mm-378 === Subject: :-0n MATH game theory in the sense of economics at all, but a book that deals with > actual games, and concrete mathematics. You could try Winning Ways, by Berlekamp, Conway, and Guy.-- === === Subject: : Re: Request for comments on antiquated algebraic topology online-book> Well I've pretty much decided not to invest too much time with Lefschetz.> However, initially, I only need to learn a small subset of algebraic> topology (AT). Specifically, the same topics as found in the first chapter> of Munkres' 'Elements of AT'. The application is to distributed computing> where the evolution of finite asynchronous distributed protocols are modeled> as high dimensional simplicial complexes (if thats the correct terminology).> combinatorial. The seminal explanatory paper on the approach can be found> at:> http://www.cs.brown.edu/people/mph/HerlihyR96/sv.pdfThat's an interesting paper. I suppose the explanations are too brieffor you? If you read that paper's explanations on topology and combineit with looking at things online like Wikipedia, I bet you'll bealright. See below also.> I have taken another look at Hatcher, and you're right - it doesn't seem> nearly as intimidating as it did before I bulked up on general topology a> bit. Hopefully I won't need to shell out the cash for Munkres' book.I think for you, it would be worth looking at the first chapter ofMunkres. Just check it out from your local university/college library. There is no need to pay up all that money for something you only need apiece of.=== === Subject: : Re: Integrals without explicit formulas> I am in a differential calculus class. Just about all the problems> involve integrals, but I have been informed that some of the integrals> in these problems have no explicit formulas. Can you give me any rules> of thumb so I know what kind of integrals to not bother trying to> integrate?Instead of no explicit formula I would say no closed form expression in elementary functions, but I think we're talking about the same thing. I don't think there are any rules of thumb for what you want. Functions that have no elementary antiderivative can look a lot like functions that do. e^(x^2) has none, xe^(x^2) does. log x does, 1 / (log x) doesn't, 1 / (x log x) does, x / (log x) doesn't, (log x) / x does. There is a way to work out which ones do & which ones don't, but it's a bit beyond the level of an intro calculus course.-- === === Subject: : Generalization of Cauchy's Functional EquationThe continuous solution of Cauchy's functional equation,f(x+y) = f(x)+f(y)is f(x) = cx. Then what is the solution of the generalized Cauchy'sfunctional equation such as,f(x+y) = f(x)+f(y)+a{f(x)f(y)}^2 ?I tried to solve the generalized Cauchy's functional equation in thefollowing site:http://139.134.5.123/tiddler2/cauchy/ cauchyequation.htmand also tried to simplify the addition theorem for Weierstrass ellipticfunction.=== === Subject: : Re: Metamath Axiom of Choice>> ...................>>As I stated above, the Metamath formulation of the Axiom of Choice is >effectively the statement that given any set x, there exists a set y >>such that for all nonempty elements w of x, w n (Us) has exactly one >>element, where s = {t in y : w in t}. >>As stated, it is false. Add the elements of x are disjoint>>and it is one of the standard forms.>>To see that it is false as stated, let x have three elements, >>{1, 2}, {1, 3}, {2, 3}. Any set intersecting all of these in>>at least one element has to contain both elements of one.>But in the statement above, s is explicitly dependent on w (the >is not the existence of a single set with one element in common>with every nonempty element.In view of possible misinterpretations of the Metamath statement of the Axiom of Choice, the statement is effectively the statement thatfor all sets x, there exists a set y such that for all elements w of x, if w is nonempty, then w n (U{t in y : w in t}) has exactly one element. At the moment, they (the Time Lords) are far from being all-powerful. That's why it's been left up to me and me and me. quote by: Patrick Troughton in The Three Doctors-------=== === Subject: : Re: looking for an eltry soln to an old problem>>> A long time ago, J.J. Sylvester posed the problem:>> if I have arbitrarily many 5 cent or 17 cent stamps,>> what is the largest denomination I cannot make?>>> In general, if we have p and q cent stamps, it turns>> out the answer is pq-p-q (granted p and q are coprime).>> I have derived a solution to the problem, but I'd like>> to teach this to my undergraduates, some of whom have a>> limited background. So my question is: is there a very nice>> & friendly proof of this fact? (For example, which avoids>> any nonobvious facts from number theory.)> You need a couple of not-quite-obvious facts.> Fact 1: if x = a p + b q, then x = (a + n q) p + (b - n p) q for> any integer n, and these are all the ways of writing x as a> multiple of p plus a multiple of q.> Fact 2: if p and q are coprime, every integer x can be written as> x = a p + b q for some (not necessarily positive) integers a,b.> This can be done using the Euclidean algorithm; or the fact that> t -> t p mod q is one-to-one on the integers in {1,...,q-1} coprime> to q, plus the pigeonhole principle.> Suppose for some x it can't be done with a, b >= 0. If> x = a p + b q is one representation with integers a,b, then> for each n we have either a + n q < 0 or b - n p < 0.> Take a' to be the largest a + n q < 0, so -q <= a' < 0 and> x = a' p + b' q with b' - p < 0.> But then x <= -p + (p-1) q = pq - p - q.> On the other hand, pq - p - q = ap + bq with a=q-1 and b=-1;> a+nq < 0 if n < 0 and b-np < 0 if n >=0.Clearer: we may normalize N = P X + Q Y so 0 <= X < Q by adding a certain integral multiple of (-Q,P) to (X,Y)CLAIM: N = P X + Q Y for some integers X and Y >= 0iff its normalization has Y >= 0. PROOF: X and Y >= 0implies normalization requires addition of (-Q,P) zeroor more times, and this preserves the condition Y >= 0.Conversely if the normalization has Y < 0 then N has norepresentation with X and Y >= 0, because to shift Y > 0requires adding (-Q,P) at least once, which shifts X < 0.Finally, since X P + Y Q is increasing in both X and Y,it is clear that the largest non-representable number Nhas normalization (X,Y) = (Q-1,-1), so N = PQ - P - Q.Notice that the proof has a vivid geometric picture: representations of N correspond to lattice points (X,Y) on the line N = P X + Q Y with negative slope = -P/Q.Normalization is achieved by shifting forward/backwardalong the line by integral multiples of vector (-Q,P)until you land in the normal strip where 0 <= X < Q-1.>> In case you're interested, the class I'm teaching>> is linear algebra, but as you can see I like to >> give puzzles to the class which are not necessarily>> related to the material (in an obvious way).Here the underlying linear structure is a Z-module,a generalization of a vector space; i.e. here the scalars are the integers so have only the structure of a ring, not a field. Unless you've already taughtsome module theory, it might be tricky to preciselyexplain the relationship to vector space theory.Finally it should be mentioned that there has been muchwritten on this classical problem. To locate such workyou should ensure that you search on the many aliases,e.g. postage stamp problem, Sylvester/Frobenius problem,Diophantine problem of Frobenius, Frobenius conductor,money changing, coin changing, change making problems,h-basis and asymptotic bases in additive number theory,integer programming algorithms and Gomory cuts,knapsack problems and greedy algorithms, etc.=== === Subject: : Re: intercept length when a random line intersect with ellipsoid The general question is that if a infinite plane with any orientationintersect with a polyconvex shell with the fixed thickness(A), thethickness of intersecting part is larger than A. So I want to know howmuch is the average thickness of the intersecting part from theviewpoint of probability? Since I'm not sure the problem can be sovledfor any polyconvex shell. So I select the simple shape, like ellipsoidshell or orthogonal parallelepiped shell with a fixed thickness.Now I want to know, do we talk the same question?Thank you for your kind reply again!CHEN Huisu> In order to answer your question, you need to be more specific aboutrandom for the line. A line in space can be specified as all point> P satisfying P=X+tV, where X is a random point in space (needing a> probability distribution) and V is a random unit vector (isotropic?> distribution), while t ranges over the entire real line.> Once these distributions (for X and V) and the parameters of the shell> are defined, you will have a tractable porblem.> Posted Via Usenet.com Premium Usenet Newsgroup Services > http://www.usenet.com=== === Subject: : Polynomoa; solutions to Pell eqnThe Pell equation and the obvious cubic generalisationX^2 - DY^2 = 1 andX^3 + DY^3 +(D^2)Z^3 - 3DXYZ = 1have simple polynomial solutions(n,1) for D = n^2 1(n^2, n, 1) for D = n^31In the square case, you can use continuedfractions to expand expression of the form, say,SQRT(an^2 + bn + c), but in the cubic caseno such option is available.Simple cases can be guessedfor exampleD = n^3 3X = n^6 3n^3 1Y = n^5 2n^2X = n^4 nD = n^3 +2, n evenX = (9n^6)/4 + (9n^3)/2 +1Y = (9n^5)/4 + 3n^2Z = 3n(3n^3 +2)/4I was wondering if there is anysystematic approach to obtainingsolutions ?=== === Subject: : Re: Polynomial solutions to Pell eqn>The Pell equation and the obvious cubic generalisation>X^2 - DY^2 = 1 and>X^3 + DY^3 +(D^2)Z^3 - 3DXYZ = 1>have simple polynomial solutions>(n,1) for D = n^2 1>(n^2, n, 1) for D = n^31>In the square case, you can use continued>fractions to expand expression of the form, say,>SQRT(an^2 + bn + c), but in the cubic case>no such option is available.>Simple cases can be guessed>for example>D = n^3 3>X = n^6 3n^3 1>Y = n^5 2n^2>X = n^4 n>D = n^3 +2, n even>X = (9n^6)/4 + (9n^3)/2 +1>Y = (9n^5)/4 + 3n^2>Z = 3n(3n^3 +2)/4>I was wondering if there is any>systematic approach to obtaining>solutions ?Not real sure about a systematic approach, but looking at a few data points forspecific D may be useful. Consider the following:D=10^3+5*10 X=62591931727331611501 D=100^3+5*100X=4910017588770392230083131681058781350001 D=1000^3+5*1000X= 489788270129887199471510415004171147924480921387800135000001 D=10000^3+5*1000000000001 D=100000^3+5*100000X= 489776026824440065292464512748268928258240960054041644806667920 0004480920000138780000001350000000001 D=1000000^3+5*1000000X= 489776025612244400640129246451200748268928002582409600005404164 480006667920000004480920000001387800000000135000000000001 D=10000000^3+5*10000000X= 489776025600122444006400012924645120000748268928000025824096000 000540416448000006667920000000044809200000000138780000000000135 00000000000001 D=100000000^3+5*100000000X= 489776025600001224440064000001292464512000000748268928000000258 240960000000054041644800000006667920000000000448092000000000013 8780000000000001350000000000000001 There is a pattern here that seems to indicate that a subset of {n^3+5*n} has apolynomial solution to P3. I'm not sure what the exact answer is, but it couldprobably be worked out, given sufficient time and interest. There are lots ofother curiosities that can be found this way.=== === Subject: : Re: Polynomial solutions to Pell eqn>The Pell equation and the obvious cubic generalisation>X^2 - DY^2 = 1 and>X^3 + DY^3 +(D^2)Z^3 - 3DXYZ = 1>have simple polynomial solutions>(n,1) for D = n^2 1>(n^2, n, 1) for D = n^31>In the square case, you can use continued>fractions to expand expression of the form, say,>SQRT(an^2 + bn + c), but in the cubic case>no such option is available.>Simple cases can be guessed>for example>D = n^3 3>X = n^6 3n^3 1>Y = n^5 2n^2>X = n^4 n>D = n^3 +2, n even>X = (9n^6)/4 + (9n^3)/2 +1>Y = (9n^5)/4 + 3n^2>Z = 3n(3n^3 +2)/4>I was wondering if there is any>systematic approach to obtaining>solutions ?Not real sure about a systematic approach, but looking at a few data points forspecific D may be useful. Consider the following:D=10^3+5*10 X=62591931727331611501 D=100^3+5*100X=4910017588770392230083131681058781350001 D=1000^3+5*1000X= 489788270129887199471510415004171147924480921387800135000001 D=10000^3+5*1000000000001 D=100000^3+5*100000X= 489776026824440065292464512748268928258240960054041644806667920 0004480920000138780000001350000000001 D=1000000^3+5*1000000X= 489776025612244400640129246451200748268928002582409600005404164 480006667920000004480920000001387800000000135000000000001 D=10000000^3+5*10000000X= 489776025600122444006400012924645120000748268928000025824096000 000540416448000006667920000000044809200000000138780000000000135 00000000000001 D=100000000^3+5*100000000X= 489776025600001224440064000001292464512000000748268928000000258 240960000000054041644800000006667920000000000448092000000000013 8780000000000001350000000000000001 There is a pattern here that seems to indicate that a subset of {n^3+5*n} has apolynomial solution to P3. I'm not sure what the exact answer is, but it couldprobably be worked out, given sufficient time and interest. There are lots ofother curiosities that can be found this way.=== === Subject: : Axioms defining a finite fieldLet (F, +, *) be a finite set with two operations and constants 0, 1such that the following rules hold:(1) a + (b + c) = (a + b) + c(2) a + 0 = a(3) for every a there's a b so that a + b = 0(4) a*(b*c) = (a*b)*c(5) a*1 = a(6) 1 is distinct from 0(7) a*(b + c) = a*b + a*c(8) (a + b)*c = a*c + b*c(9) a*b = 0 => a=0 or b=0Show that F is a field. Can one of the rules be omitted so that Fstill has to be a field? Posted Via Usenet.com Premium Usenet Newsgroup Services------------------------------------------------------ ---- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY **---------------------------------------------------------- http://www.usenet.com=== === Subject: : Re: Axioms defining a finite field=== === Subject: : Axioms defining a finite field >Let (F, +, *) be a finite set with two operations >and constants 0, 1 such that the following rules hold: >(1) a + (b + c) = (a + b) + c >(2) a + 0 = a >(3) for every a there's a b so that a + b = 0 >(4) a*(b*c) = (a*b)*c >(5) a*1 = a >(6) 1 is distinct from 0 >(7) a*(b + c) = a*b + a*c >(8) (a + b)*c = a*c + b*c >(9) a*b = 0 => a=0 or b=0 >Show that F is a field. Can one of the rules be omitted >so that F still has to be a field?let a /= 0, b0 = 0, b1 = 1A = { a bj | 0 <= j <= |F| }|A| = |F| because if a bj = a bk: a(bj - bk) = 0; bj - bk = 0; bj = bkAs F is finite, 1 in A which shows a has right multiplicative inverse a_r.This with a1 = a, shows F0 is a group under *.Did you forget a+b = b+a, ab = ba?----=== === Subject: : Re: Axioms defining a finite field===> === Subject: : Axioms defining a finite field>Let (F, +, *) be a finite set with two operations>and constants 0, 1 such that the following rules hold:>(1) a + (b + c) = (a + b) + c>(2) a + 0 = a>(3) for every a there's a b so that a + b = 0>(4) a*(b*c) = (a*b)*c>(5) a*1 = a>(6) 1 is distinct from 0>(7) a*(b + c) = a*b + a*c>(8) (a + b)*c = a*c + b*c>(9) a*b = 0 => a=0 or b=0>Show that F is a field. Can one of the rules be omitted>so that F still has to be a field?>let a /= 0, b0 = 0, b1 = 1>A = { a bj | 0 <= j <= |F| }>|A| = |F| because> if a bj = a bk: a(bj - bk) = 0; bj - bk = 0; bj = bk>As F is finite, 1 in A> which shows a has right multiplicative inverse a_r.>This with a1 = a, shows F0 is a group under *.>Did you forget a+b = b+a, ab = ba?(1) - (3) imply F is a group under +Next show a*0 = 0 for all a. This follows as usual from a*0 = a*(0 + 0).Then from (9) and (7), you can deduce that, for a in F - {0}, a*b = a*cimplies b = c (just add a*(-c) to both sides), and hence, by finiteness,there exists a^-1 with a*a^-1 = 1. Then, from (4) and (5), F - {0} is a group under *.Now we can use (7) and (8) to deduce a + b = b + a:(1 + a)*(1 + b) = 1*(1 + b) + a*(1 + b) = 1 + b + a + a*b(1 + a)*(1 + b) = (1 + a)*1 + (1 + a)*b = 1 + a + b + a*bso a + b = b + a, and we have a division ring. Now, by Wedderburn's Theorem, which is indeed nontrivial, a finite division ring is a field.I would be surprised if any of these axioms could be omitted, but to showthat none of them could, you would have to construct 9 examples, each ofwhich satisfied all hypotheses but one!If this is homework, then it is hard!Derek Holt.=== === Subject: : Re: Axioms defining a finite field===> === Subject: : Axioms defining a finite field>Let (F, +, *) be a finite set with two operations>and constants 0, 1 such that the following rules hold:>(1) a + (b + c) = (a + b) + c>(2) a + 0 = a>(3) for every a there's a b so that a + b = 0>(4) a*(b*c) = (a*b)*c>(5) a*1 = a>(6) 1 is distinct from 0>(7) a*(b + c) = a*b + a*c>(8) (a + b)*c = a*c + b*c>(9) a*b = 0 => a=0 or b=0>Show that F is a field. Can one of the rules be omitted>so that F still has to be a field?>let a /= 0, b0 = 0, b1 = 1>A = { a bj | 0 <= j <= |F| }>|A| = |F| because> if a bj = a bk: a(bj - bk) = 0; bj - bk = 0; bj = bk>As F is finite, 1 in A> which shows a has right multiplicative inverse a_r.>This with a1 = a, shows F0 is a group under *.>Did you forget a+b = b+a, ab = ba?I don't know about the a+b = b+a, but if we add that thenab = ba follows: it's a theorem that every finite divisionring is a field. (It's not quite trivial, as I recall. Possiblyit actually is trivial and just didn't seem trivial to meat the time; that was an algebra class when I was anundergraduate...)************************=== === Subject: : Re: Axioms defining a finite field===>> === Subject: : Axioms defining a finite field> Let (F, +, *) be a finite set with two operations> and constants 0, 1 such that the following rules hold:> (1) a + (b + c) = (a + b) + c> (2) a + 0 = a> (3) for every a there's a b so that a + b = 0> (4) a*(b*c) = (a*b)*c> (5) a*1 = a> (6) 1 is distinct from 0> (7) a*(b + c) = a*b + a*c> (8) (a + b)*c = a*c + b*c> (9) a*b = 0 => a=0 or b=0> Show that F is a field. Can one of the rules be omitted> so that F still has to be a field?>> let a /= 0, b0 = 0, b1 = 1>> A = { a bj | 0 <= j <= |F| }> A| = |F| because>> if a bj = a bk: a(bj - bk) = 0; bj - bk = 0; bj = bk>> As F is finite, 1 in A>> which shows a has right multiplicative inverse a_r.>> This with a1 = a, shows F0 is a group under *.>> Did you forget a+b = b+a, ab = ba?> I don't know about the a+b = b+a, but if we add that then> ab = ba follows: it's a theorem that every finite division> ring is a field. (It's not quite trivial, as I recall. Possibly> it actually is trivial and just didn't seem trivial to me> at the time; that was an algebra class when I was an> undergraduate...)No, not trivial . This is Wedderburn's theorem; Proofs from the Book givesa nice (what else?) four page proof (chapter 5) due to Witt (and mention 7or 8 more, using quite different ideas). Witt begins by proving, usinglinear algebra, that the centraliser of s , is of dimension q^(n_s) , whereq is the characteristic of F (q.1=0) and that n_s divides |F|. Then, heimbeds F in the roots of unity in C, and conclude by a clever argument onthe cyclotomic polynomials...> ************************> === === Subject: : Re: Axioms defining a finite field> No, not trivial . This is Wedderburn's theorem; Proofs from the Book gives> a nice (what else?) four page proof (chapter 5) due to Witt (and mention 7> or 8 more, using quite different ideas). Witt begins by proving, using> linear algebra, that the centraliser of s , is of dimension q^(n_s) , where> q is the characteristic of F (q.1=0) and that n_s divides |F|. Then, he> imbeds F in the roots of unity in C, and conclude by a clever argument on> the cyclotomic polynomials...There are a couple of proofs at PlanetMath.org: http://planetmath.org/?op=getobj&from=objects&id=3627 http://planetmath.org/?op=getobj&from=objects&id=4198Neither seem to be credited, but the second one looks like it could be theWitt proof you describe, or something related to it. -- === === Subject: : Re: Axioms defining a finite field===> Subject: Axioms defining a finite field>> Let (F, +, *) be a finite set with two operations>> and constants 0, 1 such that the following rules hold:>> (1) a + (b + c) = (a + b) + c>> (2) a + 0 = a>> (3) for every a there's a b so that a + b = 0>> (4) a*(b*c) = (a*b)*c>> (5) a*1 = a>> (6) 1 is distinct from 0>> (7) a*(b + c) = a*b + a*c>> (8) (a + b)*c = a*c + b*c>> (9) a*b = 0 => a=0 or b=0>> Show that F is a field. Can one of the rules be omitted>> so that F still has to be a field?> let a /= 0, b0 = 0, b1 = 1> A = { a bj | 0 <= j <= |F| }>> A| = |F| because> if a bj = a bk: a(bj - bk) = 0; bj - bk = 0; bj = bk> As F is finite, 1 in A> which shows a has right multiplicative inverse a_r.> This with a1 = a, shows F0 is a group under *.> Did you forget a+b = b+a, ab = ba?>> I don't know about the a+b = b+a, but if we add that then>> ab = ba follows: it's a theorem that every finite division>> ring is a field. (It's not quite trivial, as I recall. Possibly>> it actually is trivial and just didn't seem trivial to me>> at the time; that was an algebra class when I was an>> undergraduate...)>No, not trivial . This is Wedderburn's theorem; Proofs from the Book gives>a nice (what else?) four page proof (chapter 5) due to Witt (and mention 7>or 8 more, using quite different ideas). Witt begins by proving, using>linear algebra, that the centraliser of s , is of dimension q^(n_s) , where>q is the characteristic of F (q.1=0) and that n_s divides |F|. Then, he>imbeds F in the roots of unity in C, and conclude by a clever argument on>the cyclotomic polynomials...You have any idea whether a + b = b + a follows from the conditionsabove?>> ************************>> ************************=== === Subject: : Re: Axioms defining a finite field Adjunct Assistant Professor at the University of Montana.> (1) a + (b + c) = (a + b) + c> (2) a + 0 = a> (3) for every a there's a b so that a + b = 0> (4) a*(b*c) = (a*b)*c> (5) a*1 = a> (6) 1 is distinct from 0> (7) a*(b + c) = a*b + a*c> (8) (a + b)*c = a*c + b*c> (9) a*b = 0 => a=0 or b=0>You have any idea whether a + b = b + a follows from the conditions>above?First, from (1,2,3) we can derive that 0+a=a+0=a for all a, and thatfor all a there exists b such that a+b=b+a=0, by the usual methodsvalid for groups. Then, by 1, addition is associative. Now considera + a + b + b = a*1 + a*1 + b*1 + b*1 (by 5) = a*(1+1) + b*(1+1) (by 7) = (a+b)*(1+1) (by 8) = (a+b)*1 + (a+b)*1 (by 7) = a + b + a + b (by 8)c+(a+a+b+b) + d = c+(a+b+a+b)+dhence(c+a) + (a+b) + (b+d) = (c+a)+(b+a)+(b+d) 0 + (a+b) + 0 = 0+(b+a)+0 a + b = b + a.Unless I've missed something...-- Arturo Magidinmagidin@math.berkeley.edu=== === Subject: : Re: Axioms defining a finite field> Unless I've missed something...Doesn't look like it...so does this mean that a+b=b+a doesn't need to be inthe field axioms?It is annoying enough that the group axioms usually have a*e=e*a=a anda*a'=a'*a=e, when you only need a*e=a and a*a'=e, and I've just barelymanaged to surpress my rage at that...this field thing is going to pushme over the edge. :-)-- === === Subject: : Re: Axioms defining a finite field>...so does this mean that a+b=b+a doesn't need to be in>the field axioms?>It is annoying enough that the group axioms usually have a*e=e*a=a and>a*a'=a'*a=e, when you only need a*e=a and a*a'=e, and I've just barely>managed to surpress my rage at that...this field thing is going to push>me over the edge. :-)Easy there, big fella! Take a breath and try to see the big picture.There is a certain delight in checking for independence of axioms, andcertainly there is elegance in minimal presentations of axiom systems.But that isn't the only possible goal for a set of axioms. Indeed, onecan trim down the set of axioms for a group to 1, I think, but thatone axiom is difficult to digest and certainly fails to convey thegestalt of what a group is.So too with the axioms for fields (or vector spaces, or ...). The pointof the mathematics is to study some interesting and useful things. Thoseinteresting and useful things are given a name, as soon as we seewhat characteristics make them similar. The point of the axioms is toestablish that definition in an unambiguous way so that results can beproved which apply to all these objects. Where is the harm in presentingthe axioms in a redundant way, if it helps provide a better picture ofthe objects? Doesn't that make it easier on you, the mathematician, asyou attempt to detect the interesting features of these things?My recommendation is to keep your interest in the independence of theaxioms but to store it in the back of your mind for those times whenyou find yourself stranded on a desert island with nothing else to doto occupy your time. In the mean time, get to work understanding theobjects being described, and prove theorems about them which the rest of us can use.dave=== === Subject: : Re: Axioms defining a finite field Adjunct Assistant Professor at the University of Montana.>> Unless I've missed something...>Doesn't look like it...so does this mean that a+b=b+a doesn't need to be in>the field axioms?It doesn't have to be in the ring axioms in the presence of a 1 anddistributivity (and on this, Jacobson's algebra book agrees with me atany rate); if you know that the group is a (not necessarily abelian)group under +, a semigroup with identity under *, and thata*(b+c)=(a*b)+(a*c), (a+b)*c = (a*c) + (b*c), then commutativity of +will follow.This sort of exercise often shows up in algebra books after theintroduction of rings, as a way of explaining why one does notconsider the more general structure where we don't requirecommutativity of addition...>It is annoying enough that the group axioms usually have a*e=e*a=a and>a*a'=a'*a=e, when you only need a*e=a and a*a'=e, and I've just barely>managed to surpress my rage at that...Yes, but on the other hand, if you have a*e = a for all a, and for allb there exists b' such that b'*b = e, then you don't necessarily havea group...-- Arturo Magidinmagidin@math.berkeley.edu=== === Subject: : Re: Axioms defining a finite fieldwhy not try and do your own homework?=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum> Before you lies the Void.To toss a chuck of reality into your cranial cave...> I want to address ontology.then get real. ;-)=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum>I want to address ontology.> then get real. ;-)There _is_ a problem there. Should it beMr Ontology, Mrs Ontology...?I was about to suggest Comrade Ontologybut... ah, the English speaker's lot isnot a happy one! If only we spoke Japanese: Ontorogii-san. Done!(Why Ontorogii and not Ontorojii?Dunno. Sounds better).=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum>I want to address ontology.>then get real. ;-)> There _is_ a problem there. Should it be Mr Ontology, Mrs> Ontology...? I was about to suggest Comrade Ontology but... ah, the> English speaker's lot is not a happy one! If only we spoke Japanese:> Ontorogii-san. Done! (Why Ontorogii and not Ontorojii? Dunno. Sounds> better).Riddle of the day: are ontologists for real?=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum> Riddle of the day: are ontologists for real?Ah.... ontology is about existence. L'.90tre et len.8eant (sounds much better, doesn't it?)Can something that does not exist be ontological?Does an ontologist need to exist in order to... er... be an ontologist? (Never mind what anontologist is, I have a hunch that you couldargue the same of a communist, a capitalist,a typist, a writer, a plumber, a feather).=== === Subject: : Re: the anticlassicalist }{ vi: into the quantumFor the term to exist to mean anything at all, there must besomething that doesn't exist.It does not matter how the one is distinguished from the other or whois doing the distinguishing; it does not matter where the line isdrawn between that which exists and that which doesn't exist, how itis drawn, who is drawing it, whether it is sharp or fuzzy.What matters is that for the phrase X exists to mean anything atall, there must be some entity Y for which it is valid to state Ydoes not exist.Otherwise the word exists in the phrase X exists would expressnothing whatsoever.> Ah.... ontology is about existence. Or (in light of the above): ontology is about that which isdistinguished from that which does not exist.> Can something that does not exist be ontological?It would have to - just like something that does. For a distinction to take place requires that there is more thanpossible outcome.> Does an ontologist need to exist in order to> ... er... be an ontologist? Q: What is red and invisible?A: No tomatoes.Q: But what if they're green and invisible? A: Then they aren't ripe yet.=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum> What matters is that for the phrase X exists to mean anything at> all, there must be some entity Y for which it is valid to state Y> does not exist.> Otherwise the word exists in the phrase X exists would express> nothing whatsoever.Replace exists by sings. Same story. By snores. Samestory. By lives. Same story. > Or (in light of the above): ontology is about that which is> distinguished from that which does not exist.Yes. Carminology is about that which is distinguished fromthat which does not sing. (I was going to write cantologyby I retracted that)> Q: What is red and invisible?The bottle caps of a six-pack of Cooper's Sparkling Ale in a closed fridge. If I hadstudied medicine I might have come up witha cuter example.=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum>Riddle of the day: are ontologists for real?> Ah.... ontology is about existence. L'tre et le> nant (sounds much better, doesn't it?)How am I to know? What's that capital sigma after ?> Can something that does not exist be ontological?Ya, I imagine so. How is real reality different than reality?> Does an ontologist need to exist in order to> ... er... be an ontologist?What if a thought was that no ontologist thought?> (Never mind what an ontologist is, I have a hunch that you could argue> the same of a communist, a capitalist, a typist, a writer, a plumber, a> feather).Does the corrupt president be need to be elected to be corrupt?Riddle of the day: what do ontologists imagine reality is?=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum>Riddle of the day: are ontologists for real?>Ah.... ontology is about existence. L'.90tre et le>n.8eant (sounds much better, doesn't it?)> How am I to know? What's that capital sigma after and the theta after ?A sign your newsreader isn't using the charset directions in theheaders correctly - the post was in iso-8859-1, and those were lettersfound in the Frenchy-French, into which Jacques suffers occasionalrelapses.[...]> Riddle of the day:> what do ontologists imagine reality is?An illusion. Lunch-reality doubly so.Deswill contemplate _l'.90tre de l'.8etant_ for food.-- [T]he structural trend in linguistics which took root with theInternational Congresses of the twenties and early thirties [...] hadclose and effective connections with phenomenology in its Husserlianand Hegelian versions. -- Roman Jakobson=== === Subject: : Re: the anticlassicalist }{ vi: into the quantumalt.philosophy,sci.lang,sci.logic,sci.math,sci.physics: [...]> Des> will contemplate _l'.90tre de l'.8etant_ for food.Raisins, no doubt.Brian=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum>Des>will contemplate _l'.90tre de l'.8etant_ for food.> Raisins, no doubt.No. There's a typo there. That should be l'.8etang.L'.90tre de l'.8etang. Some fish. Obvious. Unless...the Lady of the Lake? Des, a cannibal? Ah, il faut de tout pour faire un monde.=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum>Riddle of the day:> what do ontologists imagine reality is?> An illusion. Lunch-reality doubly so.> Des will contemplate _l'.90tre de l'.8etant_ for food.Before swallowing that check to see if existence of existence exists.[T]he structural trend in linguistics which took root with the> International Congresses of the twenties and early thirties [...] had> close and effective connections with phenomenology in its Husserlian and> Hegelian versions. -- Roman JakobsonPhenomenally unphenomenal phenomena.=== === Subject: : Re: the anticlassicalist }{ vi: into the quantum-=-=-=-=-=-= into the quantum =-=-=-=-=-=-=-Before you lies the Void. Some use the word vacuum here, but the vacuum ismore rigorously used as a term of a particular subspace structure I willdescribe. Some might call it space, but that too is structure, and I wantto address something prior to structure.I want to address ontology.These are the words we build our models of reality with.If I haven't made it clear enough yet, this is my most fundamental concern.When I think about the world, I want to be precise. I want to have a skillwith the concepts. I feel many others here feel exactly the same. We enjoylearning or have some drive to learn because all these groups discussknowledgable things.Maybe I am just naive and young, but I see very smart people, some who liketo show off, some who teach, an annoying gang of aggresive types, some veryhere are interested in such things. It is usenet among the crowd thatinterprets ontologies in a rigorous science of symbology.And you all have models, built on ontologies possessing vary many logics.We even use them in everday speech or natural language. We order ourspeech to be useful.The ontology is where we define the objects and their transformations. Itis where we define a language about which we talk.We build symbolic names for the patterns we recognise. These naturallyclassify themselves into objects and transformations.Let me introduce the ontology of the quantum.In models, we start with a structure over an ontology.A structure is: - a collection of names for objects :: both directly associated to existents and allowed :: :: to vary, freely and bound :: - a language of truth {quantifiers, (co)products, negation, implication,etc.