cantor's (et al) theorems? No. It would affect the very core of how we currently do set theory. This is because there is already a proof that P(N) is uncountable. So if you could also show that it is countable, you would have a contradiction. We would have to rethink the whole way in which we think about set theory. Like Mark said, it would be somewhat akin to proving that 1+1=3. Stephen === Subject: Re: Cardinality of N and P(N) >>> My question is, what are the ramifications if it could be >>> demonstrated that P(N) is, in fact, countable; i.e., what if the >>> members of P(N) could be ordered in such a way as to be countable, >>> and thus mappable to members of N? >>> >>> Would this impact Cantor's hierarchy of transfinites (aleph-1, >>> aleph-2, etc.)? >>> >>> Would this impact the Continuum Hypothesis? > >> Yes, in a very major way. >> In fact, every set theoretic question could be answered, both >> affirmitively and negatively at the same time. > > > Wouldn't it just prove that the notion of counting infinite > sets could be incorrect, at least for power sets? Assuming that > it only applied to the countability of power sets, and could not > be extended to other non-countable sets, would this really affect > the rest of Cantor's (et al) theorems? There is in existence a proof that there is no bijection between S and P(S), for any set S. If you could prove that there IS a bijection between any single set and its power set, it would mean that the axioms of set theory are inconsistent. There cannot be proofs of both a statement and its negation in any theory with a consistent set of axioms. Given an inconsistent set of axioms, ANY statement that can be framed in the theory can be proven. So it would not just affect infinite sets, and it would not just affect the notion of power sets, or countability. It would demolish set theory, as it is currently defined. --Mark === Subject: Re: comparaison of 3 equations of multiple linear regression for 2000 : cost = -964 + 288*dms + 339*act + 697*grav + 636*nir + 594*cl for 2001 : cost = - 858 + 333*dms + 439*act + 771*grav + 512*nir + 919*cl for 2002 : cost = -1255 + 320*dms + 516*act + 834*grav + 641*nir + 813*cl I want to ask you another question : in my equations, is it possible without doing what you suggested ! but just to say that for example : number of acts (variable act in the equations) increase from 2000 to 2002 because as you see in the three equations, coefficients (339, 439 and 516 respectively) increase, the class (variable cl) increases because its coef. are passing from 594 to 919 and 813 for 2002, but decrease from 2001 to 2002. the dms changes no hard because their values are approximatively the same (288, 333 and 320). is it correct to do these conclusions by this way ? thank you very much, I think that I would have a depression soon. it's the first time that I work with biostatistics --> very hard === Subject: Re: comparaison of 3 equations of multiple linear regression > for 2000 : cost = -964 + 288*dms + 339*act + 697*grav + 636*nir + 594*cl for 2001 : cost = - 858 + 333*dms + 439*act + 771*grav + 512*nir + 919*cl for 2002 : cost = -1255 + 320*dms + 516*act + 834*grav + 641*nir + 813*cl I want to ask you another question : > in my equations, is it possible without doing what you suggested ! but just > to say that for example : number of acts (variable act in the equations) > increase from 2000 to 2002 because as you see in the three equations, > coefficients (339, 439 and 516 respectively) increase, the class (variable > cl) increases because its coef. > are passing from 594 to 919 and 813 for 2002, but decrease from 2001 to > 2002. the dms changes no hard because their values are approximatively the > same (288, 333 and 320). > is it correct to do these conclusions by this way ? It will be a bit questionable because there is nothing said about how the variables are correlated with each other. Are the standardized coefficients (beta) each very similar to the corresponding zero-level correlation? That will be true if there is not much overlap between variables, or other confounding (which can be more subtle). I would be more willing to try that sort of comparison if all the betas were within (say) 10-15% of the r's. that the big *changes* across time would be accounted for by overt policy changes. When were fees changed by the hospital administration? When did the government change certain policies on reimbursement? - Did those exist, and are they reflected by changes in coefficients? You previously mentioned huge t-tests on all variables. It might be worth paying attention to which were hugest; a t-test that is twice as large represents twice the effect. thank you very much, I think that I would have a depression soon. it's the > first time that I work with biostatistics --> very hard > There are simpler problems that one should start with.... -- Ul, wpilib@pitt.edu http://www.pitt.edu/~wpilib/index.html === Subject: Re: comparaison of 3 equations of multiple linear regression <3EeIe.21468$pH4.771025@news20.bellglobal.com> > for 2000 : cost = -964 + 288*dms + 339*act + 697*grav + 636*nir + 594*cl > > for 2001 : cost = - 858 + 333*dms + 439*act + 771*grav + 512*nir + 919*cl > > for 2002 : cost = -1255 + 320*dms + 516*act + 834*grav + 641*nir + 813*cl > > I want to ask you another question : > in my equations, is it possible without doing what you suggested ! but just > to say that for example : number of acts (variable act in the equations) > increase from 2000 to 2002 because as you see in the three equations, > coefficients (339, 439 and 516 respectively) increase, the class (variable > cl) increases because its coef. > are passing from 594 to 919 and 813 for 2002, but