mm-1030 Subject: Re: e is transcendental >O.k. then , as long there is no conžict with : >Re(e^[iPi]) = Re(-1+i[0]) = -1 That has never been in doubt. Neither has the observation that Im(e^[i pi]) = Im(-1+i[0]) = 0. The statement that caused all the problem was the statement that e^[i pi] = 0. === Subject: Re: Clean coordinates on a unit sphere >> Hi - I¹m a computer science student working in the Želd of vision and >> graphics. I¹m looking for a clean way to represent a _direction_ in 3d >> space (this can also be imagined as a position on a unit sphere). >> I don¹t like the traditional methods of using a 3d vector or spherical >> coordinates for the following reasons: >> - 3d Vector e.g. (x, y, z): This is of course very common - can be >> rotated with matrices, quaternions... but it also describes length, >> which I don¹t need. >> - Spherical coordinates e.g. (theta, phi): This is more appropriate as >> it does not describe length. But it still appears ugly to me since >> multiple coordinate pairs can describe the same point e.g. (x, pi/2). >> Is there a better way? I¹m sure I¹m not the Žrst who asked this >> question. It seems similar to the problem of creating a 2d map of earth, >> so it¹s probably been thought about before... And since there doesn¹t >> seem to be a _popular_ solution, it¹s probably hard - but that¹s what >> makes it FUN !!! >> My ideal solution would map a unit sphere with a uniform density (if >> that makes any sense to any one). Could a person use some type of >> tesselation of a sphere with an arbitrary density??? >> Any ideas are good ideas! Please keep the language simple - I¹m not as >> smart as you. >> Nathan > How about using unit vectors (x,y,z), i.e., satisfying x^2+y^2+z^2=1? > This seems clean, unless by clean you really mean efŽcient, since > (x,y,z) does contain redundant information. For efŽciency, you could > just store (x,y,sign(z)), since then z is uniquely determined by z = > sign(z)*sqrt(1-x^2-y^2). But then you¹re back some reduncency, since > on the equator x^2+y^2=1, the sign of z is automatically zero. > You say My ideal solution would map a unit sphere with a uniform > density. Do you mean that you want a bijective map f:U-->S from a > subset of the Euclidean plane to the unit sphere that preserves area, > or at least so that for (nice) subsets V of U, f satisŽes > Area(f(V))=c*Area(V) for a constant c that does not depend on V? I¹m > not quite sure how otherwise to interpret your word density. If you circumscribe a cylinder with radius r and height 2*r about the sphere, then the horizontal projection from the cylinder¹s lateral surface onto the sphere preserves area. Hence there is an area-preserving map from a rectangle with width 2*pi*r and height 2*r onto the surface of the sphere with radius r. There is still the objection that the mapping is not a bijection, but it does at least preserve area, if not cardinality. -- Dave Seaman Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Interesting problem Trivially not ! take the element ab where a in A and b in B, then it is neither in A nor in B on account of the mult. closure > I have the following problem: > A and B are two disjoint sets whose union is |R+. Both A and B are closed > under sum and multiplication. Is it possible that neither A nor B is the > VOID set? > {V} === Subject: Re: Interesting problem >Trivially not ! >take the element ab where a in A and b in B, >then it is neither in A nor in B on account >of the mult. closure perhaps that isn¹t what is meant by multiplicative closure - that would seem to be ideal closure, but I believe it can be extended to show the answer. If ab is in B, and 1/b is in B, then so is a# so 1/b is in A. Now wlog 1 is in B, and hence so is 2, thus 1/2 is in A, hence 1/2+1/2=1 is in A# >> I have the following problem: >> A and B are two disjoint sets whose union is |R+. Both A and B are closed >> under sum and multiplication. Is it possible that neither A nor B is the >> VOID set? >> {V} === Subject: Re: Interesting problem >perhaps that isn¹t what is meant by multiplicative closure - that >would seem to be ideal closure, but I believe it can be extended to >show the answer. >If ab is in B, and 1/b is in B, then so is a# contradiction assuming a in A and ab in B, that is... >so 1/b is in A. Now wlog 1 is in B, and hence so is 2, thus 1/2 is in >A, hence 1/2+1/2=1 is in A# No, you could only say thus 1/2 is in A if there was some a in A such that 2a is in B. Of course that can¹t happen. Why can¹t 1/2, 1, 2 all be in B? In fact if 1 is in B, then all positive rationals are in B. Moreover, all rational powers of rationals are in B. But there are lots of other numbers, e.g. perhaps some transcendentals, that could be in A. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Interesting problem Robert Israel ha scritto nel messaggio > No, you could only say thus 1/2 is in A if there was some a in A such > that 2a is in B. Of course that can¹t happen. Why can¹t 1/2, 1, 2 all > be in B? In fact if 1 is in B, then all positive rationals are in B. > Moreover, all rational powers of rationals are in B. > But there are lots of other numbers, e.g. perhaps some transcendentals, > that could be in A. I can show that both A and B have to be dense in |R+. The problem i found in this problem are trascendental numbers.... i have no idea about a solution... === Subject: Re: Interesting problem <1mZ_b.18593$gk.818633@news3.tin.it> sci.math, Interesting problem >> A and B are two disjoint sets whose union is |R+. >> Both A and B are closed under sum and multiplication. >> Is it possible that neither A nor B is the VOID set? >In fact if 1 is in B, then all positive rationals are in B. >Moreover, all rational powers of rationals are in B. >But there are lots of other numbers, e.g. perhaps some >transcendentals, that could be in A. Generalization: b in B ==> bQ+ = { bq | q in Q, q > 0 } subset B b in B ==> b^Q+ = { b^q | q in Q, q > 0 } subset B b in B ==> [b^Q+ * Q+]^Q+ = b^Q+ * (Q+)^Q+ subset B and similar for A. Thus as Q+ subset B, B is dense subset R+ and if a in A, aQ+ subset A shows A is dense subset R+ ---- === Subject: Re: RFI:Wanless¹ Fourth Conjecture/Dirichlet¹s Geometric Theorem >For every pair of integers a, b with a > 1 and ab <> 2 (thus leaving >out the case of the Mersenne numbers), there are inŽnitely many >primes p such that a^p + b is composite. Sorry, there¹s a typo here: it should say ab <> -2. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: measure of stuck-togetherness So the earth¹s surface is about one-quarter land. That land could be distributed across the surface of the globe in different ways - I can imagine at one extreme, all the land stuck together in one roundish continent, and at the other extreme, all the land spread out evenly, in equally-spaced tiny islands. Is there a measure, preferably just one number, that captures these differences? If so, how is it calculated? === Subject: Re: measure of stuck-togetherness >So the earth¹s surface is about one-quarter land. That land could be >distributed across the surface of the globe in different ways - I can >imagine at one extreme, all the land stuck together in one roundish >continent, and at the other extreme, all the land spread out evenly, in >equally-spaced tiny islands. Is there a measure, preferably just one >number, that captures these differences? If so, how is it calculated? I don¹t know if there¹s an accepted measure, but here¹s one for you: Partition the surface into many equal-sized regions. In each region, measure the percentage of land. Take the standard deviation across all the regions. That¹s your number. -- Matthew T. Russotto mrussotto@speakeasy.net Extremism in defense of liberty is no vice, and moderation in pursuit of justice is no virtue. But extreme restriction of liberty in pursuit of a modicum of security is a very expensive vice. === Subject: Re: measure of stuck-togetherness charset=Windows-1252 >So the earth¹s surface is about one-quarter land. That land could be >distributed across the surface of the globe in different ways - I can >imagine at one extreme, all the land stuck together in one roundish >continent, and at the other extreme, all the land spread out evenly, in >equally-spaced tiny islands. Is there a measure, preferably just one >number, that captures these differences? If so, how is it calculated? > I don¹t know if there¹s an accepted measure, but here¹s one for you: > Partition the surface into many equal-sized regions. In each region, > measure the percentage of land. Take the standard deviation across > all the regions. That¹s your number. I could be wrong, but that doesn¹t seem to work. Call each of the equal-sized regions a cell, and suppose there are n cells. Let p_k be the proportion of the kth cell-area that¹s land. Then the number you describe is m = s.d.{p_1,...,p_n}. If P is the overall proportion of the global area that¹s land, then avg{p_1,...,p_n} = P, and m**2 = (1/n) sum( (p_k - P)**2, k=1..n ). There seem to be two cases: n is either large enough to give a stable value for m (so making n even larger won¹t matter much), or n is too small (and m will be režecting n rather what it¹s supposed to be measuring). In the acceptable, stable case, it seems that n would eliminate most boundary effects, leaving a sum that comes mainly from two contributions: m**2 ~ (1/n) [ (#cells on land)(1-P)**2 + (#cells off land)(0-P)**2 ] ~ (1/n) [ n P (1-P)**2 + n (1-P) (0-P)**2 ] ~ P(1-P) In other words, either n is too small for m to measure what it¹s supposed to, or n is large enough but then m measures only the global property P(1-P). --r.e.s. === Subject: re:measure of stuck-togetherness Suppose that only the land on the surface has mass, and the rest of the earth (the oceans and the interior of the earth) is massless. We still suppose the earth holds together. In each point X in the interior of the earth measure the absolute value of the gravitaional pull F(X) and gravitational potential E(X). The following are possible measures of dispersion: 1) max F(X) 2) mean F(X) or mean F(X)^2 etc. 3) max (E(X) - E(Y)) 4) mean (E(X) - E(Y)) or mean (E(X) - E(Y))^2 etc. For uniform distribution those are all about zero. ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: measure of stuck-togetherness > So the earth¹s surface is about one-quarter land. That land could be > distributed across the surface of the globe in different ways - I can > imagine at one extreme, all the land stuck together in one roundish > continent, and at the other extreme, all the land spread out evenly, in > equally-spaced tiny islands. Is there a measure, preferably just one > number, that captures these differences? If so, how is it calculated? You¹ve received lots of suggestions, but let me just add the fractal dimension of the boundary (coastline). - Risto - === Subject: Re: measure of stuck-togetherness charset=Windows-1252 > So the earth¹s surface is about one-quarter land. That land could be > distributed across the surface of the globe in different ways - I can > imagine at one extreme, all the land stuck together in one roundish > continent, and at the other extreme, all the land spread out evenly, in > equally-spaced tiny islands. Is there a measure, preferably just one > number, that captures these differences? If so, how is it calculated? Various measures of dispersion could work. For points x and y on the surface, let dist(x,y) be the length of the shortest great-circle arc connecting x and y. Then two such measures are E dist(x,y) and sqrt(E dist(x,y)**2), where x and y are iid uniform on the land-region. As approximations, you could replace the expectations by averages with x and y suitably discretized. --r.e.s. === Subject: Re: measure of stuck-togetherness Standard deviation of Statistical mathematics, measures the difference. === Subject: Re: measure of stuck-togetherness > So the earth¹s surface is about one-quarter land. That land could be > distributed across the surface of the globe in different ways - I can > imagine at one extreme, all the land stuck together in one roundish > continent, and at the other extreme, all the land spread out evenly, in > equally-spaced tiny islands. Is there a measure, preferably just one > number, that captures these differences? If so, how is it calculated? How about how many islands there are or the area of the largest one. Depends on what you mean by differences. Bill === Subject: Re: measure of stuck-togetherness > So the earth¹s surface is about one-quarter land. That land could be > distributed across the surface of the globe in different ways - I can > imagine at one extreme, all the land stuck together in one roundish > continent, and at the other extreme, all the land spread out evenly, in > equally-spaced tiny islands. Is there a measure, preferably just one > number, that captures these differences? If so, how is it calculated? The perimeter is one such measure. === Subject: Re: The Dot and Line--A Movie They played it this morning. Unfortunately, they cannot foretell much in advance when certain shorts will be shown. I guess I¹ll just have to have my VHS recorder ready for action. >>The animator Chuck Jones directed a animated feature in 1965 called The Dot >>and Line, >>which was nominated for an Academy Award. I happened to see part of it on the >>TMC >>channel this morning. Does anyone know if it¹s on video, VHS or DVD? I went >>to Amazon >>and looked and found mention of it on one Chuck Jones feature, but a reviewer >>noted >>that it was incomplete. I¹m told it sometimes appears on The Cartoon Channel. > There is a book, from which I think the movie was made, called > The Dot and the Line, A Romance in Lower Mathematics by Norton Juster > See for exaxmple > http://www.amazon.com/exec/obidos/tg/detail/-/1587170663/103- 8735420-9207 > 022?v=glance -- Wayne T. Watson (121.015 Deg. W, 39.262 Deg. N, 2,701 feet, Nevada City, CA) Don¹t hassle me with your signs, Chuck -- Peppermint Patty, Peanuts Web Page: sierra_mtnview -at- earthlink -dot- net Imaginarium Museum: === Subject: Re: Bounds on Moments > Suppose x is a non-negative random variable. Given that the Žrst > moment of x, E(x^1) and E(x^2) are m1 and m2, respectively. What are > the bounds upon m3=E(x^3)? Why should we expect such a bound? For example, let f(x) = c/(1+x^4), where c is chosen so that the integral of f over (0,oo) is 1. Then for the probability density function f we have E(x^1) and E(x^2) < oo, but E(x^3) = oo. === Subject: Re: More transcendentals > The other transcendentals than e and pi got my quriosity perched. > So besides the obvious generalization to Lousville¹s construction, where one > has all the digits after the decimal point equal to k except the digits at > positions n! which might equal m=/=k, are there any famous decimals that are > constructed by using digits taken from a particular sequence a_n? The factorial base allows you to create a lot of transcendentals from e. The allowed coefŽcients at position n are (0 through n). Each position represents 1/n! (or n! left of radix). Every Žnite representation in base ! is rational. So we only care about inŽnite representations. Let c = e-2. c = 0.111... base ! (1/2 + 1/6 + 1/24 + ...) Let q be a rational number. q * c = transcendental. I think this means that any inŽnite base ! representation that has a repeating period can be shown to be of the form q * c, q rational. I don¹t have a proof yet. (multiplication rules for base ! are complicated.) 