} - a collection of operations taking collections of objects back into theobject domain - a collection of relations mapping collections of objects into the truthdomainMathematicians use structures all the time. Magmas, semigroups, groups,rings, fields, topologies, orders, etc. But it is very likely moreuniversal than that. As I've tried to show with the building of thecognitive maps, the ontology of natural language and our ability torecognise objects and transformations in the world abstractly through alanguage, can be structures as well.Our quantum ontology is built inside a structure known as a Hilbert space.I will not describe the full definition, but if you are familiar with linearspaces, like Euclidean space, studying Hilbert spaces is very much the studyof what generalises when we look to include infinite dimensions in a linearway, ie. what is common to the finite and infinite spaces with a certainbinary form that makes the space metrically complete.Hilbert spaces have a beautiful geometry, much inherited from their parentBanach spaces but with added structure. And this is the ontology quantummechanics first settled in on (there have been numerous extensions, like thebeautiful geometric extension of B. J. Hiley's Algebraic quantum mechanics,algebraic spinors, and Hilbert space which looks to Heisenberg algebras andhas some ties with realist interpretation theory).State is a vector or ray in a Hilbert space. We build this to anepistemology through a language of observables. The observation mechanismincludes applying projection operators to the state to get a mapping intothe interval [0, 1], from which our epistemology gets a probabilisticinterpretation.Each projection operator corresponds in a natural way to a closed linearsubspace of the Hilbert space through the operator spectrum structure of thespace.So when we want to be able to discuss making logical propositions of eventswith conjunctive and disjunctive connectives, negation, and general build asentence structure to be able to make predictions of a logical kind forempirical review.The way this operates in quantum mechanics is by looking at the naturallattice structure of the closed, linear subspaces under the identification:[A is a subspace of B] => [Lattice(A) -> Lattice(B)]Here meet and join, product and coproduct, etc. can both defined naturally,in the same way I mentioned previously, and identified with closed spans ofunions and intersection of these closed linar subspaces.With these notions, it is possible to form experiments looking for theprobability of event A and event B and event A or event B. You can findthe lattice element corresponding to the logical structure of a sentence andproject it into a probability which you can compare to experiments and theirabstractions.There is even a natural negation that corresponds to the properepistemological predictions of the theory through what is calledorthocomplementation of subspaces, where you look at the spans where innerproducts become zero.derived. Orthocomplemetation ~() obeys:~(~(a)) <-> aif a -> b, then ~(b) -> ~(a)(a / ~(a)) <-> Lattice(0)(a / ~(a)) <-> Lattice(I)where Lattice(0, 1) are the _|_, T respectively of the lattice. Besides thecommon relationships for / and / general to lattices, there isadditionally the relation(a / (b / ~(a))) <-> bwhich describes how negation on one subspace can be used to separate asecond subspace into a section whose conjunction with the former recoversthe latter. The probability of one event does not change if we look atrelated to another event and this latter event's opposite in a correlationexperiment.This defines a logic that is called orthomodular (for finite dimensionalHilbert spaces it is fully modular, ie. (a / (b / c)) <-> ((a / b) /c). It is the logic we build epistemological statements with in order tocompare with experimental observation. If you were to try to use booleanconnectives here, you would give predictions that differ from experiments.Most physicists don't pay too much attention to this, because they'velearned the formalism to convert a problem into the language of algebra,often without being pointed to the logical structure that defines thealgebra, but that is the structure of reasoning in a quantum ontology.There are natural connectives as well for implication and quantifiers.It is interesting to note that a difference between Hilbert space and thisstructure uniting ontology with epistemology for the quantum model andclassical mechanics in Euclidean spaces is captured by the geometric notionscontained in the boolean and quantum logics. For one, operators need notcommute.Now, there is a very interesting theorem out there which proves:There exist no partial Boolean algebra homomorphism from the lattice algebraof a Hilbert space into a Boolean algebra whenever the Hilbert space is morethan 2 dimensional (many Hilbert spaces for quantum systems are infinitedimensional).So, models with non Boolean structure are fairly necessary to the study ofquantum mechanics. As my other posts have tried to underline, models areeverywhere that use nonclassical logics directly. There are many modelswhere the epistemology is connected to the ontology through logics that cannot always derive true or false.And what I mean by relating a sense of absoluteness (like a form ofPlatonism) to this belief that all logic should be viewed as Boolean is thatit states often in some form that all propositions have a true or false evenif we cannot derive the answer. But that is actually a provablyinconsistent statement for quantum models. Even von Neumann's originalno go theorem for hidden variables in quantum mechanics actually pointedto the lack of a probabilistic embedding.Sure, the game you play with the symbols to understand the various logicsdescribed by many models can be taken to be many things. That's studyingsomething completely different, and I've shown that some cognitive researchwould point to a different logic being more faithful.I haven't mentioned |= yet.I am not satisfied...-- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea: prankster, fablist, magician, liar=== === Subject: : Re: No Set Contains Every Computable Natural If you want to be more general than the usual treatment,> you can do the following:> Let A be some finite alphabet (containing 0 and 1, at least),> and let A^omega be the set of infinite sequences of elements of A.> Every such infinite sequence is a possible state> for the infinite tape of a Turing machine.[...]regarding the same issue (N is recursive as a subset of N, but the setof tapes consisting of representations of natural numbers is notrecursive as a subset of all possible input tapes). It would be nicehere. Otherwise (?), he's simply diddling and wasting the time of everyoneinvolved.-- So, at this time, I'd like to assure you that I am not interested inmaking sure mathematicians worldwide get fired. I've rethought mydesire to go to Congress and try to get funding for mathematicianscut. -- James Harris is a reasonable man. Whew! === === Subject: : Re: No Set Contains Every Computable Natural>Given any set, S, of representations of natural numbers,>I can write a TM that will find a representation of a natural number>that is not in S. (No, this TM does not halt after a finite number>of steps.)> This is, of course, simply false, false, false.> If you give the set of all representations of natural numbers, then> your TM will not find any representation of a natural number not in> your set.> You're speaking nonsense again.> Of course, we must be careful here and fix a convention for when a> tape represents a set of natural numbers. Let's say that a tape> represents a set of natural numbers if it consists of blocks of ones> separated by a single space.> We must also be careful to require that your TM has the property that,> for each square x of the tape, there exists a steps t such that for all> steps s after t, the symbol on the square x at time s is the same as> the symbol on x at t. After all, a TM which repeatedly changes the> 0th square to 1 and blank and 1 and blank cannot be said to produce> any tape at all.I have given this proof several times.My TM has the property you describe.> Now, on an input tape containing 1, ,1,1, ,1,1,1, ,1,1,1,1, ,...,How about b1b11b111b1111...> there is simply no possibility that your TM, even after an infinite> number of steps, creates a tape with a representation of some natural> number not on my input tape. Every natural number is represented on> my tape. None is missing. Your tape either returns a set of numbers> already on my tape or it returns no set of numbers at all.This TM will find a representation not on your tape:It is a three state machine and I can provide a statetransition table if you like.1) Find a blank2) Find a second blank3) Backup and write a 1 on the previous blankrepeat steps (1) through (3)This TM will produce a tape that contains exactly one blank.The contiguous string of 1's preceding this blank will bea representation not on your tape.(Others have noted this is a very crude adder.)>> Obviously, a human has to decide if a symbol represents>> a natural number. No TM can do this.>>No TM can devise a convention, if that's what you mean. At least, no>TM can devise a convention in the sense that matters here.>But, a TM can extend the convention in such a way that no set>contains every symbol that represents a natural number.> No, no, no. A TM cannot extend the representation of natural numbers> for two reasons: (1) It's a fucking Turing machine, isn't it? I don't> want to argue about requirements for intentionality or whether TMs are> capable of intelligence, but this TM is not part of the negotiations> regarding our conventions.No intelligence required.My TM just adds all the numberstogether (plus 1 for each addition).A TM is too stupid to know thatthe sum is supposed to be infinity.> (2) Every natural number has a> representation in our convention, so the representation cannot be> non-redundantly extended.I just described such a TM.No tape can contain every representation of a natural number.There will always be a bigger representation.Russell- 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural>Given any set, S, of representations of natural numbers,>I can write a TM that will find a representation of a natural number>that is not in S. (No, this TM does not halt after a finite number>of steps.)>This is, of course, simply false, false, false.>If you give the set of all representations of natural numbers, then>your TM will not find any representation of a natural number not in>your set.>You're speaking nonsense again.>Of course, we must be careful here and fix a convention for when a>tape represents a set of natural numbers. Let's say that a tape>represents a set of natural numbers if it consists of blocks of ones>separated by a single space.>We must also be careful to require that your TM has the property that,>for each square x of the tape, there exists a steps t such that for all>steps s after t, the symbol on the square x at time s is the same as>the symbol on x at t. After all, a TM which repeatedly changes the>0th square to 1 and blank and 1 and blank cannot be said to produce>any tape at all.> I have given this proof several times.> My TM has the property you describe.>Now, on an input tape containing 1, ,1,1, ,1,1,1, ,1,1,1,1, ,...,> How about b1b11b111b1111...>there is simply no possibility that your TM, even after an infinite>number of steps, creates a tape with a representation of some natural>number not on my input tape. Every natural number is represented on>my tape. None is missing. Your tape either returns a set of numbers>already on my tape or it returns no set of numbers at all.> This TM will find a representation not on your tape:> It is a three state machine and I can provide a state> transition table if you like.> 1) Find a blank> 2) Find a second blank> 3) Backup and write a 1 on the previous blank> repeat steps (1) through (3)> This TM will produce a tape that contains exactly one blank.> The contiguous string of 1's preceding this blank will be> a representation not on your tape.> (Others have noted this is a very crude adder.)Thus, when the TM reaches 'n' on the tape, it will produce the number 'n*(n+1)/2+n-1', which is still further along on the tape (but already on the tape).What number 'n' does your TM ever reach for which 'n*(n+1)/2+n-1' is not already on the tape?Answer: there is no such 'n'.>> Obviously, a human has to decide if a symbol represents>> a natural number. No TM can do this.> No TM can devise a convention, if that's what you mean. At least, no>> TM can devise a convention in the sense that matters here.>>But, a TM can extend the convention in such a way that no set>contains every symbol that represents a natural number.>No, no, no. A TM cannot extend the representation of natural numbers>for two reasons: (1) It's a fucking Turing machine, isn't it? I don't>want to argue about requirements for intentionality or whether TMs are>capable of intelligence, but this TM is not part of the negotiations>regarding our conventions.> No intelligence required.> My TM just adds all the numbers> together (plus 1 for each addition).> A TM is too stupid to know that> the sum is supposed to be infinity.>(2) Every natural number has a>representation in our convention, so the representation cannot be>non-redundantly extended.> I just described such a TM.> No tape can contain every representation of a natural number.> There will always be a bigger representation.No finite tape, but as soon as you posit a tape containing a representation for every natural, you have such a tape as does not allow any bigger representations.> Russell> - 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org > Given any set, S, of representations of natural numbers,>> I can write a TM that will find a representation of a natural number>> that is not in S. (No, this TM does not halt after a finite number>> of steps.)>> This is, of course, simply false, false, false.>> If you give the set of all representations of natural numbers, then>> your TM will not find any representation of a natural number not in>> your set.>> You're speaking nonsense again.>> Of course, we must be careful here and fix a convention for when a>> tape represents a set of natural numbers. Let's say that a tape>> represents a set of natural numbers if it consists of blocks of ones>> separated by a single space.>> We must also be careful to require that your TM has the property that,>> for each square x of the tape, there exists a steps t such that for all>> steps s after t, the symbol on the square x at time s is the same as>> the symbol on x at t. After all, a TM which repeatedly changes the>> 0th square to 1 and blank and 1 and blank cannot be said to produce>> any tape at all.> I have given this proof several times.Yes, and you've been simply wrong several times.> My TM has the property you describe.>> Now, on an input tape containing 1, ,1,1, ,1,1,1, ,1,1,1,1, ,...,> How about b1b11b111b1111...>> there is simply no possibility that your TM, even after an infinite>> number of steps, creates a tape with a representation of some natural>> number not on my input tape. Every natural number is represented on>> my tape. None is missing. Your tape either returns a set of numbers>> already on my tape or it returns no set of numbers at all.> This TM will find a representation not on your tape:> It is a three state machine and I can provide a state> transition table if you like.> 1) Find a blank> 2) Find a second blank> 3) Backup and write a 1 on the previous blank> repeat steps (1) through (3)> This TM will produce a tape that contains exactly one blank.> The contiguous string of 1's preceding this blank will be> a representation not on your tape.It will produce no such tape. It will produce a tape consisting ofall 1's. This is obvious.Suppose that it contains a blank. Then that blank must occur at somesquare on the tape, say square n. It is trivial to see that, by thenth iteration of your three steps, square n is no longer blank.(After the first iteration, square 1 is not blank, square two wasnever blank, square three is not blank after the second iteration andhence is also not blank after the third, and so on.)Your proof of this claim is simply another confusion.>> No, no, no. A TM cannot extend the representation of natural numbers>> for two reasons: (1) It's a fucking Turing machine, isn't it? I don't>> want to argue about requirements for intentionality or whether TMs are>> capable of intelligence, but this TM is not part of the negotiations>> regarding our conventions.> No intelligence required.> My TM just adds all the numbers> together (plus 1 for each addition).> A TM is too stupid to know that> the sum is supposed to be infinity.If your tape contains a single contiguous block of an infinite numberof 1's, then your tape does not contain a set of natural numbers.Nothing can change that fact *except* changing what our conventions oftranslating tapes to subsets of N and back are. Your machine isinvoked only after the conventions are set and cannot change anyconventions.It is obvious that your machine does not produce a (tape representinga) set of natural numbers even though at each finite step the tape themachine is working on represents a set of natural numbers.>> (2) Every natural number has a>> representation in our convention, so the representation cannot be>> non-redundantly extended.> I just described such a TM.> No tape can contain every representation of a natural number.> There will always be a bigger representation.You have not extended the representation of N. You haven't evendescribed correctly what tape your machine produces. If you coulddescribe what your machine produces and if you were capable ofunderstanding, then you would realize that your machine does notproduce a tape representing a set of natural numbers.Of course, you have been making the same rudimentary and embarrassingmistakes regarding transfinite sets for literally years. I don'texpect you to learn a damned thing from this exchange either. Toobad.-- However, you presuppose that certain numbers *are* prime ideals,... when in fact ...* they are not... (Maybe I should look up 'primeideals' but the effort doesn't seem to be worth it. I assume someposter will get excited ... if I messed up.) --James Harris=== === Subject: : Re: No Set Contains Every Computable Natural>>> Given any set, S, of representations of natural numbers,>> I can write a TM that will find a representation of a natural number>> that is not in S. (No, this TM does not halt after a finite number>> of steps.)>>> This is, of course, simply false, false, false.>>> If you give the set of all representations of natural numbers, then>> your TM will not find any representation of a natural number not in>> your set.>>> You're speaking nonsense again.>>> Of course, we must be careful here and fix a convention for when a>> tape represents a set of natural numbers. Let's say that a tape>> represents a set of natural numbers if it consists of blocks of ones>> separated by a single space.>>> We must also be careful to require that your TM has the property that,>> for each square x of the tape, there exists a steps t such that for all>> steps s after t, the symbol on the square x at time s is the same as>> the symbol on x at t. After all, a TM which repeatedly changes the>> 0th square to 1 and blank and 1 and blank cannot be said to produce>> any tape at all.>>I have given this proof several times.> Yes, and you've been simply wrong several times.>My TM has the property you describe.>> Now, on an input tape containing 1, ,1,1, ,1,1,1, ,1,1,1,1, ,...,>How about b1b11b111b1111...>> there is simply no possibility that your TM, even after an infinite>> number of steps, creates a tape with a representation of some natural>> number not on my input tape. Every natural number is represented on>> my tape. None is missing. Your tape either returns a set of numbers>> already on my tape or it returns no set of numbers at all.>This TM will find a representation not on your tape:>It is a three state machine and I can provide a state>transition table if you like.>1) Find a blank>2) Find a second blank>3) Backup and write a 1 on the previous blank>repeat steps (1) through (3)>This TM will produce a tape that contains exactly one blank.>The contiguous string of 1's preceding this blank will be>a representation not on your tape.> It will produce no such tape. It will produce a tape consisting of> all 1's. This is obvious.> Suppose that it contains a blank. Then that blank must occur at some> square on the tape, say square n. It is trivial to see that, by the> nth iteration of your three steps, square n is no longer blank.> (After the first iteration, square 1 is not blank, square two was> never blank, square three is not blank after the second iteration and> hence is also not blank after the third, and so on.)This TM always checks to see if there is another blank on the tapebefore overwriting the previous blank. That is what step 2 does.It is easy to prove there is a blank on the output tape.> Your proof of this claim is simply another confusion.It is a three state TM.How confusing can it be?>> No, no, no. A TM cannot extend the representation of natural numbers>> for two reasons: (1) It's a fucking Turing machine, isn't it? I don't>> want to argue about requirements for intentionality or whether TMs are>> capable of intelligence, but this TM is not part of the negotiations>> regarding our conventions.>No intelligence required.>My TM just adds all the numbers>together (plus 1 for each addition).>A TM is too stupid to know that>the sum is supposed to be infinity.> If your tape contains a single contiguous block of an infinite number> of 1's, then your tape does not contain a set of natural numbers.A single contiguous block of an infinite number of 1's followed bya blank in a finite position? I never said the output of my TMcontained the set of all natural numbers. I said it would containa representation not on the initial input tape. The output containsthe representation of exactly one natural number.> Nothing can change that fact *except* changing what our conventions of> translating tapes to subsets of N and back are. Your machine is> invoked only after the conventions are set and cannot change any> conventions.Namely, each natural is represented by a contiguous string of 1'sfollowed by a blank. This TM produces such a representation.> It is obvious that your machine does not produce a (tape representing> a) set of natural numbers even though at each finite step the tape the> machine is working on represents a set of natural numbers.My TM produces a tape with one representation of a natural number.This representation is not on the initial input tape.>> (2) Every natural number has a>> representation in our convention, so the representation cannot be>> non-redundantly extended.>I just described such a TM.>No tape can contain every representation of a natural number.>There will always be a bigger representation.> You have not extended the representation of N. You haven't even> described correctly what tape your machine produces. If you could> describe what your machine produces and if you were capable of> understanding, then you would realize that your machine does not> produce a tape representing a set of natural numbers.This TM produces a tape with a contiguous string of 1'sfollowed by a blank. The output tape contains the representationof exactly one natural number.Russell- 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural>It is obvious that your machine does not produce a (tape representing>a) set of natural numbers even though at each finite step the tape the>machine is working on represents a set of natural numbers.> My TM produces a tape with one representation of a natural number.> This representation is not on the initial input tape.What is the makeup of your initial tape? Is it finite or infinite? If finite, does it end with a 0 or a 1?Only in the case of an input tape starting with a zero and ending with a zero, does your TM work. And such never contains ALL naturals.> This TM produces a tape with a contiguous string of 1's> followed by a blank. The output tape contains the representation> of exactly one natural number.Only if your input tape ends with a finite string of 1's followed by a single 0. Which means that the input tape does not contain representations of all naturals.> Russell> - 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural>It is obvious that your machine does not produce a (tape representing>a) set of natural numbers even though at each finite step the tape the>machine is working on represents a set of natural numbers.>My TM produces a tape with one representation of a natural number.>This representation is not on the initial input tape.> What is the makeup of your initial tape? Is it finite or infinite?The initial input tape is 01011011101111...The assumption is that this tape has a representationfor every natural number. This would seem to indicatethat the input tape in infinite.> If finite, does it end with a 0 or a 1?How would it end if it were infinite?Does it end with an infinite string of 1's?It might end with 00.This would kind of like an end of file mark.I guess most people would say it never ends.> Only in the case of an input tape starting with a zero and ending with a> zero, does your TM work. And such never contains ALL naturals.No, there could be an infinite string of 1's at the end of the output tape.This would be the case if the input tape ended with an infinite string of1's.The output would be a finite string of 1's followed by a 0 (or blank)followed by an infinite string of 1's.>This TM produces a tape with a contiguous string of 1's>followed by a blank. The output tape contains the representation>of exactly one natural number.> Only if your input tape ends with a finite string of 1's followed by a> single 0. Which means that the input tape does not contain> representations of all naturals.Actually, I am trying to prove the input tape doesn't contain arepresentation of every natural. So, you are right, the inputtape does not contain a representation of every natural number.Russell- Zeno was right. Motion is impossible.=== === Subject: : Re: No Set Contains Every Computable Natural> It is obvious that your machine does not produce a (tape representing>> a) set of natural numbers even though at each finite step the tape the>> machine is working on represents a set of natural numbers.>>My TM produces a tape with one representation of a natural number.>This representation is not on the initial input tape.>What is the makeup of your initial tape? Is it finite or infinite?> The initial input tape is 01011011101111...> The assumption is that this tape has a representation> for every natural number. This would seem to indicate> that the input tape in infinite.>If finite, does it end with a 0 or a 1?> How would it end if it were infinite?Are you too illiterate to understand IF?> Does it end with an infinite string of 1's?> It might end with 00.> This would kind of like an end of file mark.> I guess most people would say it never ends.>Only in the case of an input tape starting with a zero and ending with a>zero, does your TM work. And such never contains ALL naturals.> No, there could be an infinite string of 1's at the end of the output tape.> This would be the case if the input tape ended with an infinite string of> 1's.> The output would be a finite string of 1's followed by a 0 (or blank)> followed by an infinite string of 1's.The output could not exist, as the TM cold never finish.>>This TM produces a tape with a contiguous string of 1's>followed by a blank. The output tape contains the representation>of exactly one natural number.>Only if your input tape ends with a finite string of 1's followed by a>single 0. Which means that the input tape does not contain>representations of all naturals.> Actually, I am trying to prove the input tape doesn't contain a> representation of every natural. So, you are right, the input> tape does not contain a representation of every natural number.Not if it is finite. But if it is infinite it can have all naturals represented, but the TM never finishes.=== === Subject: : Re: No Set Contains Every Computable Natural <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org > This TM will find a representation not on your tape:>> It is a three state machine and I can provide a state>> transition table if you like.>>> 1) Find a blank>> 2) Find a second blank>> 3) Backup and write a 1 on the previous blank>> repeat steps (1) through (3)>>> This TM will produce a tape that contains exactly one blank.>> The contiguous string of 1's preceding this blank will be>> a representation not on your tape.>> It will produce no such tape. It will produce a tape consisting of>> all 1's. This is obvious.>> Suppose that it contains a blank. Then that blank must occur at some>> square on the tape, say square n. It is trivial to see that, by the>> nth iteration of your three steps, square n is no longer blank.>> (After the first iteration, square 1 is not blank, square two was>> never blank, square three is not blank after the second iteration and>> hence is also not blank after the third, and so on.)> This TM always checks to see if there is another blank on the tape> before overwriting the previous blank. That is what step 2 does.> It is easy to prove there is a blank on the output tape.Wrong.It is easy to prove that at each step n, there is another blank on thetape.Unfortunately, you want to talk about the output after an infinitenumber of steps. You have proved that at each finite step n, there isa blank on the tape. That is not sufficient to prove that after aninfinite number of steps, there is still a blank.Moreover, there is a very simple proof that there is no blank. Lookup at the paragraph that begins, Suppose that it contains a blank.>> Your proof of this claim is simply another confusion.> It is a three state TM.> How confusing can it be?The TM is not confusing and yet you are confused. Huh.I didn't say your TM was likely to confuse a man of averageintelligence. I merely said you were confused. Evidently, youconfused these two statements also.>>> No, no, no. A TM cannot extend the representation of natural numbers>> for two reasons: (1) It's a fucking Turing machine, isn't it? I don't>>> want to argue about requirements for intentionality or whether TMs are>>> capable of intelligence, but this TM is not part of the negotiations>>> regarding our conventions.>>> No intelligence required.>> My TM just adds all the numbers>> together (plus 1 for each addition).>> A TM is too stupid to know that>> the sum is supposed to be infinity.>> If your tape contains a single contiguous block of an infinite number>> of 1's, then your tape does not contain a set of natural numbers.> A single contiguous block of an infinite number of 1's followed by> a blank in a finite position? I never said the output of my TM> contained the set of all natural numbers. I said it would contain> a representation not on the initial input tape. The output contains> the representation of exactly one natural number.And of course you're simply wrong.You repeat the same basic mistake repeatedly hereafter. There's nopoint in my writing that you're mistaken three more times. I'vesnipped the rest.My above explanation is sufficient.-- Jesse F. HughesI thought it relevant to inform that I notified the FBI a couple ofmonths ago about some of the math issues I've brought up here. -- James S. Harris gives Special Agent Fox a new assignment.=== === Subject: : Re: No Set Contains Every Computable Natural>> This TM will find a representation not on your tape:>> It is a three state machine and I can provide a state>> transition table if you like.>>> 1) Find a blank>> 2) Find a second blank>> 3) Backup and write a 1 on the previous blank>> repeat steps (1) through (3)>> This TM will produce a tape that contains exactly one blank.>> The contiguous string of 1's preceding this blank will be>> a representation not on your tape.>>> It will produce no such tape. It will produce a tape consisting of>> all 1's. This is obvious.>>> Suppose that it contains a blank. Then that blank must occur at some>> square on the tape, say square n. It is trivial to see that, by the>> nth iteration of your three steps, square n is no longer blank.