decrease from 2001 to > 2002. the dms changes no hard because their values are approximatively the > same (288, 333 and 320). > is it correct to do these conclusions by this way ? > > thank you very much, I think that I would have a depression soon. it's the > first time that I work with biostatistics --> very hard No, you can't base such conclusions on the coefficients alone; you must also consider their standard errors. The t-test that Bruce described is the simplest way of doing this. === Subject: dist. of sum of squares of uniformly... I am trying to find a distribution but i am living difficulties: let u1,u2,u3 ... , up are distributed independently and uniformy over 0 I am trying to find a distribution but i am living difficulties: > > let u1,u2,u3 ... , up are distributed independently and uniformy > over 0 Then what is the ditribution of u1^2+u2^2+....+up^p? u1^2 would be defined in the unit square, but non-uniform. Your sum would be the convolution of said distribution. But you later added this condition: > My actual aim is to find conditional joint distribution of > u1,u2,.....,up where the condition is u1^2+u2^2+....+up^2<1 . This changes the problem COMPLETELY! The condition u1^2+u2^2+....+up^2<1 only defines the constraint in the p-space of the p-tuple of independent U(0,1)s. As such, the joint distribution is uniform inside the positive orthon of the unit p-sphere. For p = 1, f(u1) = 1, 0 < u1 < 1. For p = 2, f(u1, u2) = 4/pi u1^2 + u2^2 < 1, For p = 3, f(u1, u2, u3) = 6/pi u1^2 + ... + u3^2 < 1 etc. I think that's what you're looking for. But you don't have to know the form of the multivariate pdf to simulate data rom it. There are many different ways of simulating data from it, including the very inefficient way of simulating p-tuples of U(0,1) and discharding all those outside of the unit p-sphere. -- Bob. === Subject: Re: dist. of sum of squares of uniformly... My actual aim is to find conditional joint distribution of u1,u2,.....,up where the condition is u1^2+u2^2+....+up^2<1 . According to this aim, let i generate one number from uni(0.1), say that number z. This number will be smaller than 1.Then I behave that number z as the total (u1^2+u^2+..+up^2). Then let imagine a part of real line from 0 to z. I divide this line into p parts randomly (using p-1 numbers which are generated from uni(0,z)) . Then the length of each parts are random variables. If i find the roots of each lenght , does the resulting process manage my aim? === Subject: Re: Data Mining Algorithm results. Disclaimer: I'm no expert statistician, but that rarely shuts me up. An association rule describes the relationship between two variables X and Y. The confidence is P(Y|X). I.e. if we know that X has occurred (a customer has bought beer in the last week) what is the new probability of Y occurring (the same customer buys headache pills in the current week). The support is P(X&Y), the probability of both events occurring. You say that you understand what these are already. The lift is: P(Y|X) / P(Y). What it measures is how much the probability of Y occurring changes given that we know that X is (e.g.) true. E.g. the ratio between the probability that a customer who bought beer last week buying headache pills this week and the probability for all customers that they will buy headache pills this week. This is a useful thing to know because it is possible for an association rule to have both high confidence and support, but for there still to be no real relationship between the two variables. If X and Y are independent variables, then P(Y|X) = P(Y) as knowledge of X does not change the probability of Y. Hence in this case lift will be 1. If knowledge that (e.g) X is true raises the probability of Y occurring then lift will be > 1. If knowledge that X is true decreases the probability of Y, then lift will be < 1. Lift is useful as large (or small) numbers give a quick idea of the strength of the dependence between X and Y. E.g. a lift of 10 would indicate that the probability of Y occurring has increased 10 times if we know X. I cannot know exactly what null hypothesis the p-value you mention is being calculated for. However, in my view (see disclaimer) the most obvious null hypothesis to test would be that P(Y|X) = P(Y), i.e. that there is no signficant relationship [association] between the two variables. Ross-c === Subject: Re: n-wise correlation among binary vectors > > > > Please forgive the naivete of this question; I realize that it's > Stats 101 stuff. > > I want to know the name of the types of analysis one would use to > solve the following generic problem. > > I have a collection of M items and N properties, and an M x N matrix > of 1s and 0s representing which items have which properties. > > The problem is to find clusters of properties (i.e. column vectors > in the matrix) that are strongly and/or significantly correlated > (I realize these strength and signficance are different; I'm > interested in both, together and individually). > > I know of methods to cluster the vectors hierarchically (I think), > but I have no basis for deciding the cutoff distance for these > clusters. minimum entropy methods are somewhat in vogue for model order estimation. May be useful for this problem, with a bit of development. http://www.rennes.supelec.fr/ren/rd/ssir/publis/maxent02_puttini_marrakchi_m e.pdf rusty === Subject: Ross Stochastic Processes 2e, page 90 problem 2.6 Hi all, This is a selfstudy problem, since nobody has the solution so we have to check against each other. The problem states that a machine functions only when both two types components function. Type-A components last for an exp(u1) time; Type-B componnents last for exp(u2) time. There