1.12 * 0.111... (base !) is transcendental. 1.12 * .1 = 1+1/2+2/6 (11/6) * 1/2 = 11/12 = .122 (base !) 1.12 * .11 = 11/6 * 2/3 = 11/9 = 1.01114 (base !) It would be interesting if PI has a repeating base ! representation. number. I have no idea what this means. Russell - 2 many 2 count === Subject: Re: More transcendentals charset=iso-8859-7 Ć Russell Easterly Ūēņįćå óōļ ķÜīłķį > The factorial base allows you to create a lot of transcendentals from e. > The allowed coefŽcients at position n are (0 through n). > Each position represents 1/n! (or n! left of radix). > Every Žnite representation in base ! is rational. > So we only care about inŽnite representations. > Let c = e-2. [snip actual constructions for brevity] have been saved and will be analyzed by noodle at school. > Russell > - 2 many 2 count -- Ioannis Galidakis http://users.forthnet.gr/ath/jgal/ ------------------------------------------ Eventually, _everything_ is understandable === Subject: Re: More transcendentals charset=Windows-1252 > You might be interested in one or more of the following. -snip- > MR0965057 (90a:11082) > Loxton, J. H.(5-MCQR); van der Poorten, A. J.(5-MCQR) > Arithmetic properties of automata: regular sequences. > J. Reine Angew. Math. 392 (1988), 57--69. > 11J85 (11K16 11R04 68Q70) > Not very much is known about the decimal expansions of algebraic > irrational numbers. One might conjecture that these expansions are > random, or even normal. In this paper it is shown that they cannot be > generated by a Žnite automaton. The result appears as a corollary of a > theorem on algebraic independence. Therefore, heavy Mahler-type > machinery is used. > Reviewed by F. Schweiger I thought it might be worth mentioning that the reviewer apparently overstated what was proved in the paper when he said In this paper it is shown that they cannot be generated by a Žnite automaton. According to Allouche & Shallit (in their book Automatic Sequences, paper by Cobham ... ... which was a step towards proving that all numbers in L(k,b) are either rational or transcendental. This last assertion was approached by Cobham [1968b] (who incorrectly claimed a proof) and Loxton and van der Poorten [1982, 1988], but the general result has not been fully proved yet. Here L(k,b) is the set of all (k,b)-automatic reals. --r.e.s. === Subject: Re: BETA-BINOMIAL EXTENSION FORMULA ??? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1P2cGO15493; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P2UBi14992 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i1P2UAk02756; >I am a programmer working on a project that performs calculations on TV >ratings data. The beta-binomial extension formula is used to express reach >and frequency information (reach and frequency refer to how many people view >a schedule of commercials and how often they view them, respectively). >However, I cannot get my hands on the formula itself. Does anyone know what >PS: I have a mathematics background, but it is based more on theory than >statistics. Could this formula have something to do with a beta >distribution? > === Subject: Calculus help please! hi, i¹m having troubles trying to understand section 7.5 out of Stewart¹s single variable calculus 5e - does anyone know of any good sites ( or could possibly explain to me in a nutshell) inverse trig === Subject: ArcCos vs arcCos I recall that one of ArcCos and arcCos which is a function. I think that ArcCos is the function but I was hoping that someone can conŽrm this for === Subject: Re: ArcCos vs arcCos > I recall that one of ArcCos and arcCos which is a function. I think that > ArcCos is the function but I was hoping that someone can conŽrm this for There¹s no way to conŽrm this because there are two competing, diametrically opposite conventions. See, below, a copy of a portion of an old post of mine for details. (But in any event I can¹t see any reason to capitalize the second c in Arccos -- unless you¹re using Mathematica, that is.) David Cantrell ---------------------------------------- There is a convention according to which, if a multivalued relation is denoted in lower case, then capitalization is used to distinguish the corresponding principal-valued function. [Referring again to the entry for principal value in the _HarperCollins Dictionary of Mathematics_, The function that has as its values the principal values of a many-valued function is conventionally indicated by writing it with a capital letter; thus, Cotan^(-I) is the principal value of the inverse cotangent, and, especially in complex functions, Ln is the principal value of the natural logarithm.] ... Unfortunately, another convention is the precise reverse of that described above! [For example, in _An Atlas of Functions_, arcsin denotes the inverse sine function while Arcsin denotes the multivalued relation.] And yet another practice is to make no notational distinction whatsoever between a multivalued relation and the corresponding principal- valued function! [See, for example, the treatment of inverse trigonometric functions in the CRC Standard Mathematical Tables and Formulae, 30th ed.] === Subject: Re: ArcCos vs arcCos >> I recall that one of ArcCos and arcCos which is a function. I think that >> ArcCos is the function but I was hoping that someone can conŽrm this for >There¹s no way to conŽrm this because there are two competing, >diametrically opposite conventions. See, below, a copy of a portion of >an old post of mine for details. (But in any event I can¹t see any >reason to capitalize the second c in Arccos -- unless you¹re using >Mathematica, that is.) There¹s a fairly common convention in calculus books (when deŽning inverse trig functions - it almost never shows up outside that section) to write Cos for the restriction of cos to the interval [0,pi]. So (although I don¹t recall seeing anybody actually do it), it¹s possible the function Cos, not of cos. I don¹t know why you¹d want to capitalize the A as well, unless you just don¹t want it to look like a Java method. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: ArcCos vs arcCos >> I recall that one of ArcCos and arcCos which is a function. I think >> that ArcCos is the function but I was hoping that someone can conŽrm >There¹s no way to conŽrm this because there are two competing, >diametrically opposite conventions. See, below, a copy of a portion of >an old post of mine for details. (But in any event I can¹t see any >reason to capitalize the second c in Arccos -- unless you¹re using >Mathematica, that is.) > There¹s a fairly common convention in calculus books (when deŽning > inverse trig functions - it almost never shows up outside that section) > to write Cos for the restriction of cos to the interval [0,pi]. So > (although I don¹t recall seeing anybody actually do it), it¹s possible > the function Cos, not of cos. you, I can¹t recall having ever seen anyone actually use arcCos). I might also add that one does sometimes see Cos^{-1} used to denote the principal-valued inverse cosine function. David David > I don¹t know why you¹d want to capitalize > the A as well, unless you just don¹t want it to look like a Java method. === Subject: Re: Do Prime Algebraic Numbers even exist? I am reminded of one error and one deŽciency in my original post, so some additions: > In sci.math, Dik T. Winter > > : > Indeed, but there are *no* primes in the algebraic integers. Now you > could deŽne class 1 primes to be primes in the integers, but I fail > to see how you could deŽne class 2 primes or anything else. Not > all primes in number Želds are of the form n-th root of p with p > a prime. For instance, one of the primes in Q(sqrt(2)) is 1 + sqrt(2) > (I think). I should have thought further, as I was reminded, 1 + sqrt(2) is a unit, so no prime. To get a prime we have to know something about norms in quadratic Želds. In Q(sqrt(m)) the norm of a number a + b.sqrt(m) is a^2 - m.b^2. So the norm of 1 + sqrt(2) = -1, and hence it is a unit. You may verify that the norm so deŽned is multiplicative, also that the norm is integer if and only if a and b are both integers, or are both half of an odd integer when m = 1 mod 4. And these are precisely the algebraic integers in the Želd. To get a prime you look for a number with (integer) prime norm. In this case 3 + sqrt(2), with norm 7. > The number of units in quadratic Želds (Q(sqrt(m)) ) is as follows: > m > 0: inŽnitely many > m = -1: 4 > m = -3: 6 > m < 0: 2 in all other cases. > (again, assuming m to be square free.) > However, when m > 0 it is not easy to always Žnd a unit. What I meant here is that it may take a lot of work. There are deterministic methods to get a unit (or the fundamental unit), but the computational complexity becomes pretty big. I have done extensive calculations with discriminant < 10000. (The discriminant is equal to m when m = 1 mod 4, otherwise it is 4m.) The Žrst one I did not complete was D = 409 (m = 409) where in the fundamental unit x + y.sqrt(m), x exceeds 10^11. And there are 19 more in the range D < 1000. (Yes, I could have gone to some bignum package, but did not feel inclined to do so...) This is similar to the case of showing the gcd of two algebraic integers, even if they are both from a quadratic Želd. You may have reasons to know that there is a non-unit common factor, but showing its value is something else. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: antisymmetric relation = ~symmetric? > Recall that antisymmetric mean aRb ^ bRa => a = b, > or for all a not = b, aRb => ~bRa. > In other words this relation is never symmetric! No! Equality is both symmetric and assymmetric. Assymmetric is what you call antisymmetric. Antisymmetric is not(aRb & bRa). === Subject: Exortic Coordinate Systems. Hello All, I need to create the direction cosines for an orthogonal coordinate system for the following two geometries: 1. A cylynder of varying elliptical cross section. 2. A torid of varying elliptical cross section. I know how do derive the direction cosines for the three orthogonal directions (radial, circumferential, and longitudinal) at every point, but it is fairly complex. Before I embark on this adventure, I want to make sure this informaton is not readily available elsewhere. Since my geometry is not particularly complex (just modiŽed cylynders and toroids), I suspect that this has been done before. I am looking for references. If you know of any, plese let me know. Nanri. Ashok.R === Subject: Re: Bounds on Moments by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1P4mTh27257; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P4iBi27094 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i1P4iBd15034; >> Suppose x is a non-negative random variable. Given that the Žrst >> moment of x, E(x^1) and E(x^2) are m1 and m2, respectively. What are >> the bounds upon m3=E(x^3)? >Why should we expect such a bound? For example, let f(x) = c/(1+x^4), where >c is chosen so that the integral of f over (0,oo) is 1. Then for the >probability density function f we have E(x^1) and E(x^2) < oo, but E(x^3) = >oo. No upper bound, but there is a lower bound. For example, for any p >= 0 we have Q(t) = t (t-p)^2 >= 0 for all t >= 0, so E[Q(x)] = m3 - 2 p m2 + p^2 m1 >= 0, i.e. m3 >= 2 p m2 - p^2 m1. In particular, take p = m2/m1 to get m3 >= m2^2/m1. Conversely, consider a distribution with mass m1^2/m2 at m2/m1 and 1-m1^2/m2 at 0 (noting that 0 < m1^2/m2 <= 1): we have E[x] = m1, E[x^2] = m2 and E[x^3] = m2^2/m1, so that bound is best possible. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: the anticlassicalist }{ ii: the spectre continues sci.lang,sci.logic,sci.math: [...] >:> Do you disagree with any of the points I have made in the towards a >:> constructive education or more focus... posts? Or do you believe >:> that I have in some other way violated the constraints of the groups¹ >:> topicalities? >: You obviously have in respect of sci.lang. You also >: obviously either violated one of the Žrst principles of >: netiquette, namely, that one should become familiar with the >: actual content and customary practices of a newsgroup before >: posting to it, or deliberately posted to sci.lang something >: that you should have known was inappropriate. Quite clearly >: you *do* feel entitled to grab attention at others¹ expense. > Expense? Yes, expense. You are, for example, directly responsible for cluttering sci.lang with off-topic mathematics and complaints about the lack of physical content in your posts from sci.physics. > I took the time to write several long posts concerning a topic > relevant to all newsgroups posted to. In your opinion. The relevance is not apparent to many of those better qualiŽed than you to judge. [...] > As for sci.lang, obviously there are problems in understanding reasoning on > topological trees (and linguistic phylogeny) that have been evidenced > several times recently in that forum. Jacques erred in calling his three trees topologically equivalent (though in fact two of them are -- as unlabelled trees), but I doubt that he was trying very hard: you¹ve offered nothing to suggest that you should be taken any more seriously on the subject of linguistic phylogeny than, say, Cavalli-Sforza, who is a linguistic ignoramus. > Plus the whole cognitive origins of > language, semiotics and natural language models, etc. features of my > exposition that I¹ve been willing to discuss in depth make your statement > patently false. Bollocks. I¹ve not seen you offer anything substantive on any of those topics, which with the possible exception of the Žrst are in any case not topics of great interest to most of the regulars here, judging by the eight or so years that I¹ve been reading the group. > Oh, did you miss the thread on modality in language as well? What, Andrew Patterson¹s nonsense? That was (a) independent of your posturing and (b) obviously received largely with indifference. > Yeah, I¹m reading... > Are you? Everything? Of course not. I read the threads in which the professional linguists and knowledgeable sci.lang regulars participate. I¹ve also read some of the stuff dragged in from sci.physics. I read this bit because from the days when I used to read sci.math I recognized Keith Ramsay as someone likely to have something sensible to say. >: [...] >:> I have seen many pleas against the cross posting. >: Which is prima facie evidence that it was inappropriate. It >: is ultimately the members of a newsgroup who determine what >: is appropriate, not some document. If I want to read >: physics, I¹ll go to sci.physics; I don¹t want it cluttering >: up sci.lang. If I want to read mathematics, I¹ll go to >: sci.math. If I want to read logic, I¹ll go to sci.logic. >: If I want to read philosophy, I¹m ill. > No, it¹s evidence of nothing of the kind! None of those pleas has ever > described in any way how I have violated the topicality of their newsgroup. You still don¹t