mm-2419 === Subject: Re: does anybody know how to remove the braces of a list in mathematica so it can be sent as parameters to a function? something with Flatten perhaps? === Subject: Re: The Proper Way to put a long URL in a NG post > Constants/Miscellaneous/classification.html#Hardy (the two lines need to be reattached to make a real URL). Most of us use newsreaders that automatically wrap words, phrases, and > sentences at or near a certain column and URLs are particularly prone to > getting clobbered this way. BUT ... at least for Outlook Express, if the > URL is enclosed in angle-brackets > i.e. < ... Like so tml#Hardythen it doesn't matter how many lines the URL gets split across, it will > show up as a single link in the reader. WRONG! Check out the following page (Google Groups): (BTW, Google Groups used to have the feature that it wouldn't wrap lines that begin with a > sign.) === Subject: Convergence of a Crazy Series I need some help in proving the convergence of the series: sum_{n=2}^{infty} [cos(log n)]/[n log n]. -kira === Subject: Re: Convergence of a Crazy Series > I need some help in proving the convergence of the series: > sum_{n=2}^{infty} [cos(log n)]/[n log n]. > > -kira Another possibility might be to look at the intervals such that pi*k < log n < pi*(k+1), where cos(log n) is of constant sign. === Subject: Re: Convergence of a Crazy Series > I need some help in proving the convergence of the series: > sum_{n=2}^{infty} [cos(log n)]/[n log n]. Note int_[2,oo) [cos(log(x))]/[xlog(x)] dx converges (integrate by parts). Letting f(x) denote the integrand, show that f(n) is close enough to int_[n,n+1] f(x)dx to give convergence of sum f(n). === Subject: Re: Convergence of a Crazy Series >I need some help in proving the convergence of the series: > sum_{n=2}^{infty} [cos(log n)]/[n log n]. 1) int_{t=2}^infty cos(log t)/(t log t) dt = -Ci(log 2) exists. 2) d/dt (cos(log t)/(t log t)) = O(1/t^2) as t -> infty and thus is absolutely integrable. Use these to bound the difference between the sum and the integral. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Convergence of a Crazy Series > 1) int_{t=2}^infty cos(log t)/(t log t) dt = -Ci(log 2) exists. > 2) d/dt (cos(log t)/(t log t)) = O(1/t^2) as t -> infty and thus > is absolutely integrable. > Use these to bound the difference between the sum and the integral. difference between the sum and the integral. -kira === Subject: Re: Convergence of a Crazy Series > > difference between the sum and the integral. > Oh wait. I got it. hehe... The difference cannot be more than sum 1/t^2, which also converges. -kira === Subject: Re: Convergence of a Crazy Series > I need some help in proving the convergence of the series: > sum_{n=2}^{infty} [cos(log n)]/[n log n]. Here is an approach. Let C(x) denote the antiderivative of cos(x)/x. Show that C(x) is bounded as x->infinity. Now consider sum_{n=50}^infty (C(log n)-C(log(n-1))/[n log n]. There is some test similar to the alternating series test whose name I forget (but you prove it by summation by parts) that will show that this series converges. Next, see if this series differs from your series by a bounded amount. I am fairly sure that it does, but I admit that I haven't done the details, so it might not work. === Subject: Re: mathematica command >I need a mathematica command that does the following: >Assume X={1,2,3} >and I would like to evaluate the funcion F[1,2,3] . I mean I need a command >to remove the braces off my X so I can use them as paramenter to my function >F. I could use for example F[X[[1]],X[[2]],X[[3]]] but I hope somebody >knows a command so that I don't have to do this.. In[1]:= x={1,2,3} Out[1]= {1,2,3} In[2]:= Apply[f,x] Out[2]= f[1,2,3] === Subject: When does this equation make sense? Hi all, I learned this random sum formula from Ross' book. E(S)=E(summation of X_i where i from 1 to N, where N is another random variable)=E(X_i)*E(N) Could you please elaborate general conditions on when does this equation hold? Simple and complicated conditions all are welcome. === Subject: Re: When does this equation make sense? >E(S)=E(summation of X_i where i from 1 to N, where N is another random >variable)=E(X_i)*E(N) > >Could you please elaborate general conditions on when does this equation >hold? Simple and complicated conditions all are welcome. > If X1, X2, X3,... are independent random variables with well-defined common mean m and N is a random variable which is independent of the process X and almost surely a natural number, then E sum(i=1..N, X_i) = m EN, where we interpret 0 * infty as 0. Let X1, X2, X3,... be a sequence of random variables with common well-defined mean m. Let N be a stopping time for the process X with finite expectation. If all the X_i are all almost surely nonnegative or m is finite, then E sum(i=1..N, X_i) = m EN. (This is a generalization of Wald's equation.) -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: When does this equation make sense? >> If X1, X2, X3,... are independent random variables with >> well-defined common mean m and N is a random variable which is >> independent of the process X and almost surely a natural number, >> then E sum(i=1..N, X_i) = m EN, where we interpret 0 * infty as 0. > Actually, we can drop the condition that the X_i are independent. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: When does this equation make sense? >I learned this random sum formula from Ross' book. >E(S)=E(summation of X_i where i from 1 to N, where N is another random >variable)=E(X_i)*E(N) >Could you please elaborate general conditions on when does this equation >hold? Simple and complicated conditions all are welcome. For this to make any sense, the values of N are in the positive integers, the X_i are iid and also independent of N, and E[N] and E[X_i] exist. That should be sufficient: consider sum_{i=1}^{min(N,m)} X_i for constant m, and use the Monotone Convergence Theorem for the case X_i > 0, and then the Dominated Convergence Theorem for the general case. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: When does this equation make sense? >>I learned this random sum formula from Ross' book. > >>E(S)=E(summation of X_i where i from 1 to N, where N is another random >>variable)=E(X_i)*E(N) > >>Could you please elaborate general conditions on when does this equation >>hold? Simple and complicated conditions all are welcome. >For this to make any sense, the values of N are in the positive integers, >the X_i are iid and also independent of N, and E[N] and E[X_i] exist. >That should be sufficient: consider sum_{i=1}^{min(N,m)} X_i for >constant m, and use the Monotone Convergence Theorem for >the case X_i > 0, and then the Dominated Convergence Theorem for the >general case. I think it can be proved by more elementary methods, by noting that if m=E(X_i), then E(S) = E(E(S|N)) = E(mN) = mE(N) ---------------------------------------------------------------------------- Radford M. Neal radford@cs.utoronto.ca Dept. of Statistics and Dept. of Computer Science radford@utstat.utoronto.ca University of Toronto http://www.cs.utoronto.ca/~radford ---------------------------------------------------------------------------- === Subject: Re: When does this equation make sense? > I learned this random sum formula from Ross' book. > > E(S)=E(summation of X_i where i from 1 to N, where N is another random > variable)=E(X_i)*E(N) > > Could you please elaborate general conditions on when does this equation > hold? As written, it makes no sense. For which i do you evaluate E(X_i) in the last piece? Whatever it is that you actually meant to write, doesn't Ross say something about the conditions under which it holds? Usually, people who write books include hypotheses with their theorems. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: discriminating SO3 invariance from general O3 invariance > Given the space S of symmetric real 3x3 matrices, we operate on it with > O3 via conjugation. > I wish to discriminate SO3 conjugation classess from O3 conjugation > classess. that is - i'm looking for a function of a matrix E which is > invariant under rotations, but not under reflections (in general, at > least). > I believe i'm able to prove, with the aid of some earlier advice in > this group, that such an invariant function cannot be a polynomial in > the entries of E, of any degree. Still, I hope, the search is not lost. > is anyone familiar with such invariants? > > any thoughts or just directions would be appreciated too. > > Ofek Suppose the function f(E) is invariant under the conjugation action of all elements A of SO(3), i.e. f(AEA^{-1}) = f(E) for all A in SO(3) and all real symmetric 3x3 matrices E Suppose also that f is *not* an O(3) invariant, i.e. there exists an element B of O(3) and a real symmetric 3x3 matrix E such that: f(BEB^{-1}) != f(E). Clearly B is not an element of SO(3), i.e. it belongs to O(3)SO(3), and hence it has a determinant of -1. So -B has determinant 1 and is an element of SO(3). Then: f((-B)E(-B)^{-1}) = f(E) by our assumption that f is SO(3)-invariant but we also have f((-B)E(-B)^{-1}) = f(BEB^{-1}) != f(E). So these two assumptions are contradictory, and there can be no f with the properties we've assumed. -- Greg Egan Email address (remove name of animal and add standard punctuation): gregegan netspace zebra net au === Subject: Re: discriminating SO3 invariance from general O3 invariance Elegant and true, sadly. === Subject: Re: set of a set etc. <42DE957A.50802@netscape.net> <69mdncsaeP8wskLfRVn-1Q@comcast.com> <7aadnQiaA9-lIULfRVn-iQ@comcast.com> Sets are conceptual objects, not physical ones. The fact that a car is > physically composed of components has no bearing on what members it has, > unless you decide that it does by calling it a set of components rather than > a single entity. In other words, the set containing car is different from > the set containing the components of the car, even though a physical box > containing one car may be identical to a physical box containing the > components of the car. Sets have properties based on how they're defined -- > investigation into