are n Type-A components and m Type-B components. These components can be continuously applied without interruption. Now want to compute E[min(sum(Xi, i from 1 to n), sum(Yj, j from 1 to m))] where E is the expectation. I did it, finally got a doubly partial sum of truncated version of binomial distrubtions. symsum(symsum(gamma(i+j+1)/gamma(i+1)/gamma(j+1)*(u1/(u1+u2))^i*(u2/(u1+u2)) ^j, j, 1, m), i, 1, n) I got the following error message in Matlab: >> syms m n i j u1 u2 real >> symsum(symsum(gamma(i+j+1)/gamma(i+1)/gamma(j+1)*(u1/(u1+u2))^i*(u2/(u1+u2)) ^ j, >> j, 1, m), i, 1, n) ??? Error using ==> sym.maple Error, (in CombSumRecognition:-isAbeltype) cannot split rhs for multiple assignment Error in ==> sym.symsum at 43 r = maple('map','sum',f,[x.s '=' a.s '..' b.s]); Anybody can get the a closed-form solution? Please provide details if you === Subject: Re: Bizarritudes My calculator rounds 1/6 in 0.1616161617 but 1/66 in 0.0151515151. Space no ? === Subject: Re: Bizarritudes > My calculator rounds 1/6 in 0.1666666667 but 1/66 in > 0.0151515151. > Space no ? Or 1/6.1875:) === Subject: first settled in sci.math? Has any well known conjecture been first settled in sci.math? I don't mean just someone giving an account of their proof, presumably to be later published elsewhere. What I mean is this: Has there ever been a thread or sequence of threads, with contributions by various participants, which resulted in the resolution of a well known conjecture. And of course, I'm referring to real proofs, not the pseudo-math proofs. quasi === Subject: Re: Integral <1yLpQtRtBJiy1BHreeWttS2F1fba@4ax.com> @David Moran: I think you've that substitution in your trigonometry class, when you were solving trigonometric equations and were expressing sines, cosines and tangets using the half angle formulas (tan(x/2) etc) === Subject: Re: Troubling combinatorics > Supposed you are given 4 red balls, 4 blue balls and 4 green balls. You > want to arrange them in a row such that no 2 adjacent balls are of the > same colour. > > (i)How many ways of doing so? > > (ii) Is there a generalisation for n balls of each colour? > (iii) What if the balls are arranged in a circle? What is the relation > to (i)? > http://www2.ocn.ne.jp/~mizuryu/kadai/kadai70a.html lists the answers to (ii) up to n = 59. (At least, I think it does. I just calculated up to n = 8 and did a search on those numbers. This is what came up!) If, like me, your Japanese is ... er... sketchy, then you can see the usual hilarious machine-translated gobbledegook at Unfortunately this sequence doesn't seem to be listed at OEIS (http://www.research.att.com/~njas/sequences/). <99ad3$42e76f9e$82a1e3ad$14399@news1.tudelft.nl> <851x5kii2h.fsf@lola.goethe.zz> <5c00b$42e89baf$82a1e3ad$4772@news1.tudelft.nl> <13cd1$42e8c676$82a1e3ad$7889@news1.tudelft.nl> <6cb6f$42e8ebfc$82a1e3ad$3912@news2.tudelft.nl> > > Let's get physical now. > > It seems that you miss that set theory and mathematics are not a > narrative of the physical universe and set theory and mathematics do > not denote with words that pick out objects and even concepts of the > physical universe in the way that everyday language or physical > sciences do. For that matter, mathematics can't be tied to a particular > theory of the physical universe, since, such theories are about > contingent states-of-affairs, > > This is a negation of the history of mathematics. Mathematics has a > current state-of-affairs in relation to REALITY also. Ask any > high-school or junior-college teacher of mathematics. I think you're speaking tongue in cheek and about some other pedgogical matters? Just in case you're not, I didn't claim that it is impossible for mathematics to give a theory for a particular state of affairs in the physical world or a non-physical one, but rather that mathematics in general can't be tied to a particular state of affairs in the physical world. Even granting, for sake of argument, a platonist view, I don't think this entails that mathematics cannot rationally study theories that don't conform to a particular platonist universe. I'm happy to hear arguments to the contrary, though. But, again, if you we're just joking around, then nevermind this reply. MoeBlee <99ad3$42e76f9e$82a1e3ad$14399@news1.tudelft.nl> <851x5kii2h.fsf@lola.goethe.zz> <5c00b$42e89baf$82a1e3ad$4772@news1.tudelft.nl> <13cd1$42e8c676$82a1e3ad$7889@news1.tudelft.nl> <6cb6f$42e8ebfc$82a1e3ad$3912@news2.tudelft.nl> > > > Let's get physical now. > > It seems that you miss that set theory and mathematics are not a > narrative of the physical universe and set theory and mathematics do > not denote with words that pick out objects and even concepts of the > physical universe in the way that everyday language or physical > sciences do. For that matter, mathematics can't be tied to a particular > theory of the physical universe, since, such theories are about > contingent states-of-affairs, > > This is a negation of the history of mathematics. Mathematics has a > current state-of-affairs in relation to REALITY also. Ask any > high-school or junior-college teacher of mathematics. > > I think you're speaking tongue in cheek and about some other pedgogical > matters? No. > Just in case you're not, I didn't claim that it is impossible > for mathematics to give a theory for a particular state of affairs in > the physical world or a non-physical one, but rather that mathematics > in general can't be tied to a particular state of affairs in the > physical world. And that negates history. It is by man's nature tied to the CURRENT state of affairs in the physical world. > Even granting, for sake of argument, a platonist view, > I don't think this entails that mathematics cannot rationally