> (After the first iteration, square 1 is not blank, square two was>> never blank, square three is not blank after the second iteration and>> hence is also not blank after the third, and so on.)>This TM always checks to see if there is another blank on the tape>before overwriting the previous blank. That is what step 2 does.>It is easy to prove there is a blank on the output tape.> Wrong.> It is easy to prove that at each step n, there is another blank on the> tape.> Unfortunately, you want to talk about the output after an infinite> number of steps. You have proved that at each finite step n, there is> a blank on the tape. That is not sufficient to prove that after an> infinite number of steps, there is still a blank.Yes it is.> Moreover, there is a very simple proof that there is no blank. Look> up at the paragraph that begins, Suppose that it contains a blank.Look at the definition of my TM.Suppose that it contains a blank. Step (2) will then search foranother blank. The blank on the tape will not be overwrittenunless a second blank is found. The output tape will ALWAYScontain one blank. Even after an infinite number of steps.Russell- 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org <87ptcau3xj.fsf@phiwumbda.org >> This TM will find a representation not on your tape:>>> It is a three state machine and I can provide a state>>> transition table if you like.>>>>> 1) Find a blank>> 2) Find a second blank>>> 3) Backup and write a 1 on the previous blank>>> repeat steps (1) through (3)>>>> This TM will produce a tape that contains exactly one blank.>>> The contiguous string of 1's preceding this blank will be>>> a representation not on your tape.>> It will produce no such tape. It will produce a tape consisting of>>> all 1's. This is obvious.>>> Suppose that it contains a blank. Then that blank must occur at some>> square on the tape, say square n. It is trivial to see that, by the>>> nth iteration of your three steps, square n is no longer blank.>>> (After the first iteration, square 1 is not blank, square two was>>> never blank, square three is not blank after the second iteration and>>> hence is also not blank after the third, and so on.)>>> This TM always checks to see if there is another blank on the tape>> before overwriting the previous blank. That is what step 2 does.>> It is easy to prove there is a blank on the output tape.>> Wrong.>> It is easy to prove that at each step n, there is another blank on the>> tape.>> Unfortunately, you want to talk about the output after an infinite>> number of steps. You have proved that at each finite step n, there is>> a blank on the tape. That is not sufficient to prove that after an>> infinite number of steps, there is still a blank.> Yes it is.A withering retort! I am stunned by the force of logic.And yet, you're simply wrong.>> Moreover, there is a very simple proof that there is no blank. Look>> up at the paragraph that begins, Suppose that it contains a blank.> Look at the definition of my TM.Suppose that it contains a blank. Step (2) will then search for> another blank. The blank on the tape will not be overwritten> unless a second blank is found. The output tape will ALWAYS> contain one blank. Even after an infinite number of steps.You have proven by induction that for all n, at step n in thecomputation, there is a blank space. This does not prove that afteromega steps, there is a blank space. Your proof by bald assertiondoesn't cut it.Your proof allows for this silly argument. I can prove that, for alln, there exists a number m such that n < m. *Therefore*, there mustbe a number m bigger than every number n. (I suppose that thisreductio won't be persuasive to you, since you have put forwardsomething like this argument a million times so far.)Now, if you'll read my argument again, you will see that, on thecontrary, each blank space is overwritten within a finite number ofsteps of the computation and therefore, after omega steps, there areno blank spaces on the tape at all. For, if there was a blank space,that space must occur at some finite position, say n. However, weknow that by the n+1st iteration of your algorithm, the nth space is*not* blank. Therefore, there is no blank space on your tape after infinitely manysteps of computation. (Here is the point where you wittily rejoinder,yes, there is, so that I may admit defeat in the face of suchunassailable arguments.)-- If you see math knowledge as a tool--as a hammer--with whichyou can attack other people then ... you defeat rational discourse.I get to call my proof the Hammer. It's more powerful than *any*physical object. It is overwhelming force. -- Two JSH quotes=== === Subject: : Re: No Set Contains Every Computable Natural <3_OdnTxoJNyqVK_dRVn-hA@comcast.com> <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org <87ptcau3xj.fsf@phiwumbda.org <87fzd6ti0j.fsf@phiwumbda.org Discussion, linux)>> Look at the definition of my TM.>> Suppose that it contains a blank. Step (2) will then search for>> another blank. The blank on the tape will not be overwritten>> unless a second blank is found. The output tape will ALWAYS>> contain one blank. Even after an infinite number of steps.> You have proven by induction that for all n, at step n in the> computation, there is a blank space. This does not prove that after> omega steps, there is a blank space. Your proof by bald assertion> doesn't cut it.In fact, once again you seem to make the error of commutingquantifiers.You have proved[1] that, for all steps n, there exists a square x suchthat x is blank.You have asserted that there exists a square x such that for all stepsn, x is blank.This is simply false. This sort of reasoning[2] would yield thefollowing argument.Start with a blank tape. Execute the algorithm:(1) Write a 1 in the current square, and move right one square. Go tostate 1. Now let square 0 be the start square, 1 the square to the right of it,2 the square to the right of square 1 and so on.Clearly, for every step of the computation, there is a positive n suchthat n is blank.Nonetheless, it is obviously not the case that after an infinitenumber of steps, there is an n such that n is blank.Obviously here means obvious to anyone capable of simple logicalarguments. Since I won't presume that's true of all parties here,I'll give the argument.For any n, after the n+1st step, square n contains a 1. Thereforethere is no n such that, after an infinite number of steps, square nis blank.Footnotes: [1] I'm being charitable here.[2] I'm being facetiously charitable here.-- It has been shown that no man can sit down to write without a very profounddesign. Thus to authors in general trouble is spared. A novelist, for example,need have no care of his moral. It is there -- that is to say, it is somewhere-- and the moral and the critics can take care of themselves. --E.A. Poe=== === Subject: : Re: No Set Contains Every Computable Natural> This TM will find a representation not on your tape:> It is a three state machine and I can provide a state> transition table if you like.> 1) Find a blank 2) Find a second blank> 3) Backup and write a 1 on the previous blank> repeat steps (1) through (3)> This TM will produce a tape that contains exactly one blank.> The contiguous string of 1's preceding this blank will be> a representation not on your tape.> It will produce no such tape. It will produce a tape consisting of> all 1's. This is obvious.> Suppose that it contains a blank. Then that blank must occur at some> square on the tape, say square n. It is trivial to see that, by the> nth iteration of your three steps, square n is no longer blank.> (After the first iteration, square 1 is not blank, square two was> never blank, square three is not blank after the second iteration and> hence is also not blank after the third, and so on.)>>This TM always checks to see if there is another blank on the tape>before overwriting the previous blank. That is what step 2 does.>It is easy to prove there is a blank on the output tape.>Wrong.>It is easy to prove that at each step n, there is another blank on the>tape.>Unfortunately, you want to talk about the output after an infinite>number of steps. You have proved that at each finite step n, there is>a blank on the tape. That is not sufficient to prove that after an>infinite number of steps, there is still a blank.> Yes it is.Thus again proving that Russell does not understand what is going on here.>Moreover, there is a very simple proof that there is no blank. Look>up at the paragraph that begins, Suppose that it contains a blank.> Look at the definition of my TM.Suppose that it contains a blank. Step (2) will then search for> another blank. The blank on the tape will not be overwritten> unless a second blank is found. The output tape will ALWAYS> contain one blank. Even after an infinite number of steps.So you claim, but you also claim that the blank is in the last position, which makes the tape finite, as infinite tapes do not have lastpositions. That comes with the definition of infinite.> Russell> - 2 zany 2 count=== === Subject: : Re: No Set Contains Every Computable Natural <8JCdncK7SMPAiK7dRVn-vw@comcast.com <87hdxovh3h.fsf@phiwumbda.org> Moreover, there is a very simple proof that there is no blank. Look>up at the paragraph that begins, Suppose that it contains a blank.> Look at the definition of my TM.Suppose that it contains a blank. Step (2) will then search for> another blank. The blank on the tape will not be overwritten> unless a second blank is found. The output tape will ALWAYS> contain one blank. Even after an infinite number of steps. A TM never takes an infinite number of steps. After any stepit takes, it was some finite time. And if we are to homour you and pretend that it does take aninfinite amount of steps, then there are no 0s. Since assume thereis some 0; there must have been a 0 after it (since the string of1s after it is always finitely long) therefore that 0 was turned toa 1, contradiction. Take an analogy to the natural numbers. Your removal of 0s is like this:If, for any odd number 2k+1, there is an odd number coming after it, thenremove 2k+1 from the set. Your argument is that the remaining set would still have an odd number,but it's not hard to see that the set {1,2,...,N} up to ANY N at all wouldnever have an odd number.J=== === Subject: : Re: No Set Contains Every Computable Natural <3_OdnTxoJNyqVK_dRVn-hA@comcast.com> <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org <87ptcau3xj.fsf@phiwumbda.org A TM never takes an infinite number of steps. After any step> it takes, it was some finite time.There is no reason why we can't allow a definition that a TM convergesafter an infinite number of steps if each square on the TM's taperemains unchanged after a certain step. That is, if for all squares x, there exists a step n in thecomputation such that for all steps m > n, the value written in x at mis the same as the value written in x at n. -- Jesse F. HughesWhat I represent is the unknowable future--the power of change. Inthat sense I'm a force of Nature, a force of the Universe, a livingemodiment of change itself. --James Harris and his sense of humility=== === Subject: : Re: No Set Contains Every Computable Natural>Moreover, there is a very simple proof that there is no blank. Look>up at the paragraph that begins, Suppose that it contains a blank.>Look at the definition of my TM.Suppose that it contains a blank. Step (2) will then search for>another blank. The blank on the tape will not be overwritten>unless a second blank is found. The output tape will ALWAYS>contain one blank. Even after an infinite number of steps.> A TM never takes an infinite number of steps. After any step> it takes, it was some finite time.A number of people have stated that time is not an issuefor an idealized TM. I have shown this is equivalent toassuming that a TM can perform an infinite number ofoperations in a finite amount of time.There is no difference between assuming that a TMwill still be computing long after the stars have turnedto dust and assuming a TM can perform an infinite numberof operations.If we assume a TM requires a fixed, non-zero amount of timeto perform an operation, I can come up with a natural numberthat will takes billions of years for that TM to process.> And if we are to homour you and pretend that it does take an> infinite amount of steps,You are not humouring me. The definition of algorithmimplicitly assumes a TM can perform any number ofoperations in a finite amount of time.> then there are no 0s. Since assume there> is some 0; there must have been a 0 after it (since the string of> 1s after it is always finitely long) therefore that 0 was turned to> a 1, contradiction.Why is it a contradiction?You just said there a 0 (or a blank) following the 0that is overwritten. The output tape will always contain ablank after any number of operations.> Take an analogy to the natural numbers. Your removal of 0s is like this:> If, for any odd number 2k+1, there is an odd number coming after it, then> remove 2k+1 from the set.> Your argument is that the remaining set would still have an odd number,> but it's not hard to see that the set {1,2,...,N} up to ANY N at all would> never have an odd number.Huh?Consider a finite set (1,2,3).Using your algorithm I get (2,3).There is no odd number that comes after 3 in the set (1,2,3).Russell- 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural <3_OdnTxoJNyqVK_dRVn-hA@comcast.com> <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org <87ptcau3xj.fsf@phiwumbda.org A number of people have stated that time is not an issue> for an idealized TM. I have shown this is equivalent to> assuming that a TM can perform an infinite number of> operations in a finite amount of time.This is wrong. I don't have anything against extending the commondefinition of convergence to allow for convergence after omega steps(for *some*, but not *all* TMs and inputs), but it *is* an extension.You have not shown equivalence. I don't recall your argument, but Iwill show my psychic powers: Your argument involves swapping the orderof quantification at an essential step.Well? Was I right? Huh? Was I?-- Even if [...] a communistic regime should come [to China], the oldtradition [...] will break Communism and change it beyond recognition,rather than Communism [...] break the old tradition. It must be so. -- Lin Yutang on Socialism with Chinese characteristics in 1935=== === Subject: : Re: No Set Contains Every Computable Natural>> Moreover, there is a very simple proof that there is no blank. Look>> up at the paragraph that begins, Suppose that it contains a blank.>>Look at the definition of my TM.Suppose that it contains a blank. Step (2) will then search for>another blank. The blank on the tape will not be overwritten>unless a second blank is found. The output tape will ALWAYS>contain one blank. Even after an infinite number of steps.> A TM never takes an infinite number of steps. After any step>it takes, it was some finite time.> A number of people have stated that time is not an issue> for an idealized TM. I have shown this is equivalent to> assuming that a TM can perform an infinite number of> operations in a finite amount of time.You may have argued it, but that does not necessarily constitute a proof.> There is no difference between assuming that a TM> will still be computing long after the stars have turned> to dust and assuming a TM can perform an infinite number> of operations.In the first case, the TM may still only be able to perform a finite, though large, number of steps. Their may be no practical difference, in that no one will be around then, but there is a theoretical difference which is critically important.> If we assume a TM requires a fixed, non-zero amount of time> to perform an operation, I can come up with a natural number> that will takes billions of years for that TM to process.So?> And if we are to homour you and pretend that it does take an>infinite amount of steps,> You are not humouring me. The definition of algorithm> implicitly assumes a TM can perform any number of> operations in a finite amount of time.What definition is that? The definitions of algorithms that I am aware of all require that the algorithm be completed in a finite number of steps. Perhaps you can give a reference for someone who agrees with your definition?>then there are no 0s. Since assume there>is some 0; there must have been a 0 after it (since the string of>1s after it is always finitely long) therefore that 0 was turned to>a 1, contradiction.> Why is it a contradiction?> You just said there a 0 (or a blank) following the 0> that is overwritten. The output tape will always contain a> blank after any number of operations.If it starts with infinitely many zeros, as it must to represent ALL natural numbers, then there will always be infinitely many zeros left.> Take an analogy to the natural numbers. Your removal of 0s is like this:>If, for any odd number 2k+1, there is an odd number coming after it, then>remove 2k+1 from the set.> Your argument is that the remaining set would still have an odd number,>but it's not hard to see that the set {1,2,...,N} up to ANY N at all would>never have an odd number.> Huh?> Consider a finite set (1,2,3).> Using your algorithm I get (2,3).> There is no odd number that comes after 3 in the set (1,2,3).That is the point. In finite sets, there is a point that nothing comes after, but in an infinite set that is not the case.> Russell> - 2 zany 2 count=== === Subject: : Re: No Set Contains Every Computable Natural <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org <87ptcau3xj.fsf@phiwumbda.org There is no difference between assuming that a TM> will still be computing long after the stars have turned> to dust and assuming a TM can perform an infinite number> of operations. It's a huge difference. One is saying that after a huge amountof time T, it is computing. This is normal and fine, since T ishuge but finite. But saying that a turing machine performs an infinite numbersof steps is nonsense and goes against the standard definitions.Create your own difinitions is you want that property. In another post, you asked what the difference was in doingsomething for an arbitrarily large amount of time and doing somethinginfinitely long. This is the source of all your problems. Once you understand thatthere is a difference between these two, most of your problems willclear up. The first one is a finite computation (but for any largeamount of steps) and the second one is simply just not a computationby the definition of turing machine computations.> If we assume a TM requires a fixed, non-zero amount of time> to perform an operation, I can come up with a natural number> that will takes billions of years for that TM to process. That's totally fine. I give you some time T and you can createa number that takes longer than T steps to compute (or output orread or anything.) Our point is that for any computation you dotaking longer than this T-steps, its computation time will (and must)be bounded by some other number B.> You are not humouring me. The definition of algorithm> implicitly assumes a TM can perform any number of> operations in a finite amount of time. As long as any number is any finite number. Arbitrarily large,yes, but still finite.>then there are no 0s. Since assume there>is some 0; there must have been a 0 after it (since the string of>1s after it is always finitely long) therefore that 0 was turned to>a 1, contradiction.> Why is it a contradiction?> You just said there a 0 (or a blank) following the 0> that is overwritten. The output tape will always contain a> blank after any number of operations. You need to learn what a proof by contradiction is. Assumption: There exists a 0 at the end of computation. We use this statement and show that it contradicts itself. Say there exists a '0' at the end of computation. If it existson the tape, it MUST exist at some finitely-indexed tape cell. But then if the tape originally had all the natural numbers,then this 0 must have been followed by a string of 1s, eventuallyanother 0, and another string of 1s. Since there is this other 0,the first 0 (that we assumed exists) must have been changed to a '1,'contradicting the fact that the 0 existed.> Take an analogy to the natural numbers. Your removal of 0s is like this:>If, for any odd number 2k+1, there is an odd number coming after it, then>remove 2k+1 from the set.> Your argument is that the remaining set would still have an odd number,>but it's not hard to see that the set {1,2,...,N} up to ANY N at all would>never have an odd number.> Huh?> Consider a finite set (1,2,3).> Using your algorithm I get (2,3).> There is no odd number that comes after 3 in the set (1,2,3). You missed the fact in elementary school that if n is some oddnumber, then so is n+2. I'm not talking about a finite set, I'm talking about takingthe natural numbers and applying that process to it. Your hypotheticaltape supposedly had all the natural numbers on it. Once the processis complete, then any set you look at {1..N} will have all its oddnumbers removed. Again, make sure you learn about the difference between 'artibrarilylarge' and 'infinite.' The set {1, 11, 111, 1111, 11111, ... } will contain strings that arearbitrarily long but will not contain the infinite string 1111....J=== === Subject: : Re: No Set Contains Every Computable Natural> *sigh*... I don't know why I bother with this discussion...> Again, I won't contribute after this.>There is no difference between assuming that a TM>will still be computing long after the stars have turned>to dust and assuming a TM can perform an infinite number>of operations.> It's a huge difference. One is saying that after a huge amount> of time T, it is computing. This is normal and fine, since T is> huge but finite.> But saying that a turing machine performs an infinite numbers> of steps is nonsense and goes against the standard definitions.There are lots of ways to define a Turing machine.In Turing's original paper, he said a number was computable only ifthere exists a TM that computes that number's binary representationand DOESN'T halt.You have chosen to use a definition that you think supports your position.It doesn't matter. My argument still holds.> Create your own difinitions is you want that property.OKAn accelerated Turing machine will perform an infinite number ofoperations in a finite amount of time.> In another post, you asked what the difference was in doing> something for an arbitrarily large amount of time and doing something> infinitely long.> This is the source of all your problems. Once you understand that> there is a difference between these two, most of your problems will> clear up. The first one is a finite computation (but for any large> amount of steps) and the second one is simply just not a computation> by the definition of turing machine computations.>If we assume a TM requires a fixed, non-zero amount of time>to perform an operation, I can come up with a natural number>that will takes billions of years for that TM to process.> That's totally fine. I give you some time T and you can create> a number that takes longer than T steps to compute (or output or> read or anything.) Our point is that for any computation you do> taking longer than this T-steps, its computation time will (and must)> be bounded by some other number B.Then I will give you another number that takes longer than B.We could do this for a really long time.Which one us will come up with the biggest number?I bet I do.There are 31,556,926 seconds in a year.This is about 3 * 10^7.Assume we have have computer that performs an operationevery 10^(-43) sec. This is as fast as we can expectany physical process to occur. Let's say the universe endsin about 3 * 10^10 years.3*10^7 * 10^43 * 3*10^10 = 9 * 10^60Nearly all natural numbers are larger than 10^61.I win. I found a number a TM can't read beforethe universe freezes over.>You are not humouring me. The definition of algorithm>implicitly assumes a TM can perform any number of>operations in a finite amount of time.> As long as any number is any finite number. Arbitrarily large,> yes, but still finite.Which one of us gets to decide what is finite?>then there are no 0s. Since assume there>is some 0; there must have been a 0 after it (since the string of>1s after it is always finitely long) therefore that 0 was turned to>a 1, contradiction.>Why is it a contradiction?>You just said there a 0 (or a blank) following the 0>that is overwritten. The output tape will always contain a>blank after any number of operations.> You need to learn what a proof by contradiction is.A bad proof?> Assumption: There exists a 0 at the end of computation.Of course there is. How can there not be?My TM will never overwrite the last 0.> We use this statement and show that it contradicts itself.> Say there exists a '0' at the end of computation. If it exists> on the tape, it MUST exist at some finitely-indexed tape cell.Yep.> But then if the tape originally had all the natural numbers,That's a big if, isn't it?> then this 0 must have been followed by a string of 1s, eventually> another 0, and another string of 1s. Since there is this other 0,What other 0? There is only one 0 on the final tape.> the first 0 (that we assumed exists) must have been changed to a '1,'> contradicting the fact that the 0 existed.You just said there was this other 0.You must be talking about some intermediate version of the tape.So, the intermediate tape still contains a 0 at some finite position.> Take an analogy to the natural numbers. Your removal of 0s is likethis:>If, for any odd number 2k+1, there is an odd number coming after it,then>remove 2k+1 from the set.> Your argument is that the remaining set would still have an oddnumber,>but it's not hard to see that the set {1,2,...,N} up to ANY N at allwould>never have an odd number.>Huh?>Consider a finite set (1,2,3).>Using your algorithm I get (2,3).>There is no odd number that comes after 3 in the set (1,2,3).> You missed the fact in elementary school that if n is some odd> number, then so is n+2.I only see two odd numbers in the set (1,2,3).3+2 = a number not in my set.3+2 must not be finite.> I'm not talking about a finite set, I'm talking about taking> the natural numbers and applying that process to it. Your hypothetical> tape supposedly had all the natural numbers on it. Once the process> is complete, then any set you look at {1..N} will have all its odd> numbers removed.Supposedly is the key word here.There will be such an N. N could be quite large (it might even equal 3).Obviously, the original tape didn't contain every natural number.> Again, make sure you learn about the difference between 'artibrarily> large' and 'infinite.'Why don't you give me a TM that decides if a string is'arbitrarily large' or 'infinite.'> The set {1, 11, 111, 1111, 11111, ... } will contain strings that are> arbitrarily long but will not contain the infinite string 1111....And you have an algorithm that proves this?How do you know the input tape doesn't contain an infinite string of 1's?How do prove there is such a thing as an infinite string of 1's?I have given a proof that every string is finite as far as a TM isconcerned,even for TM's that perform an infinite number of operations.Russell- 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural <3_OdnTxoJNyqVK_dRVn-hA@comcast.com> <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org <87ptcau3xj.fsf@phiwumbda.org Assumption: There exists a 0 at the end of computation.> Of course there is. How can there not be?> My TM will never overwrite the last 0. This is the source of your error (and just about all your errors.) Thereis no last 0 on the *infinite* list, since there will always be a 0after it.> But then if the tape originally had all the natural numbers,> That's a big if, isn't it? That was your assumption and you construction. *IF* your premises held,then a lot of wrong things happen.> You missed the fact in elementary school that if n is some odd>number, then so is n+2.> I only see two odd numbers in the set (1,2,3).> 3+2 = a number not in my set.> 3+2 must not be finite. Again, don't try redefining already-used terms.> I'm not talking about a finite set, I'm talking about taking>the natural numbers and applying that process to it. Your hypothetical>tape supposedly had all the natural numbers on it. Once the process>is complete, then any set you look at {1..N} will have all its odd>numbers removed.Supposedly is the key word here.> There will be such an N. N could be quite large (it might even equal 3).> Obviously, the original tape didn't contain every natural number. Yes, it is obvious, and this is what everyone has been telling you allalong. (In case you forgot, you were the one claiming the existence ofsuch a tape) How many times do you need to be told that you will not havea tape with all the natural numbers on it?> Again, make sure you learn about the difference between 'artibrarily>large' and 'infinite.'> Why don't you give me a TM that decides if a string is> 'arbitrarily large' or 'infinite.' TMs don't decide anything about infinite things. They only accept finite-sized input. I'm telling you this again for the fourth time.> The set {1, 11, 111, 1111, 11111, ... } will contain strings that are>arbitrarily long but will not contain the infinite string 1111....> And you have an algorithm that proves this? No, that is what the notation means. Carry on the process. The set contains {1^k, for any natural number k} Perhaps you are confused about the fact that infinity is nota natural number.> How do prove there is such a thing as an infinite string of 1's? There isn't. There is no infinity in natural numbers. There arearbitrarily large numbers, though.> I have given a proof that every string is finite as far as a TM is> concerned, even for TM's that perform an infinite number of> operations. Every string that you give a TM *should* be finite in the first place. You're the one trying to feed a TM an infinite string and then trying to treat an infinite string as a finite number (The difference between 10, 110, 1110, 11110, ... , and your infinite string is that the infinite string does NOT has a last 0, like you claim.) ( Do you think that 1 - 0.999999999 has a last digit which is 1? )J=== === Subject: : Re: No Set Contains Every Computable Natural> Moreover, there is a very simple proof that there is no blank. Look> up at the paragraph that begins, Suppose that it contains a blank.>> Look at the definition of my TM.>> Suppose that it contains a blank. Step (2) will then search for>> another blank. The blank on the tape will not be overwritten>> unless a second blank is found. The output tape will ALWAYS> contain one blank. Even after an infinite number of steps.>> A TM never takes an infinite number of steps. After any step>> it takes, it was some finite time.>A number of people have stated that time is not an issue>for an idealized TM. I have shown this is equivalent to>assuming that a TM can perform an infinite number of>operations in a finite amount of time.No. You have stated that these are equivalent. You have failedto show it.>There is no difference between assuming that a TM>will still be computing long after the stars have turned>to dust and assuming a TM can perform an infinite number>of operations.Yes there is. Just as there is a difference between an arbitrarilylarge finite number and infinity.Alan-- Defendit numerus=== === Subject: : Re: No Set Contains Every Computable Natural>There is no difference between assuming that a TMwill still be computing long after the stars have turned>to dust and assuming a TM can perform an infinite number>of operations.> Yes there is. Just as there is a difference between an arbitrarily> large finite number and infinity.What is the difference between an arbitrarily large finite number andinfinity?Can you define an algorithm that will determine if a string of 1'srepresents a natural number or is infinitely long?Russell- Solution to halting problem: Ctrl Alt Del=== === Subject: : Re: No Set Contains Every Computable Natural>There is no difference between assuming that a TM>>will still be computing long after the stars have turned>>to dust and assuming a TM can perform an infinite number>>of operations.>> Yes there is. Just as there is a difference between an arbitrarily>> large finite number and infinity.>What is the difference between an arbitrarily large finite number and>infinity?One is finite and the other isn't.>Can you define an algorithm that will determine if a string of 1's>represents a natural number or is infinitely long?I think a sufficiently rigorous definition of algorithm requires thatthe input (or problem) be finite in size, so I suspect that what you areasking for is not within the scope of an algorithm. Alan-- Defendit numerus=== === Subject: : Re: No Set Contains Every Computable Natural>>There is no difference between assuming that a TM>>will still be computing long after the stars have turned>>to dust and assuming a TM can perform an infinite number>>of operations.>>> Yes there is. Just as there is a difference between an arbitrarily>> large finite number and infinity.>What is the difference between an arbitrarily large finite number and>infinity?> One is finite and the other isn't.