their physical properties can't discover properties that > aren't derivable from their definitions. Can conceptual objects have weight? The set of car parts conprising my car weighs about 2 tons. I actually do drive myself around town in that set of car parts. I guess I'm trying to understand the relationship between sets and ordinary language. It seems like there are some significant links. Can't a set of car parts be exactly the same thing as the car itself? ( Not always of course, a dismanteled car would still be the same set of car parts but wouldn't really be a car anymore.) Jasper === Subject: Re: set of a set etc. Mail-To-News-Contact: abuse@dizum.com >Can conceptual objects have weight? The set of car parts conprising my >car weighs about 2 tons. No it doesn't. The set is a concpetual object, with no physical existence, so it doesn't have any weight. A more correct statement would be The sum of the weights of the elements of the set of car parts conprising my car is about two tons. The set doesn't weigh anything, its members do. > I actually do drive myself around town in that >set of car parts. No, you drive yourself around town in an object constructed from the elements of that set. > Can't a set of car parts be exactly the same thing >as the car itself? No, for exactly the reason that you give: > ( Not always of course, a dismanteled car would >still be the same set of car parts but wouldn't really be a car >anymore.) -- Michael F. Stemper #include If we aren't supposed to eat animals, why are they made from meat? === Subject: Re: set of a set etc. > >> >> Sets are conceptual objects, not physical ones. The fact that a car >> is physically composed of components has no bearing on what members >> it has, unless you decide that it does by calling it a set of >> components rather than a single entity. In other words, the set >> containing car is different from the set containing the components >> of the car, even though a physical box containing one car may be >> identical to a physical box containing the components of the car. >> Sets have properties based on how they're defined -- investigation >> into their physical properties can't discover properties that aren't >> derivable from their definitions. > Can conceptual objects have weight? The set of car parts conprising my > car weighs about 2 tons. I actually do drive myself around town in > that set of car parts. I guess I'm trying to understand the > relationship between sets and ordinary language. It seems like there > are some significant links. Can't a set of car parts be exactly the > same thing as the car itself? ( Not always of course, a dismanteled > car would still be the same set of car parts but wouldn't really be a > car anymore.) It's not the members of the set that are conceptual, it's the set itself. Members of sets can be anything, physical objects, abstract concepts, numbers, whatever, as long as the set is well defined so that you can always determine precisely what the members of the set are. If I define a set of two members as {engine,chassis}, the set has two members, each of which is a physical object. You can't say, but the engine and the chassis make a car, so the car is also a member of the set. The set members are determined by the set definition, and they don't change based on real-world considerations like that, any more than the set {1,2,5} is equal to the set {8} because 1+2+5 = 8. A set is roughly like a list of things -- each thing is either on the list or not, and combining things on the list to make other things changes the list. If you bought the parts for a computer, and got charged the price for an assembled computer, on the grounds that the parts when put together make a computer, you'd rightly complain that the invoice for the computer parts is different from the invoice for a whole computer, even though the computer is the combination of the parts. (Not a perfect analogy, but I'm running out of ways to say this.) A set is defined by its members, and the number of members is an intrinsic characteristic of the set. (BTW, a set isn't really like a list, because of a few technical reasons, like a list can have duplicate items but a set can't, and a set can be uncountable but a list can't.) --Mark === Subject: Re: set of a set etc. <42DE957A.50802@netscape.net> <69mdncsaeP8wskLfRVn-1Q@comcast.com> <7aadnQiaA9-lIULfRVn-iQ@comcast.com> > >> >> Sets are conceptual objects, not physical ones. The fact that a car >> is physically composed of components has no bearing on what members >> it has, unless you decide that it does by calling it a set of >> components rather than a single entity. In other words, the set >> containing car is different from the set containing the components >> of the car, even though a physical box containing one car may be >> identical to a physical box containing the components of the car. >> Sets have properties based on how they're defined -- investigation >> into their physical properties can't discover properties that aren't >> derivable from their definitions. > Can conceptual objects have weight? The set of car parts conprising my > car weighs about 2 tons. I actually do drive myself around town in > that set of car parts. I guess I'm trying to understand the > relationship between sets and ordinary language. It seems like there > are some significant links. Can't a set of car parts be exactly the > same thing as the car itself? ( Not always of course, a dismanteled > car would still be the same set of car parts but wouldn't really be a > car anymore.) It's not the members of the set that are conceptual, it's the set itself. > Members of sets can be anything, physical objects, abstract concepts, > numbers, whatever, as long as the set is well defined so that you can always > determine precisely what the members of the set are. If I define a set of > two members as {engine,chassis}, the set has two members, each of which is a > physical object. You can't say, but the engine and the chassis make a car, > so the car is also a member of the set. The set members are determined by > the set definition, and they don't change based on real-world considerations > like that, any more than the set {1,2,5} is equal to the set {8} because > 1+2+5 = 8. A set is roughly like a list of things -- each thing is either > on the list or not, and combining things on the list to make other things > changes the list. If you bought the parts for a computer, and got charged > the price for an assembled computer, on the grounds that the parts when put > together make a computer, you'd rightly complain that the invoice for the > computer parts is different from the invoice for a whole computer, even > though the computer is the combination of the parts. (Not a perfect > analogy, but I'm running out of ways to say this.) A set is defined by its > members, and the number of members is an intrinsic characteristic of the > set. (BTW, a set isn't really like a list, because of a few technical reasons, > like a list can have duplicate items but a set can't, and a set can be > uncountable but a list can't.) --Mark I wasn't trying to say that the car was a member of the set of car parts but that the car WAS the set of car parts. I accept your computer example entirely. Unassembled, a set of computer parts isn't a computer but if assembled then the same set of computer parts would be a computer. Is there anything theoretically wrong with saying that a particular set of computer parts actually is a computer? Does this violate in any way the notion of a set? As with the car example I don't mean that the computer is a member of the set of computer parts even when assembled. It just seems like under some circumstances that a multiple element set can refer directly to a physical object. I also agree with your statment > The set members are determined by the set definition, and they don't change based on real-world considerations... The members of the set of computer parts don't change when the computer is assembled but afterwards it does seem to me that the set of computer parts is now a computer. I suppose that it's just that set definitions generally don't address the issue one way or the other. In that sense they are, not surprisingly, an incomplete specification of any actual situation. So: a) Is an assembled set of computer parts still a set? b) Isn't the assembled set of computer parts also a computer? object. Whew! You're making me think a lot. Jasper === Subject: Sarfatti Lectures in Warp Drive Physics 1 === Subject: Re: Dark Energy Metric Engineering Exotic Propulsion Notes 1 Note: The objective local tetrads describe physical phenomena, e.g. Bob in free float fires his rocket engine described by eu^a(Bob). Suppose Bob is in free float closed geodesic orbit around Earth, then his basic eu^a(Bob) is determined by how he needs to fire his rocket engine in order to HOVER at a fixed distance from surface of Earth without any orbital angular momentum. That gives, in that curved spacetime situation, the basic Bu^a field in eu^a(Bob) = Identity Matrix + Bu^a where the Schwarzschild metric in this HOVER LNIF representation is guv = (Iu^a + Bu^a)(Minkowski)av(Iv^b + Bv^b) g00 = 1 - 2rs/r grr = - 1/g00 Bu^a ~ buLp(Goldstone Phase)^,a When Lp -> 0 gravity is impossible. bcc Can't you make special relativity formally general covariant simply by taking 1915 GR and setting R^u_vwl = 0 for physical spacetime regardless of the distribution of matter? No, that's a contradiction because matter bends space-time. Also empty space-time with Ruvwl = 0 is globally unstable (Geroch). That is Ruv = 0 geons will form! I have also shown why it is unstable. My pre-inflation pre-Big Bang vacuum is precisely what you say above BTW, but it collapses to form the Higgs Ocean of smooth vacuum ODLRO curved space-time where the Einstein-Cartan tetrad field, in a geodesic LIF field of frames is eu^a = Identity 4x4 matrix + bu(LpP^a/ih)(Goldstone Coherent ODLRO Phase of Higgs Ocean) Lp^2 = hG/c^3 P^a is the quantum linear momentum operator in tangent space. Gravity is impossible when c -> infinity and h -> 0 even if