get it, do you? What¹s on topic is determined by the group itself, especially when it has a core of regulars. If you¹d paid the slightest attention, you¹d have recognized that sci.lang runs to historical linguistics (about which you¹ve demonstrated considerable ignorance) and sociolinguistics, with occasional forays into syntax. (Oh, and food.) > Even you didn¹t make any such description above. You just accuse and Žght > your alpha games like you are the arbiter of truth and justice. Not at all, though I fear that you are far too self-centred and convinced of your own virtue to recognize that what you call an accusation was in fact (1) a partial explanation of the reception you¹ve had, (2) implicit advice on how to get a better reception, and (3) a fairly mild expression of exasperation. In point of fact I have for the most part treated you as you deserve: apart from a brief observation and followup when you Žrst appeared, I have ignored you. But your question to Keith was too inviting an opportunity to be ignored. > Your desire to want to avoid certain discussions of an interdisciplinary > nature is easily avoided by ignoring threads you Žnd distasteful. I can > walk you through that procedure if you are having any difŽculties. The interdisciplinary nature of these Œdiscussions¹ is mostly in your own mind. What¹s been posted here has mostly been mathematics and handwaving. >:> None of them have been >:> very convincing in my opinion, >: Which is largely irrelevant. > Unfortunately, that is quite relevant. Convincing me is the only way > someone is going to get me to stop posting. I quite believe it. I was discussing the facts of the matter, to which your opinion is largely irrelevant, and not your future behavior. >:> Most of the lack of content has been from those spamming their own >:> newsgroups, not asking for intelligent discussion, just spamming with >:> insults and the like. >: Which again is a very good indication that your content was >: widely considered inappropriate. Like it or not, many >: newsgroups are communities. Outsiders are not necessarily >: unwelcome, but outsiders who barge in and presume to lecture >: from a pedestal are likely to get the rough reception that >: they¹ve earned. > No, its an indication that there are quite a lot of jerks out there who, > when faced with a topic they do not understand and do not want to > understand, Žnd solace in insults. I do like the fact that newsgroups are > communities, particularly that they are communities of wide ranges of views > about the topics they discuss. There are certainly members, such as > yourself mister Scott, who dislike the fact that others may begin a > discussion conŽdent of the knowledge that they have such a right, but > unfortunately you are in the wrong and I am in the right. And playing your > power games, with their complete absence of any rational points, just > illustrates to me that you recognise your complete lack of power in this > circumstance. I¹m afraid that it¹s you who are playing power games. You are the one thumbing his nose at the rest of us and going ŒNyaa, you can¹t make me leave¹. In this you are quite correct: Usenet is an open forum, and I wouldn¹t have it any other way. But just as you are free to shove your id.8ee Žxe in front of people¹s faces at preposterous (and singularly ineffective) length, so am I free to point out that you are doing so. >: Bluntly, you¹re a rude, arrogant bastard with the social >: intelligence of a pet rock. On top of that you write some >: of the žabbiest, most turgid prose that it¹s been my >: misfortune to read anywhere, let alone on Usenet, and >: exhibit several of the familiar stigmata of the Usenet crank >: or monomaniac. If you don¹t like your reception, mend your >: manners. > I have never been rude to anyone who was not Žrst rude to me, and then only > enough to play the alpha game they initiated to its proper closure. You are mistaken, owing to your inability to recognize the rudeness of your behavior thus far. > I don¹t > seek contentless arguments; it is others who feel inclined to provide me > with such. I am a bastard; that is true. I was born with an unmarried > mother. Irrelevant, since I use the term in its Žgurative sense, and certainly no concern of mine in any case. > I can be quite humble when speaking to others who engage in > rational critique or otherwise educate me of my errors. I see you borrowed > the use of turgidity from the other newsgroup spammer. No, I did not. I have no idea even to whom you refer. But I am hardly surprised that someone else used so obviously apt a description. [...] If you are serious about getting together those who are genuinely interested in your views, I suggest that you set up a web-based bulletin board; I¹m given to understand that this is very easy to do these days. You can then announce it in the newsgroups in which you think it might be of interest. Reply or not, as you wish; I¹ll not be responding again outside of threads with genuine linguistic content, if I even bother to read. === Subject: Re: the anticlassicalist }{ ii: the spectre continues > Jacques erred in calling his three trees topologically > equivalent (though in fact two of them are -- as unlabelled > trees) Well, in fact, I would not mind arguing that they are. But since I left the vital bit out, arguing so would smack of a post-hoc argument. > but I doubt that he was trying very hard: you¹ve > offered nothing to suggest that you should be taken any more > seriously on the subject of linguistic phylogeny than, say, > Cavalli-Sforza, who is a linguistic ignoramus. Just right. I¹ll just give a hint of what I left out, as to me it went without saying for any comparative linguist with a smattering of topology and graph theory. The arcs are not one-dimensional lines, they are (sorry, don¹t know the proper jargon) they are ribbons, paths, roads with a _width_. > Unfortunately, that is quite relevant. Convincing me is the only way > someone is going to get me to stop posting. Post, post, post, post. And post. On s¹en bat l¹oeil, Ducon. (Ack, argh, help! Ad hominem, ad hominem! --No, ad mentulam, dickhead--shit, where¹s the editor of Maledicta when we need him?) >:> Most of the lack of content has been from those spamming their own >:> newsgroups, not asking for intelligent discussion, just spamming with >:> insults and the like. At least, insults, in these circumstances, have the merit of being honest. J¹appelle un chat un chat et Rollet un fripon. As for you, gaga-lact.8ee, it¹s all mealy-mouthed innuendos and the arguments statements ex cathedra perc.8ee. > No, its an indication that there are quite a lot of jerks out there who, > when faced with a topic they do not understand and do not want to > understand, Žnd solace in insults. Well, well, well... like people who, faced with matters of comparative linguistics, Žnd solace in calling comparative linguists jerks eh? === Subject: Re: the anticlassicalist }{ ii: the spectre continues sci.lang,sci.logic,sci.math: [...] >:> Do you disagree with any of the points I have made in the towards a >:> constructive education or more focus... posts? Or do you believe >:> that I have in some other way violated the constraints of the groups¹ >:> topicalities? >: You obviously have in respect of sci.lang. You also >: obviously either violated one of the Žrst principles of >: netiquette, namely, that one should become familiar with the >: actual content and customary practices of a newsgroup before >: posting to it, or deliberately posted to sci.lang something >: that you should have known was inappropriate. Quite clearly >: you *do* feel entitled to grab attention at others¹ expense. > Expense? Yes, expense. You are, for example, directly responsible for cluttering sci.lang with off-topic mathematics and complaints about the lack of physical content in your posts from sci.physics. > I took the time to write several long posts concerning a topic > relevant to all newsgroups posted to. In your opinion. The relevance is not apparent to many of those better qualiŽed than you to judge. [...] > As for sci.lang, obviously there are problems in understanding reasoning on > topological trees (and linguistic phylogeny) that have been evidenced > several times recently in that forum. Jacques erred in calling his three trees topologically equivalent (though in fact two of them are -- as unlabelled trees), but I doubt that he was trying very hard: you¹ve offered nothing to suggest that you should be taken any more seriously on the subject of linguistic phylogeny than, say, Cavalli-Sforza, who is a linguistic ignoramus. > Plus the whole cognitive origins of > language, semiotics and natural language models, etc. features of my > exposition that I¹ve been willing to discuss in depth make your statement > patently false. Bollocks. I¹ve not seen you offer anything substantive on any of those topics, which with the possible exception of the Žrst are in any case not topics of great interest to most of the regulars here, judging by the eight or so years that I¹ve been reading the group. > Oh, did you miss the thread on modality in language as well? What, Andrew Patterson¹s nonsense? That was (a) independent of your posturing and (b) obviously received largely with indifference. > Yeah, I¹m reading... > Are you? Everything? Of course not. I read the threads in which the professional linguists and knowledgeable sci.lang regulars participate. I¹ve also read some of the stuff dragged in from sci.physics. I read this bit because from the days when I used to read sci.math I recognized Keith Ramsay as someone likely to have something sensible to say. >: [...] >:> I have seen many pleas against the cross posting. >: Which is prima facie evidence that it was inappropriate. It >: is ultimately the members of a newsgroup who determine what >: is appropriate, not some document. If I want to read >: physics, I¹ll go to sci.physics; I don¹t want it cluttering >: up sci.lang. If I want to read mathematics, I¹ll go to >: sci.math. If I want to read logic, I¹ll go to sci.logic. >: If I want to read philosophy, I¹m ill. > No, it¹s evidence of nothing of the kind! None of those pleas has ever > described in any way how I have violated the topicality of their newsgroup. You still don¹t get it, do you? What¹s on topic is determined by the group itself, especially when it has a core of regulars. If you¹d paid the slightest attention, you¹d have recognized that sci.lang runs to historical linguistics (about which you¹ve demonstrated considerable ignorance) and sociolinguistics, with occasional forays into syntax. (Oh, and food.) > Even you didn¹t make any such description above. You just accuse and Žght > your alpha games like you are the arbiter of truth and justice. Not at all, though I fear that you are far too self-centred and convinced of your own virtue to recognize that what you call an accusation was in fact (1) a partial explanation of the reception you¹ve had, (2) implicit advice on how to get a better reception, and (3) a fairly mild expression of exasperation. In point of fact I have for the most part treated you as you deserve: apart from a brief observation and followup when you Žrst appeared, I have ignored you. But your question to Keith was too inviting an opportunity to be ignored. > Your desire to want to avoid certain discussions of an interdisciplinary > nature is easily avoided by ignoring threads you Žnd distasteful. I can > walk you through that procedure if you are having any difŽculties. The interdisciplinary nature of these Œdiscussions¹ is mostly in your own mind. What¹s been posted here has mostly been mathematics and handwaving. >:> None of them have been >:> very convincing in my opinion, >: Which is largely irrelevant. > Unfortunately, that is quite relevant. Convincing me is the only way > someone is going to get me to stop posting. I quite believe it. I was discussing the facts of the matter, to which your opinion is largely irrelevant, and not your future behavior. >:> Most of the lack of content has been from those spamming their own >:> newsgroups, not asking for intelligent discussion, just spamming with >:> insults and the like. >: Which again is a very good indication that your content was >: widely considered inappropriate. Like it or not, many >: newsgroups are communities. Outsiders are not necessarily >: unwelcome, but outsiders who barge in and presume to lecture >: from a pedestal are likely to get the rough reception that >: they¹ve earned. > No, its an indication that there are quite a lot of jerks out there who, > when faced with a topic they do not understand and do not want to > understand, Žnd solace in insults. I do like the fact that newsgroups are > communities, particularly that they are communities of wide ranges of views > about the topics they discuss. There are certainly members, such as > yourself mister Scott, who dislike the fact that others may begin a > discussion conŽdent of the knowledge that they have such a right, but > unfortunately you are in the wrong and I am in the right. And playing your > power games, with their complete absence of any rational points, just > illustrates to me that you recognise your complete lack of power in this > circumstance. I¹m afraid that it¹s you who are playing power games. You are the one thumbing his nose at the rest of us and going ŒNyaa, you can¹t make me leave¹. In this you are quite correct: Usenet is an open forum, and I wouldn¹t have it any other way. But just as you are free to shove your id.8ee Žxe in front of people¹s faces at preposterous (and singularly ineffective) length, so am I free to point out that you are doing so. >: Bluntly, you¹re a rude, arrogant bastard with the social >: intelligence of a pet rock. On top of that you write some >: of the žabbiest, most turgid prose that it¹s been my >: misfortune to read anywhere, let alone on Usenet, and >: exhibit several of the familiar stigmata of the Usenet crank >: or monomaniac. If you don¹t like your reception, mend your >: manners. > I have never been rude to anyone who was not Žrst rude to me, and then only > enough to play the alpha game they initiated to its proper closure. You are mistaken, owing to your inability to recognize the rudeness of your behavior thus far. > I don¹t > seek contentless arguments; it is others who feel inclined to provide me > with such. I am a bastard; that is true. I was born with an unmarried > mother. Irrelevant, since I use the term in its Žgurative sense, and certainly no concern of mine in any case. > I can be quite humble when speaking to others who engage in > rational critique or otherwise educate me of my errors. I see you borrowed > the use of turgidity from the other newsgroup spammer. No, I did not. I have no idea even to whom you refer. But I am hardly surprised that someone else used so obviously apt a description. [...] If you are serious about getting together those who are genuinely interested in your views, I suggest that you set up a web-based bulletin board; I¹m given to understand that this is very easy to do these days. You can then announce it in the newsgroups in which you think it might be of