>Can you define an algorithm that will determine if a string of 1's>represents a natural number or is infinitely long?> I think a sufficiently rigorous definition of algorithm requires that> the input (or problem) be finite in size, so I suspect that what you are> asking for is not within the scope of an algorithm.So there is no way to determine if a string is infinite?Why do you believe some strings are finite and otherstrings are infinite if there is no possible way to tellthe difference?Russell- 2 many 2 count=== === Subject: : Re: No Set Contains Every Computable Natural So there is no way to determine if a string is infinite?Not according to the scope in this thread. Of course if you have a humanholding both ends of the tape....Actually there's another question, if you have a theoretical peice ofstring, and you have an end in each hand but the middle of it extends toan infinite distance away from you before coming back is it of fintelength? Does such a string even exist?> Why do you believe some strings are finite and other> strings are infinite if there is no possible way to tell> the difference?We can proove the exitence of an infinte set. We can't (by itdefinition) give you a peice of paper with every member listed on it asit is impossible to list the second to last member.-- If you can read this you've gone too far.Stephen Jones (Zombywuf) D5A4 5342 E7BD E524 710A E44F E997 4422 7FEC E44E=== === Subject: : Re: No Set Contains Every Computable Natural I think a sufficiently rigorous definition of algorithm requires> that the input (or problem) be finite in size, so I suspect that> what you are asking for is not within the scope of an algorithm.That depends on context. In algorithmic information theory as developed by Chaitin, forinstance, the input tape is filled with 0's and 1's from the get-go,with no restriction on the number of 1's. (On the other hand, I don't want to press the claim that Chaitin isthe model of rigor in mathematics.)-- I've been thinking about my problems with getting any kind ofadmission that my math arguments showing the core error in mathematicsare correct, so I've gone to marketing books. -- James S. Harris, on when mathematics isn't enough=== === Subject: : Re: No Set Contains Every Computable Natural>>There is no difference between assuming that a TM>>will still be computing long after the stars have turned>>to dust and assuming a TM can perform an infinite number>>of operations.>Yes there is. Just as there is a difference between an arbitrarily>large finite number and infinity.> What is the difference between an arbitrarily large finite number and> infinity?The difference is, itself, infinite.> Can you define an algorithm that will determine if a string of 1's> represents a natural number or is infinitely long?No. Has anyone claimed to be able to?=== === Subject: : Re: No Set Contains Every Computable Natural (was Church-Turing compared to Zuse-Fredkin thesis)message>>message>>>> This is the definition of a recursively enumerable language:>>> This language is recursively enumerable. Given a string, x, there>exists>a>> TM that>> will halt after a finite number of steps if x is a member of L.>> This language is NOT decidable. Consider what happens if x is an> infinite string of 1's. The TM will not halt after a finite number> of>steps.>> You can't input an infinite string to the tape. If you could, it>wouldn't be a Turing machine.>No, but you can input finite instructions which generate infinite> output.>Russell is right about Turing saying this, and I found it repeated:>>But that would be a different language. The language would then be { M>is a TM that outputs a string a finite string of 1s}. This language is>clearly undecidable. But that does not prove his point, as any language>where (language U lang_complement) is composed of arbitrary TMs andwhose>membership is based on determining non-trivial properties of those TMs,is>undecidable (a la Rice's theorem). However L is certainly decidable:>Let me define a language, L, that consists of all unary representationsof>natural numbers.>1 = one, 11 = two, 111 = three, etc.>l8r, Mike N. Christoff>> You can't input an infinite string to the tape. If you could, it>wouldn't be a Turing machine.> Harris:>No, but you can input finite instructions which generate infinite> output.>Russell is right about Turing saying this, and I found it repeated:> Chaitin et al: This is a machine that performs an unendingsymbols.> [SH: If you read the thread, Russell previously claimed this.}> Harris added in last post:I posted not to argue on Russell's side, but because I thought this> Turing ingenuity deserved some recognition.I didn't get that far. You mention this at the very end, you should havementioned it at the beginning. I assumed you were equivocating Russell'sinfinite input on a TM input tape claim with the finite instructions onthe tape idea.As to your original point, I agree that not-halting or looping caneither mean repeating a sequence of states forever, or taking an unendingnumber of steps while never repeating a state. output as L: R -> R, atleast for some inf-in/out infinite state TMs. But where Wegner gets itwrong is to imply that since we seem to be mapping reals to reals,interaction machines are superTuring. What he fails to realize is thatalthough the cardinality of the languages recognized by these machines hasgone from countable to uncountable, we have not upgraded our definitionstating a computation must take a finite number of steps. (Note, it makessense to replace 'steps' with 'time', since steps implies a particular modelof computation). Maybe call this notion limit computable, or streamcomputable. For example, although it may be able to stream-compute pi, astream TM is not superTuring unless it could output pi in finite time.Languages would probably end up being defined in terms of the output setinstead of the input set, since one can no longer accept or reject an inputfor any inf-in stream TM. This would fit perfectly in terms of TMs thatstream compute numbers like pi, since the important stuff is in the outputnot the input. I'm remembering Wegner's interaction machines. This may be a completelift> of Wegner, but here's an idea. Define a so-called 'stream' TM that canread> a possibly infinite input stream and produce a possibly infinite output> stream. The TMs being discussed would then befinite-input/infinite-output> stream TMs. So instead of 'non-halting', you would say 'streaming'.> However, a stream TM would have both an internal tape, and a write-only> output tape. Any TM that can (*)-output its stream to any finite> specified length, in finite time, can be modified to use a seperate tape> for output in an obvious way.> To be a stream TM a machine must have one or both of : infinite output or> infinite input. Stream TMs can be either infinite or finite state. TheTM> discussed would have to be infinite state. Any finite state infinite-out> stream TM has an output that eventually repeats forever, therefore its> output set must be countable; each element representable by a rational.As> a generalization of Turing's rule (*), I propose : (**) for every finite> prefix O of the output, there exists a finite prefix I of the input, such> that O appears on the output within a finite number of steps after I is> read.> In the case of a stream language L, (and I definitely recall reading about> this in a Wegner paper), we can define L: input -> output as L: R -> R, at> least for some inf-in/out infinite state TMs. But where Wegner gets it> wrong is to imply that since we seem to be mapping reals to reals,> interaction machines are superTuring. What he fails to realize is that> although the cardinality of the languages recognized by these machines has> gone from countable to uncountable, we have not upgraded our definition> stating a computation must take a finite number of steps. (Note, it makes> sense to replace 'steps' with 'time', since steps implies a particularmodel> of computation). Maybe call this notion limit computable, or stream> computable. For example, although it may be able to stream-compute pi, a> stream TM is not superTuring unless it could output pi in finite time.> Languages would probably end up being defined in terms of the output set> instead of the input set, since one can no longer accept or reject aninput> for any inf-in stream TM. This would fit perfectly in terms of TMs that> stream compute numbers like pi, since the important stuff is in the output> not the input.> Note, mapping reals to reals assumes the input can be an uncomputable> sequence. Without this, even infinite-input/infinite-state stream TMscould> only output a countable set, since the computable reals are countable.Allowing uncomputable inputs, would be akin to standard TMs withuncomputable oracles. Basically, one has to wonder about, even thetheoretical benefit, of defining the input set as being uncountable, giventhat an uncountable number of those inputs will be both uncomputable, and(more importantly in terms of an ability to do algorithmic and output setanalysis) undefinable as well.l8r, Mike N. Christoff=== === Subject: : Re: No Set Contains Every Computable Natural (was Church-Turing compared to Zuse-Fredkin thesis)message>>Note, mapping reals to reals assumes the input can be an uncomputable>sequence. Without this, even infinite-input/infinite-state stream TMs> could>only output a countable set, since the computable reals are countable.> Allowing uncomputable inputs, would be akin to standard TMs with> uncomputable oracles. Basically, one has to wonder about, even the> theoretical benefit, of defining the input set as being uncountable, given> that an uncountable number of those inputs will be both uncomputable, and> (more importantly in terms of an ability to do algorithmic and output set> analysis) undefinable as well.I realize this is gross over-posting, but I just had to add this quote fromMathworld: In fact, all theorems of calculus remain true if the field of real numbersis replaced by the field of definable numbers, sequences are replaced bydefinable sequences, sets are replaced by definable sets and functions bydefinable functions.Quite interesting.l8r, Mike N. Christoff=== === Subject: : Re: No Set Contains Every Computable Natural (was Church-Turing compared to Zuse-Fredkin thesis)> I realize this is gross over-posting, but I just had to add this quote from> Mathworld: In fact, all theorems of calculus remain true if the field of real numbers> is replaced by the field of definable numbers, sequences are replaced by> definable sequences, sets are replaced by definable sets and functions by> definable functions.> Quite interesting.I find it quite natural!!!=== === Subject: : Re: No Set Contains Every Computable Natural <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org > I don't know what it means for N to be computable. I think it's>> pretty clear what I mean when I say N is a recursive set. Evidently,>> it's not clear to you, but maybe the above helps.> Given any set, S, of representations of natural numbers,> I can write a TM that will find a representation of a natural number> that is not in S. (No, this TM does not halt after a finite number> of steps.)This is, of course, simply false, false, false.If you give the set of all representations of natural numbers, thenyour TM will not find any representation of a natural number not inyour set.You're speaking nonsense again.Of course, we must be careful here and fix a convention for when atape represents a set of natural numbers. Let's say that a taperepresents a set of natural numbers if it consists of blocks of onesseparated by a single space.We must also be careful to require that your TM has the property that,for each square x of the tape, there exists a steps t such that for allsteps s after t, the symbol on the square x at time s is the same asthe symbol on x at t. After all, a TM which repeatedly changes the0th square to 1 and blank and 1 and blank cannot be said to produceany tape at all.Now, on an input tape containing 1, ,1,1, ,1,1,1, ,1,1,1,1, ,...,there is simply no possibility that your TM, even after an infinitenumber of steps, creates a tape with a representation of some naturalnumber not on my input tape. Every natural number is represented onmy tape. None is missing. Your tape either returns a set of numbersalready on my tape or it returns no set of numbers at all.>Obviously, a human has to decide if a symbol represents>a natural number. No TM can do this.>No TM can devise a convention, if that's what you mean. At least, no>TM can devise a convention in the sense that matters here.> But, a TM can extend the convention in such a way that no set> contains every symbol that represents a natural number.No, no, no. A TM cannot extend the representation of natural numbersfor two reasons: (1) It's a fucking Turing machine, isn't it? I don'twant to argue about requirements for intentionality or whether TMs arecapable of intelligence, but this TM is not part of the negotiationsregarding our conventions. (2) Every natural number has arepresentation in our convention, so the representation cannot benon-redundantly extended.You really aren't thinking very clearly.-- Come on people!!! The US just blew up a lot of people in Iraq, don'tyou realize that a person with my exposure might just end up dead, bymysterious circumstances? --James Harris, on the dangers of proving Fermat's last theorem=== === Subject: : Re: the anticlassicalist }{ i: linguistic negation>Klingon was specifically created to be the worst language possible by>folks who knew linguistics. It was enthusiastically embraced by the>mob and it is as good as Korean or Chinese for transferring content.>Okrand says he violated a few human language universals in inventing>Klingon, since of course it wasn't meant to be a human language. Nothing>about worst language possible. Studies of how Klingon-users actually>speak might be interesting. Do you know of any? And anyhow what was this>meant to prove, even if it were all true? Sure as hell wouldn't prove>whatever you were salivating about in the previous paragraph anyway.> You can find a significant community of people who speak Klingon at> the Klingon Language Institute.> http://www.kli.org/> The archives of the KLI's email discussion forum might be useful if> you're serious about studying how Klingon is spoken.> http://www.kli.org/tlhIngan-Hol/Not me personally, no. But I can see some interest in looking at whathappens to any conlang if it is regularly used by a lot of people. Hasanybody tried to bring up a child as a native speaker? Now that would beinteresting -- though they probably wouldn't want to advertise the factto the local child welfare authorities. If Okrand really did build somecounter-universal features into it, how a child would deal with it as afirst language would be really fun to watch.Ross Clark> As for worst language possible, that would be INTERCAL. Klingon is> actually a very easy language. Compared to natural languages, it's> essentially a toy, but it's a very powerful and interesting toy.=== === Subject: : Re: the anticlassicalist }{ i: linguistic negation> ... Has> anybody tried to bring up a child as a native speaker?Yes. http://www.wired.com/wired/archive/7.08/mustread.html?pg=8I'm aware of other children who have been exposed to spoken Klingonsince they were born, but it's been inconsistent (in one case, thefather is the skilled speaker, and he's often away on extendedmilitary service).> Now that would be> interesting -- though they probably wouldn't want to advertise the fact> to the local child welfare authorities. If Okrand really did build some> counter-universal features into it, how a child would deal with it as a> first language would be really fun to watch.Language acquisition apparently works no matter how un-natural thelanguage. The broken rules of Klingon seem to be more a matter ofgoing against evolutionary pressures rather than being contrary todeep human mental features. No human language can do this is toostrong; the best we can probably do is say That wouldn't survive inan evolving human language.=== === Subject: : Re: the anticlassicalist }{ i: linguistic negation>... Has>anybody tried to bring up a child as a native speaker?> Yes. http://www.wired.com/wired/archive/7.08/mustread.html?pg=8> I'm aware of other children who have been exposed to spoken Klingon> since they were born, but it's been inconsistent (in one case, the> father is the skilled speaker, and he's often away on extended> military service).>Now that would be>interesting -- though they probably wouldn't want to advertise the fact>to the local child welfare authorities. If Okrand really did build some>counter-universal features into it, how a child would deal with it as a>first language would be really fun to watch.> Language acquisition apparently works no matter how un-natural the> language. The broken rules of Klingon seem to be more a matter of> going against evolutionary pressures rather than being contrary to> deep human mental features. No human language can do this is too> strong; the best we can probably do is say That wouldn't survive in> an evolving human language.According to a strict Chomskyan view, a language constructed contrary toUG should be un-learnable (i.e. un-acquirable in the normalfirst-language way, though of course you could learn it as anintellectual exercise). Presumably the child would re-structure it intosomething else which conformed with UG. However, from the examplesgiven, it appears as Jacques says that the counter-universal featuresare rather trivial or perhaps not counter-universal at all.Ross Clark=== === Subject: : Klingon, UG, et UGG (Re: the anticlassicalist }{ i: linguistic negation)> Presumably the child would re-structure it into> something else which conformed with UG. Presumably :-) Into Neanderthalish perhaps?They say there are lots of UGs in it. Even UGGs.And GUGGs (Generalized Universal GrammarGlossematics, or whatever fancy jargon strikesyours [*])> However, from the examples> given, it appears as Jacques says that the counter-universal features> are rather trivial or perhaps not counter-universal at all.Nothing counter-universal at all. Only one thing, mentionedin the grammatical part of the dictionary... I don'tremember it clearly, and I just don't feel likehunting for it now. If memory serves vaguely enough,there is a certain type of relative clause which,either cannot be expressed, or cannot be disambiguated.It is not there by design, however, but by oversight.It has to do with two syntactic rules clashing, so thatyou cannot properly hook your relative clause onto the main clause. Big deal. Alas, the poor Yorick whom I knew ...Alas, that poor Yorick (I knew him)... will do just asnicely, thank you very much.[*] fancy, of course.=== === Subject: : Re: the anticlassicalist }{ i: linguistic negation> In message <1035esdt30h4d38@corp.supernews.com>, galathaea>Of course, you also snipped all the references to math and physics,> Did I blink? I haven't seen any physics yet.You should have realised that anything full of -ism and -ist which is spewedto sci.lang and sci.logic is not going to contain any physics.Franz=== === Subject: : Re: the anticlassicalist }{ i: linguistic negation <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com , Franz Heymann >> In message <1035esdt30h4d38@corp.supernews.com>, galathaea>>Of course, you also snipped all the references to math and physics,>> Did I blink? I haven't seen any physics yet.>You should have realised that anything full of -ism and -ist which is spewed>to sci.lang and sci.logic is not going to contain any physics.(Nor language; I can't speak for sci.logic ;-) I'm well aware of that. What I was wondering was whether the _OP_ realises that there's a difference between these isms and physics, and what it consists of.-- ard Herring=== === Subject: : Re: the anticlassicalist }{ i: linguistic negation: >You should have realised that anything full of -ism and -ist which isspewed: >to sci.lang and sci.logic is not going to contain any physics.: : (Nor language; I can't speak for sci.logic ;-):: I'm well aware of that. What I was wondering was whether the _OP_: realises that there's a difference between these isms and physics, and: what it consists of.Installment vi details the properties of the closed linear subspace latticeof a Hilbert space, detailing the relationship with observables andprediction in quantum systems. This builds off of the mathematical theoryYou do understand the physical content, do you not?-- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea: prankster, fablist, magician, liar=== === Subject: : Re: the anticlassicalist }{ i: linguistic negation <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.comIn message <1039dr5kpjloif1@corp.supernews.com>, galathaea >: >You should have realised that anything full of -ism and -ist which is>spewed>: >to sci.lang and sci.logic is not going to contain any physics.>: >: (Nor language; I can't speak for sci.logic ;-)>:>: I'm well aware of that. What I was wondering was whether the _OP_>: realises that there's a difference between these isms and physics, and>: what it consists of.>Installment vi >>Before you lies the Void...>details the properties of the closed linear subspace lattice>of a Hilbert space, detailing the relationship with observables and>prediction in quantum systems. This builds off of the mathematical theory>You do understand the physical content, do you not?Haven't seen any.Dirac compressed quantum mechanics into ~300 pages of terse elegance, and didn't use ontology once.-- ard Herring=== === Subject: : Re: the anticlassicalist }{ : mitchism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.com> <+No$awZ0HMNAFw0n@baesystems.com> In message <1039dr5kpjloif1@corp.supernews.com>, galathaea>: >You should have realised that anything full of -ism and -ist which is>spewed>: >to sci.lang and sci.logic is not going to contain any physics.>: >: (Nor language; I can't speak for sci.logic ;-)>:>: I'm well aware of that. What I was wondering was whether the _OP_>: realises that there's a difference between these isms and physics, and>: what it consists of.>Installment vi>>Before you lies the Void...details the properties of the closed linear subspace latticeof a Hilbert space, detailing the relationship with observables and>prediction in quantum systems. This builds off of the mathematical theory>You do understand the physical content, do you not?> Haven't seen any.> Dirac compressed quantum mechanics into ~300 pages of terse elegance,> and didn't use ontology once.Obviously, Dirac knew the tastes of the audience to whom he was writing. Theman had to get published, didn't he? Taxes needed to be paid. The wife andchildren needed to eat. Little things like that have often gotten in the wayof principle.But one must not to say Dirac did not have ontology in mind....Therefore, there is plenty of room for improvementand for seeking to discover a new and logically superiorfoundation for the theory. I suspect that it is simpleAristotelian logic (in the context of Platonic ontology)that is missing from the theory, and this is due to thepainful divorce of physics from philosophy.Fragmented people think fragmented thoughts thatnever add up and ultimately don't make sense, whateverillicit success may initially obtain. The deepest urge inmy being is to understand the principles of physicsfrom the first principles of logic. Without that ourclaim to knowledge is nought but a pretense, and oursouls are divided against themselves, leaving usculturally schizophrenic and socially insane. http://www.geocities.com/saint7peter/ DiracslectureonQFT.htmlWhat a great legacy you physics clowns leave everyone. Since Aristotlecomplained of fatalists in his own work, I doubt your kind of 'ism' offers muchhope for change. Moreover, Hobbes observed that the utility of mathematicsresides in building weapons of war. Surely, one cannot deny that physics--thepristine science of non-isms--has its ideological protectors.Methinks the Daleks doth protest too loudly.In any case, whatever Dirac did or did not do has little in common with thosepeople who advertise physics in the modern information age. Ontology flashesbetween every word like the old subliminal buy popcorn frames.:-)mitch----If one were actually paying any attention to anything but the choir and itslaw-giving preachers, one might actually have a clue to how mathematicalphysics is linked into philosophy by Heyting algebras.In Distributive Lattices Balbes and Dwinger define a closure algebra asfollows:A Boolean algebra L with an additive closureoperator ^c in which 0^c=0 is called a closurealgebra. An element (a e L) is called open if(a-bar e L^c). The set of open elements isdenoted by L^oThey make that definition in order to establish a representation theorem forHeyting algebras:If L is a closure algebra, then L^o is a Heytingalgebra under the partial ordering of L and is alsoa D01-subalgebra of LMore remarkable is the fact that every Heytingalgebra can be represented this wayNow, what one needs to recognize immediately is that the Heyting algebra hereis entangled relative to consequence relations in the closure algebra. InProtoalgebraic Logics by Janusz Czelakowski you will find the remark,Thus any consequence is a closure operator ona sentential language. Following logical tradition,the separate term 'consequence operation' isreserved to denote such closure operations.Since Mr. Herring is personally being such a pissant, perhaps he can explainwhat is meant by physical content anyway? Does that mean using words andphrases that depend on Cauchy's (?) epsilon-delta justification for thederivative or Robinson's non-standard analysis justifying naive use ofdifferentials? His apparent interest in Dirac extends to I let other peopledo my 'isms' so I can act the fool... just like the well-received FranzHeymannThe development of the limit statement and derivative in terms of open setsintimately links truth in physics with Heyting algebras. Cech's association ofa nerve with open sets links truth in physics with simplicial homology. Andthe work in foundations with topos have associated the logic of open sets withconsistent logic. In other words, the mathematicians have been justifyingphysicists' crap while they play at predicting how the machines they build willbehave in the experiments in which they imagine them to be used.The problem is understanding the complexity between the Heyting algebras andthe Boolean algebras. We get some sense of that with Kuratowski 14 setproblem,Consider the collection of all subsets A of thetopological space X. The operations of closureA ---> A-barand complementationA ---> X-Aare functions from this collection to itself.(a) Show that starting with a given set A, onecan form no more than 14 distinct sets by applyingthese two operations successively.Unfortunately, while that problem helps us a little bit with respect to closureoperations and complementation between closed sets and open sets, the number 14here is significant.For example, in tense logic we have Benthem reporting on the Hamblin FourteenTenses Theorem:Of course, questions of correspondence andaxiomatization are not the only queries arisingin this semantics. For instance, an obvious questionis, on any given temporal structure, what happensto the potentially infinite number of 'tenses' that is,sequences of operators F, P, G, H in our language.A good exercise to become familiar with thepeculiarities of our formalism is to prove Hamblin'sFourteen Tenses Theorem: On the real time axis, there are only 14 logically distinct sequences of operators.We cannot simply draw pictures if we are to understand what is happening here.I could go through a great deal of complexity, but no one would understand thateither. The fact of the matter is that problems must be reduced to what Fregecalled 'judgeable content.' The appropriate algebra for this is a de Morganalgebra,[Balbes and Dwinger]The reader should note that the operation ~ is, ingeneral, not set theoretic complementation.... there is a one-to-one correspondence betweenall complete de Morgan fields of subsets of a setX and the involutions on X.In other words, one first needs a restriction to statements capable ofrefutation and confirmation before embarking on an empirical science groundedin an envelope of refutations. So, where we understand logic with respect toconsequence relations, the modern Boolean logic and Heyting logic decompose thesituation to reflect that empirical sciences operate according to datasemantics. These are very close to intuitionistic logics.In any case, the next technical concept of interest the free Boolean algebras.This is because they satisfy the countable chain condition,[Balbes and Dwinger]Definition 20Let a be an element in a lattice L. A non-emptysubset (S subset L) is a-disjoint provided thatxy=awhenever x, y are distinct elements of S. S iscalled disjoint if L has a 0 and S is 0-disjoint. Alattice which has no uncountable disjoint subsetsis said to satisfy the countable chain condition.Then,The countable chain condition derives its namefrom the fact that in a complete Boolean algebra L,the countable chain condition holds if and only ifevery well ordered chain in L is countable. Theseconditions are not, in general, equivalent. However,in the course of this section, we will show that bothconditions hold in F_B(X) and F_D(X)F_B(X) and F_D(X) are the free Boolean algebras and the free distributivelattices.Here are two of the exercises that follow the remarks quoted above,Let X be a countable infinite set. Show that 2^Xsatisfies the countable chain condition but containsa chain isomorphic with the reals.Let Y be a set such that |Y|=2^(Aleph_0) and let Lbe the Boolean algebra of finite and cofinite subsetsof Y. Show that every chain in L is countable but thatL does not satisfy the countable chain condition.The first question should get us back to physics somewhat. I do believe theystill think in terms of real spaces, except perhaps with loop quantum gravity(they think in terms of tetrhedral simplexes and are wondering why some oftheir objects seem to require recursive definition. Go figure.)In fact, it tells us a little more. One of the questions in the metaphysics ofphysics asks why the mathematics reduces to probabilistic models built onHilbert spaces. Well, if we simply look at the discussion of descriptive settheory in Set Theory by Jech, we discover that there is a minor problem withLebesgue measure,[Jech]Although both 'null' and 'meager' mean in a sense'negligible,' the following exercise shows that thereal line can be decomposed into a null set and ameager set:Exercise 39.12. There is a null set of reals whosecomplement is meager[Let q_1, q_2, ... be an enumeration of the rationals.For each n>=1 and k>=1, let I_nk be the openinterval with center q_n and length 1/(k*2^n). LetD_k=bigcup_(n=1 to oo) I_nk andA = bigcap_(k=1 to oo) D_kEach D_k is open and dense, and m(D_k)<=1/k.Hence A is null and R-A is meager]With respect to the second problem from Balbes and Dwinger, we can concludethat the Boolean algebra of finite and cofinite subsets is not a free Booleanalgebra. Indeed, they go on to write,Every free Boolean algebra satisfies the countablechain condition.Every chain in a free Boolean algebra is countable.Recalling from above that the Heyting algebra representations are associatedwith closure spaces, we might get some sense of what is involved here by citingSchmidt's theorem,[Czelakowski]For a closure system C the following conditions areequivalent: (i) C is finitary (ii) C is inductive(iii) The union of every non-empty directed subset of C belongs to CIf C is a finitary closure system, then (C, subset) is analgebraic lattice and the sets J(X), X-finite are compactelements of this lattice.Well, J(X) refers to the closure operation, and, the difference betweenCzelakowski's closure systems and the closure algebras of Balbes and Dwinger isan additivity condition. Czelakowski's definition of finitary and inductiveused in the Schmidt's theorem is straightforward,[Czelakowski]The notions of a closure operator and of a closuresystem are thus coextensive. A closure operator Jon a set A is called finitary, if for any (X subset A)and (a e A)if (a e J(X)), then (a e J(X_f) for some finiteset (X_f subset X)A closure system is finitary if the correspondingclosure operator is finitary....A family C of subsets of A is said to be inductiveif the *set-theoretic union* of every non-emptychain in C belongs to C (A chain is being understoodhere as a chain with respect to set-theoretic inclusion).In case anyone missed my long discourses on Tore Langholm's Partiality, Truth,and Persistence, the notion of a neighborhood assignment with which heanalyzes truth definitions is contrary to the description of finitary closureoperators,For a given model M, a neighborhood assignmentis a function eta from the finite subsets of |M| tosubsets of |M| satisfying the following,1. (D subset eta(D))2. if (D_0 subset D_1) then (eta(D_0) subset eta(D_1))Since Langholm is investigating first-order logic relative to strong Kleenetruth, however, this does make sense. In the current context, closure systemsrefer to loci separated from the loci in which Langholm is interested.Apparently, it takes no more than seven alternations of closure andcomplementation to achieve the separation needed to satisfy the compactnesscondition for (X-finite) in a finitary closure system.If we simply ignore the closure/complementation relationship, we are reflectedelsewhere.As per the statement above, the representation for a Heyting algebra admits itsinterpretation as a D01-subalgebra. According to Balbes and Dwinger, there areno projectives in D and the only projective in D01 is 2 ({0,{0}}). Withrespect to world semantics (whatever) the 2-universal models in http://www.illc.uva.nl/Publications/ResearchReports/MoL-2001- 09.text.pdfare asssociated with 7 graphs generating 26 Heyting algebras. So, there isthe magic (Dalek) number of 26 again.Do the Daleks see what comes of running around telling people one knows what istrue and false in the universe? Refuting other people's belief systems is notthe same as truth. Ahh,... but such GREAT BOMBS!For my part, I am guessing that the concept of a question for datasemantics/strong Kleene truth seems to reside with the free de Morgan algebraon one generator, 1 | | * / / a ~a / / * | | 0The proof concerning projectives in D and D01 involves a map from 3 into 2x2, 2 -------> (1,1) | / | / 1 ---> (1,0) (0,1) | / | / 0 -------> (0,0)0 ---> (0,0)1 ---> (1,0)2 ---> (1,1)But, when 3 is taken as a de Morgan algebra, 1=(~1). So, a comparable diagramin the class of de Morgan algebras to the free algebra on one generator wouldsuggest dissociation of a Shannon bit into yes and no distinguishablefrom 0 and 1. It is a simple and meaningful construct.I mentioned Hamblin's tense logic above in comparison with Kuratowski's 14 settheorem. Malinowski specifically notes that tense logic is a subclassicallogic to which presupposition via negation applies, http://www.uni.torun.pl/~jacekm/bimatrix.pdfMoreover, that process of discerning a presumed consequence only works where deMorgan laws hold. Otherwise, there is simply no logic known (to Malinowski)that enables one to investigate presupposition. That is why I believe aquestion in quantum mechanics (per Mackey's seventh axiom) must be understoodwith respect to idempotents in the sense of de Morgan.This kind of thing is already appearing in quantum topology with theformalization of idempotent Jones-Wentzel projectors.Are the Daleks confused yet?Welcome to Cantor's paradise where ALL means ALL.lol:-)mitch=== === Subject: : Re: the anticlassicalist }{ : mitchism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.com>> In message <1039dr5kpjloif1@corp.supernews.com>, galathaea>>: >You should have realised that anything full of -ism and -ist which is>>spewed>>: >to sci.lang and sci.logic is not going to contain any physics.