G =/= 0. The physical tetrads describe LNIFs created by the application of non-gravity forces! These physical tetrads have nothing to do with mere relabeling of addresses. The objective local tetrads describe physical phenomena, e.g. Bob in free float fires his rocket engine described by eu^a(Bob). Or if Bob is already LNIF then the GCT Xu'^u(Bob) is Bob changing his rocket motor thrust. Clearly such a theory is not in itself general relativistic, and neither is there any Andersonian action-reaction principle. Nevertheless, it is formally general covariant. I think this is a standard argument. It's wrongly conceived even if it is standard, i.e. congealed error. It's wrong because you need to add non-physical degrees of freedom to do it. More on that later. The equivalence principle in strong form rules supreme. It has no kind of fault nor flaw within its limited domain of validity that will break down. Everything is correspondence and metaphor. The gravity g-force in reaction to any non-gravity force pushing the inertial detector off a timelike geodesic is locally equivalent to a non-tensor gravity field. Whether or not there is also a tidal curvature geodesic deviation tensor field is completely irrelevant to the idea of Einstein's equivalence principle. The alleged Ruffini's objection is really a Red Herring. The experimental methods to detect the gravity field and the curvature field are independent of each other. timelike geodesics. You measure gravitational field in space only by firing rocket engines to say hover at a fixed distance from Earth or maybe a Black Hole! No action without direct reaction, without the two-way relation (Bohm & Hiley): Einstein overthrew the conception we had previously formulated of a homogeneous space-time that pre-existed phenomena, and in which these phenomena happen to insert themselves without altering it. The special theory of relativity respected this conception [of a rigid unbending absolute 4D space-time that does not respond to the stuff in it, just like the micro-quantum BIT field does not react to the IT field in it in order to have signal locality in the midst of nonlocal entanglements] ... the fundamental hypothesis of Einstein is not, as many people have believed, that it is possible to formulate the laws of physics in all systems of coordinates - which would merely be a tautology - but that in every sufficiently small region the laws of special relativity are valid in the first approximation. Here Cartan seems to refute Kretschmann's 1917 objection to Einstein's GR that all physical theories can be made generally covariant. However, in order to do that requires adding in non- physical gauge freedom redundant pseudo-degrees of freedom i.e. excess baggage. More on that anon. === Subject: Re: Sarfatti Lectures in Warp Drive Physics 1 > === > Subject: Re: Dark Energy Metric Engineering Exotic Propulsion Notes 1 http://www.freefarts.com/farts.html Move cursor over blinkers to hear Sarfatti lecture. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: Questions wrt computable order of numbers. |This concerned me a bit because I like to have an order function that |would eventually complete on whatever computable numbers I choose to |implement in a program (it makes me feel better knowing, even if |completion is after the big party at the end of the Universe). In January we had a discussion of some related issues, in a thread titled, Arithmetic on computable reals. In my posting <20050115194113.16716.00000044@mb-m14.aol.com> I describe a way in principle of representing all the computable reals in such a way that they can be added, multiplied, and compared. Note that this isn't constructive. If someone hands you a computable real in some standard format, you won't necessarily be able to represent it the way I describe. There's just a pure existence proof that it can be. It also is completely impractical because of the amount of data that one would need for each number. There are some good subsets of the computable numbers, though. It's always possible to add, multiply, and compare algebraic numbers, for instance. If you want to do that, algorithms for doing it could be dug up for you. Keith Ramsay === Subject: Re: Questions wrt computable order of numbers. > This concerned me a bit because I like to have an order function that > would eventually complete on whatever computable numbers I choose to > implement in a program Yes, that would be a useful property. > Are there convenient greater subsets of the computables (how about all > algebraic numbers) for which it is always possible (assuming pre-detection > of the equal case) to guarantee that the combined error of two > approximations will eventually be less than the difference so it > will be possible to establish the order after a finite number of steps? If you've got a finite equality test (an oracle), then any computable numbers can be ordered. First test for equality. By assumption, this can be done in finite time. If they're equal, you're done. Otherwise, use any enumeration of the rationals until you find one between them. Any computable number can be compared with a rational in a finite number of steps. There are infinitely many rationals in that interval, and they all have a finite index in the enumeration. - Tim === Subject: Re: Questions wrt computable order of numbers. > at: http://en.wikipedia.org/wiki/Computable_numbers it says that the order relations of computable numbers are not computable. Wikipedia has a lot of junk in it. You should find a better reference. >> 2. No continuum has been discovered in physics -- everything seems >> to change in finite units called quanta. That's the real world. Mathematics is not interested in the real world. Apart from that, the > _probabilities_ of quantum changes appear to be quite well modeled > with continuous distributions. Kastrup, Kriemhildstr. 15, 44793 Bochum Mathematics can't be interested in anything. People show interest. What you mean is mathematicians aren't interested in the real world. === Subject: Re: A question about planar graphs (graph theory) This is what I'm really thinking about: 1. Given a connected simple planar graph, as I construct its dual, the dual is not only a planar graph, but also looks just like some real-life map with borders. 2. However, not every realizable map's dual is a simple planar graph. As an counter-example, see: http://www.geocities.com/gangxu_csu/dual-ex1.jpg So it seems that simple planar graphs can NOT represent all realizable maps as their duals. I'm not quite familiar with this topic, so if i'm already wrong at this point, please point me out. When we arbitrarily draw a map, which has to look like a real-life map in any possible ways, it seems that they always satisfy the relation: e <= 3f -6 here e is the edge/border number and f is the face/region number, f>=3. Note that, since we can represent the map in dual form, so map's faces correspond to dual's vertices, as being consistent with that in my initial question. Now question is: if the above relation does hold for any realistic maps, and simple planar graphs are not enough to represent realistic maps, what kind of general planar graphs do and, so they satisfy above relation (with f substituted by v)? any references already studies this problem? so I can simply steal it, ;). Gary === Subject: Re: Number Theory problem: seeking Calculus example [LONG] >A colleague teaching first-semester Calculus asked my help to >construct an example of a rational function which could be graphed >by his students (he hopes) with traditional tools. Specifically, >he wanted a function f(x) = p(x)/q(x) where p is a cubic >polynomial, and q is a quadratic polynomial; the characteristics >he wanted were that f should have poles, critical points, and >inflection points at INTEGER points in the domain. In order to >make this useful to the students -- and a sporting challenge to us -- >we add the constraint that there be _four_ critical points >(which is the maximum number for such an f ). > >I have succeeded in finding such an f with two integer poles, >one integer inflection point, and two integer critical points, >but the other two (real) critical points in my example are not >rational integers. Here is the simplest of the examples I had when I first posted: f = (x^3 +374*x-15390)/(x^2-20^2) i.e. 3 x + 374 x - 15390 18 (43 x - 855) f = ------------------ = x + ----------------- (x - 20) (x + 20) (x - 20) (x + 20) That may be simple enough for calculus students. But since my first post, I went one better: I can make all seven interesting points rational! The better function is f = (x^3+157613963*x+332031930516)/(x-1771)/(x+1771) i.e. 438012 (367 x + 758043) f := x + ----------------------- (x - 1771) (x + 1771) with poles at +-1771, critical points at -10373, -3289, -989, and 14651, and with an inflection point at -4389 . Let me describe how I found this example. As someone else noted, we can use affine transformations on the two axes to reduce the general function of this type to one of the generic form f = (x^3+bx+c)/(x^2-1). Other canonical forms can be used, but they're all two-parameter families of functions, that is, the function we're looking for will be represented as a nice point on a _surface_. Call this Surface #1 for now; since b and c are unrestricted, this surface is just the affine plane. So what makes a point (b,c) on this plane nice? Well, we want there to be three rational numbers making f'(x)=f'(y)=f''(z)=0. That gives three equations in x,y,z,b,c and we can eliminate b and c to get a single equation in x,y,z , which describes another surface in R^3 (Surface #2). We want a point on this surface with rational coordinates: 0 = (x+y)*(x^2+y^2*x^2+y^2-3) + (-6*y^3*x+12*y*x+6-6*y^2*x^2-6*y*x^3)*z +3*(x+y)*(x^2+y^2*x^2+y^2-3)*z^2 + (4*y*x-2*y^3*x-2*y^2*x^2+2-2*y*x^3)*z^3 The mapping that recovers the point on Surface #1 from a point on Surface #2 is b = x*y*(x^2+x*y-3+y^2)/(x*y-1), c = -1/2*(x+y)*(y^2*x^2+x^2-3+y^2)/(x*y-1) To my surprise, I found that Surface #2 is _rational_, that is, it may be parameterized by rational functions