interest. Reply or not, as you wish; I¹ll not be responding again outside of threads with genuine linguistic content, if I even bother to read. === Subject: Re: convolution of measures Since we¹re talking about convolution of measures, is there any Fourier stuff we can do (since this would transform convolution into multiplication in the Fourier transform space)...? >Suppose M and N are distributions (measures), and it¹s given that >M * N = d_1, >where * indicates convolution and d_1 indicates the delta measure >concentrated at 1. How do you show that either M = d_c or N = d_c for >some constant c? This should probably follow from some really simple >manipulations with convolutions, but I lamentably don¹t see how to do >it. > Here is a solution which works for Žnite measures; perhaps it can be > modiŽed by approximation to all measures? > The problem is equivalent to showing that if X and Y are independent > random variables and X + Y is almost surely 1, then either X or > Y is almost surely constant. Actually, they both are, since > Var(X + Y) = Var(X) + Var(Y) = 0. > -- > Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: convolution of measures >Since we¹re talking about convolution of measures, is there any Fourier >stuff we can do (since this would transform convolution into multiplication >in the Fourier transform space)...? Yes, you could look at it via the Fourier transform. I was going to mention this, decided not to bother, since working out all the details would be more work than, say, working out the details in what Edgar suggested. But since you ask: First we need to clarify what sort of measures we¹re talking about; it¹s been pointed out that the assertion becomes false with some sorts of measures. Let¹s assume that they¹re probability measures (this _must_ be what the OP meant, else M = 2 d_0, N = d_0/2 would be a trivial counterexample...) Now we have (M^)(N^) = 1 and |M^| <= 1, |N^| <= 1. It follows that |M^| = 1, and it follows from _that_ that M = d_c So how does that last bit follow? One could give a slightly fussy direct argument... ah: there¹s a simple elegant argument from a not-quite-trivial fact: If M is a complex measure a result of Wiener shows that sum_{x in R} |M({x})|^2 = lim_{T->inŽnity} int_{-T}^T |M^(t)|^2 dt. (I leave it to you to Žgure out which ^¹s denote the Fourier transform and which ones denote exponentiation. I think this result is in Stein&Weiss Fourier Analysis on Euclidean Spaces or Stein Singular Integrals; the corresponding result for the circle is in Katznelson An Introduction to Harmonic Analysis, p. 42.) Now if M is as above then we have sum_{x in R} M({x}) <= 1 sum{_x in R} M({x})^2 = 1; the only way this can happen is if there exists c with M({x}) = 1 for x = c, 0 for x <> c. Then since M is a probability measure it follows that M = d_c. >>Suppose M and N are distributions (measures), and it¹s given that >>M * N = d_1, >>where * indicates convolution and d_1 indicates the delta measure >>concentrated at 1. How do you show that either M = d_c or N = d_c for >>some constant c? This should probably follow from some really simple >>manipulations with convolutions, but I lamentably don¹t see how to do >>it. >> Here is a solution which works for Žnite measures; perhaps it can be >> modiŽed by approximation to all measures? >> The problem is equivalent to showing that if X and Y are independent >> random variables and X + Y is almost surely 1, then either X or >> Y is almost surely constant. Actually, they both are, since >> Var(X + Y) = Var(X) + Var(Y) = 0. >> -- >> Stephen J. Herschkorn herschko@rutcor.rutgers.edu ************************ David C. Ullrich === Subject: linear operator Let T be a linear operator on an inner product space V and suppose |T(x)| = |x| for all x. Prove that T is 1-1. I started off with suppose T(x) = T(y) (and want to conclude that x=y) then |T(x)| = |T(y)|. Then |x| = |y|. The thing is that just because |x| = |y| we can¹t conclude x=y (consider x=(1,0,0) and y= (0,0,1) for example). There must be something else I can use. I guess it is the fact that T preserves the norm but I do not see how to use this. Any hints will be appreciated. === Subject: Re: linear operator > Let T be a linear operator on an inner product space V and suppose |T(x)| = > |x| for all x. Prove that T is 1-1. > I started off with suppose T(x) = T(y) Then T(x-y) = 0 ... === Subject: Re: linear operator > Let T be a linear operator on an inner product space V and suppose |T(x)| = > |x| for all x. Prove that T is 1-1. > I started off with suppose T(x) = T(y) (and want to conclude that x=y) > then |T(x)| = |T(y)|. Then |x| = |y|. The thing is that just because |x| = > |y| we can¹t conclude x=y (consider x=(1,0,0) and y= (0,0,1) for example). > There must be something else I can use. I guess it is the fact that T > preserves the norm but I do not see how to use this. Any hints will be > appreciated. Since T preserves norm, when is |Tx|=0? What does this imply about the null space of T? === Subject: Re: projective geometry axioms > Why not combine the Žrst two axioms of projective geometry? It seems > like any proof that involved either two would be upheld with an =1 . > I¹ll list them all here from mathworld: > 1. If A and B are distinct points on a plane, there is at least one > line containing both A and B. > 2. If A and B are distinct points on a plane, there is not more than > one line containing both A and B. > 3. Any two lines in a plane have at least one point of the plane > (which may be the point at inŽnity) in common. > 4. There is at least one line on a plane. > 5. Every line contains at least three points of the plane. > 6. All the points of the plane do not belong to the same line That sort of mess is a long way out of date. In the early 20th century various such axioms were proposed, usually something like: (1) Any two points both lie on a unique line, (2) Any two lines both lie on a unique point, (3), (4), ... Non-triviality conditions such as your 4, 5, 6 above. But in 1943 Marshall Hall produced a much simpler non-triviality condition: (3) There exist four points of which no three are collinear, i.e. there exists a quadrangle. This is a purely existential statement about Žnitely many points, as against the older universal-existential things like your 5. (1), (2), (3) above are now widely regarded as the basic axioms for plane projective geometry. An important special case satisŽes also the Desargues condition. An important special case of _that_ satisŽes also the Pappus condition (which implies Desargues, by Hessenberg¹s Theorem). But you may not be interested in those reŽnements just yet. Ken Pledger. === Subject: Re: Goading Godel > Most of the above passage is utter nonsense. Your concept of > inŽnities needs some *serious* reŽning as it¹s so vague that it¹s > basically meaningless. For example, if you want to show that an > arbitrary inŽnite set can¹t be mapped to a Žnite set, just consider > the set of all real numbers, which cannot be put into correspondence > with the set of integers. Since every Žnite set can be considered as > a subset of the integers (there exists a bijective mapping), this > shows that you can¹t map ALL inŽnite sets to Žnite representations. Sure you can. Map every inŽnite set to 1. Or 2. Or even 3. I suspect you had something else in mind. Perhaps *into* a Žnite set? You deŽnitely didn¹t mean *onto,* as that is incredibly easy to do. Œcid Œooh === Subject: Re: e is transcendental by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1P6o2U04174; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P6fvi03850 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i1P6fvp20699; >>O.k. then , as long there is no conžict with : >>Re(e^[iPi]) = Re(-1+i[0]) = -1 >That has never been in doubt. Neither has the observation that >Im(e^[i pi]) = Im(-1+i[0]) = 0. >The statement that caused all the problem was the statement that >e^[i pi] = 0. Actualy stated e^[ipi]=i0=0 [REF: A) e^[ipi]=-1 the real part solution and B) e^[ipi]=i[0] , or e^[ipi]=0 the imaginary part solutio.] Panagiotis Stefanides >David McAnally > Despite anything you may have heard to the contrary, > the rain in Spain stays almost invariably in the hills. === Subject: Re: e is transcendental >O.k. then , as long there is no conžict with : >Re(e^[iPi]) = Re(-1+i[0]) = -1 >>That has never been in doubt. Neither has the observation that >>Im(e^[i pi]) = Im(-1+i[0]) = 0. >>The statement that caused all the problem was the statement that >>e^[i pi] = 0. >Actualy stated e^[ipi]=i0=0 Which is exactly the same statement in the long run. And I have not yet seen you retract the statement, as you should have by now. >[REF: > A) e^[ipi]=-1 the real part solution and > B) e^[ipi]=i[0] , or e^[ipi]=0 the imaginary part solutio.] Arguments which you have now acknowledged as invalid (they are either both valid or both invalid, and the false conclusion in B from the true premise that exp[i pi] = -1+i[0] demonstrates that the argument used in B is invalid). David McAnally At the moment, they (the Time Lords) are far from being all-powerful. That¹s why it¹s been left up to me and me and me. quote by: Patrick Troughton in The Three Doctors ------- === Subject: Does Euclidean geometry exist as a physical entity, or does only Riemannian and Lobachevskian > I am going to have to revamp File 103 on FLT, and File 120 of 3 and > only 3 geometries and File 125 of two proofs of the Riemann > Hypothesis in my website of www.iw.net/~a_plutonium/ > I did not do much mathematics after 1997 and recently when I reviewed > my Riemann Hypothesis proof I realized that it is the p-adics that are > on the 1/2 Real line which means that lines are curved when out at > inŽnity. There are no straightlines. I have to change and revise my > Poincare Conjecture proof also. Physics is duality and not triality. If I go by that presumption then I have to concede that there are only really 2 geometries and not 3. Of the three known geometries of Riem, Loba, and Eucl, I would bet that Euclidean is the nonexistant one. The P-adics create a nice point by point Riemannian geometry and they naturally curve back around. But I have trouble forming Lobachevskian geometry with a one to one correspondence with algebraic numbers. Perhaps the Doubly-InŽnites? Perhaps the negative Reals. But then I have trouble with the Reals for they seem to be Euclidean geometry. But are they really? Euclidean geometry is zero curvature. But the number 0 exists in p-adics and doubly-inŽnites. The complex-numbers I can rule out as just a gimmick that gives added dimensions. So I am faced with 3 number sets of P-adics, Reals, and Doubly-InŽnites. If physics is the Žnal word on this that duality exists but triality is nonsense, then I am going to have to Žnd out what 2 and only 2 geometries exist and that one of them is a mental illusion for human minds. Just as Newtonian absolute space and absolute time was just a mental illusion. The most perfect match is P-adics to Riemannian Geometry because the P-adics are all positive numbers which Riem geometry deals with only positive quantities and they naturally curve back around such as in 10-adics the number ....99998 is equivalent of -2 and then ....99999 is equivalent to -1. So it is a beautiful matchup. But I have trouble in Žnding a number set to matchup with Lobachevskian Geometry. It cannot be the Reals because they have both negative and positive but Loba requires only negative. So are the Doubly-InŽnites intrinsically negative quantities similar to the fact that P-adics are intrinsically positive quantities? Archimedes Plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: cracks in Euclidean Geometry and why Reals are fake Re: Does Euclidean geometry exist as a physical entity, or does only Riemannian and Lobachevskian > I am going to have to revamp File 103 on FLT, and File 120 of 3 and > only 3 geometries and File 125 of two proofs of the Riemann > Hypothesis in my website of www.iw.net/~a_plutonium/ > > I did not do much mathematics after 1997 and recently when I reviewed > my Riemann Hypothesis proof I realized that it is the p-adics that are > on the 1/2 Real line which means that lines are curved when out at > inŽnity. There are no straightlines. I have to change and revise my > Poincare Conjecture proof also. > > Physics is duality and not triality. If I go by that presumption then > I have to concede that there are only really 2 geometries and not 3. > Of the three known geometries of Riem, Loba, and Eucl, I would bet > that Euclidean is the nonexistant one. The P-adics create a nice point > by point Riemannian geometry and they naturally curve back around. But > I have trouble forming Lobachevskian geometry with a one to one > correspondence with algebraic numbers. Perhaps the Doubly-InŽnites? > Perhaps the negative Reals. > But then I have trouble with the Reals for they seem to be Euclidean > geometry. But are they really? Euclidean geometry is zero curvature. > But the number 0 exists in p-adics and doubly-inŽnites. The > complex-numbers I can rule out as just a gimmick that gives added > dimensions. > So I am faced with 3 number sets of P-adics, Reals, and > Doubly-InŽnites. If physics is the Žnal word on this that duality > exists but triality is nonsense, then I am going to have to Žnd out > what 2 and only 2 geometries exist and that one of them is a mental > illusion for human minds. Just as Newtonian absolute space and > absolute time was just a mental illusion. > The most perfect match is P-adics to Riemannian Geometry because the > P-adics are all positive numbers which Riem geometry deals with only > positive quantities and they naturally curve back around such as in > 10-adics the number ....99998 is equivalent of -2 and then ....99999 > is equivalent to -1. So it is a beautiful matchup. > But I have trouble in Žnding a number set to matchup with > Lobachevskian Geometry. It cannot be the Reals because they have both > negative and positive but Loba requires only negative. So are the > Doubly-InŽnites intrinsically negative quantities similar to the fact > that P-adics are intrinsically positive quantities? Another major crack in Euclidean Geometry is that Real Numbers really do not coincide with Euclidean Geometry, do they? Perhaps I am making a statement or perhaps I am asking a question, for I am only at the beginning of this inquiry. If we look at the P-adics, they are all positive numbers, even the zero point can be said to be positive and so they are ideal for coinciding with Riemannian Geometry with its *positive curvature*. P-adics are ideal for representing Riemannian Geometry, or, making a 1-to-1-correspondence. So one can say that P-adics are the points of Riemannian Geometry and ideally such because the p-adics have a natural curvature to them for as we start in the 10-adics with 0 and then next is .000001 and next is .000002 and going way out it comes back to ...99997 which we can conceive of as -3 then ....999998 which is -2 and excitedly ....999 which is -1 and Žnally back to our starting point of 0. So the P-adics can be said to be the actual algebraic points of Riemannian Geometry. But now inspecting Reals