study > theories that don't conform to a particular platonist universe. I'm > happy to hear arguments to the contrary, though Sorry, but what's the platonist view? karl m > Daryl McCullough said: > >Daryl McCullough said: > >> So, you agree that for *finite* sets, two sets have the same > >> bigulosity if and only if there is a bijection between the two? >> But that no longer holds for infinite sets? >> Then how is bigulosity an improvement over cardinality? >Because Bigulosity takes into account the nature of the bijection >in order to determine a precise relative size of infinity. > > In other words, bigulosity is whatever you want it to be, and so > you have a lot more flexibility. Just make it up as you go along. > > A = the set of natural numbers { 0, 1, 2, ... } B = the set of > base ten numerals { 0, 1, 2, ... } C = the set of base > two numerals { 0, 1, 10, 11, 100, ... } > > A and B have the same bigulosity. A and C have the same bigulosity. > But B and C do *not* have the same bigulosity (clearly C has a > smaller bigulosity than B). That's bigulosity for you... > > -- Daryl McCullough Ithaca, NY > If you consider your numerals to have N digits, then base ten has > 10^N elements and base 2 has 2^N elements So that TO is declaiming that the number of naturals is dependent on the base in which they are to be represented? Those mushrooms TO is nibbling must be really potent! <87hderrxf7.fsf@phiwumbda.org> > > Daryl McCullough said: > >Daryl McCullough said: > >> So, you agree that for *finite* sets, two sets have the same > >> bigulosity if and only if there is a bijection between the two? >> But that no longer holds for infinite sets? >> >> Then how is bigulosity an improvement over cardinality? >Because Bigulosity takes into account the nature of the bijection >in order to determine a precise relative size of infinity. > > In other words, bigulosity is whatever you want it to be, and so > you have a lot more flexibility. Just make it up as you go along. > > A = the set of natural numbers { 0, 1, 2, ... } B = the set of > base ten numerals { 0, 1, 2, ... } C = the set of base > two numerals { 0, 1, 10, 11, 100, ... } > > A and B have the same bigulosity. A and C have the same bigulosity. > But B and C do *not* have the same bigulosity (clearly C has a > smaller bigulosity than B). That's bigulosity for you... > > -- Daryl McCullough Ithaca, NY > > > If you consider your numerals to have N digits, then base ten has > 10^N elements and base 2 has 2^N elements > > So that TO is declaiming that the number of naturals is dependent on the > base in which they are to be represented? > > Those mushrooms TO is nibbling must be really potent! Mushrooms are part of both, food and medicine. You need an expert. karl m > David Kastrup said: > > No, I understand what you mean by countable. My issue with your > uncountability of the powerset of the naturals is that you can > easily form a bijection between the members of the powerset and the > set of infinite bit strings denoting set membership in the subsets, > > Yes. > > and between those bit strings and whole numbers, > > No. > > and the only reason this bijection is rejected is because of the > refusal to allow infinite whole numbers in the set of whole numbers, Infinite whole numbers need not be any part of the set of natural numbers. > despite the fact that an infinite set of whole numbers requires > infinite whole numbers, > > Whining does not make it so. It requires arbitrarily large numbers, > but none of them need to be infinite. > I am not whining and you are not correct. TO is whining and DK is correct. Two strikes in one short sentence! > > so you don't consider the infinite bitstrings to represent whole > numbers. It's like all the wrong choices have been made, almost on > purpose, in order to make some grand distinction which really isn't > there. > > Well, it's like you try to fix something that you perceive as a > problem by making it much worse. > > If you allow infinite whole numbers, as is required, > > Whining won't make it so. > And making statements like that doesn't make it any less so. It requires assuming infinite naturals exist and are required to prove that they exist and are required. Without that assumption, there is no need for them at all. The Peano properties do very nicely without them. <84c62$42df5560$82a1e3ad$10805@news1.tudelft.nl> <988f4$42e63112$82a1e3ad$21199@news1.tudelft.nl> <85wtndllpr.fsf@lola.goethe.zz> <85hdehlixy.fsf@lola.goethe.zz> <853bq0gju4.fsf@lola.goethe.zz> > It requires assuming infinite naturals exist and are required to prove > that they exist and are required. Without that assumption, there is no > need for them at all. The Peano properties do very nicely without them. It refers to the axiom of infinity. karl m === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? > String Theory is missing all of this. > > Srting Theory DOES NOT predict the stand model a priori. > > String Theory is reverse-engineered, and still there are these silly > little details which prevent it from coming close to the Standard > Model. So was QM and many other things in physics - it in no way changes the fact it has predictions or postulates. But even if you believe it was reverse engineered to fit the standard model it was not reverse engineered to predict gravitons. That it made this prediction without it being deigned to contain it was one of the reasons interest in it blossomed - along with the proof it was anomaly free. Come to think of it that is another prediction of the theory - it is anomaly free. > > This is well know. > > Moving Dimensions Theory at least has a postulate: Moving dimensions in total nonsense as has been demonstrated many times on