>>: >>: (Nor language; I can't speak for sci.logic ;-)>>:>>: I'm well aware of that. What I was wondering was whether the _OP_>>: realises that there's a difference between these isms and physics, and>>: what it consists of.>>>Installment vi>>>Before you lies the Void...>>details the properties of the closed linear subspace lattice>>of a Hilbert space, detailing the relationship with observables and>>prediction in quantum systems. This builds off of the mathematical theory>>>You do understand the physical content, do you not?>> Haven't seen any.>> Dirac compressed quantum mechanics into ~300 pages of terse elegance,>> and didn't use ontology once.>Obviously, Dirac knew the tastes of the audience to whom he was writing.The Nobel committee?> The>man had to get published, didn't he? Taxes needed to be paid. The wife and>children needed to eat. Little things like that have often gotten in the way>of principle.I think you may be confusing him with Hawking. I doubt if the royalties on PQM, which is hardly a dumbing-down for popular tastes, would have kept him in hot dinners.>But one must not to say Dirac did not have ontology in mind....Therefore, there is plenty of room for improvement>and for seeking to discover a new and logically superior>foundation for the theory. I suspect that it is simple>Aristotelian logic (in the context of Platonic ontology)>that is missing from the theory, and this is due to the>painful divorce of physics from philosophy.Fragmented people think fragmented thoughts that>never add up and ultimately don't make sense, whatever>illicit success may initially obtain. The deepest urge in>my being is to understand the principles of physics>from the first principles of logic. Without that our>claim to knowledge is nought but a pretense, and our>souls are divided against themselves, leaving us>culturally schizophrenic and socially insane.> http://www.geocities.com/saint7peter/ DiracslectureonQFT.htmlAre you sure all those words are Dirac and not Mutnick? The way it's laid out, the separation of quotes and comment isn't entirely clear.http://www.geocities.com/saint7peter/index.html>Since Mr. Herring is personally being such a pissant, perhaps he can explain>what is meant by physical content anyway? Does that mean using words and>phrases that depend on Cauchy's (?) epsilon-delta justification for the>derivative or Robinson's non-standard analysis justifying naive use of>differentials?No, it means that at some point you have to hook into what physicists *observe*. Will it tell us the energy levels of a hydrogen atom and predict the Lamb shift? If you're interested in developing a better mathematical formalism, that's fine, go ahead. Just don't tell us it's physics.> His apparent interest in Dirac extends to I let other people>do my 'isms' so I can act the fool... just like the well-received Franz>Heymann>The development of the limit statement and derivative in terms of open sets>intimately links truth in physics with Heyting algebras. Cech's association of>a nerve with open sets links truth in physics with simplicial homology. And>the work in foundations with topos have associated the logic of open sets with>consistent logic.... and it would help your case if you could make it read less like the hermeneutics of quantum gravity.[snip again ]-- ard Herring=== === Subject: : Re: the anticlassicalist }{ : mitchism for Daleks: No, it means that at some point you have to hook into what physicists: *observe*. Will it tell us the energy levels of a hydrogen atom and: predict the Lamb shift? If you're interested in developing a better: mathematical formalism, that's fine, go ahead. Just don't tell us it's: physics.Actually, all I did was point to the connection between the foundationalobjects of the theory and the observational objects it predicts. I calledthem the ontology and the epistemology of the model in a rigorous way toaccord somewhat with common usage, but any other names would work. But theformalism is all about observational propositions (what is the likelihoodthat A and B happen?, if C happens, what does that imply for D?, wherethe letters stand for quantum events like spontaneous decay). Anywhere youhave time series of quantum events or concurrent systems, you implicitly orexplicitly use the logic I mention, often in an algebraic setting.It _is_ physics. I _am_ interested in developing a better mathematicalformalism, but alas cannot take credit for this one. This one has beenstudied by physicists now for 3 quarters of a century.-- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea: prankster, fablist, magician, liar=== === Subject: : Re: the anticlassicalist }{ : mitchism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.com>Since Mr. Herring is personally being such a pissant, perhaps he can explain>what is meant by physical content anyway? Does that mean using words and>phrases that depend on Cauchy's (?) epsilon-delta justification for the>derivative or Robinson's non-standard analysis justifying naive use of>differentials?> No, it means that at some point you have to hook into what physicists> *observe*. Will it tell us the energy levels of a hydrogen atom and> predict the Lamb shift? If you're interested in developing a better> mathematical formalism, that's fine, go ahead. Just don't tell us it's> physics.It is not about the formalism. It is about whether assumptions in the formalismthat make it coherent are even being respected.Robert Hermann discusses a particular negative result that was ignored for sometime:Dirac and his successors among the physicists werealways vague (purposely and wisely so, it turned out)about which Lie algebras fo classical observables thiswas supposed to apply to. The question was onlytreated in the beginning of a reasonable mathematicalway by L. van Hove in 1951. He showed that onecould not completely 'quantize' a classical system bythe simple-minded way suggested by the work of thepioneers. [...]Of course, van Hove's negative general result did notconstrain the physicists from using Dirac's idea to setup quantum systems which seemed to be an adequateversion of the classical ones. What they did in practicewas to choose not *all* such classical observables,but a 'physically reasonable' set, and represent it byoperators, with Poisson bracket -- so far as it wasdefined within the set -- going over to operatorcommutator.Your claim translates into a definition for physics as nothing more than achaotically organized aggregate of partial results. If that is the case, you shouldnot be relying on mathematical formalism for justifications--as in the many times Ihave seen the statement it works. If mathematics is a symbolic tool with noinherent connection to the material and the natural language descriptions arenonsensical, then you are just idiots generating random information.:-)mitch=== === Subject: : Re: the anticlassicalist }{ i: linguistic negationgalathaea> linguistic negationgalathaea> It is not impossible to do that.Notice: my english, franch and spanish knowledge is limited and possibly wrong.FYIin italian (derived from Latin),double negation is grammatically correct and commonly used,semantically sometimes is double meaning (1- the double negationis considering as an enforcement of the concept is being negated,2- the boolean not not x = x is just only a possible interpretation.an example:non ho visto niente= i didn't see anything (right whole meaning translation)but literally (local mean, such as word by word / mean by mean)is i didn't see nothingthe strange is that sometime the logical meaning is biforked andtreated as a paradox, but it isn't paradox, it's bad logicalgrammar definitionnon voglio offendere nessuno= i want not offend someone (right whole translation 1)literally:i would not offend nobody ( ~circa~ right translation 2)which means the opposite [if the last one is not correct in english,then the meaning i tried to translate was i want offend someone]All the two opposite translations/interpretations are right.there is another courious difference in logical keywords:with and without, [economically abbreviated by w, w/o and w/?]in Italian (with-con, without-senza) are the two unique translations,the meaning of the two word is opposite (and complementary), butsenza(whithout) doesn't contain con(with). This mean that italianlanguage choosed to create a new word to express an opposite concept.I noticed because.. Italian was my first language, in that time...(my parent remember me) i used the undefined con senza(wrong)in my first try to express `without' concept)in spanish con / sin (similar to italian)in franch avec / avec pas (or pas avec)Saluti,MARco mar.i.am-- x(t),y(t) = th(3t-34.5)*e^[-(3t-34.5)^2]/2-4.3+e^(-1.8/t^2)/(.8*atg(t-3)+2 )(t-1.8)-.3th(5t-42.5),(1.4e^[-(3t-34.5)^2]+1-sgn[|t-8.5|-.5]* 1.5*|sin(pi*t)|^[2e^(-(t-11.5)^2)+.5+e^(-(.6t-3.3)^2)])/(.5+t) +1 ; 0Okrand says he violated a few human language universals in inventing>Klingon, since of course it wasn't meant to be a human language. > I've got the Klingon dictionary (which is actually a handbook),> I've got the tapes, I've got the later Klingon Berlitz Guide> (not its true title, it's the Klingon dictionary again,> adapted for galactic hitchhikers, can't remember its real> title),...(_Conversational Klingon_ and _Klingon for the Galactic Traveler_)> I've been through the lot, I haven't come across > anything even remotely weird.The standard example of weirdness is the division of color words. Nohuman language ever splits things the way Klingon does, groupingred/orange in one word and yellow/green/blue in another.There's also some major weirdness in the inventory of sounds. There'sa standard English /t/ but no corresponding /d/; the retroflex voicedstop that takes its place has no unvoiced counterpart. The weird partis not in what sounds are used, but rather in what sounds are missing.=== === Subject: : Re: Klingon, anti-universals etc. (Re: the anticlassicalist }{ i: linguistic negation)> The standard example of weirdness is the division of color words. No> human language ever splits things the way Klingon does, grouping> red/orange in one word and yellow/green/blue in another.Doesn't strike me as weird. I know a language whichhas no word for orange, no word for blue, which hasthree words for yellow and green, but does not differentiatebetween them by colour, but by brightness (or perhaps bysaturation, I never quite figured it out).> There's also some major weirdness in the inventory of sounds. There's> a standard English /t/ but no corresponding /d/; the retroflex voiced> stop that takes its place has no unvoiced counterpart. The weird part> is not in what sounds are used, but rather in what sounds are missing.You have led a linguistically sheltered life. The Lolovuevue languageof Oba Island has only voiced, prenasalized stops ([mb], [Ng], [Ngmb]),butno [p], no [k], no [kp], let alone a [b], a [g], or a [gb]. But it doeshave a /t/ which is always a straight [t]. And it does not have a [nd].As for the language of Shark Bay, it has two s's, but only one ts, andits speakers cannot tell the difference between s and sh.There is nothing weird in the Klingon inventory, not major, not minor.=== === Subject: : Re: Klingon, anti-universals etc. (Re: the anticlassicalist }{ i: linguistic negation)>Okrand says he violated a few human language universals in inventing>Klingon, since of course it wasn't meant to be a human language.> I've got the Klingon dictionary (which is actually a handbook),> I've got the tapes, I've got the later Klingon Berlitz Guide> (not its true title, it's the Klingon dictionary again,> adapted for galactic hitchhikers, can't remember its real> title), I've been through the lot, I haven't come across> anything even remotely weird. If you want something really> weird, try Lojban instead.> (No apologies for cross-posting to alt.phil, sci.m, sci.l,> sci.p, someone out there started this , not me)Yp, I've got the dictionary, too, and my recollection from reading itquite a few years ago is that there was nothing glaringly weird aboutit. I was just quoting something Okrand said in an interview. Someoneonce told me that he had based it on some indigenous language ofnorthern California, which would not be implausible, since many graduatestudents at Berkeley in those days worked on those languages. But I havenot heard anything more specific than that, so it can't be too obvious,even if he got some ideas from there.Ross Clark=== === Subject: : I got low score on math test, please advise me and take a lookmy website states my case and has jpg files of the four pages of the testplease take a look and advise me or give me opinionshttp://www.cho.us=== === Subject: : Re: I got low score on math test, please advise me and take a look> my website states my case and has jpg files of the four pages of the test> please take a look and advise me or give me opinions> http://www.cho.usYou got a 77 on your first calc test. A brilliant woman, advanceddegrees in Divinity, self-taught in Greek, told me that the only thingthat kept her out of Phi Beta Kappa was a 69.5 average in first-yearGerman (instead of a 70). You are not badly off.If you want to remediate a W in Calc I, work with someone who is doingwell in the course and then take it over. Do exercises -- somethinglike Calculus on Web at www.math.temple.edu -- and you increase yourchances.David Ames=== === Subject: : Re: I got low score on math test, please advise me and take a lookA 77 is not that bad in a begining Calc class. Why would you want todrop?Just study more and practice more problems for the next test, you canbring the grade up.Dropping a class is going to make your college years slower, becauseyou are going to have to take this class sometimes , and who knowsyour next teacher might be harder.Plus, write more legible on the next test.=== === Subject: : Re: I got low score on math test, please advise me and take a look> my website states my case and has jpg files of the four pages of the> test please take a look and advise me or give me opinions> http://www.cho.usAside from the all the politics of whether the grade was fair and if thetest should be returned in time, I have a few comments. First, for all the instructors out there, how many of you take of 1/4 ofa point? To me, if you make the problem worth 8 points, anything lessthan perfection is 7. And, when I am assigning point values toproblems, I usually look over the problem and pick out the two or threeconcepts being tested in that question and give each of those concepts avalue of 1 or 2 or even 3 points. I just can't igaine trying to gradewith fractional points. But, that is just my grading style and I'm notreally saying anything is wrong with giving fractional points,I justfind it uncommon. Secondly, if I am grading and I cannot follow the work (perhaps becauseit is not neatly written), I have a hard time giving full credit. Thatmight sound harsh, but I have often found that students write two orthree possible answers for the problem in their mess and never clearlymark which one they want me to grade. Also, judging from your scans ofthe page, it looks like you had the back of each page left blank (whichis pure speculation on my part). I despise it when students try tosqueeze in something in the margins, leaving the whole back side of apage blank. You aren't going to use that page for anything else, whysqueeze things together? Now, if you really want to get your instructormad, you could ask him (or her) to please write the comments on yourtest neater. But that would serve no purpose otehr than to point youout as a snot-nosed brat. Okay, about the grading. Personally, I don't know what topics werestressed in class so I don't feel like I can make a judgement call. Forexample, I gave a test in Trigonometry II on Tuesday (which I am in theprocess of grading and won't be given back until Monday). For this testthe material I stressed was changing degrees to radians, law of sines,and law of cosines. On one question I asked about the area of atriangle. If a student accidentally used the formula area = base *height, I will not take off many points probably one out of 10 points,because they were not being tested on geometry (and had they asked methe correct formula, I would have told them, since that wasn't the topicthey were being tested on). Or, maybe another example, if someonecomputed 2+3=6 in part of a larger problem, multiplying instead ofadding, and they correctly carried that mistake through the problem(that is if they had used 5 they would have gotten the correct answer) Iwould probably only take off one point out of 10. Those are my thoughts. Your grade, suck it up. If you ask theinstructor to look over the test again, he or she might find some placeswhen you should hae lost more points that you did. At least, if someonequibles over two or three points I say I will look it over, and I do andmake sure I *took off* all the points I should have. - Tim-- Timothy M. BrauchGraduate StudentDepartment of MathematicsWake Forest Universityemail is:news (dot) post (at) tbrauch (dot) com=== === Subject: : Re: I got low score on math test, please advise me and take a lookTim,I had run into a few professors throughout my college years who were asarrogant and utterly unthoughtful (in the analytical way, not in the loveydovey way) as you are; luckily, I do not find quite so many in graduateschool. Allow me to explain: Your students are purchasing a service fromyou, if the course title is Calculus and the description in the handbook is'Introduction to Calculus. Topics include convergent sequences, limits,differentiation [... insert additional Calculus topics here ...]' then yourstudents are purchasing a course in Calculus. Your comments about markingstudents wrong when it takes additional time to read their answer and yourpersonal vendetta against students squeezing work in the margin when thereis a back page are fine for your own time - in fact, you can even explain toyour students that you, personally, like this and don't like that. But thesecond that you take points off of a student's grade for work that you don'tlike, you are commiting fraud and breaking the law in the worst way.You see, not only have you, mid course, decided that Calculus will no longerbe the topic of discussion, but (addressing the 'in the worst way' comment)your misleading of the students and change of topic may very well destroy astudent's whole future. Imagine the scenario where a student is geneticallypredispositioned to write in margines, but is Godel smart in mathematicallogic and wants to study at Princeton University. A grade of 'F' in the'Test taking, my way' class that the student inadvertently signed up for(entitled 'Calculus') could very well destroy these chances.I don't expect you to understand the above argument, but perhaps you'llunderstand this and copy the behavior. Whenever I ran into a professor whothought so highly of themselves that they dared to dictate the format that astudent could write their test answers, above and beyond the standard 'showyour work, write so that I can read it and circle your final answer' (butnever 'look to see if there is a back page, if so: do not write in marginif not: blah blah blah' etc.), they were always very untallentedtheoretically. That is, they would never be able to understand the'theoretical' scenario that I proposed above, they would never be able tounderstand the fact that they are 'theoretically' changing the course topicand commiting fraud (false advertising). This lack of tallent invariablycarried over into mathematics (or computer science, depending upon thecourse) as well. I never ran into a very smart professor who did this sortof thing. Perhaps you'll want to be thought of as smart, too, and get downfrom your high horse.Please note, I think that this 'Spockie Hendrick' guy is a pathetic whineyexcuse for a student who refuses to read his handbook which, almostcertainly, states that a student can't drop a corse past the drop date forany reason whatsoever. This post has nothing to do with 'Spockie Hendrick'... This is only addressed to Tim and his arrogant thought processes thattell him that his way of test taking is right and must be spread around theworld.>my website states my case and has jpg files of the four pages of the>test please take a look and advise me or give me opinions>http://www.cho.us> Aside from the all the politics of whether the grade was fair and if the> test should be returned in time, I have a few comments.> First, for all the instructors out there, how many of you take of 1/4 of> a point? To me, if you make the problem worth 8 points, anything less> than perfection is 7. And, when I am assigning point values to> problems, I usually look over the problem and pick out the two or three> concepts being tested in that question and give each of those concepts a> value of 1 or 2 or even 3 points. I just can't igaine trying to grade> with fractional points. But, that is just my grading style and I'm not> really saying anything is wrong with giving fractional points,I just> find it uncommon.> Secondly, if I am grading and I cannot follow the work (perhaps because> it is not neatly written), I have a hard time giving full credit. That> might sound harsh, but I have often found that students write two or> three possible answers for the problem in their mess and never clearly> mark which one they want me to grade. Also, judging from your scans of> the page, it looks like you had the back of each page left blank (which> is pure speculation on my part). I despise it when students try to> squeeze in something in the margins, leaving the whole back side of a> page blank. You aren't going to use that page for anything else, why> squeeze things together? Now, if you really want to get your instructor> mad, you could ask him (or her) to please write the comments on your> test neater. But that would serve no purpose otehr than to point you> out as a snot-nosed brat.> Okay, about the grading. Personally, I don't know what topics were> stressed in class so I don't feel like I can make a judgement call. For> example, I gave a test in Trigonometry II on Tuesday (which I am in the> process of grading and won't be given back until Monday). For this test> the material I stressed was changing degrees to radians, law of sines,> and law of cosines. On one question I asked about the area of a> triangle. If a student accidentally used the formula area = base *> height, I will not take off many points probably one out of 10 points,> because they were not being tested on geometry (and had they asked me> the correct formula, I would have told them, since that wasn't the topic> they were being tested on). Or, maybe another example, if someone> computed 2+3=6 in part of a larger problem, multiplying instead of> adding, and they correctly carried that mistake through the problem> (that is if they had used 5 they would have gotten the correct answer) I> would probably only take off one point out of 10.> Those are my thoughts. Your grade, suck it up. If you ask the> instructor to look over the test again, he or she might find some places> when you should hae lost more points that you did. At least, if someone> quibles over two or three points I say I will look it over, and I do and> make sure I *took off* all the points I should have.> - Tim> -- > Timothy M. Brauch> Graduate Student> Department of Mathematics> Wake Forest University> email is:> news (dot) post (at) tbrauch (dot) com=== === Subject: : Re: I got low score on math test, please advise me and take a lookA bit unclear what's happening. Would a W be counted as units attempted? Ifyou met your deadline, then the lack of proper results for successfullydropping and deleting enrollment records for the course is merited. Your truely bigger problem is learning the material and gaining useful credit. That will require first, RESTUDY on your own, and then COURSE REPETITION toactually study a second (or third) time. Lack of concentrated, repeatedpractice and study prevents some fairly normal people from succeeding inMathematics. G C=== === Subject: : Re: I got low score on math test, please advise me and take a look> my website states my case and has jpg files of the four pages of the test> please take a look and advise me or give me opinions> since I took the test last Monday I feel 48 hours was ample time to > return the test beck on Wednesday especially with the withdraw without > a w date near. Furthermore the test could have been made available by > Tuesday, that is one week and one day from the test date.> I feel I should be allowed to withdraw without a W under these > circumstances.Frankly, you're wrong on a number of levels.It would be a matter of courtesy if the professor would get theexam back quickly so that these decisions could be made, buthe is under no professional obligation to do so, and 48 hoursis not a very long time. Here's a few issues:1. It's not fair if wealthier students purchase higher GPA'sbecause they can afford to take lower course loads and spend6 years getting a 4-year degree. It's even more unfair ifthey get to spend several weeks test driving their coursesand throw away the ones they are getting bad grades in.It's even more worse that, as you are doing, they throw awaythe courses they are getting _average_ grades in. This isthe main reason there is a W deadline. Probably you hadother feedback upon which to make your decision.2. Education is not about working the system to manufacturethe highest possible GPA, but about learning and maturing. TheGPA is a (rough) measurement of how much learning and maturinghas taken place. If you succeed in beating the system byartificially elevating your GPA, you won't have any idea whether you've learned anything or matured properly. 3. It takes a long time to evaluate an exam properly. I knowprofessors who grade (and brag about it) stacks of 70 or 100math exams overnight, but watching them in action, the techniqueis strictly slash and burn. I usually take a week, even ifthe stack is only 10 or 20 high. I read the exams for patternsin the overall understanding of the class to see if I need togo back and re-cover something, and I think about other adjustmentsto the curriculum. Besides evaluating each student, I need toevaluate myself and the flow of the course.I would like you to notice two things about your own case:A: You complain that the test was not evaluated accurately.B: You complain that it was not evaluated quickly enough.These are very contradictory wishes. 4. In the course of my career, I've had the un-pleasure of havingto proctor math placement exams to incoming freshmen. These testsare multiple-choice, fill in the bubble and the only result of thetests is to place students in Basic Math / Intermediate Algebra /College Algebra / Precalculus / Calculus. The idea is to evaluatethe preparation of the student and put him in the course he's ready for. For some crazy, unknown reason, these 18-year-olds cheat on thesetests. There's no GPA involved here. The only penalty for successfully cheating is to be placed in a math course which is too hard for you. It seems to me that a person of such low character as would cheat, would also figure out that the thingone wants to do in this situation is to purposely get a low score in order to be placed in an easy course and get a cheapA. But every Summer, there they sit, with their baseball capspulled down over their eyes trying to peek across the aisle........and I just let 'em.And the point of this nearly irrelevant story is that sometimesa guy needs to sit back and blink a couple of times and saywhat am I doing here? You're taking Calculus for a reasonand you got a test score that was lower than you wanted. I'ma bit surprised that your immediate _reflex_ is to drop the class.Most students' immediate reflex would have been to begin studyingharder. What is the reason you are taking Calculus? If that reason isimportant (such as part of an engineering major) then puttingit off is likely a bad idea. My story is supposed to show thathigh school students have a sort of mentality about things whichcontrols their decisions. It's important to get high scores _period_. So I'll cheat on the placement test. Likewise,undergraduates have some kneejerk reactions to situationsthat cause them to make decisions without thinking.You think you deserved an 84 on the test, but you got a 77. Ihave very little data to work with, but I can't help but thinkthat you could make up the 7 points and not drop the class atall.Bart=== === Subject: : Re: I got low score on math test, please advise me and take a look>my website states my case and has jpg files of the four pages of the test>please take a look and advise me or give me opinionsWell, before looking at the test I see whining about how you shouldbe allowed to drop after the drop date because it took more than48 hours for the instructor to return the test. My opinion on thisis (i) asking us for our opinion on policies at your school is utterlystupid, (ii) whining about the policies is stupid. _Is_ there a rulesomewhere stating that tests administered shortly before thedrop date must be returned by the drop date? I didn't think so._Did_ the instructor promise to return the test by the drop date?I didn't think so.I don't know about the rules where you are, but _here_ there_is_ a rule that states explicitly the test wasn't returned on timeshall _not_ be a valid reason to be allowed to drop past theofficial date.Good luck with that.Now glancing at the test:Can't see page 1: I get a page-size jpg, but it's blank exceptfor the to inch or so. Same problem with page 2.I can see page 3. First thing I notice is the complaint aboutneatness - expecting someone to be able to read that sortof scrawl is totally unreasonable. Let's see:problem 8: 9.75 points out of 10. I'm not sure why he tookoff a quarter point, possibly because he couldn't read it.Let's say to be generous you have another 1/4 point coming there.problem 9: 4.5 out of 5. I would have given you fewer pointsthan that, because there's a bit of explanation missing(you need to point out that the function is not even_defined_ at the point in question, because thatdenominator vanishes). Also your sketch is _not_ whatthe function looks like - where did that extra humpcome from? I'd give you at _most_ 3 points, probably less;that 1.5 down, so my score is 1.25 below what you gotso far.problem 10: 1 out of 5. You have 0 out of 5 coming onthat one - it _said_ using properties of limits and Idon't see a single property of limits mentioned, anywherein your solution. So far my estimate is 2.25 below what you got.problems 11, 12: you got full credit.problem 13: you missed 1.25 points. I can't quite tell fromthe jpg, it looks like he took off 1/4 point for writing f'(x)where you meant f'(a). I can't see where the other pointdeducted is, but I would have taken off more than aquarter point for the error I see, so we'll call this one even.problem 14: It looks like you lost 1.5 points for sayingv'(t) = -2. What you said is wrong; v'(t) = 2t - 6, it'sv'(2) that equals -2. 