of two variables. It took me a while to discover this. I first used the symmetry between x and y to rewrite the equation in terms of s = x+y and p = x*y, which must also be rational. The new equation is: (s+6*z+2*z^3+3*z^2*s)*p^2 + (-2*s-6*z^2*s-6*z*s^2-2*z^3*s^2+12*z+4*z^3)*p + (6*z-3*s+2*z^3-9*z^2*s+s^3+3*s^3*z^2) = 0 This equation describes a new surface in R^3 (Surface #3 in (s,p,z)-space). Geometrically, Surface #2 is a 2-to-1 branched covering of Surface #3 under the map (x,y,z) --> (x+y,x*y,z); rational points are sent to rational points on the new surface, and conversely the pre-image of (s,p,z) is rational iff s, p, AND sqrt(s^2-4p) are rational. This new surface is easier to describe. Since the equation is a quadratic in p, we can compute a rational p for any s and z as long as the discriminant is a square, but that discriminant turns out to be VERY easy: it's 4(s^2-4)(s^2)(z^2-1)^3, which is a square iff (s^2-4)/(z^2-1) is. So we can parameterize the rational points (s,p,z) using the rational points (s,v,z) on the surface s^2-4 = v^2 (z^2-1). (We're up to Surface #4 now.) This is now just quadratic in each of the variables, and a description of the solution set is obtained by very classical methods. A parameterization of Surface #4 is s = -2*(-m^2-v^2+m*v^2)/(v-m)/(v+m), z = -(m^2+v^2-4*m)/(v-m)/(v+m) where m and v are arbitrary. Sure enough, for these pairs s,z we can indeed compute a _rational_ p using the quadratic equation describing Surface #3: p = (-3*v^5+4*m*v^2+4*m^3+12*m^3*v+12*v^3*m-3*m*v^4-12*m^2*v^3-3*m^3*v^2+2* m^2*v^4+2*m^3*v^3+2*v^5*m-8*m^2*v-3*m^4*v)/(v+m)^2/(m^2*v+4*m-4*m*v-m*v^2+v^ 3 ) (There is another solution p to the same quadratic, of course, but it turns out that that point on the surface is in the image of this parameterization already -- it arises by replacing v with -v. ) So this formula and the one for s give a parameterization of Surface #3 in terms of parameters m and v. Now we can attempt to work back to its two-fold cover, Surface #2 -- each of these points determines two points (x,y,z) on the originial surface; we just have to solve the quadratic X^2 - s*X + p to get the values of x and y. But we do want x and y to be rational and that requires that the discriminant s^2-4p be a rational square. This puts an additional constraint on the two parameters v and m. Well, again we are lucky: s^2-4p turns out to be a product of several _square_ factors times just (m^2*v+4*m-4*m*v-m*v^2+v^3)*(v^2-m*v+m^2)*v so that we will obtain a rational point (x,y,z) on the original surface iff THIS quantity is a rational square. A succession of substitutions makes this quantity look even nicer; for example, we notice that if we write m = v*n then there is a factor of v^4, which is a square; so what must be a square now is the other factor, 4*n*(n^2-n+1) - 4*n*(n^2-n+1)*v + (n^2-n+1)^2*v^2 I then substituted v=w/(n^2-n+1), then w = u + 2n, then u = r(2n-2) to conclude that what must be square is just n + r^2. So we write n = q^2-r^2 and then work backwards through the substitutions to get a parameterization of the pairs (m,v) that make the quantity at the end of the last paragraph become a square: v = -2*(-r*q^2+r^3+r-q^2+r^2)/(q^4-2*q^2*r^2+r^4-q^2+r^2+1) m = 2*(-r*q^2+r^3+r-q^2+r^2)*(-q+r)*(r+q)/(q^4-2*q^2*r^2+r^4-q^2+r^2+1) We can substitute these values of v and m into our parameterizations for s, p, and z to get a parameterization of those points in Surface #3 that make s^2-4p be a square, that is, this is a parameterization of the points in the image of the 2-to-1 map from Surface #2. So finally we can lift the parameterization back to Surface #2. We substitute into the equation X^2 - s X + p = 0 and discover that the roots X (i.e. the coordinates x,y on the original surface) are {x,y} = { (2*r+q^7-3*q^5*r^2-2*r*q^6+6*r^3*q^4-6*r^5*q^2-q+2*q*r^3-q*r^6-2*q^4+6*q^2*r ^ 2-4*r^4+2*r^7+3*r^4*q^3-2*r*q^3)/(r^2-1-q^2)/q/(q^4-2*q^2*r^2+r^4-q^2+r^2+1) , -(2*r-q^7+3*q^5*r^2-2*r*q^6+6*r^3*q^4-6*r^5*q^2+q-2*q*r^3+q*r^6-2*q^4+6*q^2* r^ 2-4*r^4+2*r^7-3*r^4*q^3+2*r*q^3)/(r^2-1-q^2)/q/(q^4-2*q^2*r^2+r^4-q^2+r^2+1) }. (The second is equal to the result of replacing q by -q in the first.) Taking one of these for x, the other for y, and the previous z = (r^5-r^4-2*q^2*r^3+2*q^2*r^2-r^2+r+q^4*r-q^4+q^2)/ ((-r*q^2+r^3+r-q^2+r^2)*(r^2-1-q^2)) we now have a parameterization of Surface #2. That is, we can take any rational values of p and q and obtain in this way the rational coordinates x,y (the critical points) and z (the inflection point) of a function of our generic form f = (x^3+bx+c)/(x^2-1). It is trivial to compute b and c from this information. The function f which I gave at the very beginning corresponds to {q=5/4, r=3/4} . As I say, I was surprised that Surface #2 was rational, i.e. I didn't expect to be able simply to write out 2-parameter formulas which will yield all the functions f with two rational critical points and a rational inflection point. It was certainly not guaranteed up-front that this would be possible. By the way, the parameterization is far from unique. In this case, for example, things look better if we parameterize in terms of h=r+q, k=r-q ; they then get a bit better if we replace k with a/h : x = -(h^3+3*h*a-6*a^2*h^2-2*a^3+h^3*a^3+3*a^4*h)/h/(a-1)/(-h^2+a)/(a^2+a+1), y = (3*h^2+a-2*h^3*a-6*h*a^2+3*h^2*a^3+a^4)/(a-1)/(-h^2+a)/(a^2+a+1), z = (h^2+a-2*h*a-2*h*a^2+a^2*h^2+a^3)/(h^2*a+a^2+h^2+a+2*h*a)/(a-1) I don't know how much more simply Surface #2 can be parameterized. Now, for each h and a we can use the parameterization to determine the values of x and z; if we use these in the equation describing the original surface, we then get a cubic polynomial in y. One of the roots of this polynomial is also given by the parameterization. But there are THREE roots to this cubic, and each of them corresponds to one of the critical points in your graph. So what we would REALLY like is a set of parameters h,a which makes _all three_ of these roots rational. We have a parameterization which gives one of these roots, y. We can divide out to get a quadratic to solve for the other two roots. These other two roots will be rational too, iff the discriminant of this quadratic is a rational square. Well, we can compute that discriminant; it is a product of some squares times [*] -(2*h*a^2-h^2*a-a^2-h^2-a+2*h)*(h^2+a-4*h*a^2+h^2*a^3+a^4)* (4*h^2+2*h^3+2*h*a+h^4+2*h^2*a+a^2-4*h^3*a-4*h*a^2-h^4*a-6*a^2*h^2-a^3 -4*h^3*a^2-4*h*a^3+a^2*h^4+2*h^2*a^3+a^4+2*h^3*a^3+2*a^4*h+4*a^4*h^2) (The long factor further factors in Q(1^(1/3)) but I don't see how that helps.) So we want a rational pair h,a that makes this become square. In principle this is ANOTHER algebraic-surface question. That is, the equation ([*])=(square) describes a surface (Surface #5) and we just need a rational point on it. But unlike the previous cases, this surface does not seem to be parameterizable by rational functions. It's a much more complicated surface! I don't know how to determine what rational points are on it. It's possible that the whole surface is parameterizable (I doubt that!), but that's not necessary for us. Plus, we have to be a little careful -- there are some cases that will easily show this expression to be square, such as if h=0 But when traced back through all the substitutions, we find in these easy cases that the quartic whose roots are the critical points of f will have roots where the poles are, or multiple roots, etc.; or else these values of h,a will lead to a division by zero in one of the many rational maps I have used (I have been very glib about the domains and codomains of all these mappings! Birational equivalence of varieties is weaker than algebraic bijection!) So we are left looking for rational points on an algebraic surface which do not lie in a collection of curves embedded on the surface. I don't know how to do this sort of thing in general. My only real trick is to look at _other_ curves embedded in the surface, and look to see where they have rational points. For example, we can hold the value of h to some constant; then [*] becomes a polynomial in one variable which we want to be a square. That describes an algebraic curve (actually the intersection of the surface and the plane h = constant), and since it has the form (polynomial)=(square), it's a hyperelliptic curve. These things are known to have only a finite number of rational points (Faltings' Theorem), except a few cases when the curve has genus 0 or 1 (here requiring h=0, h=1, or one of fourteen algebraic irrationals). So for any fixed value for h there's not much gold to be found -- and there's not even a very good way to do the prospecting. (The elliptic curve corresponding to h=1 has only the rational points with a=-1, -2, and -1/2, which are all bad.) But as it turns out the computer package Magma has algorithms which can look efficiently for rational points on a hyperelliptic curve (without a guarantee that it will find all points). So it's not too much trouble (see postscript) to try a few values of h and ask Magma for what rational points it can find on that curve. We then compare to the known bad values of a for this h, and if any rational points are left, we have a winner! So I tried h=2,3,5,6,9 and had no success. But Magma did find some unexpected rational when h=4 (a=20/17), h=7 (a=-1/2), h=10 (a= 34/7). The example I listed at the outset corresponds to h=7,a=-1/2, and the one corresponding to h=4,a=20/17 is -f(-x) for that same f. The rational point with h=10 leads to something really new, though: 15479572 (1237 x - 137840725) x + ----------------------------- (x - 112255) (x + 112255) There are other embedded curves one could investigate, of course. The curves a = constant look promising (for a different from 0, 1). For example, a=-1 gives an elliptic curve with, unfortunately, no rational points except h=+1, h=-1 (both bad). Other real values of a give a hyperelliptic curve. Some seem to have no useful rational points but e.g. a=5 gives points with h=-11/7 and h = -35/11, both of which give the function -f(-x) where f is the function I gave at the outset. (There are various