sci.physics.relativity. You would be better off addressing the problems with your ideas than trying to find them in legitimate ones. Bill > > The fourth dimesnion is expanding relative to the three stationary > spatial dimensions. > > http://physicsmathforums.com > === Subject: Re: Update: Objections to Cantor's Theory > Natural numbers are based on the contradiction between ZERO and ONE. > They came from nature. You begin to teach mathematics to babies by > teaching the difference between having ZERO and having it all. It sounds as if you have never raised children. I have raised four of my own, and I started their understanding of numbers by counting things. The idea of zero comes along much later than the notion of starting with one thing and going to the next and the next. The first kinds of numbers my kids learned were ordinals, not cardinals. Bob Kolker === Subject: Re: Update: Objections to Cantor's Theory <8764v3a2jb.fsf@phiwumbda.org> <69133$42e5f433$82a1e3ad$16995@news1.tudelft.nl> > Natural numbers are based on the contradiction between ZERO and ONE. > They came from nature. You begin to teach mathematics to babies by > teaching the difference between having ZERO and having it all. > > It sounds as if you have never raised children. I have raised four of my > own, and I started their understanding of numbers by counting things. > The idea of zero comes along much later than the notion of starting with > one thing and going to the next and the next. The first kinds of numbers > my kids learned were ordinals, not cardinals. What makes you so sure we're dealing with children end of things and not the dimentia end? karl m === Subject: Re: Update: Objections to Cantor's Theory What makes you so sure we're dealing with children end of things and > not the dimentia end? karl m That is easy. All my kids were accomplished counters by not later than 18 months and they all learned how to add and read by three years. So tell us sport, how many kids have you raised? Bob Kolker > === Subject: Re: Update: Objections to Cantor's Theory <8764v3a2jb.fsf@phiwumbda.org> <%7dFe.179715$x96.1139@attbi_s72> <8bydnSZC_YE1YnjfRVn-3w@comcast.com> > The validity of a bijection for infinite sets is not contained in any > of the ZFC-axioms. 'The validity of a bijection for infinite sets' doesn't refer to anything. Bijections aren't things that are valid or not valid. Bijections are functions, not sentences. MoeBlee === Subject: Re: Update: Objections to Cantor's Theory > > > Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. > > Is the set of integers finite or infinite? No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. According to what definition in what axiom system do sets or their members have to conform to any physical standards? === Subject: Re: Update: Objections to Cantor's Theory > There is much ambiguity in translating the meaning of such and such > idea into formal language. It is ridiculous to believe that a computer > can verify Cantor's proofs in any objective way, i.e., independent of > the intentions of its instructor. I don't claim that translating intuitive concepts into formalism is algorithmic. And I don't know anyone in set theory who has ever claimed such a thing. But proof checking can be algorithmic. And in basic set theory, including the basic uncountability results, the chain of proofs and definitions leading to the theorems is not so long that a human being can't check, line by line, symbol by symbol if required, that the proof is correct. > I do not object to axioms. I do object to the introduction of ad hoc > notions. The notion of function and, in particular, bijection is not > based upon any axioms. I addressed your concern about ad hoc in my previous reply to you. Also, bijection is rigorously defined in the context of primitives and axioms. > Which axiom gives meaning to function? The axioms combined (not even including the axiom of infinity) provide for a defintion of 'function'. Consult any textbook on the subject to see how this is done. It's admirably clever. > The intuition of mathematicians leads to the interpretation that > equivalence is established by a bijection, even for infinite sets. Why? This is discussed in almost any textbook on the subject. Bijection puts things in 1-1 correspondence. Putting things in 1-1 correspondence is ubiquitous in everyday mathematical activity and thought. So, for infinite sets, 1-1 correspondence seems as good a candidate as any to give a formal defintion that reflects the concept of 'same size'. If claims are made that for infinite sets this has philosophical or physical implications, then you may dispute such claims. But that's not the sense in which mathematicians usually understand or use the definition, which ultimately reverts to a formal language. And if you find none of this agreeable to you, then you are welcome to construct an alternative theory. You've mentioned matters that are discussed in many textbooks on the subject. These are basics of which one can inform himself simply by picking up a book on the subject. MoeBlee === Subject: Re: Update: Objections to Cantor's Theory <3kmq5pFv8gm0U1@individual.net> <3knfrmFv4gfkU2@individual.net> <3ks78uFvhhrvU1@individual.net> > > Between every pair of irrational numbers there is a rational umber. > This means that in fact there are not more irrationals than rationals. > > Would you mind filling in the argument, for those of us > who have inadequate brains to do it for ourselves? > > If these brains do not immediately grasp the meaning, then, I am > afraid, there is no hope that elaborate explanations will be of much > help. Nevertheless I will try my best: > > If there were more irrationals than rationals in a linear