with the geometry of Euclid, it just simply does not Žt together does it. Because Euclid geometry is 0 curvature and the only number in the Reals that obeys curvature is a single point which is zero itself. The positive Reals disobey Euclidean Geometry because they are positive signifying Riem geom and not Euclidean and likewise the reverse for negative Reals for they signify Lobachevskian geom. So one is left with the conclusion that the Reals never represented the geometry entailed by Euclidean Geometry. Do the Real Number system represent any geometry??? I suspect not. I suspect the Reals are as mythical or imaginary as was Newtonian Mechanics of absolute space and absolute time. Humans have minds that can dream up things which really have no physical reality such as ghosts, witches, and Newtonian absolute space and absolute time. Are the Real Numbers another dream-up thing which has no physics reality? I suspect so. But I am troubled with what numbers correspond to Lobachevskian geometry. I would like to think that the negative REals suit the Loba geometry, but that leaves the nasty question of the positive Reals. I think I can draw some clues as what the numbers that make up Lobachevskian Geometry from the P-adics making up Riemannian Geometry. If I start with this claim: P-adics == Riemannian Geometry and accept it as fully true, then there are some other numbers called Doubly-InŽnites. Doubly-InŽnites are what the name implies. You see, p-adics are inŽnite leftward strings. Doubly-InŽnites would then be numbers that are both inŽnite leftward but also inŽnite rightwards. In some sense, the Real Numbers should be doubly-inŽnite. Perhaps that is the reason the Reals are fake and nonphysical just as the NaturalNumbers = Žnite-integers was a fake and dreamed-up illusions. TEST: the test of the above would be to show that the Doubly-InŽnites are numbers that are all negative in sign value. What I mean is that the P-adics are all positive (even ....9999 is positive but it can represent -1). So, if I can show that Doubly-InŽnites are all negative, then I will have shown a vast amount of knowledge and understanding. Because if I can show the Doubly- InŽnites are all negative in sign value then these numbers are what compose Lobachevskian geometry. And that where I have: All-P-adics == Riemannian Geometry I will also have All-Doubly-InŽnites == Lobachevskian Geometry This would then conclude that REals as a number system were a fake entity and would imply that the centuries of gaps and holes found in Reals such as the myriad types of differentiation and integration Lebesgue integral to name one is because the Reals are a hodge-podge-mess just as Newtonian absolute space and absolute time was a hodge-podge-mess that saddled Quantum Mechanics. Driver Motivation for the above: what drives me to many of these conclusions is that Quantum Mechanics is duality based and not threesome. Because Physics is twosome, then geometries should be twosome and not threesome. Therefore, geometry should have 2 and only 2 indepedent geometries. So, I have a choice of 2 of these geometries that really exist (1) Euclid (2) Riemannian (3) Lobachevskian. My choice is Riem and Loba. And since there are only two entails that Algebraically in Mathematics there exists only two real Number Systems. I am fully conŽdent that P-adics are one true number system. I suspect Doubly-InŽnites is the second truly existing number-system. Hence I suspect the Reals to be a fake system just as Newtonian Absolute Space and Absolute Time was a fake concept. Research: All I need is hard-core evidence that Doubly-InŽnites are all negative numbers. If I can get that evidence, then all of my above thoughts would be conŽrmed. Archimedes Plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies (www.iw.net/~a_plutonium) website of the science of AP under revision what used to be my old science website www.newphys.se/elektromagnum/physics/LudwigPlutonium from years 1993 === Subject: Re: as usuall , help is needed!!! > Use the chain rule to get dy/dt in terms of dy/dx. > -Michael. > dear Michael > if your means is [dy/dx]=[dy/dt]*[dt/dx] > i can not solve with it if you can then so0lve Yep, but it helps to reverse it: [dy/dt] = [dy/dx]*[dx/dt] You need the second derivative too: [d^2y/dt^2] = [dx/dt]^2 * [d^2y/dx^2] + [d^2x/dt^2] * [dy/dx] Now, for x=tan(t) you have [dx/dt] = 1 + x^2. I hope you can do the rest yourself. -Michael. === Subject: Re: inequality Jerome Davies > I trying to establish the inequality: > (x^a - 1)/(x^b - 1) <= a/b If x>1 and 00. Since a/b = ka/kb and k>0, we can assume k=1. Rearranging it, we want (e^a - 1)/a <= (e^b -1)/b. This is clear from the power series for e^x, but to see it another way, compare the slope of the line from (0,1) to (a,e^a) to the slope of the line from (0,1) to (b,e^b) and remark that all the derivatives of e^x are positive everywhere. The hypothesis 0Let S be the real numbers (0,1) >Since it is a set of real numbers it is partially ordered and every chain is >obviously bounded by 1 >Yet (0,1) does not have a maximal element. >What am I missing here? The chain (0,1), for example, does *not* have a least upper bound *in S.* -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Zorn¹s Lemma Question >>Let S be the real numbers (0,1) >>Since it is a set of real numbers it is partially ordered and every chain is >>obviously bounded by 1 >>Yet (0,1) does not have a maximal element. >>What am I missing here? >The chain (0,1), for example, does *not* have a least upper bound *in S.* More relevant to a question about Zorn¹s lemma is the fact that (0,1) does not have an upper bound in S. ************************ David C. Ullrich === Subject: Q11 format What is Q11 Format ? === Subject: a linear algebra thought All, I have a question on linear algebra. Let V be a Žnite dimensianl vector space, S be a noninvertable linear transformation on V. I wonder if the Ker(S) is perpendicular to Range(S), or ker(S)+Range(S) is direct sum and Ker(S)+Range(S)=V. === Subject: Re: a linear algebra thought > All, I have a question on linear algebra. Let V be a Žnite dimensianl > vector space, S be a noninvertable linear transformation on V. Let V be 2 dimensional with basis b_1, b_2. Let S(b_1) = 0 and S(b_2) = b_1, then Ker(S) = Range(S). >I wonder if the Ker(S) is perpendicular to Range(S), or >ker(S)+Range(S) is direct sum and Ker(S)+Range(S)=V. === Subject: Re: a linear algebra thought > All, I have a question on linear algebra. Let V be a Žnite dimensianl > vector space, S be a noninvertable linear transformation on V. I > wonder if the Ker(S) is perpendicular to Range(S), or ker(S)+Range(S) > is direct sum and Ker(S)+Range(S)=V. Perhaps that I misunderstood your questions, but consider V = R^2 and S(x,y) = (y,0). Then ker(S) = Range(S). I suppose that this answers your questions. Jose Carlos Santos === Subject: Re: Genetics and Math-Ability > > > I think most people would agree that human reasoning is far from > perfect. How can an inconsistent reasoning ability give a survival > advantage? Evolution proves that some inconsistent systems are > better than others. > > Restricting to only those conclusions that are 100% certain > and inescapable would lead to a pityful small number > of conclusions. > The requirement wasn¹t for being that could reason to a correct > conclusion, but that the resoning was consistent. > Both abilities (drawing conclusions beyond the trivial and > drawing them fast) will account for the survival advantage. > DeŽnately. There was an AI research team (I forget where or when) who > developed a robot. This robot had cameras, wheels and a very > soŽsticated AI that should allow it to navigate around a room. When it > was switched on, it simply sat still and did nothing. Examining the > robots datapath they found it was sitting there quietly analizing > everything before making a descision as to what to do. They added more > code which should enable it to selectively ignore the unimportant parts, > when it was switched on it simply sat still again. Examining the robots > datapath again showed it was quietly analysing everything so it could > decide what to ignore. > The moral, if human reasoning was forced to be consistent we¹d be stuck > in same situation as the robot. Well, they are stuck in the same situation as robots. We¹ve always, throughout all of human history, restricted both Mathematicians and Lawyers to the study of simple machines. Since the only statement that¹s ever been true about machines is: Simple is, as simple does. === Subject: re:Alternative ways to solve a quadratic equation Here¹s a simple method which I came up with before I learnt quadratic equations: Write the equation in the form (x + b)x = c Find A and B so that A + B = x + b and A - B = x Obviously A = x + b/2, B = b/2 The equation becomes A^2 - B^2 = c, or A^2 = c + b^2 / 4 ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P5KUi29951 === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P02Ri02705 Originator: Žshbowl@conservatory.com (james) >But other methods do exist. >I will give a method based on Galois theory. William, you¹ve been folderized. === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P00Ki02172 Originator: Žshbowl@conservatory.com (james) >required us to Žnd a totally new method to solve a quadratic equation >with. Totally new? That would be quite a feat. Oh, not totally new as in, hasn¹t been discovered before. Your chances of Žnding a method that the Greeks didn¹t know are pretty slim :-) >It can involve the quadratic formula but has to be different. I¹d look for a quadratic identity using trig substitution, or maybe Žnd roots with the Newton Method. Sounds like your assignment is supposed to be an exercise in Žnding identities for the quadratic formula. If you come up with something totally new here *DON¹T* give it to your professor, submit a paper to James watched Good Will Hunting one time too many M. === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1ONkTi00513 > required us to Žnd a totally new method to solve a quadratic equation > with. It can involve the quadratic formula but has to be different. I assume you mean totally new to you as opposed to totally new to mathematics. You might look for some techniques for solving higher order equations. Or here is one very simple and inefŽcient one called the half interval search - once you have established that a root lies between point A and B, because of a sign change. Evaluate the function at (A + B)/2. If it is the same sign as A this becomes the new A otherwise the new B. Keep going until you are tired or happy. Or you might try to Žnd some kind of analog computer. That is, instead of using mathematics to represent a physical phenomenon (the usual way of doing things) Žnd some sort of physical representation of the math. and go measure it. Bill === Subject: re:2 and only 2 geometries where Euclidean is like Newton¹s abs No offense, but you sound like a crank. ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: A probability question I have a probability question that looks deceptively easy but somewhat I just can¹t prove it. There are three events A, B and C. Events A and C are independent. Events B and C are independent. Prove or disprove P(A intersect B | C) = P(A intersect B). Yeoster === Subject: Re: A probability question >There are three events A, B and C. Events A and C are independent. >Events B and C are independent. Prove or disprove P(A intersect B | C) >= P(A intersect B). Consider rolling a 4-sided die with the following events: A: rolling 1 or 2 B: rolling 1 or 3 C: rolling 2 or 3 Rob Johnson take out the trash before replying === Subject: Re: A probability question >I have a probability question that looks deceptively easy but somewhat >I just can¹t prove it. >There are three events A, B and C. Events A and C are independent. >Events B and C are independent. Prove or disprove P(A intersect B | C) >= P(A intersect B). Big hint: Consider a pair of dice and the event of a seven being rolled. Cite sources of assistance on submitted assignments. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: plotting hyperelliptic curves!! could someone tell how to plot hyperelliptic curves using MAPLE, Mathematica, Matlab..anything for example..something like v^2=u^5 - 5*(u^3) + 4*u + 3, over real numbers thanx in advance for all the help.. OP. === Subject: Re: plotting hyperelliptic curves!! charset=iso-8859-1 > could someone tell how to plot hyperelliptic curves using MAPLE, > Mathematica, Matlab..anything > for example..something like > v^2=u^5 - 5*(u^3) + 4*u + 3, over real numbers > thanx in advance for all the help.. > OP. Another way in maple that works well with many algebraic curves is: with(algcurves): f:=v^2-(u^5-5*u^3+4*u+3); plot_real_curve(f,u,v); Jim Buddenhagen === Subject: Re: plotting hyperelliptic curves!! > could someone tell how to plot hyperelliptic curves using MAPLE, > Mathematica, Matlab..anything > for example..something like > v^2=u^5 - 5*(u^3) + 4*u + 3, over real numbers > thanx in advance for all the help.. With Maple, load the plots package and then do: implicitplot(v^2=u^5-5*u^3+4*u+3,u=-1..3,v=-4..4); Mathematica also has an ImplicitPlot command. Jose Carlos Santos === Subject: Re: branch of log z > |What exactly is to choose a branch of log z for complex z? I¹d > |appreciate if someone help me picture this. > The basic problem is ... log or ln changes an arrow z of length 1 of a plane into a rotational vector ln z of length @ perpendicular to the plane - this is the height in the parking garage, but as an axial vector-arrow it can be translated freely in R3. @ can not exceed 360 degrees or 2*pi, if it is created this way. Rotating an arrow w in a plane at any height is different from this - it¹s Žrst changing an rotational arrow, picture them as screws, of length beta perpendicular to the plane into an arrow of this plane. This arrow is showing the difference of direction of the grove on top of the screw, by e^(i*beta)=exp(i*beta). Second is multiplying this with w. Now an approach to branch of log z or ln z is, that it is the set or manifold of parallel axial-arrows, differing in length less than 2*pi - if we retrict to length z=1 for a start. Feel like a snail, reaching the Žrst level in the parking garage. Did it get it ? Hero by the way, links to f(z) and planegraph, my two favourite programms for visualisation, are on my website. === Subject: Re: Trying to unify axioms. > Nothing. > > You forever spewing fucking imbecile, an axiom by deŽnition is > irreducible and unprovable. > > False. > > First off, an axiom isn¹t necessarily irreducable, unless you want to > claim that reducing it makes the axiom not reducable and therefore not > an axiom. Sometimes we have an axiom that we Žnd out can be reduced > into other axioms. Does that therefore disprove the axiom, or destroy > the usefulness of the axiom, or of taking it as such? No it doesn¹t. > > Second, axioms are not unprovable. They CAN be proven. For an example > of this, consider the law of identity. Can you prove it? If not, then > how do we even know it¹s true? We do know it¹s true, and it IS an > axiom, so that just proves that axioms are not unprovable. > > (...Starblade Riven Darksquall...) > An axiom is unprovable by deŽnition. Usually, if an axiom can be > shown to be provable based on other axioms or postulates, it no longer > remains an axiom and is technically a theorem, which by deŽnition is > provable. And of course, nothing is ever provable in and of itself, > which is why a single isolated statement can only be true by > deŽnition. Then explain the human mind. Which axioms did we start with? If I can explain it to you using reason, then does it no longer become an axiom? (...Starblade Riven Darksquall...) === Subject: Re: Trying to unify axioms. >> An axiom is unprovable by deŽnition. Usually, if an axiom can be >> shown to be provable based on other axioms or postulates, it no longer >> remains an axiom and is technically a theorem, which by deŽnition is >> provable. And of course, nothing is ever provable in and of itself, >> which is why a single isolated statement can only be true by >> deŽnition. >Then explain the human mind. Which axioms did we start with? If I can >explain it to you using reason, then does it no longer become an >axiom? Poincare has a lot to say about that in his book Science and Hypothesis, $4 at Amazon. Basically, you can¹t explain it using reason. You can¹t explain things like width to someone that hasn¹t shared some of your experiences. -- Very well, he replied, I allow you cow¹s dung in place of human excrement; bake your bread on that. -- Ezekiel 4:15 === Subject: Re: Trying to unify axioms. >> An axiom is unprovable by deŽnition. In logic, an axiom is provable by deŽnition. === Subject: Re: Trying to unify axioms. > An axiom is unprovable by deŽnition. > In logic, an axiom is provable by deŽnition. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate. If an axiom was provable, what would you call the statements from which the axiom is deduced? -- Don¹t try to teach a pig how to sing. You¹ll waste your time and annoy the pig. === Subject: Re: Trying to unify axioms. > If an axiom was provable, what would you call the statements from which > the axiom is deduced? In logic, an axiom is provable because it is deducible from itself. === Subject: Re: Trying to unify axioms. > If an axiom was provable, what would you call the statements from which > the axiom is deduced? > In logic, an axiom is provable because it is deducible from itself. Only one line can be drawn parallel to a given line through an exterior point. Everybody knows that from high school geometry. It is obviously true as Euclid¹s Fifth Postulate (here restated as Playfair¹s Axiom). There is only one glitch: It is empirically wrong. There are no lines parallel to a given line on the Earth¹ surface. You cannot accurately navigate or survey with Euclid. A mile square on the Earth¹s surface bounds more than a square mile. All triangles on Earth¹s surface have their interior angles sum to more than 180 degrees (as much as 540 degrees!). Given a circle drawn on the surface of the Earth, the ratio of the circumference to the diameter is always less than pi. By trivial demonstration, you are full of shit. A axiom is a stated unprovable assumption, It is indefensible for being an axiom - and can be falsiŽed by a single reproducible counterdemonstration. A refrigerator is not a refrigerator if its motor is run backwards to create an oven. All acceptible theories of gravitation must give the same predictions to the extreme limits of experimental error because they all describe the same unique reality. There are only two exceptions: 1) They can disagree about observations that have not been made (e.g., Planck energy regimes), and 2) They can disagree about the Equivalence Principle - that all local bodies fall identically in vacuum. An unmade observation cannot be a constraint. The Equivalence Principle has not been exhaustively tested for violation. Like Euclid, the most elegant and comprehensive internally self-consistent and exhaustively empirically conŽrmed axiomatic system is only as strong as its weakest axiom. Given just one reproducible counterdemonstration, it all comes crashing down. Philosophy is crap. Humanity languished in pestilence, disease, poverty, famine, Žlth, and ignorance given 5000 years of continuously reŽned philosophies. If you want to žush away the crap, you need an engineer. If you wish to wash your hands of it, you need a chemist. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Trying to unify axioms. X-SessionID: G97%b-1332-Y4-4092@news.uchicago.edu X-Hash-Info: post-Žlter,v:1.4 X-Hash: 75279e15 71115414 b0873e09 11a69091 4553e7d6 >> If an axiom was provable, what would you call the statements from which >> the axiom is deduced? >> In logic, an axiom is provable because it is deducible from itself. >Only one line can be drawn parallel to a given line through an >exterior point. Everybody knows that from high school geometry. It >is obviously true as Euclid¹s Fifth Postulate (here restated as >Playfair¹s Axiom). There is only one glitch: It is empirically wrong. Nonsense. >There are no lines parallel to a given line on the Earth¹ surface. Again, nonsense. What you call lines on Earthj surface are *not* straight lines. They¹re curved. The Euclidean postulate is about *straight lines. The claim that Euclidean geometry cannot deal with circles would sure get an amused reaction from the Greek geometers. But you do *not* deal with circles by pretending that they¹re straight lines. If I take a square and calculate its area assuming that it is a triangle (base times height divided by two), I¹ll be off by a factor of two. The failing will be mine, *not* Euclid¹s. You do have good posts, at times. This was *not* one of them. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: Trying to unify axioms. X-SessionID: Ws8%b-1596-Y4-4432@news.uchicago.edu X-Hash-Info: post-Žlter,v:1.4 X-Hash: 70db310b 6b02a255 14a64a13 b4229be9 49b7d8ab > If an axiom was provable, what would you call the statements from which > the axiom is deduced? >> In logic, an axiom is provable because it is deducible from itself. >>Only one line can be drawn parallel to a given line through an >>exterior point. Everybody knows that from high school geometry. It >>is obviously true as Euclid¹s Fifth Postulate (here restated as >>Playfair¹s Axiom). There is only one glitch: It is empirically wrong. >> Nonsense. >>There are no lines parallel to a given line on the Earth¹ surface. >> Again, nonsense. What you call lines on Earthj surface are *not* >> straight lines. They¹re curved. The Euclidean postulate is about >> *straight lines. The claim that Euclidean geometry cannot deal with >> circles would sure get an amused reaction from the Greek geometers. >> But you do *not* deal with circles by pretending that they¹re straight >> lines. >> If I take a square and calculate its area assuming that it is a >> triangle (base times height divided by two), I¹ll be off by a factor of >> two. The failing will be mine, *not* Euclid¹s. >> You do have good posts, at times. This was *not* one of them. >A straight line (geodesic path) on the surface of a sphere is a Great >Circle. Nope. A geodesic path on the surface of a sphere is a Great circle. This is not, (repeat, ***not***) a Euclidean straight line. You can use a geometry in which these lines are deŽned as straight, but this is a different geometry. The straight line joining the North and South pole is the line going from one to the other *through* the center of the Earth. You¹ll say but this is not on the surface? There is ***no*** deŽnition in Euclidean geometry to the effect that the straight line joining to points on an arbitrary surface must be imbedded in the surface. A geodesic line, yes, a straight line, no. Elliptic (Bolyai-Lobechevsky, positive curvature) geometry >then proceeds to spec. Euclid (except those parts independent of the >Fifth Postulate) doesn¹t work on the surface of a sphere, or an >ellipsoid in general. You progress from the idiotic to the moronic. Again, did it occur to you that the sphere is described by x^2 + y^2 + z^2 = R^2 in Euclidean geometry. >Hyperbolic (Riemann, negative curvature) surfaces similarly fail >Euclid. It doesn¹t fail anything, it is just a different geometry. A mile square encompasses than a square mile, a triangle¹s >interior angles always sum to less than 180 degrees, and the ratio of >circumference to diameter of all circles is greater than pi. >In both cases, a straight line is deŽned as the path taken by a >light ray in vacuum in that 2-D space. You¹re totally confused. That¹s what¹s coming from learning geometry from coffee table books. A geometry is a mathematical system. It is a set of axioms and deŽnitions plus the conclusions that can be drawn from these. Get a different set of axioms and deŽnitions and you¹ve a different geometry. You can have as many as you wish. And the fact that geometry A differs in its axioms and results from geometry B *does not* mean that either A or B have been falsiŽed. This concept *does not* apply here. There is no empirical veriŽcation of mathematical theories. Euclid is Žne, and Lobatchevsky is Žne, and Bolyi is Žne. These are just different geometris, that¹s all. Now, if you want to use a speciŽc geometry (or, in fact, any mathematical theory) to model some observable aspect of reality, then it is a different story. What you need to do then is to establish some mapping between concepts of the theory and elements of said observable aspect of reality. And then, once you¹ve done it, you can check whether the relationships between the observables follow the relationships between the theoretical entities they map into. If this is true, Žne. And if it isn¹t true? If it isn¹t true, all it means is that *for the speciŽc mapping* you selected, the mathematical theory you¹ve chosen does not apply as a physical model (of whatever it is you¹re modeling). It says *nothing* about the validity of the mathematical theory per se (the very notion is ridiculous) only on its applicability as physical model, for the phenomena required, and using the selected mapping. This is *all*. Am I getting through to you? Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: Trying to unify axioms. >> If an axiom was provable, what would you call the statements from which >> the axiom is deduced? >> In logic, an axiom is provable because it is deducible from itself. >Only one line can be drawn parallel to a given line through an >exterior point. Everybody knows that from high school geometry. It >is obviously true as Euclid¹s Fifth Postulate (here restated as >Playfair¹s Axiom). There is only one glitch: It is empirically wrong. > Nonsense. >There are no lines parallel to a given line on the Earth¹ surface. > Again, nonsense. What you call lines on Earthj surface are *not* > straight lines. They¹re curved. The Euclidean postulate is about > *straight lines. The claim that Euclidean geometry cannot deal with > circles would sure get an amused reaction from the Greek geometers. > But you do *not* deal with circles by pretending that they¹re straight > lines. > If I take a square and calculate its area assuming that it is a > triangle (base times height divided by two), I¹ll be off by a factor of > two. The failing will be mine, *not* Euclid¹s. > You do have good posts, at times. This was *not* one of them. A straight line (geodesic path) on the surface of a sphere is a Great Circle. Elliptic (Bolyai-Lobechevsky, positive curvature) geometry then proceeds to spec. Euclid (except those parts independent of the Fifth Postulate) doesn¹t work on the surface of a sphere, or an ellipsoid in general. Hyperbolic (Riemann, negative curvature) surfaces similarly fail Euclid. A mile square encompasses than a square mile, a triangle¹s interior angles always sum to less than 180 degrees, and the ratio of circumference to diameter of all circles is greater than pi. In both cases, a straight line is deŽned as the path taken by a light ray in vacuum in that 2-D space. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Trying to unify axioms. > A axiom is a stated > unprovable assumption, It is indefensible for being an axiom - and > can be falsiŽed by a single reproducible counterdemonstration. FalsiŽability, indefensible - this has nothing to do with it. It¹s simply a fact that in logic, every axiom of a theory T is provable in T. === Subject: Re: Trying to unify axioms. > A axiom is a stated > unprovable assumption, It is indefensible for being an axiom - and > can be falsiŽed by a single reproducible counterdemonstration. > FalsiŽability, indefensible - this has nothing to do with it. It¹s > simply a fact that in logic, every axiom of a theory T is provable in T. Only one line can be drawn parallel to a given line through an exterior point. Everybody knows that from high school geometry. It is obviously true as Euclid¹s Fifth Postulate (here restated as Playfair¹s Axiom). There is only one glitch: It is empirically wrong. There are no lines parallel to a given line on the Earth¹ surface. Pookie pookie. The are an inŽnite number of lines that can be drawn parallel to a given line through an exterior point on a hyperbolic surface. Euclid is incomplete for his Žfth postulate. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Trying to unify axioms. charset=utf-8 ėÄ Uncle Al ė[CapitalEth]ėŅģŖ[EDouble Dot]±ģĆė[Micro] ģ°ģ.b3ėĄ ė.b9ėØėøģ·[EDo ubleDot].b9ė± > Pookie pookie. The are an inŽnite number of lines that can be drawn > parallel to a given line through an exterior point on a hyperbolic > surface. Euclid is incomplete for his Žfth postulate. What¹s your fucking point already? Maybe that the Ancient Greeks should have done better than that? Lessee: They are sitting in the sun 2,500 years ago and they decide to Žnd EVERYTHING is a game. Maybe they should have invented relativity instead. Wait, analyze and utilize the ergosphere of a couple of rotating black holes which they could not otherwise observe because they didn¹t have the appropriate apparatus and deduce how to have energy for free by putting those black holes in orbit around Earth by throwing beach pebbles in them. Maybe they should have dreamt of the Benzene molecule, so they could drive a Honda. Heck, why didn¹t they invent Hyperbolic Geometry, those fucking idiots? When was the last time you had your head OUT of your ass? > -- > Uncle Al -- Ioannis Galidakis http://users.forthnet.gr/ath/jgal/ ------------------------------------------ Eventually, _everything_ is understandable === Subject: Re: Trying to unify axioms. > Only one line can be drawn parallel to a given line through an > exterior point. Everybody knows that from high school geometry. It > is obviously true as Euclid¹s Fifth Postulate (here restated as > Playfair¹s Axiom). There is only one glitch: It is empirically > wrong. There are no lines parallel to a given line on the Earth¹ > surface. True or false is irrelevant. Every axiom of a theory T is provable in T. This is just a matter of terminology. === Subject: Re: Trying to unify axioms. > If an axiom was provable, what would you call the statements from which > the axiom is deduced? > In logic, an axiom is provable because it is deducible from itself. What a slick way to do away with critical thinking. Tommy Aquinas would be proud. Isn¹t this the basis of religious belief? === Subject: Re: Trying to unify axioms. > What a slick way to do away with critical thinking. You might as well say that 1+1=2 does away with critical thinking. === Subject: Re: Trying to unify axioms. >> What a slick way to do away with critical thinking. > You might as well say that 1+1=2 does away with critical thinking. Revisit my example. If we assume that this is a refrigerator, it follows that this is a refrigerator. That is, R->R. Or (R ^ R) v (-R ^ -R). Classic tautology, it¹s true whether R is true or false. So which is it, true or false? R can¹t resolve that on its own. So choose a postulate: 1a) R 1b) -R Once you¹ve chosen one we can say until the cows come home that R->R or (-R)->(-R), but nothing can be decided until you pick. 1+1=2, on the other hand, is not just a restatment of a postulate. There¹s actually a chain of reasoning involved that comes to a conclusion that¹s different from any of the postulates. -- Irony: Small businesses want relief from the žood of spam clogging their in-boxes, but they fear a proposed national ŒDo Not Spam¹ registry will make it impossible to use e-mail as a marketing tool. === Subject: Re: Trying to unify axioms. > 1+1=2, on the other hand, is not just a restatment of a postulate. > There¹s actually a chain of reasoning involved that comes to a conclusion > that¹s different from any of the postulates. Really? I thought 2 was in practice nothing more than a shorthand notation for 1+1, at least in traditional formal arithmetic. OG. === Subject: Re: Trying to unify axioms. >> 1+1=2, on the other hand, is not just a restatment of a postulate. >> There¹s actually a chain of reasoning involved that comes to a conclusion >> that¹s different from any of the postulates. >Really? I thought 2 was in practice nothing more than a shorthand >notation for 1+1, at least in traditional formal arithmetic. > OG. Number theorists are wacky. I¹ve seen the reasoning twice and never remember how it goes, but it¹s something about deŽning addition as the union of sets of empty sets. -- The preferred method of entering a building is to use a tank main gun round, direct Žre artillery round, or TOW, Dragon, or HellŽre missile to clear the Žrst room. -- THE RANGER HANDBOOK U.S. Army, 1992 === Subject: Re: Trying to unify axioms. Gregory L. Hansen says... >Revisit my example. If we assume that this is a refrigerator, it follows >that this is a refrigerator. >That is, R->R. >Or (R ^ R) v (-R ^ -R). >Classic tautology, it¹s true whether R is true or false. So which is it, >true or false? R can¹t resolve that on its own. If R is an axiom, then the axioms are sufŽcient to resolve the issue. Look, this is just a terminological issue. We can deŽne a proof from a set of axioms T to be a sequence of statements S_1, S_2, ..., S_n such that for each j, S_j is either an axiom, or follows from earlier statements in the sequence by a valid rule of inference. We then say that S is deducible from T if there is a proof such that S is the last statement in the proof. By this deŽnition, it is trivially true that every axiom of T is deducible from T. This isn¹t in any way contrary to the informal statement that an axiom is something that is assumed with proof. The confusion comes from the fact that there are conceptually two stages involved in formulating a theory: (1) Deciding what statements to include as axioms, and (2) Once we have the axioms, Žguring out what statements are logically deducible from *them*. In stage 1, we pick our axioms mean when they say that an axiom is something assumed without proof---it is placed into the axiom set T without the need for any proof. But if a statement is in the axiom set, it is trivially deducible from *those* axioms. You could, I suppose, distinguish between the *proper* consequences of a theory and the *improper* consequences---a proper consequence is a statement that is not an axiom, but is logically deducible from the axioms. However, that¹s really not a very useful distinction. The *same* theory can be described using different but equivalent axioms. Statements that are axioms in one description may be theorems in another description. For a trivial example, you could let x+0 = x be an axiom of arithmetic. Or you could prove it by induction from the axioms 0+x = x (y+1)+x = (y+x)+1 The mathematical attitude towards axioms is that they are simply a way to describe a theory. But nothing important about a theory depends on the way it is described. -- Daryl McCullough Ithaca, NY === Subject: Re: Trying to unify axioms. <0q4%b.127219$FO1.2443626@weber.videotron.net> Discussion, linux) > What a slick way to do away with critical thinking. >> You might as well say that 1+1=2 does away with critical thinking. > Revisit my example. If we assume that this is a refrigerator, it follows > that this is a refrigerator. > That is, R->R. > Or (R ^ R) v (-R ^ -R). > Classic tautology, it¹s true whether R is true or false. So which is it, > true or false? R can¹t resolve that on its own. Huh? Which is what? R -> R is clearly true, regardless of the truth value of R. What¹s the issue? > So choose a postulate: > 1a) R > 1b) -R > Once you¹ve chosen one we can say until the cows come home that R->R or > (-R)->(-R), but nothing can be decided until you pick. Absolute nonsense. No matter whether R is true or false, *both* R -> R and ~R -> ~R are true. > 1+1=2, on the other hand, is not just a restatment of a postulate. > There¹s actually a chain of reasoning involved that comes to a > conclusion that¹s different from any of the postulates. Oh? What are the postulates from which you derived 1+1=2? -- Jesse Hughes Besides, discoverers are too proud to kiss butt. Indiana Jones would never kiss some academic¹s ass to get published, and neither will I. --James Harris === Subject: Re: Trying to unify axioms. > What a slick way to do away with critical thinking. > You might as well say that 1+1=2 does away with critical thinking. It does if you proclaim it and fail to deŽne the meaning of addition, the concept of number, etc., and then show how 1+1=2 follows from the deŽnitions. Hey, by your way of deŽnition, I could proclaim 1+1=4 as an axiom and get away with it. === Subject: Re: Trying to unify axioms. <0q4%b.127219$FO1.2443626@weber.videotron.net> Discussion, linux) > Hey, by your way of deŽnition, I could proclaim 1+1=4 as > an axiom and get away with it. Of course you could. Why not? -- Jesse F. Hughes Radicals are interesting because they were considered Œradical¹ by modern mathematics depends on. --Another JSH history lesson === Subject: Re: Trying to unify axioms. > Hey, by your way of deŽnition, I could proclaim 1+1=4 as > an axiom and get away with it. In a theory in which 1+1=4 is an axiom it is, trivially, provable that 1+1=4. === Subject: Re: Trying to unify axioms. > Hey, by your way of deŽnition, I could proclaim 1+1=4 as > an axiom and get away with it. > In a theory in which 1+1=4 is an axiom it is, trivially, provable > that 1+1=4. And therefore it is an entertaining but useless endeavor. === Subject: Re: Trying to unify axioms. > And therefore it is an entertaining but useless > endeavor. Entertaining, useless, has nothing to do with it. You might as well comment on what is entertaining or useless in 1+1=2. It¹s just a matter of accepted terminology in logic. === Subject: Re: Trying to unify axioms. >> If an axiom was provable, what would you call the statements from which >> the axiom is deduced? > In logic, an axiom is provable because it is deducible from itself. That doesn¹t even make sense. If we assume this is a refrigerator, it follows that this is a refrigerator. That¹s not a deduction, it¹s a restatement of the assumption. -- For every problem there is a solution which is simple, clean and wrong. -- Henry Louis Mencken === Subject: Re: Trying to unify axioms. >That doesn¹t even make sense. It makes perfect sense. It¹s a simple observation about how provable is used in logic. === Subject: Re: Trying to unify axioms. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Trying to unify axioms. >> Nothing. >> >> You forever spewing fucking imbecile, an axiom by deŽnition is >> irreducible and unprovable. >False. >First off, an axiom isn¹t necessarily irreducable, unless you want to >claim that reducing it makes the axiom not reducable and therefore not >an axiom. Sometimes we have an axiom that we Žnd out can be reduced >into other axioms. Does that therefore disprove the axiom, or destroy >the usefulness of the axiom, or of taking it as such? No it doesn¹t. >Second, axioms are not unprovable. They CAN be proven. For an example >of this, consider the law of identity. Can you prove it? If not, then >how do we even know it¹s true? We do know it¹s true, and it IS an >axiom, so that just proves that axioms are not unprovable. >(...Starblade Riven Darksquall...) > If an axiom were reducible or proveable, it would be a conclusion, not an > axiom. The axioms would become the axioms used to reach that conclusion. So, then, the axioms in the system of the human mind are human perception and automatic thought processes, and the ability to learn? The fact is, the very idea that we need axioms is false. What we need are principles. Those can be proven but that does not mean they are therefore reducable to other facts. (...Starblade Riven Darksquall...) === Subject: Re: Trying to unify axioms. > Nothing. > > You forever spewing fucking imbecile, an axiom by deŽnition is > irreducible and unprovable. >>False. >>First off, an axiom isn¹t necessarily irreducable, unless you want to >>claim that reducing it makes the axiom not reducable and therefore not >>an axiom. Sometimes we have an axiom that we Žnd out can be reduced >>into other axioms. Does that therefore disprove the axiom, or destroy >>the usefulness of the axiom, or of taking it as such? No it doesn¹t. >>Second, axioms are not unprovable. They CAN be proven. For an example >>of this, consider the law of identity. Can you prove it? If not, then >>how do we even know it¹s true? We do know it¹s true, and it IS an >>axiom, so that just proves that axioms are not unprovable. >>(...Starblade Riven Darksquall...) >> If an axiom were reducible or proveable, it would be a conclusion, not an >> axiom. The axioms would become the axioms used to reach that conclusion. >So, then, the axioms in the system of the human mind are human >perception and automatic thought processes, and the ability to learn? >The fact is, the very idea that we need axioms is false. What we need >are principles. Those can be proven but that does not mean they are >therefore reducable to other facts. The neat thing about building a math with axioms, is that you can Žrst build one, then go back and change just one axioms slightly. Go through the exercise of building the math again, and see the differences between the Žrst and the second build. It¹s fun to do. The neat thing about math is one doesn¹t have to include a reality check. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Trying to unify axioms. > Nothing. > > You forever spewing fucking imbecile, an axiom by deŽnition is > irreducible and unprovable. >> >>False. >> >>First off, an axiom isn¹t necessarily irreducable, unless you want to >>claim that reducing it makes the axiom not reducable and therefore not >>an axiom. Sometimes we have an axiom that we Žnd out can be reduced >>into other axioms. Does that therefore disprove the axiom, or destroy >>the usefulness of the axiom, or of taking it as such? No it doesn¹t. >> >>Second, axioms are not unprovable. They CAN be proven. For an example >>of this, consider the law of identity. Can you prove it? If not, then >>how do we even know it¹s true? We do know it¹s true, and it IS an >>axiom, so that just proves that axioms are not unprovable. >> >>(...Starblade Riven Darksquall...) >> >> >> If an axiom were reducible or proveable, it would be a conclusion, not > an >> axiom. The axioms would become the axioms used to reach that > conclusion. >So, then, the axioms in the system of the human mind are human >perception and automatic thought processes, and the ability to learn? >The fact is, the very idea that we need axioms is false. What we need >are principles. Those can be proven but that does not mean they are >therefore reducable to other facts. > The neat thing about building a math with axioms, is that you can > Žrst build one, then go back and change just one axioms slightly. > Go through the exercise of building the math again, and see the > differences between the Žrst and the second build. > It¹s fun to do. The neat thing about math is one doesn¹t have > to include a reality check. > /BAH > Subtract a hundred and four for e-mail. Yup. Good point. (...Starblade Riven Darksquall...) === Subject: Re: Trying to unify axioms. Hi Gregory L. Hansen, You mentioned, If an axiom were reducible or provable, it would be a conclusion, not an axiom. . But if you kept reducing axioms ad inŽnitum, all the while keeping all axioms everywhere consistent ... then you¹d be a masochist, and your result would be convoluted. In short, you¹d be a string theorist. If an axiom works it works ... that¹s good enough. No need to further reduce it. Physicalism is the only theory of everything, and it contains not one equation. Matter is the only reality. Time is perfectly spatial. The future is perfectly Žxed but imperfectly known. === Subject: Re: Trying to unify axioms. > Hi Gregory L. Hansen, You mentioned, > If an axiom were reducible or provable, > it would be a conclusion, not an axiom. . > But if you kept reducing axioms ad inŽnitum, > all the while keeping all axioms everywhere consistent ... > then you¹d be a masochist, > and your result would be convoluted. > In short, you¹d be a string theorist. *Laughs* That¹s a good one. > If an axiom works it works ... that¹s good enough. > No need to further reduce it. Good point. > Physicalism is the only theory of everything, > and it contains not one equation. > Matter is the only reality. > Time is perfectly spatial. > The future is perfectly Žxed but imperfectly known. Actually that¹s not true. Nature is Žxxed, but nature is always changing. In that way we can say nature IS progress. The future is not set, but the means by which we must get to the future is essentially set. However, it also posesses degrees of freedom. (...Starblade Riven Darksquall...) === Subject: Re: Trying to unify axioms. >Hi Gregory L. Hansen, You mentioned, > If an axiom were reducible or provable, > it would be a conclusion, not an axiom. . >But if you kept reducing axioms ad inŽnitum, > all the while keeping all axioms everywhere consistent ... > then you¹d be a masochist, > and your result would be convoluted. That¹s true, but some stopping points are more useful than others. For instance, you might want to stop at something related to physically meaningful measurements, say the invariance of the speed of light, for example. If your postulates go beyond the measurable, you start to just make things up. > In short, you¹d be a string theorist. >If an