symmetries here; for example, each f corresponds to any of six pairs (x,y) of the four critical points, each of which may be computed in terms of h,a, and the square root of [*]. We can get -f(-x) by replacing h and a by their reciprocals. If (h,a) makes [*] be square, so does (1/h, 1/a). Etc.) I checked all integral a between -100 and +100. For every a there are rational points which must be rejected at h = a^2, 1/a, -a-1, -a/(1+a), 0 . When a = -b^2 there are points at h = b*(b^3-1)/(b^3+1), -b*(b^3+1)/(b^3-1) which are likewise bad, and similarly when a=b^4, h=-b^2. Apart from these the rational points I found are a = 5, h = { -35/11, -11/7 } a = -2, h = { -14, 1/7 } a = -22, h = { -1001/724, -2574/3383, 1448/91, 3383/117 } a = -42, h = { -8883/2641, 5282/423 } a = -68, h = { -221/55, 220/13 } a = -70, h = { -4914/979, 4895/351 } a = -74, h = { -122/23, 851/61 } The points come in pairs which determine the same f. For a = -22 there are two apparently unrelated functions f. I haven't checked non-integral a at all. Possibly these points lie on an embedded rational (i.e. parameterizable) curve; if there's a pattern in this table, I'm not seeing it. Well, here's a tantalizing tidbit: the examples I gave have partial-fractions decompostions which involve nontrivial powers: they have the form (a1 x / a2^2) + a3^3 / (x - a0) + a4^3 / (x + a0) . dave PS -- the Magma code used was: P:=PolynomialRing(RationalField()); a:=-20; C := HyperellipticCurve(-(2*x*a^2-x^2*a-a^2-x^2-a+2*x)*(x^2+a-4*x*a^2+x^2*a^3 +a^4)*(4*x^2+2*x^3+2*x*a+x^4+2*x^2*a+a^2-4*x^3*a-4*x*a^2-x^4*a-6*a^2*x^2 -a^3-4*x^3*a^2-4*x*a^3+a^2*x^4+2*x^2*a^3+a^4+2*x^3*a^3+2*a^4*x+4*a^4*x^2)); Pt:=Points(C : Bound := 10000); H:={zz[1]/zz[3] : zz in Pt | zz[3] ne 0 } diff { a^2, 1/a, -1-a, -a/(1+a), 0}; a,H; === Subject: Re: Number Theory problem: seeking Calculus example [LONG] > But since my first post, I went one better: I can make all seven > interesting points rational! The better function is > > f = (x^3+157613963*x+332031930516)/(x-1771)/(x+1771) > i.e. > 438012 (367 x + 758043) > f := x + ----------------------- > (x - 1771) (x + 1771) > > with poles at +-1771, critical points at -10373, -3289, -989, and 14651, > and with an inflection point at -4389 . > > Let me describe how I found this example. This is too good for sci.math - write it up for publication. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: 0.99999 is not 1 > Briefly put, .999... = 1.000... because there are too many problems > otherwise, although it's not entirely inconsistent to hypothesize > that .999... != 1.000..., but one then has to work with expressions > such as the following. If we still assume that R is a field (and that .999... and 1.000... > are members thereof), then we get this strange little number > d = 1.000... - 0.999... . (If we extend the concept of field > we still have some odd problems with 'd'.) Briefly put, if d is this difference, then 10d = 10.000... - 9.999... > by multiplying by 10 -- but if we subtract 9 - 9 = 0 from the > right side we get the rather interesting equality 10d = 10.000... - 9.999... = 1.000... - 0.999... = d. That looks as silly as ordinal arithmetic! 0 + omega = omega 1 + omega = omega 2 + omega = omega 3 + omega = omega 4 + omega = omega > We can also compute d/10 = 0.1000... - 0.0999...; adding 0.9 - 0.9 > to the right side, we get a similarly strange equality d/10 = 0.1000... - 0.0999... = 1.0000... - 0.9999... = d. There are three ways to attempt a path around this. [1] d = 0. This gives us standard mathematics. [2] Define arithmetic differently for infinitesimals, so that > 10d = d/10 = d. I'll admit at this point I've no idea > what the details look like, since d - d could be just > about anything. One might end up with something akin > to little-o notation, but I for one can't be sure. [3] Assume 1 - 10d = 0.999...9990. This quickly gets ridiculous, > especially if one computes things like > (1-d)^2 = 0.999...99800...001 or > (1-d)^2 - (1-d)^3 > = 0.999...99800...001 - 0.999...99700...00299...999 = ??!? Which one(s) change ordinal arithmetic to something cleaner? Why do you suppose mathematics went the way of accepting the silly ordinal arithmetic and rejecting silly infinitesmals? It would appear with a cursory look that some of your possible theories would reject transfinite ordinals as nonsensical but introduce unacceptable infinitesimals. Soon there would be a huge anti-Ghost crowd that would be called crackpots on the usenet groups. I'll bet there are people who would give up their right arm to see that! === Subject: Re: 0.99999 is not 1 [...] > > a parody. Parody? It's a parody of something? (Quickly Google > sci.logic) Damn, I wasted my time! > > > You don't know how good that makes me feel! > > :-) > > P.S. any chance you could still post your critique? It went like this: [begin] > objections to the notion that 0.999... is equal to unity, > which I plan to contribute to the Wikipedia. I would be interested in Here's some feedback. I'll leave the determination of whether it's intelligent to you. Overall, it doesn't tell me what the main objections to 0.99999... = 1 are. It also doesn't provide either a justification (accepted as true by the other) for accepting it. What it presents is either an argument from authority or from popularity: pure mathematicians almost unanimously accept.... This is puzzling because it would have been easy to present at least a sketch of a proof that 0.99999... = 1. > *** > > While the pure mathematicians almost unanimously accept > 0.9999.... is one, there are lots of intelligent people who > believe it to be an absurdity. Typically, these people > are non-experts in pure mathematics, but they are people > who have who have found mathematics to be of great practical > value in science and technology, and who like to view > mathematics itself as a science. These anti-recurrists see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. They claim that that paradigm includes all > of the mathematics which has the potential to be applied to > the task of understanding phenomena in the real world (e.g. > in science and engineering). The relation between this philosophy and the rejection of .99.. = 1 is not made clear. The nature of anti-recurrism isn't made clear either, references to computation notwithstanding. The name suggests that it rejects some aspect of repeated decimals or of some larger class that subsumes them. Is this appearance accurate or not? > In the contemporary mainstream mathematical literature, there > is almost no debate over the validity of 0.99999 = 1. > It was the advent of the internet which revealed just how > prevalent the anti-recurrist view still is; there seems to be a > never-ending heated debate about this notion in the Usenet > newsgroups sci.math and sci.logic. Typically, the > anti-recurrists accuse the pure mathematicians of living in a > dream world, and the mathematicians respond by accusing the > anti-recurrists of being imbeciles, idiots and crackpots. This seems to group better with the first paragraph. The second and fourth also seem to group better. > > It is plausible that in the future, mathematics will be split > into two disciplines - scientific mathematics (i.e. the science > of phenomena observable in the world of computation), and > philosophical mathematics, whereas the notion that 0.99999 = 1 is > merely one of the many possible theories. This paragraph suggests one thing but actually says something else. At first glance it suggests that mathematics will be split according But then it's suddenly talking about something only very loosely related, that mathematics would somehow split into scientific mathematics and philosophical mathematics. It is not made clear why this division of mathematics is considered plausible. It is not even made clear what philosophical mathematics is. It sounds like philosophers opining about maths; if so, it's not at all clear why this should have co-equal standing with maths itself. [end] And bear in mind this was before I saw what it was a parody of. -- Tom Breton, the calm-eyed visionary === Subject: Re: 0.99999 is not 1 > > > > [...] > >a parody. Parody? It's a parody of something? (Quickly Google >sci.logic) Damn, I wasted my time! > >> >>You don't know how good that makes me feel! >> >>:-) >> >>P.S. any chance you could still post your critique? > > > It went like this: > Of course, all I did was take petry's posting, and change a few words to which your critique pointed out so beautifully. I have to admit that I completely made up the phrase anti-recurrist - I wasn't even sure that I spelled it correctly. I certainly felt it was an ugly word, but since I was simply dashing off a parady, I didn't spend to much time on it. Let me also say that your critique was very kindly worded, and of course === Subject: Re: Kaprekar's constant >My question is, do there exist, i &j, such that K(i,j) has a single >cycle that runs through all the j-digit numbers in base i. No. Since f(x) = f(y) if x and y are rearrangements of the same digits, > this can't happen. Also, there are always numbers x such that f(x) = 0, namely those > where all the digits are the same. > Hi Professor Gsax === Subject: tautochrone=brachistochrone Hi All As we all know that the cycloid is the solution for both Brachistochrone as well as the Tautochrone problem for uniform gravity.. Do there exist non-uniform gravitational fields, in which the tautochrone & brachistochrone are the same curve. My guess is No. i.e. no matter how you define the non-uniform gravitational field to be varying, you will never find the same curve as solution to both the problems.. if anyone knows a counter-example, I would appreciate it. Gsax === Subject: Re: conditional probability On 21 Jul 2005 13:26:09 -0700, Butch Malahide >> If the only information I know is that P(A), P(B), P(C), P(A and B), P(A and >> C) and P(B and C), (A, B and C are not independent) is there a way to >> compute P(A and B and C? > >P(A and B and C) = P(A and B) + P(A and C) + P(B and C) > - P(A) - P(B) - P(C) + P(A or B or C). > >So you can compute P(A and B and C) if you