order, then > at least two irrationals must exists without a rational between them. > > That is an assumption not supported by any facts. > > So while it may be aaxiom in WM's axiomatic system, it need not be, and > most often is not, an axiom in anyone else's. Here's the definition of EQUIVOCATE that you may want to file away to use next time this STATE reoccurs: E*quivo*cate (?), v. i. [imp. & p. p. Equivocated (?); p. pr. & vb. n. Equivocating.] [L. aequivocatus, p. p. of aequivocari to be called by the same name, fr. L. aequivocus: cf. F. .8equivoquer. See Equivocal, a.] To use words of equivocal or doubtful signification; to express one's opinions in terms which admit of different senses, with intent to deceive; to use ambiguous expressions with a view to mislead; as, to equivocate is the work of duplicity. karl m === Subject: Re: Update: Objections to Cantor's Theory > >Then pray tell what the 'true' definition of more is. >What does it mean for a set to have more elements than >another set? > > Apparently it depends on one's profession > > Are there professions other than mathematics where the concept > of comparing infinite set sizes come up? Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. Non-responsive! Does WM suggest the existence of a PROFESSION which COMPARES infinite sizes, other than mathematics? === Subject: Re: Update: Objections to Cantor's Theory > >>Then pray tell what the 'true' definition of more is. >>What does it mean for a set to have more elements than >>another set? > > Apparently it depends on one's profession but since mathematicians > make up a minority of all people, its silly to think that any meaning > they make up should be accepted by all people. Undoubtedly when > non-mathematicians are called cranks because they disagree with > arcane, and admittedly (by mathematicians) informal meanings of > common words, then mathematicians are the people that are wrong. > > You still have not said what it means for a set to have > more elements than another set. It should be an easy > question. However I suspect you are just trolling. > Cardinality is that general property which by means of our general > capability of thinking can be attached to a set if the order of its > elements and their properties are not taken into account. (Cantor, > Collected works p. 282, my translation) This capability of thinking may be applied in the following form: > Given two infinite sets A and B with elements a e A and b e B. The > union of these sets does exist. If the elements can be put into an > order < (not necessarily a well-order) such that in this order > 1) there are all elements a e A and b e B > 2) there are never two elements b,b' e B without an element a e A > between them with respect to < > then the cardinality Card(B) of B is not larger than the cardinality > Card(A) of A: > Card(B) =< Card(A). This axiom, and it is no more that an assumption made without factual support, is not supported by anything except WM's faith in it. Wm cannot prove it except by assuming either it or some equivalent statement. Since WM elsewhere objects to allowing such assumptions, he violates his own principles to propose it. === Subject: Re: volume and weight >> >> Think you'll still be around to gloat then? >> >> Don >> > > Why do you think I am gloating? That would imply I am correct and you > are wrong. However, no I don't think I will be around after you - > unless you are older than your posts make you seem. lot of folks think that the next best thing to knowledge, is knowing where to find it: After seventy-six years in the big parade of life, I've learned a few important things: One is that everything isn't all written down and what is isn't necessarily all right. Assuming there must be some confusion over units (often a problem for Shead), I sought to clarify the issue, but without success. So he is now probably either 75 or 77. I think Shead imagines that the older he gets, the more mass his arguments carry. === Subject: Re: volume and weight >>> >>> Think you'll still be around to gloat then? >>> >>> Don >>> >> >> Why do you think I am gloating? That would imply I am correct and you >> are wrong. However, no I don't think I will be around after you - >> unless you are older than your posts make you seem. > > of folks think that the next best thing to knowledge, is knowing > where to find it: After seventy-six years in the big parade of life, I've > learned a few important things: One > is that everything isn't all written down and what is isn't necessarily > all right. > > Assuming there must be some confusion over units (often a problem for > Shead), I sought to clarify the issue, but without success. > > So he is now probably either 75 or 77. I think Shead imagines that the > older he gets, the more mass his arguments carry. Very reassuring. Now I know I will still be around to gloat when his crackpot approach to mass/weight dies out. === Subject: Re: volume and weight <42e952dd$0$6484$cc9e4d1f@news-text.dial.pipex.com> >> >> Think you'll still be around to gloat then? >> >> Don >> > > Why do you think I am gloating? That would imply I am correct and you > are wrong. However, no I don't think I will be around after you - > unless you are older than your posts make you seem. > > lot of folks think that the next best thing to knowledge, is knowing > where to find it: After seventy-six years in the big parade of life, > I've learned a few important things: One > is that everything isn't all written down and what is isn't > necessarily all right. > > Assuming there must be some confusion over units (often a problem for > Shead), I sought to clarify the issue, but without success. > > So he is now probably either 75 or 77. I