axiom works it works ... that¹s good enough. > No need to further reduce it. Well, system of axioms, really. >Physicalism is the only theory of everything, > and it contains not one equation. > Matter is the only reality. > Time is perfectly spatial. > The future is perfectly Žxed but imperfectly known. -- Are those morons getting dumber or just louder? -- Mayor Quimby === Subject: Re: Trying to unify axioms. > Nothing. > > You forever spewing fucking imbecile, an axiom by deŽnition is > irreducible and unprovable. > False. > First off, an axiom isn¹t necessarily irreducable, unless you want to > claim that reducing it makes the axiom not reducable and therefore not > an axiom. Sometimes we have an axiom that we Žnd out can be reduced > into other axioms. Does that therefore disprove the axiom, or destroy > the usefulness of the axiom, or of taking it as such? No it doesn¹t. Uh, actually, axioms are irreducible. > Second, axioms are not unprovable. They CAN be proven. For an example > of this, consider the law of identity. Can you prove it? If not, then > how do we even know it¹s true? We do know it¹s true, and it IS an > axiom, so that just proves that axioms are not unprovable. The Law of Identity cannot be proven. It is taken to be an axiom of the Žrst-order language with equality, and is only supported by our intuition. Try proving the axiom of commutativity. Oh wait, here¹s a counter-example: Matrix multiplication isn¹t commutative! All axioms do is place constraints on what sorts of objects are being studied. Œcid Œooh === Subject: derivative in the sense of distributions Is it true that a function in L^2 is always differentiable in the sense of distributions? In other words, for any f in L^2, does it exists a function g in L^1_loc such that: - = for all v in C_0^infty I could Žnd it in any book. Is there a good reference? Diogo === Subject: Re: derivative in the sense of distributions > Is it true that a function in L^2 is always differentiable in the sense of > distributions? In other words, for any f in L^2, does it exists a function > g in L^1_loc such that: > - = for all v in C_0^infty As David pointed out, every L^2 function (like every distribution) has a distibutional derivative, but the latter need not be in L^1 locally. For example, let f be the characteristic function of [0,1]. Then - = v(0) - v(1). So the distributional derivative of f is the singular measure delta_0 - delta_1, and there¹s no way this is going to be given by a g in L^1_loc. === Subject: Re: derivative in the sense of distributions >Is it true that a function in L^2 is always differentiable in the sense of >distributions? Yes. >In other words, for any f in L^2, does it exists a function >g in L^1_loc such that: > - = for all v in C_0^infty No. Your in other words is wrong. Any element of L^2 has a derivative in the sense of distribiutions, but this derivative need _not_ be a function in L^1_loc. >I could Žnd it in any book. Is there a good reference? _Any_ reference developing the theory of distributions from a mathematical point of view contains the following two facts: (i) any L^2 function deŽnes a distribution (ii) every distribution has a derivative, in the sense of distributions. (On the other hand, you¹re not going to Žnd a proof that the derivative of an L^2 function must be in L^1_loc in most books, because it¹s false.) >Diogo ************************ David C. Ullrich === Subject: e^(pi* i) I just read the e is transcendental thread and thought I would throw my hat in the ring. A few years ago while a freshman in college, I came up with this outrageous proof that took me a few days to Žnally prove incorrect. I leave it to you to prove false. I think it¹s pretty cool even if it is wrong :-) e^(pi*i) = -1 (e^(pi*i))^2 = (-1)^2 e^(2*pi*i) = 1 ln(e^(2*pi*i)) = ln(1) 2*pi*i = 0 This is where you say to yourself... WHOOPS! How can 2*pi*i be 0? Since 2, pi, and i are all constants and NONE are 0, we know from our multiplication laws that if a*b*c = 0 then a = 0 or b = 0 or c = 0 which is not the case here. So what happened? Enjoy! === Subject: Re: e^(pi* i) Anthony a .8ecrit dans le message de > I just read the e is transcendental thread and thought I would throw > my hat in the ring. A few years ago while a freshman in college, I > came up with this outrageous proof that took me a few days to > Žnally prove incorrect. I leave it to you to prove false. I think > it¹s pretty cool even if it is wrong :-) > e^(pi*i) = -1 In complex analysis, the principal logarithm of a complex number z is deŽned by: Log z=log |z| + i Arg z. Therefore Log[e^(2*pi*i)]= 0 + i Arg(2*pi)=0. Your mistake is that you take the principal log in the right side and another log in the left side! > ln(e^(2*pi*i)) = ln(1) > This is where you say to yourself... WHOOPS! > How can 2*pi*i be 0? Since 2, pi, and i are all constants and NONE > are 0, we know from our multiplication laws that if a*b*c = 0 then a = > 0 or b = 0 or c = 0 which is not the case here. > So what happened? > Enjoy! === Subject: Re: e^(pi* i) > I just read the e is transcendental thread and thought I would throw > my hat in the ring. A few years ago while a freshman in college, I > came up with this outrageous proof that took me a few days to > Žnally prove incorrect. I leave it to you to prove false. I think > it¹s pretty cool even if it is wrong :-) > e^(pi*i) = -1 This is where you say to yourself... WHOOPS! > How can 2*pi*i be 0? Since 2, pi, and i are all constants and NONE > are 0, we know from our multiplication laws that if a*b*c = 0 then a = > 0 or b = 0 or c = 0 which is not the case here. > So what happened? What happened is that you do not say what ln is nor what properties does it have. Jose Carlos Santos === Subject: easy topology problem... hello........ function f : X -> Y is continuous <=> for any B in Y , f^(-1){Fr(B)} in Fr{f^(-1)(B)} ------------------------------------------------ um......i can¹t prove.......... help...me, please. === Subject: Re: easy topology problem... > function f : X -> Y is continuous > <=> for any B in Y , f^(-1){Fr(B)} in Fr{f^(-1)(B)} > ------------------------------------------------ > um......i can¹t prove.......... Me neither; this is false. Take f:R --> R deŽned as f(x) = x^2 and take B=[0,1[. Then f^(-1)(Fr(B)) = f^(-1)({0,1}) = {-1,0,1} and Fr(f^(-1)([0,1[)) = Fr(]-1,1[) = {-1,1}. My guess is that what you meant was f^(-1){Fr(B)} contains Fr{f^(-1)(B)}. Am I right? Jose Carlos Santos === Subject: simple differentiable question. F(x,y,z) = (ax, by, cz) is differentiable <=> del F = (a,b,c) is exist -------------------------------- it¹s right?? I am anxious to this..... um.....let me advice...please...thank you. === Subject: Re: simple differentiable question. > F(x,y,z) = (ax, by, cz) is differentiable > <=> del F = (a,b,c) is exist > -------------------------------- > it¹s right?? > I am anxious to this..... > um.....let me advice...please...thank you. If F is a functions from R to R (reals) then F is differentiable iff delF is not zero... === Subject: Re: simple differentiable question. > F(x,y,z) = (ax, by, cz) is differentiable > <=> del F = (a,b,c) is exist del * F = (a,b,c) === Subject: Re: simple differentiable question. > F(x,y,z) = (ax, by, cz) is differentiable > <=> del F = (a,b,c) is exist > del * F = (a,b,c) in ASCII, it may be better to say div F and avoid confusion. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: simple differentiable question. >> F(x,y,z) = (ax, by, cz) is differentiable >> <=> del F = (a,b,c) is exist >> del * F = (a,b,c) >in ASCII, it may be better to say div F and avoid confusion. And even better to say div F = a + b + c ? --Lynn === Subject: Re: A slight generalization of Waring¹s problem > I_l = {n s.t. n=x_1^k+...+x_l^k does *not* have a solution}, >Now I wonder if anything is known about about the slightly more >general problem of Žnding how many k-th powers are required for the >representation of *most* numbers, i.e. of Žnding the least l for >which > (1/n)*card I_l intersects {1...n} -> 0 >as n->infty. I got no answers, any thoughts?!? Michele -- > Comments should say _why_ something is being done. Oh? My comments always say what _really_ should have happened. :) - Tore Aursand on comp.lang.perl.misc === Subject: Re: A slight generalization of Waring¹s problem > I_l = {n s.t. n=x_1^k+...+x_l^k does *not* have a solution}, >Now I wonder if anything is known about about the slightly more >general problem of Žnding how many k-th powers are required for the >representation of *most* numbers, i.e. of Žnding the least l for >which > (1/n)*card I_l intersects {1...n} -> 0 >as n->infty. > I got no answers, any thoughts?!? > Michele Mathworld does make brief mention of this problem, with a couple of results the appropriate number of kth powers. === Subject: Re: the anticlassicalist }{ : mitchism for Daleks [...] > As I keep trying to point out, my focus is on education with these > series of posts. The theory of models and their logics is crucial > to all science. Oh, no it isn¹t! > Indeed, it rigorously formalises the notion of a theory. A philosopher of science may have a professional duty to give a toss about such a thing, but a hardworking biochemist or oceanographer by no means does. > Many models across disciplines have a Heyting structure, and these > are often unknown to practitioners in the various Želds, so I use > that as my archetype for education reform. There are many cases > where the unfamiliarity with the logical structure of the models > used presents confusion to the students early on. Hello childrens! Today we¹re going to learn about counting! But Žrst we will need a few set-theoretical preliminaries so as to avoid any confusion about the structure of the natural number system. Des has no Bourbaki today -- [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 Jakobson === 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> <6IvurhFpHeNAFwU6@baesystems.com> <103cn32aodr3m3a@corp.supernews.com> <103n8ds9mudn455@corp.supernews.com> In message <103n8ds9mudn455@corp.supernews.com>, galathaea >: OK. So if I¹m already unknowingly using this logic, what is the point >: you¹re trying to make? Believe me, it hasn¹t yet emerged from all the >: verbiage. >Well, from a practical point of view, understanding the logic of >orthomodular lattices that arises in propositions on a Hilbert space can >prevent a lot of early conceptual difŽculties in the use of quantum >mechanics. Really? My experience is that the difŽculties arise because quantum objects don¹t behave intuitively, i.e. like the objects of everyday experience. No amount of bafžegab about orthomodular lattices will change that. [snip another few hundred words...] >Extend to quantum Želd theories, strings, and such, and maybe it gets >clearer why I see the necessity to alter education of logic in physics >curricula. No. No clearer. >And this does not even touch upon the theory of causal sets and other >directions in physics where such an education would be beneŽcial. -- Richard Herring === Subject: simple question for cluster point. i saw a proposition increasing function have a cluster point(=limit point) at most --------------------- um........i have a doubt. i think that it must to exchage increasing function => increasing sequence or domain of increasing function is countable point. i think....... if f(x) = x for x in R, limit point is R. thus, proposition is not perfect. my thinking is right?? let me advice...please.......thank you === Subject: Solving a set of multivariate polynomial equations in Z/2Z I have a set of equations of the form sum of products of subsets of the variables equal to 0 or 1 in Z/2Z. Something like: x0.x1.x3 + x2.x3.x4 + x1 = 1 x1.x2.x3 + x3.x5 = 0 [etc] This is in Z/2Z, i.e. you can also see that as the variables being booleans, * = and, + = xor. I have 64 variables and 100-150 equations of maximal order 20 or so, and I¹m trying to solve the system. My questions are: - does this problem have a well known name? It¹s very hard to Žnd call it exactly. - are there known methods/algorithms/heuristics that work reasonably well, i.e. with which I can hope for an answer in max a week of cpu time? OG. === Subject: Re: Solving a set of multivariate polynomial equations in Z/2Z >I have a set of equations of the form sum of products of subsets of >the variables equal to 0 or 1 in Z/2Z. Something like: > x0.x1.x3 + x2.x3.x4 + x1 = 1 > x1.x2.x3 + x3.x5 = 0 > [etc] >This is in Z/2Z, i.e. you can also see that as the variables being >booleans, * = and, + = xor. I have 64 variables and 100-150 equations >of maximal order 20 or so, and I¹m trying to solve the system. >My questions are: >- does this problem have a well known name? It¹s very hard to Žnd > call it exactly. You¹re looking for a SAT Solver. The problem you have is basically a boolean satisŽability problem. Your example: can you assign thruth values to P0, P1, ... such that (((P0 / P1 / P2) xor (P2 / P3 / P4)) xor P1) / not ((P1 / P2 / P3) xor (P3 / P5)) / etc. ? SAT Solvers (eg Chaff, SatZoo) like to have their input in conjunctive normal form and use some version of Davis-Putnam to cehck satisŽability (so you may have to translate to CNF). Maybe you should take a look at or . Peter -- Peter van Rossum, | Universal law of linearity: for all Dept. of Mathematics, New Mexico | f : R -> R and for all x, y in R: State University, Las Cruces, NM, USA. | f(x + y) = f(x) + f(y) === Subject: distributional derivative in L^2 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1PCZpG31596; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1PBECi25139 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i1PBECt31601; Is it true that a function in L^2 is always differentiable in the sense of distributions? In other words, for any f in L^2, does it exists a function g in L^1_loc such that: - = for all v in C_0^infty I couldn¹t Žnd it in any book. Is there a good reference? Diogo === Subject: Re: distributional derivative in L^2 > Is it true that a function in L^2 is always differentiable in the sense of > distributions? In other words, for any f in L^2, does it exists a function g in L^1_loc such that: > - = for all v in C_0^infty > I couldn¹t Žnd it in any book. Is there a good reference? You (or someone else) has already posted that same question (under the name derivative in the sense of distributions) and David C. Ullrich has already provided an answer. Jose Carlos Santos === Subject: Re: I¹m guessing this is an easy problem - but I can¹t solve it! Finally found the solution myself... using (a^3 - b^3) = (a-b)(a^2 + ab + b^2) > Working through calculus... Stuck on a problem of factorisation: Please > help! > question is: > Žnd lim u--> 1 f(u) > where f(u) = (u^4 - 1) / > (u^3 -1) > Obviously - I cannot just replace u with 1 as this will give me a division > by zero. I tried replacing (u^2 - 1) (u^2 +1) on top as this seemed most > logical - but I can¹t Žgure out what to do with the bottom now. > I¹m probably going to kick myself for not Žguring this one out - but I¹ve > spent hours on this and I¹m still lost. > Help! === Subject: determinant X-Filename: sci.math/A103P8K7 1. Let A be a nxn real matrix such that A^t = -A,