1.5 points seems reasonable to me.Well there you have it. I can only see half the test; onthe half that I can see you probably would have got afew points _less_ than you did if I'd been grading the test.My advice is next time be more careful about the details.And when you're asking to use your calculator to sketcha graph and you see a parabola draw something that_looks_ like a parabola! I mean on that one you justhad to copy what you saw on the screen...>http://www.cho.us************************=== === Subject: : Re: I got low score on math test, please advise me and take a lookX-ID: buRBAoZZ8ep-0neGpdIfA21ofva91P3Udt3Zut1ttQkdUU8eCml20c schrieb:>problem 9: 4.5 out of 5. I would have given you fewer points>than that, because there's a bit of explanation missing>(you need to point out that the function is not even>_defined_ at the point in question, because that>denominator vanishes).Am I missing something here? How can you say that a function isdiscontinuous at a point, where it's not defined? That function iscontinuous at each point of its domain - Unless of course you definef(2) = 3.01 or something like that.Thomas=== === Subject: : Re: I got low score on math test, please advise me and take a look> schrieb:>>problem 9: 4.5 out of 5. I would have given you fewer points>>than that, because there's a bit of explanation missing>>(you need to point out that the function is not even>>_defined_ at the point in question, because that>>denominator vanishes).>Am I missing something here? How can you say that a function is>discontinuous at a point, where it's not defined? That function is>continuous at each point of its domain - Unless of course you define>f(2) = 3.01 or something like that.Well yes. The meaning of continuous seems to shift a littlebetween calculus and mathematics on exactly this point...Regardless, _my_ point was just that he didn't give anadequate explanation of why that point is not includedin the set where f is continuous.>Thomas************************=== === Subject: : Re: I got low score on math test, please advise me and take a lookI would just stick to it. Any way 77% is just about 85%. Just be sure tostudy things when they come up and well now that you have had that test youshould make it your job to get a full understanding of every ting up to thatpoint. If you dont get something i am sure there is some one around that canhelp you.Personally i have only ever done subjects that are core to my degree. orthat i wanted to do so i have never think about getting out of it.In most uni's a score of 75-85 gives you a distinction. which is good.maths is not about grades its about understanding and fun! (i know its morethen that aswell)stephen> my website states my case and has jpg files of the four pages of the test> please take a look and advise me or give me opinions> http://www.cho.us=== === Subject: : [OT]Re: No Set Contains Every Computable Natural> --Come on people!!! The US just blew up a lot of people in Iraq, don't> you realize that a person with my exposure might just end up dead, by> mysterious circumstances?> --James Harris, on the dangers of proving Fermat's last theoremThese quotes are classic :) Note that in the 'shaquille o'niel in your backyard' quote, its funnier if you put his actual intention in. ie: not an'internet personality', but 'a gifted thinker who's better at math thanyou'. :)l8r, Mike N. Christoff=== === Subject: : Re: question about periodic functioncontinued..However, the bounds are +/- 2 as it is product of two cosinefunctions, and not sqrt(2) as erroneously stated above. Theapproximation procedure is valid for time/frequency purpose only, notamplitude!=== === Subject: : Re: question about periodic function> I am having trouble empirically determining the period of the followingfunction:y = sin(2*pi*.018*t) + cos(2*pi*.02*t)It is a situation of superposition of two waves with a slightlydiffering frequencies (.018,.02). One can hear beats in a tuning forkmathematical sense it is strictly not a periodic function with periodT as the definition f(x+T)=f(x) is not satisfied. But the followingapproximation is accepted in electronics engineering wave formanalysis and acoustics.Let 2 pi f t =w t be the angular argument. sin(w1 t) + cos (w2 t) = cos(pi/2- w1 t) + cos (w2 t) = 2 cos(pi/4+(w2-w1)t) cos(pi/4-w t) where w's w1~w2~w areapproximately equal.w2-w1 is neglected in comparision to pi/4, making it sqrt(2)cos(pi/4-w t).So it has maximum value sqrt(2) at wt=pi/4 and regular(high frequency) time period 2 pi/w and beating time period 2pi/(w1-w2) appearing as the envelope in the compound function graph.Works out to 1/.019 and 1/.002 or 52.6 and 500 seconds of regular(hf)and beat periods respectively.=== === Subject: : Re: question about periodic function>> I am having trouble empirically determining the period of the following>function:What do you mean empirically? Why not do it mathematically?>y = sin(2*pi*.018*t) + cos(2*pi*.02*t)>It is a situation of superposition of two waves with a slightly>differing frequencies (.018,.02). One can hear beats in a tuning fork>mathematical sense it is strictly not a periodic function with period>T as the definition f(x+T)=f(x) is not satisfied.Nonsense. It is periodic with period T = 500, because both the sin andthe cos components are.> But the following>approximation is accepted in electronics engineering wave form>analysis and acoustics.>Let 2 pi f t =w t be the angular argument. >sin(w1 t) + cos (w2 t) = cos(pi/2- w1 t) + cos (w2 t) >= 2 cos(pi/4+(w2-w1)t) cos(pi/4-w t) where w's w1~w2~w are>approximately equal.>w2-w1 is neglected in comparision to pi/4, making it sqrt(2)>cos(pi/4-w t).So it has maximum value sqrt(2) at wt=pi/4 and regular>(high frequency) time period 2 pi/w and beating time period 2>pi/(w1-w2) appearing as the envelope in the compound function graph.>Works out to 1/.019 and 1/.002 or 52.6 and 500 seconds of regular(hf)>and beat periods respectively.No approximation is needed.sin(2*pi*.018*t) + cos(2*pi*.02*t) = cos(2*pi*.02*t) - cos(2*pi*(.018*t+.25)) = -2*sin(2*pi*(.019*t+.125))*sin(2*pi*(.001*t-.125)) = 2*sin(2*pi*.019*(t-125))*sin(2*pi*.001*(t-125))So it's exactly your familiar scenario of carrier and beats, butstarting at a different time.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2=== === Subject: : Pascals TriangleThere was an interesting puzzle in the Sunday Times in the UK recently thatset me thinking about the issue that the puzzle raised.The idea is that of a set of points arranged in the familiar Pascal'striangle format (rows 1, 2, 3, ... etc. containing 1, 2, 3, ... points in atriangular format) after which the issue is that of the total number ofequilateral triangles that can be found in this set of points.This itself is not hard but the subsequent question seems more difficult -for such an arrangement with N rows, what is the minimum number of pointsthat need to be blocked in order to reduce this total number of equilateraltriangles to zero?I wondered if anyone here might have come across this or have some insightsabout a general solution. Brian Gladman=== === Subject: : Re: Pascals Triangle> There was an interesting puzzle in the Sunday Times in the UK recentlythat> set me thinking about the issue that the puzzle raised.> The idea is that of a set of points arranged in the familiar Pascal's> triangle format (rows 1, 2, 3, ... etc. containing 1, 2, 3, ... points ina> triangular format) after which the issue is that of the total number of> equilateral triangles that can be found in this set of points.For N in 1..10, I get 0, 1, 5, 13, 27, 48, 78, 118, 170, 235, which can beexpressed asfloor((N - 1) (N + 1) (2 N - 1) / 8)and appears in Sloane's database:http://www.research.att.com/cgi-bin/access.cgi/as/ njas/sequences/eisA.cgi?Anum=A002717> This itself is not hard but the subsequent question seems more difficult -> for such an arrangement with N rows, what is the minimum number of points> that need to be blocked in order to reduce this total number ofequilateral> triangles to zero?> I wondered if anyone here might have come across this or have someinsights> about a general solution.> Brian GladmanThe second problem can be formulated as a set covering problem and solvedusing integer programming:minimize sum [i in 1..N, j in 1..i] x[i,j]subject tox[i1,j1] + x[i2,j2] + x[i3,j3] >= 1 for each equilateral triangle ((i1,j1),(i2,j2), (i3,j3))x[i,j] in {0,1} for all (i,j)Solving this IP for N in 1..11 yields 0, 1, 2, 4, 7, 9, 14, 18, 23, 29, 36.This sequence does not appear in Sloane's database.Rob Pratt=== === Subject: : Re: Pascals TriangleRob Pratt escribi.97 en el mensaje>> There was an interesting puzzle in the Sunday Times in the UK>> recently that set me thinking about the issue that the puzzle raised.>> The idea is that of a set of points arranged in the familiar Pascal's>> triangle format (rows 1, 2, 3, ... etc. containing 1, 2, 3, ...>> points in a triangular format) after which the issue is that of the>> total number of equilateral triangles that can be found in this set>> of points.> For N in 1..10, I get 0, 1, 5, 13, 27, 48, 78, 118, 170, 235, which> can be expressed as> floor((N - 1) (N + 1) (2 N - 1) / 8)> and appears in Sloane's database:http://www.research.att.com/cgi-bin/access.cgi/as/ njas/sequences/eisA.cgi?An> um=A002717It seems that you only count triangles with sides paralel to array sides.But there are other equilateral triangles with vertices in the points of theequilateral triangular array with its sides in another orientation.Exactly, there areComb(n, 4) = n(n + 1)(n - 1)(n + 2)/24in a triangular array of order n. For n = 1 to 10,0, 1, 5, 15, 35, 70, 126, 210, 330, 495http://math.smsu.edu/~les/POW03_01.htmlhttp:// www.research.att.com/projects/OEIS?Anum=A000332-- Best regards,Ignacio Larrosa Ca.96estroA Coru.96a (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com=== === Subject: : Re: Pascals Triangle> Rob Pratt escribi.97 en el mensaje>> There was an interesting puzzle in the Sunday Times in the UK>> recently that set me thinking about the issue that the puzzle raised.>>> The idea is that of a set of points arranged in the familiar Pascal's>> triangle format (rows 1, 2, 3, ... etc. containing 1, 2, 3, ...>> points in a triangular format) after which the issue is that of the>> total number of equilateral triangles that can be found in this set>> of points.>For N in 1..10, I get 0, 1, 5, 13, 27, 48, 78, 118, 170, 235, which>can be expressed as>floor((N - 1) (N + 1) (2 N - 1) / 8)>and appears in Sloane's database:http://www.research.att.com/cgi-bin/access.cgi/as/ njas/sequences/eisA.cgi?An>um=A002717> It seems that you only count triangles with sides paralel to array sides.> But there are other equilateral triangles with vertices in the points ofthe> equilateral triangular array with its sides in another orientation.> Exactly, there are> Comb(n, 4) = n(n + 1)(n - 1)(n + 2)/24> in a triangular array of order n. For n = 1 to 10,> 0, 1, 5, 15, 35, 70, 126, 210, 330, 495> http://math.smsu.edu/~les/POW03_01.html> http://www.research.att.com/projects/OEIS?Anum=A000332Hi Ignacio,Thank you for your input - I must admit that I also missed the rotatedtriangles in my search algorithm. Do you have any insights into the minimumnumber of grid points that need to be removed in order to eliminate allequilateral triangles?It seems possible that this might actually be easier to solve with therotated triangles since all the points (except one) on one side of allnon-rotated sub-triangles always need to be removed.best regards, Brian Gladman=== === Subject: : Re: Pascals Triangle>Rob Pratt escribi.97 en el mensaje> There was an interesting puzzle in the Sunday Times in the UK> recently that set me thinking about the issue that the puzzle raised.> The idea is that of a set of points arranged in the familiar Pascal's> triangle format (rows 1, 2, 3, ... etc. containing 1, 2, 3, ...> points in a triangular format) after which the issue is that of the> total number of equilateral triangles that can be found in this set> of points.>For N in 1..10, I get 0, 1, 5, 13, 27, 48, 78, 118, 170, 235, which>can be expressed as>>floor((N - 1) (N + 1) (2 N - 1) / 8)>>and appears in Sloane's database:>http://www.research.att.com/cgi-bin/access.cgi/as/ njas/sequences/eisA.cgi?An>um=A002717>It seems that you only count triangles with sides paralel to arraysides.>But there are other equilateral triangles with vertices in the points of> the>equilateral triangular array with its sides in another orientation.>Exactly, there are>Comb(n, 4) = n(n + 1)(n - 1)(n + 2)/24>in a triangular array of order n. For n = 1 to 10,>0, 1, 5, 15, 35, 70, 126, 210, 330, 495>http://math.smsu.edu/~les/POW03_01.html>http:// www.research.att.com/projects/OEIS?Anum=A000332> Hi Ignacio,> Thank you for your input - I must admit that I also missed the rotated> triangles in my search algorithm. Do you have any insights into theminimum> number of grid points that need to be removed in order to eliminate all> equilateral triangles?> It seems possible that this might actually be easier to solve with the> rotated triangles since all the points (except one) on one side of all> non-rotated sub-triangles always need to be removed.> best regards,> Brian GladmanOkay, the problem can still be formulated as a set covering problem. Onceall equilateral triangles (even the rotated ones) have been considered, forN in 1..11 we have0, 1, 2, 4, 7, 11, 16, 22, 28, 35, 44as the minimum number of grid points to be removed. This sequence is not inSloane's database.Rob Pratt=== === Subject: : Re: Pascals TriangleIgnacio Larrosa Ca.96estro escribi.97 enel mensaje> It seems that you only count triangles with sides paralel to array> sides. But there are other equilateral triangles with vertices in the> points of the equilateral triangular array with its sides in another> orientation.> Exactly, there are> Comb(n, 4) = n(n + 1)(n - 1)(n + 2)/24Obviously, it must beComb(n + 2, 4) = n(n + 1)(n - 1)(n + 2)/24> in a triangular array of order n. For n = 1 to 10,> 0, 1, 5, 15, 35, 70, 126, 210, 330, 495> http://math.smsu.edu/~les/POW03_01.html> http://www.research.att.com/projects/OEIS?Anum=A000332-- Best regards,Ignacio Larrosa Ca.96estroA Coru.96a (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com=== === Subject: : Re: Pascals Triangle>This itself is not hard but the subsequent question seems moredifficult ->for such an arrangement with N rows, what is the minimum number ofpoints>that need to be blocked in order to reduce this total number of>equilateral triangles to zero?>I wondered if anyone here might have come across this or have some>insights about a general solution.> Brian Gladman> The second problem can be formulated as a set covering problem and solved> using integer programming:> minimize sum [i in 1..N, j in 1..i] x[i,j]> subject to> x[i1,j1] + x[i2,j2] + x[i3,j3] >= 1 for each equilateral triangle((i1,j1),> (i2,j2), (i3,j3))> x[i,j] in {0,1} for all (i,j)> Solving this IP for N in 1..11 yields 0, 1, 2, 4, 7, 9, 14, 18, 23, 29,36.> This sequence does not appear in Sloane's database.> Rob Prattprogram ran out of steam at about N = 15 and I hence wondered whether therewas a more efficient technique.Is there a standard reference describing this technique? Brian Gladman=== === Subject: : Re: Pascals Triangle> There was an interesting puzzle in the Sunday Times in the UK recently that> set me thinking about the issue that the puzzle raised.> The idea is that of a set of points arranged in the familiar Pascal's> triangle format (rows 1, 2, 3, ... etc. containing 1, 2, 3, ... points in a> triangular format) after which the issue is that of the total number of> equilateral triangles that can be found in this set of points.Pascal's triangle is 1 1 1 1 2 1 1 3 3 11 4 6 4 1and does not contain points, as such; it contains numbers.Row 2 does not contian a 2, but sums to 2. Row 3 does not contain a 3 orsum to 3. You've completely lost me.Phil-- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp2nd and 3rd bug found after 10 more minutes: gethost.cBoth non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL)=== === Subject: : Re: Pascals Triangle>There was an interesting puzzle in the Sunday Times in the UK recentlythat>set me thinking about the issue that the puzzle raised.>The idea is that of a set of points arranged in the familiar Pascal's>triangle format (rows 1, 2, 3, ... etc. containing 1, 2, 3, ... pointsin a>triangular format) after which the issue is that of the total number of>equilateral triangles that can be found in this set of points.> Pascal's triangle is> 1> 1 1> 1 2 1> 1 3 3 1> 1 4 6 4 1> ...> and does not contain points, as such; it contains numbers.> Row 2 does not contian a 2, but sums to 2. Row 3 does not contain a 3 or> sum to 3.> You've completely lost me.I was using the Pascal triangle analogy only to descibe the layout ofpoints:> .> . .> . . .> . . . .That is, replace each of the numbers in the Pascal triangle with a point andthen count the total number of equilateral triangles (both 'up' and 'down'and all sizes) in the resulting grid of points. The question then is 'whatis the minimum number of points that need to be removed in order to reducethe number fo such triangles to zero? Brian Gladman=== === Subject: : Re: Sets That Resemble Derivatives Somewhat by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1JDEKP11801;>Define:>1) (A-->B) = (AB')' = A' U B>for any two sets A, B, where ' is complement and Adjacent letters>are intersected. This is the set analog of logical a-->b for a,>b propositions, the latter defined as ~(a ^ ~b) = ~a V b withBefore replying to the other comments on my previous posting, whichI prefer to do tomorrow morning (night debates aren't my favoriteway of going to sleep at age 65, whether positive or negative), Iwould like to point out some things about complements.1) A U A' = UniverseThis says intuitively that a set (A) and/or its complement (thepart of the Universe outside the set) comprises the Universe. Morespecifically, it comprises the Mathematical Universe, not necessar-ily the Physical Universe.If we translated the idea behind this into Real or Complex Analysis,or even Tensor Analysis and so on, we would get something like:1') A variable and/or a function and its change constitute Universeof study or Knowledge about the variable/function.This is what Garrett Birkhoff of Harvard meant by saying thatDifferential Equations contain causation/causality. Of course, incausation/causality the change is usually with respect to time, buteven in spatial change (partial derivatives in general) it takestime in a sense to scan different spatial regions - even on alight wave, where time is supposed to be instantaneous, sometemporal notion of the order of attention to different placescan be introduced, since for one thing simultaneous attention tomany different places contains some paradoxes depending on howintense the attention is. I'll mention one more aspect:2) (A')' = AThis doesn't correspond to differentiation, but I don't think itis the most critical feature of differentiation. It is more like d^2 in exterior algebra. Intuitively speaking, when A is aset and ' is complement, we could translate this as:2') The change in a change of A is the change back to A.We would, I admit, sacrifice second order derivatives and higherin this scenario, but we already know that higher order equationscan often be expressed as systems of first order equations. Ithink that the advantages outweigh the orders so to speak, andwe may eventually find a generalization of the usual derivativeorder in some other ideas. I especially like the fact that theRiccati Differential Equation (with of course its Algebraic version)which underlies Growth-Expansion-Contraction processes and events(e.g., in biology - see the Logistic and Simple Positive or Neg-ative Exponential Differential Equations which are special casesof the Riccati Equation, and the generalization to a Bernoulliequation) is first order. Quantitative Biology is a new and veryvaluable branch of Arxiv.org. I'll try to discuss the differencesbetween Curvilinear One-Direction-At-A-Time motion and SimultaneousMany-Directions-At-A-Time Growth-Expansion-Contraction Motion(including radiation, expansion of the Physical Universe as awhole, etc. - possibly with different rates and amounts in different directions, relating to (optimal) control also) later.Osher Doctorow=== === Subject: : Re: Sets That Resemble Derivatives Somewhat[Snip: in Doctorowese A' is the completement of A, A+B is the unionof A and B and AB is the intersection of A and B]> 2) (A')' = A> This doesn't correspond to differentiation,Yet another thing that doesn't correspond to differentiationis that AB' + A'B =/= (AB)'.Indeed I am at a loss reading these posts to find any analogy betweencomplementation and differentiation (save that Doctor Osherow usesthe same notation for both).-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9Francis Wheen, _How Mumbo-Jumbo Conquered the World_=== === Subject: : Re: Sets That Resemble Derivatives Somewhat>[Snip: in Doctorowese A' is the completement of A, A+B is the union>of A and B and AB is the intersection of A and B]>>> 2) (A')' = A>>> This doesn't correspond to differentiation,>Yet another thing that doesn't correspond to differentiation>is that >AB' + A'B =/= (AB)'.>Indeed I am at a loss reading these posts to find any analogy between>complementation and differentiation (save that Doctor Osherow uses>the same notation for both).Evidently you've forgotten that derivatives satisfy (f + g)' = (f')(g').************************=== === Subject: : Re: Parity Check Matrix of a Systematic Linear Block Code windows-nt)>> The generator matrix of a systematic linear block code has the>> form G = [Ik : P]. How can it be shown that the parity check>> matrix is of the form H = [-P^T : In-k]?> With great ease.> If a vector (a : b) is supposed orthogonal to the code generated> by G then (a : b)G^t = 0 so a + b P^t = 0 or a = -b P^t etc.> Are you in the IEEE?Yes. Why do you ask?-- % Randy Yates % Ticket to the moon, flight leaves here today %% Fuquay-Varina, NC % from Satellite 2%%% 919-577-9882 % 'Ticket To The Moon' %%%% % *Time*, Electric Light Orchestrahttp://home.earthlink.net/~yatescr=== === Subject: : Are the derivatives of abs[(x-a)^3] different for x>a and xa and xa and x given abs[(x-a)^3], where abs[ ] means taking the absolute value, and> a is a constant, are the first, second, third derivatives (with> respect to x) different for x>a and x I forgot how to write down the derivative of an absolute function in a> formal way, I mean, not using if else, but using something likesign(x-a).> I think the first derivative would look like:> 3(x-a)^2 * sign(x-a)Take a = 0 for convenience. Let f(x) = |x|^3. Then f(x) = x^3, x >= 0, f(x) = -x^3, x <= 0. So f'(x) = 3x^2 for x >= 0, f'(x) = -3x^2, x <= 0, f''(x) = 6x for x >= 0, f''(x) = -6x for x <= 0. And f'''(0) doesn't exist. You can of course use the sgn function to express these if you like. All the claims about derivatives at 0 need to be checked using the basic definition of a derivative.=== === Subject: : Re: Are the derivatives of abs[(x-a)^3] different for x>a and x given abs[(x-a)^3], where abs[ ] means taking the absolute value, and> a is a constant, are the first, second, third derivatives (with> respect to x) different for x>a and x I forgot how to write down the derivative of an absolute function in a> formal way, I mean, not using if else, but using something likesign(x-a).The usual method is to define sgn(x) in terms of the heaviside functionH(x) = { 0, x < 0; 1, x > 0; 0.5, x = 0 }, and write formallydH/dx = delta(x) where delta(x) = { 0, x != 0 } is the Dirac delta'function' (integral of delta(x) over any interval containing 0 is 1 bydefinition, and 1/2 if 0 is an endpoint).sgn(f(x)) = 2H(f(x)) - 1|f(x)| = f*sgn(f)d(sgn(f(x))/dx = 2*delta(f(x))*f'(x)d(sgn(f)f)/dx = sgn(f)df/dx + 2*f*f'*delta(f)Terms involving delta(f) vanish if one avoids points where |f| = 0.> I think the first derivative would look like:> 3(x-a)^2 * sign(x-a)Yes.> But then I don't know how to proceed to the second derivative.You have to exclude points where |f| = 0 from the domain to proceedfurther; d/dx(delta(x)) is not well-defined.D(sgn(f)Df) = sgn(f)(D^2)f + (Df)*(D(sgn(f))) (D = d/dx) = sgn(f) d^2f/dx^2 if |f| != 0.Similarly, D^n(sgn(f)f) = sgn(f)*(D^n)f if |f| != 0.-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will find these people and we will bring them to justice.=== === Subject: : Re: Are the derivatives of abs[(x-a)^3] different for x>a and x The usual method is to define sgn(x) in terms of the heaviside function> H(x) = { 0, x < 0; 1, x > 0; 0.5, x = 0 }, and write formally> dH/dx = delta(x) where delta(x) = { 0, x != 0 } is the Dirac delta> 'function' (integral of delta(x) over any interval containing 0 is 1 by> definition, and 1/2 if 0 is an endpoint).This is a joke, right?=== === Subject: : Re: Are the derivatives of abs[(x-a)^3] different for x>a and x> The usual method is to define sgn(x) in terms of the heaviside function>> H(x) = { 0, x < 0; 1, x > 0; 0.5, x = 0 }, and write formally>> dH/dx = delta(x) where delta(x) = { 0, x != 0 } is the Dirac delta>> 'function' (integral of delta(x) over any interval containing 0 is 1 by>> definition, and 1/2 if 0 is an endpoint).> This is a joke, right?-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9Francis Wheen, _How Mumbo-Jumbo Conquered the World_=== === Subject: : Re: Are the derivatives of abs[(x-a)^3] different for x>a and xThe usual method is to define sgn(x) in terms of the heaviside function>H(x) = { 0, x < 0; 1, x > 0; 0.5, x = 0 }, and write formally>dH/dx = delta(x) where delta(x) = { 0, x != 0 } is the Dirac delta>'function' (integral of delta(x) over any interval containing 0 is 1 by>definition, and 1/2 if 0 is an endpoint).> This is a joke, right?No. The OP requested a formal solution, so I provided a formal solution.You may be misreading my compact notation for H(x) = 0 if x < 0, 1 ifx > 0 or 0.5 if x = 0, in which case I have just clarified it for you, oryou may not fully understand the properties of the Dirac delta.H(0) = 1/2 is a definition made to ensure that the formal statementdH/dx = delta(x) is consistent with the definition of the integralof delta. d/dx might not denote the classical analytic derivative, butit had still better be true that it is the inverse of indefiniteintegration and that it satisfies the Liebnitz product rule and the chainrule. If it didn't, the formalism would be useless.The knock-on effect that sgn(0) = 0 seems fairly logical: there is noreason why sgn(0) should be +1 or -1.The fact that H(0) = int_-oo^0 delta(x) dx = 1/2 follows trivially fromthe observations (i) delta(x) is an even function and (ii) the integral ofdelta over the whole real line is 1 (by definition).What is wrong is my statement that d/dx of delta(x) is undefined; it canbe multiplied by any differentiable f(x) and integrated by parts to yield-f'(0) if the domain of integration includes 0, and 0 if 0 lies outsidethe domain, so in this sense it is a well-defined distribution. Having 0on the boundary is fatal in this case (the boundary contribution isf(0)*delta(0), which is undefined). In fact, all derivatives of delta(x)are well-defined distributions:(d^n/dx^n delta(x)): f(x) |-> (-1)^n*d^n/dx^n(f)(0).However, whereas delta(x) has some meaning outside of integrals, the sameis not true of its derivatives, which is why I didn't bother quoting termsinvolving delta'(x) or higher derivatives.See http://en.wikipedia.org/wiki/Dirac_delta_function andhttp://en.wikipedia.org/wiki/Distribution-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will find these people and we will bring them to justice.=== === Subject: : Re: Are the derivatives of abs[(x-a)^3] different for x>a and xYou have to exclude points where |f| = 0 from the domain to proceed>further; d/dx(delta(x)) is not well-defined.Um, you just derivated the distribution related to the Heaviside stepfunction. How can derivating the Dirac delta distribution suddenly notbe defined?-- === === Subject: : Re: Advice on future with Math>I'm an upperclass math major that was/is planning on attempting a>masters or phd in math. Here lately, I've wondered if this is a great>idea.>My first question is this: How much should you study (reading the>material, working problems) for, say, an abstract algebra or advanced>calculus junior/senior level class? I talked to a graduate student>that said he studied for a couple of his first year grad classes at>least 4 hours a day on weekdays and about 8 hours on Saturday and>Sunday. Needless to say, I do nothhing like that. Should I be doing>that or is that just an insane amount of time to be spending on it?>It seems that if you're good at something, it shouldn't require so>much work.> I might be an anomaly, but I don't spend much more than 2 hours a day> actually sitting down with pencil and paper working on math. However,> what I am doing is almost every minute I am awake and every single> minute I am asleep, I am thinkingabout math, or one of the problems I am> working on. Occasionally I'll grab some scratch paper and jot something> down. Most of the time, not.> Okay, maybe saying every minute is a little exaggeration, but not much.> Seriously, though, I spend the best part of my day, everyday thinking> about math. If I am watching TV or reading the newspaper, or talking> with someone, in the back of my head is something like, What is the> solution to the PDE a*u_x + b*u_y + c*u_z = 0?> Also, it has been my experience that if you have to spend insane amounts> of time studying math, then maybe continuing on is not for you. I am> not trying to discourage you, and I would hope you at least try for a> year, but if you are stuggling and spending 8 hours a day in the books,> you might want to look for something else.Well, I disagree. In my experience, it is very important to practice,practice, and practice. This is of course not particular to math. So, yes, Ispent a large amount of time on math, many hours a day. It does take hardwork studying math, unless you're born a genious, but for now I assumeyou're just like the rest of us :-) You've really got to *like* math. If itis a pain doing the assignments and reading the books, then you should havesecond thoughts.Perhaps you should think about why you're interested in math. To me, math islike a tool, kind of like a swiss army knife. It can do all kinds of weirdthings; or rather, *I* can do weird and interesting stuff using this tool,i.e. math. While I was an undergraduate, I had a different attitude. It wasan intellectual challenge: I was *determined* to prove (to myself, if notanyone else) that I could solve the problems. So basically, I just keptgoing and going and going, until I had solved a problem.In high school, math was very easy for me, and the only homework I did wasthe written assignments. When I reached graduate level, it suddenly becamequite difficult. Sometimes I had to think about a proof for days or weeks,before I found the right path, or even understood the solution. It helped meimmensely that I was studying with some likeminded people.> Also, there are many different branches of math. Personally, I am of> the discrete/combinatorics and numerical methods persuasions. Analysis,> topology and PDE, I only took those classes becaues they were required.> And, I struggled through them.Good point, I completely agree. No-one is good at everything, so you mustexpect that some branches give you the kick and interest you, while otherseither bore you or are too difficult to grasp (at the moment).>At times I have difficulty understanding things that, after finding>out where I went wrong, seem pretty simple. I assume others don't>have this problem because normally they can respond right away with>what some math concept meant or, less often, help me where I am having>a problem understanding part of a proof. Is it normal to not follow a>proof or just a sign that I'm just weak mathematically (we're again>talking junior/senior level advanced calc/abstract algebra here)?> If after finding out where you went wrong things are still difficult,> you are, well, screwed. The right answers always look easy. Then you> think, Who the heck thought of doing it that way? However, that is> not to say you shouldn't be able to follow a proof. I read many papers> and I can't always follow the proofs givne in them. Often, they skip> steps or say somethign is obvious when it isn't (unless you know som> obscure theorem). But, if the proof is in your text, you should, for> the most part, be able to follow it.Let me just repeat: Practice, practice, and practice. Here's a suggestion.Dig out some textbooks that you used a couple of years ago, and read themagain. You should expect two benefits: The first is repetition, the secondis a realisation that things aren't as difficult now as they were then.