think Shead imagines that the > older he gets, the more mass his arguments carry. I was born on Nov. 14, 1927 and have acquired a lot of wisdom since; most of it is gone. but not completely forgotten, just harder to recall. What are your problems? Like they say too soon old, and too late smart;^! You'd better get a wiggle on if you expect to 'ketchup'. Quit wasting your time; it's the only time you'll get. Don === Subject: Re: Groups > Could someone please supply the definition of global group and p-local group. I've seen them mentioned alot, but never defined. Jim Where have you seen these before specifically? === Subject: Determining weight An object's weight (w) is the net force (f) that it exerts on a weight-scale or other support, divided by the acceleration of free fall (g) at that location: Mathematically it can be shown as: Weight=f/a/g; or more concisely as w=(fg/a). NOT as w=mg, because m=w/g=f/a. A quick, easy way to determine an object's weight is with a spring scale like those in the produce section of a grocery market, or a steelyard type of scale like doctors use; if they are made, and calibrated to show weight; which is the force due to gravity at Earth's surface. Put your old balance scales away with all of your other antiques. While you are at it, what _is_ the volume and weight of a slug of water at 39.2 degrees F, and atmospheric (sea level) pressure? Don === Subject: Re: Determining weight > An object's weight (w) is the net force (f) that it exerts on a > weight-scale or other support, divided by the acceleration of free fall > (g) at that location... Not in the lingo of physics and mathematics, Shead! Weight http://scienceworld.wolfram.com/physics/Weight.html Inertia http://scienceworld.wolfram.com/physics/Inertia.html The resistance to change in state of motion which all matter exhibits. It's a concept, Shead, not a number with units, not a ratio. Newton's First Law http://scienceworld.wolfram.com/physics/NewtonsFirstLaw.html Also called the law of inertia, Newton's first law states that a body at rest remains at rest and a body in motion continues to move at a constant velocity unless acted upon by an external force. Newton's Second Law is about inertial mass http://scienceworld.wolfram.com/physics/NewtonsSecondLaw.html A force F acting on a body gives it an acceleration a which is in the direction of the force and has magnitude inversely proportional to the mass m of the body: F = ma Inertia is an intrinsic property of mass. Most of what follows is quoted from http://www.physlink.com/ae305.cfm Gravitational Mass F = GmM/r^2 Inertial Mass F = ma Acceleration a = dv/dt 1) Inertial mass. This is mainly defined by Newton's law, the all-too-famous F = ma, which states that when a force F is applied to an object, it will accelerate proportionally, and that constant of proportion is the mass of that object. In very concrete terms, to determine the inertial mass, you apply a force of F Newtons to an object, measure the acceleration in m/s^2, and F/a will give you the inertial mass m in kilograms. 2) Gravitational mass. This is defined by the force of gravitation, which states that there is a gravitational force between any pair of objects, which is given by F = G m1 m2/r^2 where G is the universal gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them. This, in effect defines the gravitational mass of an object. As it turns out, these two masses are equal to each other as far as we can measure. Also, the equivalence of these two masses is why all objects fall at the same rate on earth. The only difference that we can find between inertial and gravitational mass that we can find is the method. Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass. This is typically done with some sort of balance scale. The beauty of this method is that no matter where, or what planet, you are, the masses will always balance out because the gravitational acceleration on each object will be the same. This does break down near supermassive objects such as black holes and neutron stars due to the high gradient of the gravitational field around such objects. Inertial mass is found by applying a known force to an unknown mass, measuring the acceleration, and applying Newton's Second Law, m = F/a. This gives as accurate a value for mass as the accuracy of your measurements. When the astronauts need to be weighed in outer space, they actually find their inertial mass in a special chair. The interesting thing is that, physically, no difference has been found between gravitational and inertial mass. Many experiments have been performed to check the values and the experiments always agree to within the margin of error for the experiment. Einstein used the fact that gravitational and inertial mass were equal to begin his Theory of General Relativity in which he postulated that gravitational mass was the same as inertial mass and that the acceleration of gravity is a result of a valley or slope in the space-time continuum that masses fell down much as pennies spiral around a hole in the common donation toy at your favorite chain store. Useful references for Shead http://scienceworld.wolfram.com/physics/Inertia.html http://scienceworld.wolfram.com/physics/MomentofInertia.html http://scienceworld.wolfram.com/physics/Mass.html http://scienceworld.wolfram.com/physics/Momentum.html http://scienceworld.wolfram.com/physics/NewtonsLaws.html http://scienceworld.wolfram.com/physics/Weight.html === Subject: Re: My progress on expression simplifier > >>But if it's for mathematicians, who expect sqrt(2) to be a single >>>positive number, >> >>They do? Are you speaking for ALL mathematicians? > Yes. I'm not one myself but even I know what sqrt(2) is. > You