also know P(A or B or C); >otherwise you can get upper and lower bounds on P(A and B and C) from >0 <= P(A or B or C) <= 1. You can do better on the lower bound: P(A or B) <= P(A or B or C) P(B or C) <= P(A or B or C) P(A or C) <= P(A or B or C) The terms on left can all be computed based on the given info, hence: max( P(A or B), P(B or C), P(A or C) ) <= P(A or B or C) <= 1 quasi === Subject: Re: A Correction in Set Theory >Quote from some other post/reply to TO: > >> >> that you are mixing up the order of quantifiers: >> >> (1) forall b, exists s in S, >> (if b is a finite bound, then length(s) > b) >> >> (2) exists s in S, forall b >> (if b is a finite bound, then length(s) > b) >> >> Statement (1) says that the *set* S has no finite bound. >> Statement (2) says that S contains an *element* that has >> no finite bound. Those are two different statements. I don't get it. I have no idea what you guys are talking about and I'm a bit slow so - can someone run above quote sloooowly by me? Darko >> Daryl McCullough >> Ithaca, NY >> > > >F. === Subject: about_sup Suppose we have a set of elements in an ordered group a(i) ,indexed by a totally ordered set. The set has the property that a(i)<=a(j) whenever i I am reading John Rice's book on statistics: chapter 9 Hypothesis testing. He mentioned about p-value here and there but I really don't know a. What is p-value? This is the probability that a test statistic is at least as extreme as the value you got from your data, assuming you have the proper distribution model. > b. How to compute p-value? This depends on the distribution, but usually it is done through numeric integration or by table lookup (which is basically using other's numeric integrations) Some models (such as a discrete distribution) can be done explicitly, rather than by approximation > c. What are the significance of p-value? What are the uses of p-value? If the p value is small, it indicates that either you got a rare event, or at least one of your assumptions is incorrect (the distribution is a different model, or your null-hypothesis is wrong. === Subject: Re: what is the use of the p-value in hypothesis testing... > I am reading John Rice's book on statistics: chapter 9 Hypothesis testing. > > He mentioned about p-value here and there but I really don't know > > a. What is p-value? > b. How to compute p-value? > c. What are the significance of p-value? What are the uses of p-value? > > Can anybody enlighten me? > > > > http://www.tufts.edu/~gdallal/pval.htm === Subject: Help! Difficulty in understanding order-statistics vs. quantile plot... I am trying to understand how to skectch the empirical distribution quantile vs. theoratical distribution quantile ... In the following webpage, the theoratic distribution is normal distribution. The data ordered statistics is double exponential, which was plot against it. http://www.itl.nist.gov/div898/handbook/eda/section3/normprp3.htm I don't understand this sentence: The double exponential distribution is symmetric, but relative to the normal it declines rapidly and has longer tails. I think it is of the order exp(-x), compared to exp(-x^2), relative to the normal it should decline slowly and has fatter tails... am I right? I don't understand why the plot is S-like... I also don't understand the sentences For long tails, the first few points show increasing departure from the fitted line below the line and last few points show increasing departure from the fitted line above the line. -----------------------Quotes from the webpage------------------ First, the middle of the data may show an S-like pattern. This is common for both short and long tails. In this particular case, the S pattern in the middle is fairly mild. Second, the first few and the last few points show marked departure from the reference fitted line. In the plot above, this is most noticeable for the first few data points. In comparing this plot to the short-tail example in the previous section, the important difference is the direction of the departure from the fitted line for the first few and the last few points. For long tails, the first few points show increasing departure from the fitted line below the line and last few points show increasing departure from the fitted line above the line. For short tails, this pattern is reversed. === Subject: Re: Euclidean Postulates II > What postulate is more general? > a.- For a point external to a straight line can pass a straight line > and only one straight line. I don't understand a. > Or > b.- A triangle can be translated and/or rotated without changing its > sides or its angles. My hunch is that they're independent of each other. === Subject: Re: Euclidean Postulates II Yes. I forget something. The postulate (Playfair form) is: a.- In a plane, for a point external to a straight line pass a parallel to it and only one. I want to know if that form of Euclide's fifth postulate is less powerful than (Thibaut form): b.- A triangle can be translated and/or rotated in the plane without changing its sides or its angles. Powerful in the sense that it can replace more than one postulate of the Hilbert's system of euclidean geometry. (Naturally, supposing that this system is complete) === Subject: Re: A missing factor to solve a functional equation . > Here is the idea , let us look at -1) (1-1/x)*f(x) = f(3*x)*(1-3/x) , > f real, strictly monotonous , x > 0 > pairing factors from both sides this way > f(3*x) = 1-1/x > f(x) = 1-3/x we've got compatible equations > so we obtain a solution f(x)=1-3/x to generalize ... -2) (1-4*x^2)*f(x) = 2*f(x)*(1-x^2) Same domain and range as before? If f(x) /= 0, then 1 - 4x^2 = 2(1 - x^2) 0 = 1 + 2x^2 Thus for all x, f(x) = 0 === Subject: Re: Movement of a circle in contact with a rotating ellipse. Yes, that is correct. The center of the ellipse will be moving as well. And, yes, it moves in order to maintain tangency with the two circles. The lower circle then moves to maintain contact with the ellipse, but this can only move along a vertical line. Lee === Subject: Re: Movement of a circle in contact with a rotating ellipse. >Yes, that is correct. The center of the ellipse will be moving as >well. And, yes, it moves in order to maintain tangency with the two >circles. The lower circle then moves to maintain contact with the >ellipse, but this can only move along a vertical line. > >Lee OK, let's say we have the dimensions of the wheels, the location of the upper wheels and the constraint on location of the lower wheel that it only moves vertically. So you want to know the position of the center of the lower wheel in terms of what? If your answer is time then you need to specify how the ellipse moves with respect to time. Or maybe in terms of the angle of rotation of the ellipse? Or what variable? --Lynn === Subject: Re: Movement of a circle in contact with a rotating ellipse. >>Yes, that is correct. The center of the ellipse will be moving as >>well. And, yes, it moves in order to maintain tangency with the two >>circles. The lower circle then moves to maintain contact with the >>ellipse, but this can only move along a vertical line. >OK, let's say we have the dimensions of the wheels, the location of >the upper wheels and the constraint on location of the lower wheel >that it only moves vertically. So you want to know the position of the >center of the lower wheel in terms of what? If your answer is time >then you need to specify how the ellipse moves with respect to time. >Or maybe in terms of the angle of rotation of the ellipse? Or what >variable? To start you off: write the equation of the ellipse as A(t) (x-x_0)^2 + 2 B(t) (x-x_0)(y-y_0) + C(t) (y-y_0)^2 = 1 where (x_0, y_0) are the coordinates of the centre of the ellipse, A(t) = cos(t)^2/a^2 + sin(t)^2/b^2, B(t) = sin(t) cos(t)(1/a^2 - 1/b^2), C(t) = sin(t)^2/a^2 + cos(t)^2/b^2 where t is the rotation angle and a,b are the semi-major and semi-minor axes of the ellipse. The equation of the first circle is (x-x_1)^2+(y-y_1)^2=r_1^2 where (x_1,y_1) are the coordinates of the centre and r_1 is the radius of that circle. The resultant of the two equations with respect to x is a quartic polynomial in y (an ellipse and a circle generally intersect in up to four points). For the ellipse and circle to be tangent, the discriminant of the quartic must be 0 (this is necessary but not sufficient). That discriminant will be a rather complicated polynomial in x0, y_0, and the parameters. The same polynomial with (x_1,y_1,r_1) replaced by (x_2,y_2,r_2) gives a condition for the ellipse to be tangent to the second circle. Solve the system of the two discriminants = 0 to get x_0 and y_0. It won't be pretty... Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Questions about co-np! F is a set of combinations of vertices in G. Each element of F should be a vertex cover of G which cannot be reduced. Which means if a vertex is removed from this element will makes it no longer a vertex cover of G. So on input , we need to verifer if F is a set of all minimal vertex cover of G. So I have problems to verify if F contains all such vertex cover in polynomial time regarding to ||. > I have a question, I have a undirected graph G and a set F of > combinations of vertices in G. > A minial vertex cover of G is a vertex cover whose size is not > reducable which means removal of any vertex from the vertex cover > will make it no longer a vertex cover. It's different with smallest > vertex cover. > eg, F={{v1,v2,v3},{v3,v4,v5}} > > Are you talking about removing sets (in F) or elements of the sets in > F? > > Is the size of the cover the number of sets in it? > > Now I have a language L={|F is set of all minimal vertex covers > of G} > Assume, I know {|G has a vertex cover of size k} is in NP. > How can I prove it's in coNP? > Maybe I can prove the complement of L is in NP, then L is in coNP. > But I have no idea how to do it in poly-time. Can anyone give me some idea on it? > > What does the statement F is not a vertex cover mean? Can you use the > information which makes F not a vertex cover, and verify it in > polynomial time? > > === Subject: Re: Questions about co-np! > F is a set of combinations of vertices in G. Combinations of vertices are better known as sets. Your example didn't give what G is; I'm assuming it's {v1,v2,v3,v4,v5}. > Each element of F should be a vertex cover of G which cannot be > reduced. So you're saying that {{v1,v2,v3}, {v3,v4,v5}} is minimal, because if you take out {v1,v2,v3} or {v3,v4,v5}, you don't have a vertex cover of G. I got confused with taking vertices out of the combinations of vertices, because in that case, F isn't minimial; you can remove v3 from either combination of vertices. > Which means if a vertex is removed from this element will > makes it no longer a vertex cover of G. So on input , we need to verifer if F is a set of all minimal > vertex cover of G. So, F is really a set, all of whose elements are vertex covers? In this case, your example is really F = {{{v1,v2,v3}, {v3,v4,v5}}} (a set of vertex covers of G), and you should say that F is THE set ..., not F is A set ..., since the set F would be unique. > So I have problems to verify if F contains all such > vertex cover in polynomial time regarding to ||. (1) To show that this problem is in co-NP (I'm not so sure this isn't homework, so I'll outline it), you need to show that you can give a short proof of F NOT being the set of all minimal vertex covers of G, provided F is not the set of all minimal vertex covers of G. (2) By definition, this means that you need to generate some minimal vertex cover which is not in F, and prove that you can check to see whether this alleged minimal vertex cover is really a minimal vertex cover in polynomial-time. (3) Note that if you can prove that an arbitrary alleged vertex cover C is a real vertex cover in polynomial time, then you can prove it's a _minimal_ vertex cover. (Hint: Check to see whether C S_i is a vertex cover, where C = {S_1,S_2,...,S_k}. What answers would you get for a real minimal vertex cover?) You're welcome. > I have a question, I have a undirected graph G and a set F of > combinations of vertices in G. > A minial vertex cover of G is a vertex cover whose size is not > reducable which means removal of any vertex from the vertex cover > will make it no longer a vertex cover. It's different with smallest > vertex cover. > eg, F={{v1,v2,v3},{v3,v4,v5}} Are you talking about removing sets (in F) or elements of the sets in > F? Is the size of the cover the number of sets in it? Now I have a language L={|F is set of all minimal vertex covers > of G} > Assume, I know {|G has a vertex cover of size k} is in NP. > How can I prove it's in coNP? > Maybe I can prove the complement of L is in NP, then L is in coNP. > But I have no idea how to do it in poly-time. Can anyone give me some idea on it? What does the statement F is not a vertex cover mean? Can you use the > information which makes F not a vertex cover, and verify it in > polynomial time? > > === Subject: Difficult Integral - who can help? Hello together, during my calculations for a physical problem I derived the following integral that must be solved: Integral[z]:= int_{-infinity}^{z} dz' Exp[-i pi (z'^2 - i alpha z')] * int_{-infinity}^{z'} dz'' Exp[+i pi (z''^2 - i alpha z'')] As you see its a double-integral. Please note: z is real i = sqrt(-1) alpha is real and also alpha > 0 I solved the integral for z=infinity. So I have Integral[infinity] which gives 1/2. But I also need Integral[z]. I asked lots of people and tried some tricks, but nothing worked for this integral. Perhaps you can help. Bye, Mark === Subject: Re: Difficult Integral - who can help? >Hello together, > >during my calculations for a physical problem I derived the following >integral that must be solved: > >Integral[z]:= > >int_{-infinity}^{z} dz' Exp[-i pi (z'^2 - i alpha z')] * >int_{-infinity}^{z'} dz'' Exp[+i pi (z''^2 - i alpha z'')] > >As you see its a double-integral. (That's the warning for those who think they've already drunk too much..) > >Please note: > >z is real >i = sqrt(-1) >alpha is real and also alpha > 0 > >I solved the integral for z=infinity. So I have Integral[infinity] >which gives 1/2. > >But I also need Integral[z]. I asked lots of people and tried some >tricks, but nothing worked for this integral. >.. At least one of the integrals ought be traceable with the substitution z'+z'' = t z'-z'' = s The Jacobian is 1/2 [I guess, since z'=(t+s)/2 and z''=(t-s)/2 ]. -> (1/4) Int(t=-infty..2z) Int(s=0..?) exp[i*pi*t*s + pi*alpha*s] where ? is some upper limit which is a linear function of t that I am too lazy to figure out (2z-t, perhaps). Since the original (z',z'') area of integration was some sort of infinite triangle with the upper right corner at (z'=z''=z), one must write down how the new variables (t,s) (diagonals and antidiagonals) are limited by the old main diagonal and the old upper ceiling at z'=z. So to solve for the inner integral over s, one only needs the antiderivative of exp(const*s) ds. The key idea is to observe that z''^2-z'^2=(z''+z)(z''-z) and also the term with the alpha's is in z'-z'' . === Subject: Re: Transcendental Dimensions > > >> >> Dimensions of fractals often have the form (log p / log q) for some >> integers p and q. If the ratio isn't obviously a rational then it's >> irrational and hence transcendental. If log p / log q is a rational >> then p^n = q^m with m > 1, which tends to be pretty obvious. That's a >> fairly uncommon relationship, requiring that p and q be different >> powers of a common base. >> >> Irrational is not the same as transcendental. >> (A number can be irrational without being transcendental.) > > Yes -- but this is not a general case. Look up Gelfond-Schneider > Theorem. I don't need to look up anything. The poster appeared to claim that log(p)/log(q) is transcendental because it is irrational. That is not a valid argument. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Transcendental Dimensions > I don't need to look up anything. Of course you don't. It's just that if you don't, you risk looking like a brash fool. Your choice. If you had followed the thread, you would have seen my previous post quoting the Gelfond-Schneider theorem: a^b is transcendental if a is algebraic and not 0 or 1, and b is an algebraic irrational. If log(p)/log(q) were irrational and algebraic, then q^(log(p)/log(q)) would be transcendental. But q^(log(p)/log(q)) = p, which is algebraic. Therefore log(p)/log(q) cannot be irrational and algebraic. Hence the claim to which you responded. If a number of that particular form is irrational, then it is indeed transcendental. - Tim === Subject: Re: Transcendental Dimensions > If a number of that particular form is irrational, then it is indeed > transcendental. Please excuse the parts of that post preceding that sentence. They were needlessly uncivil in tone. My sincere apologies. It should have sufficed for me to simply clarify that I was talking only of numbers of the form log(p)/log(q) for integer p and q. The quoted implication for numbers of that type is a consequence of the Gelfond-Schneider theorem. It certainly isn't true for all numbers. - Tim === Subject: Re: Transcendental Dimensions <3k9dkvFtc0egU3@individual.net> Hi Jose > It's very easy to create transcendental numbers. What is hard is to > prove that a specific number is transcendental. I think you mean using the Gelfond theorem? Yet it is quite surprising that it took mathematicians so much effort to come up with theorems that could be used to produce examples of Transcendental numbers, since the first one produced by Liouville Gsax === Subject: Re: Transcendental Dimensions > Yet it is quite surprising that it took mathematicians so much effort > to come up with theorems that could be used to produce examples of > Transcendental numbers, since the first one produced by Liouville It's fairly easy to prove that e is transcendental, I think. It might even have been proved before Liouville's theorem was published? -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Transcendental Dimensions >> Yet it is quite surprising that it took mathematicians so much effort >> to come up with theorems that could be used to produce examples of >> Transcendental numbers, since the first one produced by Liouville >It's fairly easy to prove that e is transcendental, I think. >It might even have been proved before Liouville's theorem was published? No. Hermite proved e is transcendental in 1873. Liouville's results on transcendental numbers were published in 1844 and 1851. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Spherical geometry Proofs/derivations Trying to plod through Gravity by Jim Hartle and I'm stuck (for a week) on 2 proofs. No. 1 The sum of the interior angles, S of a spherical is S = pi + A/a^2 where A is the area of the triangle and a is the radius of the underlying sphere. No.2 Derive a formulae for the area of a triangle in terms of r, the (spherical distance from its centre to its circumference) and a (as above) Starters, pointers would be most welcome. Zinc zincnews123@tiscali.c123o.u123k To reply to address don't click. Cut and paste, then delete all 123's ------------------------------------ === Subject: Re: why did the Apollo missions all land in the same area? > >> Hey, Mark did not know the moon is yellow color and so smooth up >> close. Mark saw a moon rock in a museum and it was not yellow. > >That's to be expected if you store cheese for too long outside of a >vacuum. > I usually store dust in mine. Has anybody noticed that dd defines the same area of the moon to be a hemisphere? This kid is definitely geometry-challenged. I would recommend that he spend the next week crawling around on his hands and knees to get from here to there. It may help rearrange some brain circuitry. /BAH Subtract a hundred and four for e-mail. === Subject: Re: why did the Apollo missions all land in the same area? !