>I have a bad tendency to be hot/cold when it comes to math. It seems>like I'll have the highest/near highest score on an exam and then>really screw up on the next exam, getting something in the bottom half>of the grades. I wish that I could attribute it to something like>when I first started school: I would do no work until after I screwed>up on the first exam, and then work harder after that and do quite>well. Now it seems that I'll do well on the first exam and then screw>up on the next one. I really don't think that I will have studied>much less for the second one but I guess it could be a possibility.That was my first thought. However, begin hot/cold should not frighten you;we can all have out down periods. Don't look at the results, but considerinstead what it's like studying. Is it fun or is it tedious? Again, don'tfocus on isolated bad days, we all have them. Think about the big picture.Look back at last semester, what was it like taking the math courses?>When I do bad on the other exam, I begin to wonder if I just got>lucky on the problems that were given on the good exam and would have>done lousy if a few different problems were picked.Argh! You must believe in yourself. Whenever you do good in an exam, giveyourself a pat on the shoulder. You've deserved it!> Okay, those are my thoughts. Take them and do what you will with them.> But, I tyhink if you reall ylike math, and are not a burnt out senior> who is just looking forward to graduation, you should at least try a> semester working towards a master degree. If you find out it's not for> you, well, you learned and you didn't really waste any of your life.> But, if you don't try, you will always be kicking yourself.I completely agree.-Michael.=== === Subject: : Re: Advice on future with Math> I'm an upperclass math major that was/is planning on attempting a> masters or phd in math. Here lately, I've wondered if this is a great> idea. > My first question is this: How much should you study (reading the> material, working problems) for, say, an abstract algebra or advanced> calculus junior/senior level class? I talked to a graduate student> that said he studied for a couple of his first year grad classes at> least 4 hours a day on weekdays and about 8 hours on Saturday and> Sunday. Needless to say, I do nothhing like that. Should I be doing> that or is that just an insane amount of time to be spending on it? > It seems that if you're good at something, it shouldn't require so> much work. I might be an anomaly, but I don't spend much more than 2 hours a day actually sitting down with pencil and paper working on math. However, what I am doing is almost every minute I am awake and every single minute I am asleep, I am thinkingabout math, or one of the problems I am working on. Occasionally I'll grab some scratch paper and jot something down. Most of the time, not.Okay, maybe saying every minute is a little exaggeration, but not much. Seriously, though, I spend the best part of my day, everyday thinking about math. If I am watching TV or reading the newspaper, or talking with someone, in the back of my head is something like, What is the solution to the PDE a*u_x + b*u_y + c*u_z = 0?Also, it has been my experience that if you have to spend insane amounts of time studying math, then maybe continuing on is not for you. I am not trying to discourage you, and I would hope you at least try for a year, but if you are stuggling and spending 8 hours a day in the books, you might want to look for something else.Also, there are many different branches of math. Personally, I am of the discrete/combinatorics and numerical methods persuasions. Analysis, topology and PDE, I only took those classes becaues they were required. And, I struggled through them.> At times I have difficulty understanding things that, after finding> out where I went wrong, seem pretty simple. I assume others don't> have this problem because normally they can respond right away with> what some math concept meant or, less often, help me where I am having> a problem understanding part of a proof. Is it normal to not follow a> proof or just a sign that I'm just weak mathematically (we're again> talking junior/senior level advanced calc/abstract algebra here)?If after finding out where you went wrong things are still difficult, you are, well, screwed. The right answers always look easy. Then you think, Who the heck thought of doing it that way? However, that is not to say you shouldn't be able to follow a proof. I read many papers and I can't always follow the proofs givne in them. Often, they skip steps or say somethign is obvious when it isn't (unless you know som obscure theorem). But, if the proof is in your text, you should, for the most part, be able to follow it.> I have a bad tendency to be hot/cold when it comes to math. It seems> like I'll have the highest/near highest score on an exam and then> really screw up on the next exam, getting something in the bottom half> of the grades. I wish that I could attribute it to something like> when I first started school: I would do no work until after I screwed> up on the first exam, and then work harder after that and do quite> well. Now it seems that I'll do well on the first exam and then screw> up on the next one. I really don't think that I will have studied > much less for the second one but I guess it could be a possibility. > When I do bad on the other exam, I begin to wonder if I just got> lucky on the problems that were given on the good exam and would have> done lousy if a few different problems were picked.You have to remember that all math is comprehensive. You couldn't do modern algebra if you didn't know how to add and multiply. The biggest thing that I have found that helped me was relating one subject to another. This is especially easy when the subjects are closely related, like topology and analysis. But try it for subjects that seem less related. The triangle inequality in linear algebra is the (exact) same one you learn about in functional analysis. Once you start doing this, it becomes easier to learn, because you aren't learning all new stuff, you are just review things, popssibly with different names. The solution to the heat equation is just an inverse problem, but using forawrd modelling (which makes it easier).> Other times, I wonder if I am just not good at proofs, thus making me> about the crappiest grad school candidite out there. I sit and stare> at the homework problems and often can't get anywhere. After I see an> answer, it doesn't look too bad but it just seems that they're> impossible at times. I had a bad proofs class that was designed for> just about anyone wanting to take it, so it was really easy. I'm not> sure if I can use that excuse at this point though. My pre-calculus> education was really bad since I was from a really poor school (I > often have no clue what certain things that were assumed as> 'knowledge' from high school) but again this doesn't seem to matter> too much in the proof-based classes.Proof writing takes skill, alot of skill. It also take practice, even more practice than skill. But, the most important thing of all is that writing good proofs require you to have seen well written proofs. If you are taking modern (abstract) algebra, go to the library and check out a few other books on it. Start looking the QA section. Often times, different books will have very different proofs for the same theorems. Often times, in one book they prove theorem A and use that to prove theorem B, then another book proves theorem B and uses it to prove theorem A. Try looking in a few different analysis books for the proofs of the following theorems:Axiom of Choice (okay, this is a given)Nested Interval PropertyBolzano-WierestrassCauchy CriterionMontone Convergence Theorem.All of these theorems are equivalent (more or less) so the order they are proven, and which is used to prove the other, is very much up to the author. In fact, on an analysis test, one of our questions was to prove one of the 4 theorems, assuming a different one is true, not the same way the text did it. Our text did AoC => NIP => BW => CC and AoC => MCT. I think I showed CC => MCT on the test> Well, that was kind of long-winded. I hope that I got the point> across. Any advice would be helpful. Okay, those are my thoughts. Take them and do what you will with them. But, I tyhink if you reall ylike math, and are not a burnt out senior who is just looking forward to graduation, you should at least try a semester working towards a master degree. If you find out it's not for you, well, you learned and you didn't really waste any of your life. But, if you don't try, you will always be kicking yourself. - Tim-- Timothy M. BrauchGraduate StudentDepartment of MathematicsWake Forest Universityemail is:news (dot) post (at) tbrauch (dot) com=== === Subject: : Re: Teaching philosophyteaching philosophy. I think that gives me a good idea of the genre.What about the length of the statement? Teaching Philosophy: Les fleurs qui pousse de la merde by Allan Adler Evil is the root of all money. Hence if funding has been foundfor a position, it necessarily cometh of evil. Yet, though mushroomsnourish themselves on excrement, they in turn nourish people and, inlike manner, our fate as educators is to be a kind of mushroom. Let metherefore describe some recipes in which this mushroom has been used.[details omitted for the purpose of this posting](I know, cut the first paragraph...)Allan Adlerara@zu.ai.mit.edu************* ** Disclaimer: I am a guest and *not* a member of the MIT Artificial ** Intelligence Lab. My actions and comments do not reflect ** in any way on MIT. Moreover, I am nowhere near the Boston ** metropolitan area. ** *************=== === Subject: : Re: Teaching philosophy> Evil is the root of all money. No, this is a common misconception. The love of evil is the rootof all money.=== === Subject: : Re: Teaching philosophy> teaching philosophy. I think that gives me a good idea of the genre.> What about the length of the statement?> Teaching Philosophy: Les fleurs qui pousse de la merde> by Allan Adler> Evil is the root of all money. Hence if funding has been found> for a position, it necessarily cometh of evil. Yet, though mushrooms> nourish themselves on excrement, they in turn nourish people and, in> like manner, our fate as educators is to be a kind of mushroom. Let me> therefore describe some recipes in which this mushroom has been used.> [details omitted for the purpose of this posting]> (I know, cut the first paragraph...) Use the first as a cover sheet. Since Philosophy is known to be orthogonal to money. And state plainly in the second paragraph that whether or not it's orthogonal to intelligence depends on crucial ongoing experiments in Bejing. Hence the education commitee needs to organize a Buddhist revival movement in order to effectively prove that mathemathematics is truly a logic system, rather than simply a faint remnent of the after-glow of the exploding super-galaxy, PI-ZETA-4* in the Constellation of SETI-Alpha. > Allan Adler> ara@zu.ai.mit.edu> ntelligence Lab. My actions and comments do not reflect *=== === Subject: : Re: JSH: Apology to Ramsay, why I post> So I apologize to Keith Ramsay for questioning his honesty here.> Why not apologise to those who did repost that result for your perusal?Because that's obsessive behaviour, and we all know that's wrong :-)V.-- email: lastname at cs utk eduhomepage: cs utk edu tilde lastname=== === Subject: : Re: JSH: Apology to Ramsay, why I postsays...>It's simpler to just post to all the newsgroups that I posted before>an apology for questioning Keith Ramsay's honesty. It seems he did in>fact post a non-unit algebraic integer factor of (1+sqrt(-167))/2 and>7, so I was wrong.>So I apologize to Keith Ramsay for questioning his honesty here.Take the next step in the learning feedback loop: remember this incident andnext time someone disagrees with you don't automatically accuse them of lying.=== === Subject: : Re: JSH: Apology to Ramsay, why I postNora Baron> You denied and ignored the theory. You refused to learn> it. The arithmetic, however, you were ultimately able to handle and> could not deny. That is the level at which you are able to operate.Bear in mind that Harris lives through his computer. E.g. for him afunction is something done in Java or some other computer language. Harrishas no interest in mathematics, for if he did he would know by now what analgebraic integer is, for example.> ... Someone said it previously: you have to be force-fed the truth,> little bits at a time ...I have a different theory. Harris is willing to deny anything indefinitely,but he will admit that this-or-that is true when he fears that his audienceis losing interest in his latest error or fiction. He needs an audience, buthe has no interest in mathematical facts. This unusual apology of his ismotivated only by the unusually heavy pounding that he has been takinglately. He needs that audience. He has nothing else.>...> Perhaps tomorrow you will> post something bragging about your prime counting theorem and your> partial difference equation, trying to draw attention away from this> major failure...Probably.> But I am fairly certain you will be back on this topic as well...I have no doubt of it at all. He will get away from it for a while, but itwill gall him day and night, and when he reckons his latest tantrums areforgotten, he will start the same thing all over again, just as you say.-- Larry Profiler Extraordinaire Hammick=== === Subject: : Re: oil of bitter almonds: Woehler and Liebigpaper on Liebig's treatment of significant figures. I'll take a lookat it.Allan Adlerara@zu.ai.mit.edu************* ** Disclaimer: I am a guest and *not* a member of the MIT Artificial ** Intelligence Lab. My actions and comments do not reflect ** in any way on MIT. Moreover, I am nowhere near the Boston ** metropolitan area. ** *************=== === Subject: : Re: JSH: Splitting field, algebraic integer factors> Previously I posted that if you can't *see* the factors between> irrational algebraic integers then they're not there, but more> correctly the situation is that two algebraic integers have to be> members of the same splitting field to have a non-unit algebraic> integer in common.> (I say more correctly as there may be some terminology issues here> because what mathematicians currently call a splitting field is not> a true field, but something close, like the field of rationals. But> that's another issue for another time.)> That's a nifty and powerful result. Why am I the one who had to> discover it?Well I was wrong. I should have known in retrospect, but it was yetanother situation where I kind of decided that I wanted something andthought I had it, only to find out later I was wrong.There's no point to hanging on to wrong things though.Luckily, mathematics is supposed to be an arena where the truthmatters.James Harris=== === Subject: : Re: JSH: Splitting field, algebraic integer factors>Previously I posted that if you can't *see* the factors between>irrational algebraic integers then they're not there, but more>correctly the situation is that two algebraic integers have to be>members of the same splitting field to have a non-unit algebraic>integer in common.>(I say more correctly as there may be some terminology issues here>because what mathematicians currently call a splitting field is not>a true field, but something close, like the field of rationals. But>that's another issue for another time.)>That's a nifty and powerful result. Why am I the one who had to>discover it?> Well I was wrong. I should have known in retrospect, but it was yet> another situation where I kind of decided that I wanted something and> thought I had it, only to find out later I was wrong.Bump the OOPS counter again.> There's no point to hanging on to wrong things though.Since when? What's changed?> Luckily, mathematics is supposed to be an arena where the truth> matters.> James HarrisDavid Moran=== === Subject: : Re: JSH: Splitting field, algebraic integer factors[snipped unneeded restatement of errors already claimed and reclaimed too many times]> Well I was wrong. I should have known in retrospect, but it was yet> another situation where I kind of decided that I wanted something and> thought I had it, only to find out later I was wrong.> There's no point to hanging on to wrong things though.> Luckily, mathematics is supposed to be an arena where the truth> matters.> James HarrisAnd, unluckily for JSH's egomania, the truth will out.=== === Subject: : Call for participation (Omphalos)(Apologies if you receive this message more than once)CALL FOR PARTICIPATION************************************************* **OMPHALOS CONTEXT-FREE LANGUAGE LEARNING COMPETITION*************************************************** http://www.irisa.fr/Omphalos/--------------------------------- ------------------Intro-----Omphalos is the first context-free language learning competition thattask is to infer a model of a context-free language from unstructuredexamples (both positive and negative) and to use that model to label aset of test sentences as being either in or out of the language.The Competition Task--------------------We have generated some context-free grammars. For each grammar we havelabeled a set of sentences indicating whether or not those sentencescan be generated from that grammar. The goal of the competition is toinfer a model of each language (such as a grammar) using the trainingdata. You then need to tag new sentences, indicating whether thesentences are in the language or not. ---------------More Information----------------For more information, check the website:http://www.irisa.fr/Omphalos/or contact the organizers at:omphalos@irisa.frOmphalos is being organized and administered by:Brad Starkie, Fran?ois Coste, and Menno van Zaanen === === Subject: : Solving Equation with lnI was working on a math problem at work and ran into something I don'tknow how to solve and was hoping someone could give me a clue on thisfairly simple problem.Original Equation:227.11 = 5.8{ln(38/x)} + (38-x)Simplfied Equation:189.11 = 5.8*{ln(38/x)} - xof course you can reduce this down to32.60 = ln(38/x) - x/5.8and further (I think) to 1.44 x 10^14 = 3.8/x - e^(x/5.8)but I am stuck after this. Is it possible to solve for X?-Andrew V. Romero=== === Subject: : Re: Solving Equation with ln> I was working on a math problem at work and ran into something I don't> know how to solve and was hoping someone could give me a clue on this> fairly simple problem.> Original Equation:> 227.11 = 5.8{ln(38/x)} + (38-x)Well, there is a solution with x positive, but *really really* close tozero. In that case you could replace the last term 38-x simply with 38.Then you can solve for x and very that it indeed is small.> Simplfied Equation:> 189.11 = 5.8*{ln(38/x)} - x> of course you can reduce this down to> 32.60 = ln(38/x) - x/5.8> and further (I think) to> 1.44 x 10^14 = 3.8/x - e^(x/5.8)This is incorrect. It should be 1.44 x 10^14 = 3.8/x * e^(-x/5.8)> but I am stuck after this. Is it possible to solve for X?Not using elementary functions.-Michael.=== === Subject: : Re: Solving Equation with ln> I was working on a math problem at work and ran into something I don't> know how to solve and was hoping someone could give me a clue on this> fairly simple problem.> Original Equation:> 227.11 = 5.8{ln(38/x)} + (38-x)> Simplfied Equation:> 189.11 = 5.8*{ln(38/x)} - x> of course you can reduce this down to> 32.60 = ln(38/x) - x/5.8> and further (I think) to > 1.44 x 10^14 = 3.8/x - e^(x/5.8)> but I am stuck after this. Is it possible to solve for X?> -Andrew V. RomeroNot in terms of the standard elementary functions. But youcan solve in terms of the Lambert W function...x = W(38*exp(-(189.11)/5.8)/5.8)*5.8 = 2.627466197*10^(-13);=== === Subject: : Re: Solving Equation with lnHumm, I am not familier with the Lambert W function, is there a website you would recommend so that I can read up on it or is thisfunction too complex for the non-math major type person. Andrew V. Romero>I was working on a math problem at work and ran into something I don't>know how to solve and was hoping someone could give me a clue on this>fairly simple problem.>Original Equation:>227.11 = 5.8{ln(38/x)} + (38-x)>Simplfied Equation:>189.11 = 5.8*{ln(38/x)} - x>of course you can reduce this down to>32.60 = ln(38/x) - x/5.8>and further (I think) to >1.44 x 10^14 = 3.8/x - e^(x/5.8)>but I am stuck after this. Is it possible to solve for X?>-Andrew V. Romero> Not in terms of the standard elementary functions. But you> can solve in terms of the Lambert W function...> x = W(38*exp(-(189.11)/5.8)/5.8)*5.8 = 2.627466197*10^(-13);=== === Subject: : Re: Solving Equation with ln> Humm, I am not familier with the Lambert W function, is there a web> site you would recommend so that I can read up on it or is this> function too complex for the non-math major type person.I'll let you decide whether it's too complex for your taste. Briefly,it's just the inverse of the function f(W) = W*e^W. (Although that inverseis multivalued, you only need to use the principal branch for solving yourequation.) See formore information, including series expansions, etc.David>I was working on a math problem at work and ran into something I>don't know how to solve and was hoping someone could give me a clue>on this fairly simple problem.>>Original Equation:>227.11 = 5.8{ln(38/x)} + (38-x)>>Simplfied Equation:>189.11 = 5.8*{ln(38/x)} - x>>of course you can reduce this down to>32.60 = ln(38/x) - x/5.8>>and further (I think) to>1.44 x 10^14 = 3.8/x - e^(x/5.8)>>but I am stuck after this. Is it possible to solve for X?>>-Andrew V. Romero>Not in terms of the standard elementary functions. But you>can solve in terms of the Lambert W function...>x = W(38*exp(-(189.11)/5.8)/5.8)*5.8 = 2.627466197*10^(-13);=== === Subject: : Re: errors in an argument>I recently attended some talks by a creationist talking about>how bad evolution is.Anyway, at one point in his presentation, he attempted>to prove that life could not arise by chance. Yes, creationist do that a lot. They tend to incorperate a lot ofthings which have little to do with evolution in their 'evolution isbad, so we must be right' speeches. In this case he was trying to show that abiogenesis was impossible.The method he chose is not new. In fact it is so standard that thereis this:http://www.talkorigins.org/faqs/abioprob/=== === Subject: : Re: errors in an argument> My favorite is birds : How can one evolve into flying ? So many> things have to happen in order to create functional wings it seems> impossible any awkward animal developing in this direction would> survive it's cost, or any failing flying attempts. The evolutionist> arguments sound totally absurd : they ran fast or jumped from tree> to tree and got an advantage slowly developing wings. Sounds like> absolute voodoo to me. >>That depends on the size of the animal. If the animal is small>>enough, the terminal velocity can result in a non-lethal impact with>>the ground. (I believe the threshold is around the size of a mouse or>>a cat.) Incremental levels of control over the landing point then>>have an obvious benefit. >>-- >>Daniel W. son>>panoptes@iquest.net>>>href=http://members.iquest.net/~ panoptes/>http://members.iquest.net/~>>panoptes/ 039 53 36 N / 086 11 55 W> That's highly unlikely. As far as I know the first flying creatures> they talk about are dinosaurs, no less.20to 24 size the big ones came along a lot later And quite big ones as well. In> any case, a functional wing on a mouse, that allows even such a> manuver is still something very hard to produce simply by mutation.Two mutation produce flaps of skin between digits, goes back to the Fish days other Transforms also in the genome from the fish ansestor > This is a very hard one to crack for the theory of evolution. I still> didn't hear a good solution for it.-- If its Monday then I am a fool but not ignorant.=== === Subject: : Re: errors in an argument by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1JDELd11866;>> As far as I know the first flying creatures they talk about are >> dinosaurs, no less. And quite big ones as well.>That's a comment on your knowledge. Some dinosaurs were small; a quick>search turned up some with a mass below 10 kg. And all the references I>found talk about small dinosaurs as the starting point for the evolution>of wings.>-- >Daniel W. son>panoptes@iquest.net>http:// members.iquest.net/~panoptes/039 53 36 N / 086 11 55 WI admit that my knowledge of the research of small dinosaurs isn't very deep, so you may be right, but that keeps my general point intact. The way it seems to me (and I'm not talking as an expert in aerodynamics either) is that even for extremely small and light creatures, the ability to develop a functional limb that will allow it even the smallest maneuverability by mutation benefit is unlikely. Remember that it has to be aerodynamic to some extent, and to be evolutionary beneficial. It's very hard for me to imagine how such a thing would look like.=== === Subject: : Re: errors in an argument> I admit that my knowledge of the research of small dinosaurs isn't very> deep, so you may be right, but that keeps my general point intact.A little more research turns up a wingspan of 0.5 m for Archaeopteryx.That puts it at the approximate size of a housecat, a critter known tosometimes survive a fall even if it reaches terminal velocity. (Notethat sometimes survive is an ideal condition to drive naturalselection.)> The way it seems to me (and I'm not talking as an expert in aerodynamics> either) is that even for extremely small and light creatures, the ability> to develop a functional limb that will allow it even the smallest> maneuverability by mutation benefit is unlikely. Remember that it has to> be aerodynamic to some extent, and to be evolutionary beneficial. It's> very hard for me to imagine how such a thing would look like.Imagine this:http://www.animalnetwork.com/critters/profiles/ flyingsquirrel/default.aspKeep in mind that embryos of many species typically have webbing duringdevelopment; it would not take much of a mutation for the webbing topersist.A little research also turns up an explanation of feathers that hasnothing to do with their eventual use in flight. -- Daniel W. sonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/039 53 36 N / 086 11 55 W=== === Subject: : Re: A newbie's question -- about real numberX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith Theoretically NO.What theory are you using? His r2 is 0.88 + sigma^i=3_00 9*10^{-i},and sigma^i=3_00 9*10^{-i} converges to 10^{-2} = 0.01, so r2 = 0.88 +0.01 = 0.89, which is his r1.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to spamtrap@library.lspace.org=== === Subject: : Re: tensors for totsX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith How far can one get in understanding the structure of tensors at a>point on a manifold in terms of our old school friends column>vectors, row vectors and matrices?You'll get bogged down fairly quickly.>Even in general relativity all we have at each point on the >manifold is a real vector space with four components,No. You also have, e.g., g and R. You could write the components of gas a matrix, but that would be horribly misleading. R is a lost cause;you really need to forget about matrices and just think of it as atensor. at 02:28 PM, nulldev00@aol.com (Edward Green) said:>Or for that matter, say we want v* to>be the dual vector corresponding to v:You need a metric in order to do that.>orthogonal transformationsMeaningless without a metric.>I also hope not to get hung up on any terminological problems>between math and physics, perhaps by saying things like we'd like>this object to be invariant, instead of using terms likecovariant and contravariant, to start.First, you'd need to define what you mean by invariant. Second, youwon't get very far if you don't distinguish covariant fromcontravariant, and both from mixed.>Anyway, to take one more baby step, if I have not exhausted my>credits, am I correct in thinking that a reasonable way to introduce>a metric tensor into babyland would be to define the inner product>of a row vector and a column vector, in a particular coordinate>system, to be a bilinear form with a matrix sandwiched in the>middle, that matrix our metric?No, you don't need a metric for that. You need a metric tensor todefine the inner product of two row vectors, the inner product of twocolumn vector and the transpose operation that takes row vectors intocolumn vectors and vice versa.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to spamtrap@library.lspace.org=== === Subject: : Re: tensors for totsHow far can one get in understanding the structure of tensors at a>point on a manifold in terms of our old school friends column>vectors, row vectors and matrices?> You'll get bogged down fairly quickly.>Even in general relativity all we have at each point on the >manifold is a real vector space with four components,> No. You also have, e.g., g and R. You could write the components of g> as a matrix, but that would be horribly misleading. R is a lost cause;> you really need to forget about matrices and just think of it as a> tensor.> at 02:28 PM, nulldev00@aol.com (Edward Green) said:>Or for that matter, say we want v* to>be the dual vector corresponding to v:> You need a metric in order to do that.>orthogonal transformations> Meaningless without a metric. But, since a metric is nothing but a linear vector space, it's also the reason that Einstein called philosophers idiots, more often than he called Heisenburg an idiot.>I also hope not to get hung up on any terminological problems>between math and physics, perhaps by saying things like we'd like>this object to be invariant, instead of using terms likecovariant and contravariant, to start.> First, you'd need to define what you mean by invariant. Second, you> won't get very far if you don't distinguish covariant from> contravariant, and both from mixed. Since invariant means only one thing: Einstein index convention, it's simple. It's simpler than Turing machines.Anyway, to take one more baby step, if I have not exhausted my>credits, am I correct in thinking that a reasonable way to introduce>a metric tensor into babyland would be to define the inner product>of a row vector and a column vector, in a particular coordinate>system, to be a bilinear form with a matrix sandwiched in the>middle, that matrix our metric?> No, you don't need a metric for that. You need a metric tensor to> define the inner product of two row vectors, the inner product of two> column vector and the transpose operation that takes row vectors into> column vectors and vice versa. No you need you don't need a metric tensor. Since it's doesn't matter what the hell tensors are to *General Relavity*. Since GR *assumes* that gravitional metrics are equivalent to curved space-time.=== === Subject: : unsolved constrained correlation problemI posted originally the following problem on sci.stat.math,but I received solutions working only for the independant case.We consider two random vectors X and Y, having the same distribution,such that the variance matrix V is diagonal. The common distributionis fixed, not the random vectors.For simplicity, we assume that X and Y are vectors in the plane.It is pointed out that r(X1,X2)=0 but X1 and X2 could be dependant.We consider a joint distribution of (X,Y) such that r(X1,Y1)=0 andr(X2,Y2)=1. Does such a joint distribution always exists and why ?Michel Petitjean, Email: petitjean@itodys.jussieu.frITODYS (CNRS, UMR 7086) ptitjean@ccr.jussieu.frhttp://petitjeanmichel.free.fr/ itoweb.petitjean.html=== === Subject: : Nonlinear PDE HelpHere's a partial differential equation that I'm almost positivehas a unique solution, although I don't have any idea how to goabout finding it. Does anyone have suggestions?We want to find f(x,y) subject to the following two conditions 1. f(x,0) = g(x) 2. (df/dx)^2 + (df/dy)^2 = 1where g(x) is some known differentiable function of xand where d/dx and d/dy are supposed to be partial derivatives.I know of a solution for a specific case: g(x) = square-root(1 + x^2).Then a solution is f(x,y) = square-root(x^2 + (y+1)^2)However, I don't know how to leverage this particular solution toget a solution for an arbitrary g(x).--Daryl McCulloughIthaca, NY=== === Subject: : Re: Nonlinear PDE Help>Here's a partial differential equation that I'm almost positive>has a unique solution, although I don't have any idea how to go>about finding it. Does anyone have suggestions?>We want to find f(x,y) subject to the following two conditions> 1. f(x,0) = g(x)> 2. (df/dx)^2 + (df/dy)^2 = 1>where g(x) is some known differentiable function of x>and where d/dx and d/dy are supposed to be partial derivatives.>I know of a solution for a specific case: g(x) = square-root(1 + x^2).