mean, Like I'm not a doctor but I play one on TV? http://everything2.com/index.pl?node_id=1377058 > What are you talking about? _You_ used the word positive and I > pointed out to you that there no such positive number. The point, as you probably realize, is that you cannot choose one root by choosing the positive one, because sometimes all the roots are complex. You might also look up sarcasm in a dictionary. To reiterate: any program that determines a single value for a radical, as you seem to suggest, by choosing the one that is positive, is doomed when problems get just slightly more general. > As Christopher Creutzig pointed out, you CAN pick one of the roots in the complex plane as a representative. Yet it would generally be a mistake to go to far with that. Most CAS now have an expression which looks like RootOf(x^8-1=0, ) > Try, for example, r=sin(pi/8)+i*cos(pi/8). (i = sqrt(-1). >>It doesn't matter which sqrt of -1, by the way...) Raise r to the 8th power >>and see if you can get -1. See if you can figure out the other >>7 roots, > Why don't _you_ do that, and tell which of the eight numbers is > positive? I don't need to. I have a CAS that gets right answers for such problems . Maybe yours doesn't, because it doesn't care? :) RJF PS. This is a serious issue in CAS design; being flippant about it may be amusing, but there are numerous papers demonstrating that you can get into serious trouble if you make wrong choices, and that making the right choice sometimes requires fairly deep analysis. > === Subject: Re: Vertex Coloring Planar Graphs > > Another thread to file under Cranks -- 4CT (Alleged Proofs Thereof) > > Maybe, but I think it's a little bit early to say this. > Signs of incipient crankitude are present, but no proof > positive yet. Yes, I may have jumped the gun on this one. === Subject: Re: Vertex Coloring Planar Graphs > [...] > I do not understand your example! What is the point of coloring C5 > (with v removed)? The point is that, in order to color a graph, you can't just look at a vertex and its neighbors; a graph coloring is a global thing which can't be done locally. > Tell me something more about arXiv and the false proof of the 4CT? I'd rather not. But the author of the false proof is Lorenz Friess, someone who should know better because he's worked with graph coloring before. (I'm currently translating a paper of his from German to English which gives a lower bound on the number of colorings of a planar graph with every face except for one a triangle.) > The point is: > > Every vertex in a maximal planar graph is surrounded by a cycle graph. > Since a vertex may be adjacent only to the vertices of its cycle graph: > the only way that a vertex can be adjacent to four different colors is > if its cycle graph requires four colors. And there is no way to force > a cycle graph to need four colors. > > The flaw in your reasoning is: The cycle itself won't force more than 4 > colors, but the rest of G may! > > A clear example of this principle is to consider a pentagon C5 (5 > vertices in a row). If you look at an arbitrary vertex v, its neighbors > are non-adjacent. So that means you can color them the same color, then > color v with a second color. It thus appears you've managed to 2-color > C5. > > However, if you look beyond the neighbors of v, you'll see that, in > order to color C5 (with v removed), the two neighbors of v have to > receive _different_ colors. > > Don't worry; you're not the only person to have made this mistake in > the last year and submit a paper. But at least your mistake isn't at > arXiv. > > > Specifically, G is a simple loopless maximal planar graph. > > PA is a (theoretically) perfect vertex coloring algorithm. If Ch(G) > = 4, then PA will always achieve a 4-coloring. If PA cannot achieve > a 4-coloring, then Chi(G) = 5. > > Or is at least 5. > > Of course, PA will always achieve a coloring with 4 or fewer colors, > by the Four Color Theorem. But perhaps what you are doing is attempting > to prove the 4CT? > > PA operates on a simple principle. There is a pallete of four colors. > If possible, a vertex will be colored with a preselected color. If and > only if, > a preselected color cannot be assigned to a vertex, then that vertex > will be colored 'O'! If G has at least one 'O' vertex, then Chi(G) = > 5. > > Hypothesis; Every 'O' vertex in G will be adjacent to exactly four > different colors! > > [1] And possibly other 'O' vertices. > > [2] However, you may be able to color an 'O' vertex v by recoloring > part of the graph G, using a technique like Kempe chains, which results > in the neighbors of v having at most 3 colors, whereupon you can color > v. So, in short, your Primitive Algorithm might not find a 4-coloring > that exists, because it processed the vertices in the wrong order. > > Well, I guess. A vertex can't be adjacent to vertices with more than > four different colors, as there are only four different colors > (unless you count O as a color), and if a vertex is adjacent > to vertices with three or fewer colors, you can give it a color > from the 4 preselected colors. > > What's your point? > > Another thread to file under Cranks -- 4CT (Alleged Proofs Thereof) > > === Subject: Re: Vertex Coloring Planar Graphs > I think that Proginoskes was referring to me as a candidate for > crankdom. Or were you standing up for me? > > I am only trying to get some insights into the coloring of planar > graphs. > Is that so 'cranky'? Not yet. After all, you're not _insisting_ that your short proof is correct. The general argument against a short proof is that if there was a short proof of the 4CT, then someone would have found it by now. (The problem was introduced officially in 1853.)