>Then a solution is> f(x,y) = square-root(x^2 + (y+1)^2)(if you're not including the point [0,-1] in the region you're interested in)Geometrically, your differential equation says that gradient(f) has length 1 everywhere. In principle, you should be able to get solutions bystarting with a more-or-less arbitrary curve C (either closed, or going to infinity at both ends), on which you take f = c for some constant c, and defining f(x,y) = c + dist((x,y), C) on one side of the curve, = c - dist((x,y), C) on the other side of the curvewhere dist((x,y), C) is the minimum distance from (x,y) to C. Of course,depending on the curve, this may be non-differentiable at some points(which have more than one closest point on C). Two easy special casesare where C is a circle (as in your example) or a straight line.By the way, the latter example shows that in general f(x,0) = g(x) does not uniquely determine the solution: consider f(x,y) = y and f(x,y) = -y, both with g(x) = 0.I don't know if there are other solutions (besides circles and straight lines) with nice simple formulas: for most curves, calculating the distance from a point to the curve is not very easy.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2=== === Subject: : Re: Nonlinear PDE Help>>Here's a partial differential equation that I'm almost positive>>has a unique solution, although I don't have any idea how to go>>about finding it. Does anyone have suggestions?>>We want to find f(x,y) subject to the following two conditions>> 1. f(x,0) = g(x)>> 2. (df/dx)^2 + (df/dy)^2 = 1>>where g(x) is some known differentiable function of x>>and where d/dx and d/dy are supposed to be partial derivatives.>Geometrically, your differential equation says that gradient(f) has length >1 everywhere. In principle, you should be able to get solutions by>starting with a more-or-less arbitrary curve C (either closed, or >going to infinity at both ends), on which you take f = c for some >constant c, and defining >f(x,y) = c + dist((x,y), C) on one side of the curve, > = c - dist((x,y), C) on the other side of the curve>where dist((x,y), C) is the minimum distance from (x,y) to C. Of course,>depending on the curve, this may be non-differentiable at some points>(which have more than one closest point on C). Two easy special cases>are where C is a circle (as in your example) or a straight line.>By the way, the latter example shows that in general f(x,0) = g(x) >does not uniquely determine the solution: consider f(x,y) = y and >f(x,y) = -y, both with g(x) = 0.>I don't know if there are other solutions (besides circles and >straight lines) with nice simple formulas: for most curves, >calculating the distance from a point to the curve is not very easy.On second thought, there's a geometric construction to get the curve Cgiven g. Of course we need to assume |g'(x)| <= 1 everywhere. Consider the family of circles centred at (x,0) with radius |g(x)|. For g(x) >= 0 take the envelope of these in the upper half plane, and for g(x) <= 0take the envelope in the lower half plane. Of course you could reflectit across the x axis and get another solution. Since the family of circles has implicit equation (x-s)^2 + y^2 = g(s)^2, an equation ofthe envelope is obtained (in principle) by eliminating s from the system (x-s)^2 + y^2 - g(s)^2 = 0 d/ds((x-s)^2 + y^2 - g(s)^2) = 2 (s-x) - 2 g'(s) g(s) = 0Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2=== === Subject: : Re: Nonlinear PDE HelpRobert Israel says...>>We want to find f(x,y) subject to the following two conditions>> 1. f(x,0) = g(x)>> 2. (df/dx)^2 + (df/dy)^2 = 1>>where g(x) is some known differentiable function of x>>and where d/dx and d/dy are supposed to be partial derivatives.>Geometrically, your differential equation says that gradient(f) has length >1 everywhere. In principle, you should be able to get solutions by>starting with a more-or-less arbitrary curve C (either closed, or >going to infinity at both ends), on which you take f = c for some >constant c, and defining >f(x,y) = c + dist((x,y), C) on one side of the curve, > = c - dist((x,y), C) on the other side of the curvethe problem comes up in Special Relativity. If you have arigid ruler (or as rigid as possible in SR) and one endtravels in a straight-line path in the direction the ruleris oriented according to position at time t = g(t)then a point on the ruler that is initially a distance x fromthe end of the ruler will travel a path given by position at time t = f(x,t)where f is the solution to f(0,t) = g(t)and (df/dt)^2 + (df/dx)^2 = 1--Daryl McCulloughIthaca, NY=== === Subject: : equivalent expressionsI'm sure this is just a print error, but a Foundations book I haveasks the following:[Q] Give an example of two equivalent [propositional] expressions thatdo not have the same truth table.Is there something deep I'm missing here, because what I see is acontradiction. Perhaps my misunderstanding lies in what they mean bytruth table.Any help is greatly appreciated.R=== === Subject: : Re: equivalent expressions> I'm sure this is just a print error, but a Foundations book I have> asks the following:> [Q] Give an example of two equivalent [propositional] expressions that> do not have the same truth table.> Is there something deep I'm missing here, because what I see is a> contradiction. Perhaps my misunderstanding lies in what they mean bytruth table.> Any help is greatly appreciated.> RHow about two expressions which are both always true (or both always false) but which have different numbers of variables?=== === Subject: : Re: equivalent expressions>I'm sure this is just a print error, but a Foundations book I have>asks the following:>[Q] Give an example of two equivalent [propositional] expressions that>do not have the same truth table.>Is there something deep I'm missing here, because what I see is a>contradiction. Perhaps my misunderstanding lies in what they mean bytruth table.Seems like nonsense to me...Hmm, unless they meant something silly like this:P & ~P and Q & ~Q are equivalent, but they havedifferent truth tables because there's a P columnin one table and no P column in the other?>Any help is greatly appreciated.>R************************=== === Subject: : Re: equivalent expressions>I'm sure this is just a print error, but a Foundations book I have>asks the following:>[Q] Give an example of two equivalent [propositional] expressions thatdo not have the same truth table.>Is there something deep I'm missing here, because what I see is a>contradiction. Perhaps my misunderstanding lies in what they mean bytruth table.> Seems like nonsense to me...> Hmm, unless they meant something silly like this:> P & ~P and Q & ~Q are equivalent, but they have> different truth tables because there's a P column> in one table and no P column in the other?Or maybe something silly like P -> P and (P V Q) -> (P V Q) are equivalent, but they have different truth tables because one has two rows and the other, four?-- === === Subject: : Re: Collatz Conjecture : Symmetry question.http://arxiv.org/abs/math.NT/0312309I think it's clear that it will never be proven, because there is justnot enough time to prove it.Craig=== === Subject: : Re: Collatz Conjecture : Symmetry question.> http://arxiv.org/abs/math.NT/0312309> I think it's clear that it will never be proven, because there is just> not enough time to prove it.> CraigDo you mean proven true? It could be proven false with a single counter-example.=== === Subject: : Re: Collatz Conjecture : Symmetry question.>http://arxiv.org/abs/math.NT/0312309>I think it's clear that it will never be proven, because there is just>not enough time to prove it.>Craig> Do you mean proven true? It could be proven false with a single > counter-example.Yes.=== === Subject: : Re: Collatz Conjecture : Symmetry question.> Do you mean proven true? It could be proven false with a single > counter-example.The Collatz conjecture is: The Collatz sequence is allways finite and ends in 1.How then, can it be a counter-example?Luis R.=== === Subject: : Re: Collatz Conjecture : Symmetry question.>>Do you mean proven true? It could be proven false with a single >>counter-example.> The Collatz conjecture is: The Collatz sequence is allways finite and ends in 1.> How then, can it be a counter-example?By coming up with a number n that does not reach a power of two. One of two things will be true: The collatz sequence for this n will be unbounded, or it will cycle and never hit a power of 2. If such an n can be found, the Collatz conjecture is false. So the falsity of the conjecture might be proven in a finite number of steps.Bob Kolker=== === Subject: : Re: Collatz Conjecture : Symmetry question.>>> Do you mean proven true? It could be proven false with a single>> counter-example.> The Collatz conjecture is: The Collatz sequence is allways finite and> ends in 1. How then, can it be a counter-example?A non-trivial cycle would be a (finitely checkable) counterexample.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9Francis Wheen, _How Mumbo-Jumbo Conquered the World_=== === Subject: : Re: Collatz Conjecture : Symmetry question.>> Do you mean proven true? It could be proven false with a single>> counter-example.>The Collatz conjecture is: The Collatz sequence is allways finite and>ends in 1. How then, can it be a counter-example?> A non-trivial cycle would be a (finitely checkable) counterexample.You can also say that every Collatz sequence is infinite and cycles onthe trivial loop 1. Why is 1 a trivial loop? Because every 3x+C systemloops at C. Thus, 3x+3 loops at 3, 3x+5 loops at 5, etc. But whereas3x+1, 3x+3, 3x+9 (or any where C is a power of 3) have only the singletrivial loop (as far as anyone knows), non-power-of-3 Cs have multipleloops. So the general conjecture any sequence of a 3x+C system isinfinite and cycles on C is false when C=5.The 3x+5 trivial loop: 80 40 20 10 5 20 10 5 (loop at 5)A 3x+5 counterexample:44 22 11 38 19 62 31 98 49 152 76 38 19 (loop at 19)=== === Subject: : Re: Collatz Conjecture : Symmetry question.> I have a program that can build each level from the previous one.I also have written this program. I am using an arbitrary precisionlib to allow large numbers. I have two applications, one calculatesjust end points, and the other traverses the tree. We should comparenotes and maybe work together and release a simple tool for others touse. Just a thought.=== === Subject: : Re: Collatz Conjecture : Symmetry question.>I have a program that can build each level from the previous one.> I also have written this program. I am using an arbitrary precision> lib to allow large numbers. I have two applications, one calculates> just end points, and the other traverses the tree. We should compare> notes Sure, I'll see if I can dig up my program. The final version used textfiles to hold the level data. I stopped at Level 84 because the textfile, at 3.3GB was getting too big to fit on a single CD when zipped.> and maybe work together and release a simple tool for others to> use. Just a thought.Simple, yes. Usefull? That remains to be seen.But I've got some other stuff that might be interesting. I finallysolved my Big Problem (multi-generation sequence vectors) and am working on a web page to document it. I can post some of it herealong with the programs if you're interested.=== === Subject: : Re: Collatz Conjecture : Symmetry question.===> === Subject: : Re: Collatz Conjecture : Symmetry question.>> I have a program that can build each level from the previous one.>>> I also have written this program. I am using an arbitrary precision>> lib to allow large numbers. I have two applications, one calculates>> just end points, and the other traverses the tree. We should compare>> notes >Sure, I'll see if I can dig up my program. The final version used text>files to hold the level data. I stopped at Level 84 because the text>file, at 3.3GB was getting too big to fit on a single CD when zipped.>> and maybe work together and release a simple tool for others to>> use. Just a thought.>Simple, yes. Usefull? That remains to be seen.>But I've got some other stuff that might be interesting. I finally>solved my Big Problem (multi-generation sequence vectors) and am >working on a web page to document it. I can post some of it here>along with the programs if you're interested.I originally did this using a list in memory, but I ran out of memory.Here's the text file based version written in Python (which also does Big Arithmetic). The program reads the previous level out of a file,doubles every number and if a number is both == 1 (mod 3) AND== 0 (mod 2), it spawns a new branch by using the inverse 3x+1rule.# usage: python lread.py n# where n is level to process# reads file named Ln.txt# and outputs next level#import syslevel = sys.argv[1]filein = 'L' + level + '.txt'f = open(filein,'r')s = 'begin'while s != '': if s != 'begin': n = long(s) print n*2 p3 = divmod(n,3) if (p3[1]==1): p2 = divmod(n,2) if (p2[1]==0): print (n-1)/3 s = f.readline()f.closeStart by creating a text file named L5.txt that contains just16Running the program using L5.txtpython lread.py 5produces the following output:325To build up successive levels, redirect the output to a filepython lread.py 5 > L6.txtOf course, you can batch file the low levels since they go quick.It gets slow by the time you get to Level 70. And it starts eating up disk space: 65 File(s) 448,531,354 bytesThe one good thing is that you only need to keep the last levelon hand. All the previous ones can be archived. I originally wanted to go all the way to Level 100, but since Level 84took 3.3 GB, I lost interest at that point.=== === Subject: : Re: Collatz Conjecture : Symmetry question.===> === Subject: : Re: Collatz Conjecture : Symmetry question.>> I have a program that can build each level from the previous one.>> I also have written this program. I am using an arbitrary precision>> lib to allow large numbers. I have two applications, one calculates>> just end points, and the other traverses the tree. We should compare>> notes >Sure, I'll see if I can dig up my program. The final version used text>files to hold the level data. I stopped at Level 84 because the text>file, at 3.3GB was getting too big to fit on a single CD when zipped.>> and maybe work together and release a simple tool for others to>> use. Just a thought.>Simple, yes. Usefull? That remains to be seen.But I've got some other stuff that might be interesting. I finally>solved my Big Problem (multi-generation sequence vectors) and am >working on a web page to document it. I can post some of it here>along with the programs if you're interested.> I originally did this using a list in memory, but I ran out of memory.> Here's the text file based version written in Python (which also does > Big Arithmetic). The program reads the previous level out of a file,> doubles every number and if a number is both == 1 (mod 3) AND> == 0 (mod 2), it spawns a new branch by using the inverse 3x+1> rule.> # usage: python lread.py n> # where n is level to process> # reads file named Ln.txt> # and outputs next level> #> import sys> level = sys.argv[1]> filein = 'L' + level + '.txt'> f = open(filein,'r')> s = 'begin'> while s != '':> if s != 'begin':> n = long(s)> print n*2> p3 = divmod(n,3)> if (p3[1]==1):> p2 = divmod(n,2)> if (p2[1]==0):> print (n-1)/3> s = f.readline()> f.close> Start by creating a text file named L5.txt that contains just> 16> Running the program using L5.txt> python lread.py 5> produces the following output:> 32> 5> To build up successive levels, redirect the output to a file> python lread.py 5 > L6.txt> Of course, you can batch file the low levels since they go quick.> It gets slow by the time you get to Level 70. And it starts eating > up disk space:> 65 File(s) 448,531,354 bytes> The one good thing is that you only need to keep the last level> on hand. All the previous ones can be archived. I originally > wanted to go all the way to Level 100, but since Level 84> took 3.3 GB, I lost interest at that point.Hi Mensanator,You originally calculated these starting seeds for higher levelswhich I could not do because of my algorithm which found the levelrequested and thus all preceeding levels. So it was very slow findingall seeds for levels >20. What is surprising here is, your level counts do not match your oldlevel counts as shown below. Number of starting seeds for each level starting @ level 6 in the Collatz tree. Level # of seedsLevel 6 2 Level 7 2 Level 8 4Level 9 4Level 10 6Level 11 6Level 12 8 Level 13 10 Level 14 14 Level 15 18 Level 16 24 Level 17 29 Level 18 36Level 19 44 Level 20 58 Level 21 72 Level 22 91 Level 23 113 Level 24 143 Level 25 179 Level 26 227 Level 27 287 Level 28 366 Level 29 460Level 30 578 Level 31 732 Level 32 926 Level 33 1174 Level 34 1489 Level 35 1879 Level 36 2365 Etc.Am I interpeting somthing wrong here?Dan=== === Subject: : Re: Collatz Conjecture : Symmetry question.===>> === Subject: : Re: Collatz Conjecture : Symmetry question.>> I have a program that can build each level from the previous one.> I also have written this program. I am using an arbitrary precision> lib to allow large numbers. I have two applications, one calculates> just end points, and the other traverses the tree. We should compare> notes >Sure, I'll see if I can dig up my program. The final version used text>>files to hold the level data. I stopped at Level 84 because the text>>file, at 3.3GB was getting too big to fit on a single CD when zipped.>> and maybe work together and release a simple tool for others to> use. Just a thought.>Simple, yes. Usefull? That remains to be seen.>But I've got some other stuff that might be interesting. I finally>>solved my Big Problem (multi-generation sequence vectors) and am >>working on a web page to document it. I can post some of it here>>along with the programs if you're interested.>I originally did this using a list in memory, but I ran out of memory.>Here's the text file based version written in Python (which also does >Big Arithmetic). The program reads the previous level out of a file,>doubles every number and if a number is both == 1 (mod 3) AND>== 0 (mod 2), it spawns a new branch by using the inverse 3x+1>rule.># usage: python lread.py n># where n is level to process># reads file named Ln.txt># and outputs next level>#>import sys>level = sys.argv[1]>filein = 'L' + level + '.txt'>f = open(filein,'r')>s = 'begin'>while s != '':> if s != 'begin':> n = long(s)> print n*2> p3 = divmod(n,3)> if (p3[1]==1):> p2 = divmod(n,2)> if (p2[1]==0):> print (n-1)/3> s = f.readline()>f.close>Start by creating a text file named L5.txt that contains just>16>Running the program using L5.txt>python lread.py 5>produces the following output:>32>5>To build up successive levels, redirect the output to a file>python lread.py 5 > L6.txt>Of course, you can batch file the low levels since they go quick.>It gets slow by the time you get to Level 70. And it starts eating >up disk space:> 65 File(s) 448,531,354 bytes>The one good thing is that you only need to keep the last level>on hand. All the previous ones can be archived. I originally >wanted to go all the way to Level 100, but since Level 84>took 3.3 GB, I lost interest at that point.> Hi Mensanator,> You originally calculated these starting seeds for higher levels> which I could not do because of my algorithm which found the level> requested and thus all preceeding levels. So it was very slow finding> all seeds for levels >20. > What is surprising here is, your level counts do not match your old> level counts as shown below.> Number of starting seeds for each level starting @ level 6 in the Collatz tree.> Level # of seeds> Level 6 2 > Level 7 2 > Level 8 4> Level 9 4> Level 10 6> Level 11 6> Level 12 8 > Level 13 10 > Level 14 14 > Level 15 18 > Level 16 24 > Level 17 29 > Level 18 36> Level 19 44 > Level 20 58 > Level 21 72 > Level 22 91 > Level 23 113 > Level 24 143 > Level 25 179 > Level 26 227 > Level 27 287 > Level 28 366 > Level 29 460> Level 30 578 > Level 31 732 > Level 32 926 > Level 33 1174 > Level 34 1489 > Level 35 1879 > Level 36 2365 > Etc.> Am I interpeting somthing wrong here?Yes. The list I just posted is the size of the text files in bytes.L6.txt still has two seeds but it uses 7 bytes of disk space.I was pointing that out in case anyone wants to try running mylittle program (don't say you weren't warned).You can't get the count directly from the file size. Unlike Unix/Linux,there's no text file line counting utility in Windows. You can easily track the number of seeds generated and print that, but the count willend up in the file when doing simple re-direct.If one was ambitious, one could add a file write routine in place of theprint statements. That way you could print the count without it ending up in the level file.Or you could write a program that counts lines in a text file.> Dan=== === Subject: : Re: Collatz Conjecture : Symmetry question.First: Won't there be duplicates in the large files?Second: The two divmod's can be combined into one.-Michael.=== === Subject: : Re: Collatz Conjecture : Symmetry question.===> === Subject: : Re: Collatz Conjecture : Symmetry question.>>First: Won't there be duplicates in the large files?There shouldn't be- if the Collatz Conjecture is true- if my program is correctly generating the tree- but I haven't checkedThe program doesn't make any attempt to organize the numbersinto proper branches, just count them. But that can easily bedone by converting the numbers to binary.Every odd number is the start of a branch and every even numberhas an odd seed pattern in binary that is the same as the oddnumber of the start of its branch.The text file L6.txt has just two numbers 32 and 5. In binary, these are100000101Note the odd seed pattern of 16 is 1. At each subsequent level,these binary patterns get a 0 appended to them. At level 16they will have become10000000000000001010000000000All numbers that have the same odd seed pattern are on thesame branch. Since every level has a different number of 0s,there can't be any duplication across levels. And IF each number on a given level has a different odd seed pattern,there can't be any duplication within a level.The purpose of the mod 3, mod 2 requirement is to ensurethat the newly spawned numbers (those generated by (n-1)/3)create a new odd seed pattern that hasn't appeared before (in which case, neither it nor any of its descendants will bea duplicate of any other number). Any number == 1 (mod 3) will give you an integer when you apply (n-1)/3, but you cannot have two consecutive odd numbers in a Collatz Sequence, so odd numbers are not allowed to spawn new branches, otherwise you would get duplications.>Second: The two divmod's can be combined into one.How? My thinking was to do the divmod 3 first since it willfail the test two out of three times allowing the divmod 2test to be skipped.>-Michael.--MensanatorAce of Clubs=== === Subject: : Re: Got a speeding ticket and need to fight backwanted-suresh>Good one, chief. I hear the WB network is hiring comedy writers.>>WB is also looking for lawyers who can play sidewalk Santas for their>christmas in july specials. Btw, WB is also looking for statistcians>who can prove that christmas is supposed to start in july.> It took you six days to come up with *that*? Pathetic.> Doug=== === Subject: : Re: To users of GMP> The FFT code in GMP currently contains a flaw that leads to> inaccurate results.Can you please give some order of magnitude for the number of bitsbelow which it can be safely assumed that the bug will not occur(maybe just because the FFT code is not even called).> More information can be found at http://www.swox.com.Could not find more thanA bug in the FFT multiply code that can cause miscomputation has been found. Until we can provide a fix, and until we have performed extensive further testing of the code, all users are urged to recompile GMP using the configure option --disable-fft.TIA, Francois Grieu=== === Subject: : Re: To users of GMPCc: Francois Grieu > The FFT code in GMP currently contains a flaw that leads to>> inaccurate results.> Can you please give some order of magnitude for the number of bits> below which it can be safely assumed that thm.>Could not find more thanA bug in the FFT multiply code that can cause miscomputation>has been found. Until we can provide a fix, and until we have>performed extensive further testing of the code, all users are>urged to recompile GMP using the configure option --disable-fft.>TIA,>Francois Grieu> There is a new random number generator developed which can be found at> andomb.obj> which makes the span lengths proportional to the size of the output.> More information about the random code can be found at> .> The new random code triggered the bug in the FFT code almost immediately> after testing began.So what versions of GMP have this bug? I'm using the Windows gmpy module in Python which appears to be based on version 4.0. Does that have the bug?=== === Subject: : To users of GMPThe FFT code in GMP currently contains a flaw that leads to inaccurateresults. The bug was discovered with the new mpz_rrandomb routine earlier.It is recommended that users of GMP recompile the distribution and use theoption --disable-fft. More information can be found at http://www.swox.com.Thank you.=== === Subject: : smallest eigenvalue of Laplacianthere is a theorem that says that an eigenfunction corresponding tothe smallest eigenvalue of the Laplacian on some bounded domain has nozeros inside that domain.Is the reverse implication also true? Meaning: Does everyeigenfunction without zeros inside the domain belong to the smallesteigenvalue?Thank you in advance for any helpful comments.Yours sincerely,Tobias N.8ahring-- See me at tn-home.de/Tobias/=== === Subject: : Re: smallest eigenvalue of Laplacian>there is a theorem that says that an eigenfunction corresponding to>the smallest eigenvalue of the Laplacian on some bounded domain has no>zeros inside that domain.>Is the reverse implication also true? Meaning: Does every>eigenfunction without zeros inside the domain belong to the smallest>eigenvalue?The Laplacian is a self-adjoint operator, so eigenfunctions for differenteigenvalues are orthogonal. Two functions that both have constant signscan't be orthogonal, unless one is 0.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2=== === Subject: : Re: smallest eigenvalue of Laplacian> there is a theorem that says that an eigenfunction corresponding to> the smallest eigenvalue of the Laplacian on some bounded domain has no> zeros inside that domain.> Is the reverse implication also true? Meaning: Does every> eigenfunction without zeros inside the domain belong to the smallest> eigenvalue?> Thank you in advance for any helpful comments.> Yours sincerely,> Tobias N.8ahringWhat a great question. Someone told me that it has been recently shown that in two dimensions, the eigenfunction corresponding to the second eigenvalue has zeroes, indeed, the set of zeroes forms a nodal line that splits the domain in half. Presumably if you look at this paper, it might have references to the result you are looking for. I think that this is the paper:MR1152231 (93g:35100)Melas, Antonios D.(1-UCLA)On the nodal line of the second eigenfunction of the Laplacian in $ Rsp 2$.J. Differential Geom. 35 (1992), no. 1, 255--263.35P05 (35J05 58G25) eigenfunction of the Laplacian with zero boundary condition for a bounded domain $Omegasubseteqbold R^2$ does not have a closed nodal line. This was asked by S.-T. Yau for $Omega$ a bounded convex domain in $bold R^2$. Twenty years ago, Payne proved the conjecture provided the domain $Omegasubseteqbold R^2$ is symmetric with respect to one line and convex with respect to the direction vertical to this line. Also, C.-S. Lin proved the conjecture provided the domain $Omegasubseteqbold R^2$ is smooth, convex, and invariant under a rotation with angle $2pi p/q$, where $p$ and $q$ are positive integers. Recently D. Jerison [Internat. Math. Res. Notices 1991, no. 1, 1--5; MR 92d:35210] proved the conjecture for long thin convex sets. Without any assumption on the smoothness of $Omega$ he showed that the nodal line has to intersect $partialOmega$ in exactly two points.We prove the conjecture when $Omega$ is a bounded convex domain in $bold R^2$ with $C^infty$ boundary.=== === Subject: : Re: smallest eigenvalue of Laplacian>> there is a theorem that says that an eigenfunction corresponding to>> the smallest eigenvalue of the Laplacian on some bounded domain has no>> zeros inside that domain.>> Is the reverse implication also true? Meaning: Does every>> eigenfunction without zeros inside the domain belong to the smallest>> eigenvalue?>> Thank you in advance for any helpful comments.>> Yours sincerely,>> Tobias N.8ahring> What a great question.Robert Israel's answer was much better than mine!=== === Subject: : Re: If We Replaced Each Prime With -1...> Let c(k) = the sum of the exponents in the prime factorization of k.> So, if we exchanged each prime in the prime factorization of k with> (-1),> we would get (-1)^c(k).> What I am wondering, however, is> what is C(m) => sum{k=1 to m} (-1)^c(k)> asymptotical towards?The function (-1)^c(k) is usually denoted lambda(k) and is known asLiouville's lambda function. Its partial sums (up to m) are o(m); thisis equivalent to the prime number theorem. The stronger statement thatits partial sums are O(m^{1/2+epsilon}) for each fixed epsilon > 0(and m->oo) is equivalent to the Riemann hypothesis.You may have seen the same results stated for the Mobius function. Onecan relate the partial sums of lambda to those of mu and thereby goback and forth between these results as follows. As you note,> C(m) also equals:> sum{k=1 to m} floor(sqrt(m/k)) *mu(k),and this can be put in the alternate form C(m) = sum_{k <= sqrt(m)} M(m/k^2),where M(x):=sum(mu(n), n<=x); one can also verify the related identity M(m) = sum_{k <= sqrt(m)} mu(k) C(m/k^2). and similarly for the estimate mentioned in connection with theRiemann hypothesis. On the other hand, neither C(m) = o(m^{1/2}) orM(m) = o(m^{1/2}) can hold (as m->oo); this is because either wouldimply zeta had no zeros on Re(s) = 1/2.You may be interested in Chapter 4 of my online notes which presentsan elementary proof of the PNT (in the form M(x) = o(x) as x->oo) andwhich discusses some of these points. See http://www.princeton.edu/~ppollack/notes/Hope this helps,Paul=== === Subject: : Re: If We Replaced Each Prime With -1...>Let c(k) = the sum of the exponents in the prime factorization of k.>So, if we exchanged each prime in the prime factorization of k with>(-1),>we would get (-1)^c(k).>What I am wondering, however, is>what is C(m) =>sum{k=1 to m} (-1)^c(k)>asymptotical towards?> The function (-1)^c(k) is usually denoted lambda(k) and is known as> Liouville's lambda function. Its partial sums (up to m) are o(m); this> is equivalent to the prime number theorem. The stronger statement that> its partial sums are O(m^{1/2+epsilon}) for each fixed epsilon > 0> (and m->oo) is equivalent to the Riemann hypothesis.> You may have seen the same results stated for the Mobius function. One> can relate the partial sums of lambda to those of mu and thereby go> back and forth between these results as follows. As you note,>C(m) also equals:>sum{k=1 to m} floor(sqrt(m/k)) *mu(k),> and this can be put in the alternate form> C(m) = sum_{k <= sqrt(m)} M(m/k^2),> where M(x):=sum(mu(n), n<=x); one can also verify the related identity> M(m) = sum_{k <= sqrt(m)} mu(k) C(m/k^2). > and similarly for the estimate mentioned in connection with the> Riemann hypothesis. On the other hand, neither C(m) = o(m^{1/2}) or> M(m) = o(m^{1/2}) can hold (as m->oo); this is because either would> imply zeta had no zeros on Re(s) = 1/2.> You may be interested in Chapter 4 of my online notes which presents> an elementary proof of the PNT (in the form M(x) = o(x) as x->oo) and> which discusses some of these points. See> http://www.princeton.edu/~ppollack/notes/> Hope this helps,> PaulAfter I posted, I entered the first few terms of the C-sequence into:http://www.research.att.com/~njas/sequences/index.html# Lwhich sent me to:http://mathworld.wolfram.com/LiouvilleFunction.htmlBut thanks for the reply!Leroy Quet=== === Subject: : Re: Teaching philosophy> I need to write a description of my teaching philosophy. In order to> do so accurately, I think I need to write at greater length and in> greater detail on this topic than any hiring committee will want to read.> Moreover, I need to present my views accurately but, somehow, in such> a manner as not to vitiate the consideration of my application.teaching philosophy a significant number of mathematicians wouldembrace:teach with depth and require students to learn by setting highstandards.teaching philosophy majority of bureaucrats and managers plagueingacademic institutions want to see:full commitment to teach in a diverse environment with provenexcellence in teaching. (whatever the hell that means...)teaching philosophy majority of mathematicians practice:teach one or two sections per class in order to meet an extensivenon-realistic curriculum realised by either a bureaucratic committteeor a dictator.> That being the case, can someone please tell me what my teaching> philosophy is?