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Has anybody noticed that dd defines the same area of the > moon to be a hemisphere? This kid is definitely geometry-challenged. I would recommend > that he spend the next week crawling around on his hands > and knees to get from here to there. I had the impression that he already was doing that. > It may help rearrange some brain circuitry. One thing after the other. Blind activism does not achieve anything. -- Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: why did the Apollo missions all land in the same area? > >> Has anybody noticed that dd defines the same area of the >> moon to be a hemisphere? >> >> This kid is definitely geometry-challenged. I would recommend >> that he spend the next week crawling around on his hands >> and knees to get from here to there. > >I had the impression that he already was doing that. I doubt it. He has shown to not have rudimentary knowledge about geometry. Crawling is supposed to help with 3-D perspective. It seemed to connect circuits between the brain hemispheres but this is a very subjective observation. > >> It may help rearrange some brain circuitry. > >One thing after the other. Blind activism does not achieve anything. In the USA, there is a concerted effort to admire those who dignigrate math. dd is an example of the parroting. /BAH Subtract a hundred and four for e-mail. === Subject: Re: why did the Apollo missions all land in the same area? <85u0ioggoy.fsf@lola.goethe.zzIs it possible for a vacuum to contain cheese? The real question is, can cheese contain a vacuum? -Mark Martin === Subject: Re: why did the Apollo missions all land in the same area? > today, he saw a map > of where all the Apollo misisons landed. Why did all the missions seem > to land in the same > general patch of moon? To give conspiracy buffs something new to debate about. === Subject: Upperbounds on the actualdensity of odd perfect numbers over the integers I am a masteral student of mathematics currently undergoing research on the celebrated Odd Perfect Number Conjecture. Has there been any mathematical results relating to upper bounds for the actual density of odd perfect numbers over the integers? Any information along these lines would be most helpful. === Subject: Re: Upperbounds on the actualdensity of odd perfect numbers over the integers > I am a masteral student of mathematics currently undergoing research > on the celebrated Odd Perfect Number Conjecture. Has there been any mathematical results relating to upper bounds for > the actual density of odd perfect numbers over the integers? Any > information along these lines would be most helpful. Yes. I settled the Odd Perfect Number Conjecture last week. You'll have to choose another thesis topic. Not really. Since you're writing the thesis, you should do some of the research for yourself and not just ask whether there are some results. You can start by searching with MathSciNet, an index of mathematical papers since about, oh, 1947. http://www.ams.org/mathscinet/search/ === Subject: Re: Upperbounds on the actualdensity of odd perfect numbers over the integers === Subject: Re: Upperbounds on the actualdensity of odd perfect numbers over the integers We know that the sum of their reciprocals converges. === Subject: Re: Upperbounds on the actualdensity of odd perfect numbers over the integers === Subject: Product of fields in (Von Neumann) Regular rings. Given a von Neumann regular commutative ring R with unity (vNr are just rings such that for all x, xyx=x for some y.), there is a characterization that says R_M (R localized with maximal ideal M) is a field for all maximal ideal of R. Thus by knowing this, I think we can produce a canonical (injective) map from R to products of R_M (M over all maximal ideals), by taking the product of the localizations. Question is, is there and example of one vNr ring such that this map is not surjective? In case of finite field products, this map is bijective.. but I cant think of any other example. I can think of arbitrary product of fields, but I dont know how to handle them. === Subject: f(n,x,y) = (x^n - y^n) / (x - y) : can you divide it? Let n be any positive integer. Let x, y be 2 real numbers such that x is not equal to y. Let f(n,x,y) = (x^n - y^n) / (x-y) . Can you come up with a factorized polynomial expression for f(n,x,y) by dividing through? Special cases: f(0,x,y) = 0 f(1,x,y) = 1 f(2,x,y) = x+y f(3,x,y) = x^2 + xy + y^2 f(4,x,y) = (x+y) (x^2 + y^2) ... What is the general expression for f(n,x,y)? K. Onyee === Subject: Re: f(n,x,y) = (x^n - y^n) / (x - y) : can you divide it? >Let n be any positive integer. >Let x, y be 2 real numbers such that x is not equal to y. >Let > f(n,x,y) = (x^n - y^n) / (x-y) . > >Can you come up with a factorized polynomial expression for f(n,x,y) by >dividing through? > >Special cases: >f(0,x,y) = 0 >f(1,x,y) = 1 >f(2,x,y) = x+y >f(3,x,y) = x^2 + xy + y^2 >f(4,x,y) = (x+y) (x^2 + y^2) >... >What is the general expression for f(n,x,y)? > The general (unfactorized) expression is sum{x^(n-1-k)*y^k} with the summation being on k, from 0 to n-1. Factorization is left as an exercise for the reader. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: f(n,x,y) = (x^n - y^n) / (x - y) : can you divide it? You seem to be looking at the divided difference for f(z):=z^n. (f(x)-f(y)/(x-y). You can see discussion of how to express this, and how to compute this rapidly, not necessarily by factoring, in this paper http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf For example, 6 multiplies, 3 adds suffices to compute.. (x^5 - y^5)/(x - y) = x^4 + x^3 y + x^2 y^2 + xy^3 + y^4 = x .87 (x + y) .87 (x^2 + y^2 ) + (y^2 )^2. You might restrict your future questions to fewer newsgroups. RJF > >>Let n be any positive integer. >>Let x, y be 2 real numbers such that x is not equal to y. >>Let >> f(n,x,y) = (x^n - y^n) / (x-y) . >> >>Can you come up with a factorized polynomial expression for f(n,x,y) by >>dividing through? >> === Subject: Re: f(n,x,y) = (x^n - y^n) / (x - y) : can you divide it? > Let n be any positive integer. > Let x, y be 2 real numbers such that x is not equal to y. > Let > f(n,x,y) = (x^n - y^n) / (x-y) . > > Can you come up with a factorized polynomial expression for f(n,x,y) by > dividing through? > > Special cases: > f(0,x,y) = 0 > f(1,x,y) = 1 > f(2,x,y) = x+y > f(3,x,y) = x^2 + xy + y^2 > f(4,x,y) = (x+y) (x^2 + y^2) > ... > What is the general expression for f(n,x,y)? > > K. Onyee > Well, f(n,x,1) is known as the cyclotomic polynomial, and given f(n,x,1) you can easily write f(n,x,y). To completely factor f(n,x,1) with complex coefficients, use the linear factors: f(n,x,1) = product((x - exp(2 Pi i k / n)), k=1..n-1) For example: f(4,x,1) = (x+1)(x+i)(x-i), and thus f(4,x,y) = (x+y)(x+iy)(x-iy) -- http://www.math.ohio-state.edu/~edgar/ === Subject: Re: f(n,x,y) = (x^n - y^n) / (x - y) : can you divide it? >> Let n be any positive integer. >> Let x, y be 2 real numbers such that x is not equal to y. >> Let >> f(n,x,y) = (x^n - y^n) / (x-y) . >> Can you come up with a factorized polynomial expression for f(n,x,y) by >> dividing through? >> Special cases: >> f(0,x,y) = 0 >> f(1,x,y) = 1 >> f(2,x,y) = x+y >> f(3,x,y) = x^2 + xy + y^2 >> f(4,x,y) = (x+y) (x^2 + y^2) >> ... >> What is the general expression for f(n,x,y)? >> >> K. Onyee >Well, f(n,x,1) is known as the cyclotomic polynomial, and No. The nth cyclotomic polynomial is the monic polynomial whose roots are the _primitive_ nth roots of unity. It's only f(n,x,1) if n is prime. In general, f(n,x,1) is the product of the kth cyclotomic polynomial of x for all divisors k of n except 1. That's the factorization into polynomials irreducible over the rationals. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: f(n,x,y) = (x^n - y^n) / (x - y) : can you divide it? >Let n be any positive integer. >Let x, y be 2 real numbers such that x is not equal to y. >Let >f(n,x,y) = (x^n - y^n) / (x-y) . > >Can you come up with a factorized polynomial expression >for f(n,x,y) by >dividing through? > >Special cases: >f(0,x,y) = 0 >f(1,x,y) = 1 >f(2,x,y) = x+y >f(3,x,y) = x^2 + xy + y^2 >f(4,x,y) = (x+y) (x^2 + y^2) >... >What is the general expression for f(n,x,y)? > >K. Onyee polynomial division gives (x^n - y^n)/(x - y) = sum_i=0^(n-1) x^i * y^(n-1-i) Best wishes Torsten. === Subject: Re: f(n,x,y) = (x^n - y^n) / (x - y) : can you divide it? Let n be any positive integer. > Let x, y be 2 real numbers such that x is not equal to y. > Let > f(n,x,y) = (x^n - y^n) / (x-y) . Can you come up with a factorized polynomial expression for f(n,x,y) by > dividing through? Special cases: > f(0,x,y) = 0 > f(1,x,y) = 1 > f(2,x,y) = x+y > f(3,x,y) = x^2 + xy + y^2 > f(4,x,y) = (x+y) (x^2 + y^2) > ... > What is the general expression for f(n,x,y)? x^(n-1) + x^(n-2).y + x^(n-3).y^2 + ... + x.y^(n-2) + y^(n-1) === Subject: Re: Fixed-point algorithm for exponential function fitting for an embedded application > samples of CORDIC algorithm for most common functions such sin, cos, > atan2, ln...? I have only one link to C source handy, http://people.csail.mit.edu/hqm/imode/fplib/cordic_code.html The other links I've got: http://my.execpc.com/~geezer/embed/cordic.htm http://www.dspguru.com/info/faqs/cordic2.htm http://www.opencores.org/projects.cgi/web/cordic/overview http://archive.chipcenter.com/circuitcellar/august00/c0800rp48.htm Googling for 'cordic c source' can bring more. Hope this will help. Vadim === Subject: Quaternionic Derivative In fixed point iteration, one uses the relation |f'(z_0)|<1, =1, > 1 to conclude that the fixed point z_0 is attracting, resp. indifferent, resp. repuslive. Does anyone know which version of the Quaternionic Derivative one should use when one is tries to determine the corresponding situation for a quaternion fixed point z_0 if f is a quaternion function? -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: Quaternionic Derivative > In fixed point iteration, one uses the relation |f'(z_0)|<1, =1, > 1 to > conclude that the fixed point z_0 is attracting, resp. indifferent, resp. > repuslive. > Does anyone know which version of the Quaternionic Derivative one should use > when one is tries to determine the corresponding situation for a quaternion > fixed point z_0 if f is a quaternion function? > Consider a complex transformation as a map from R^2 to R^2. Its derivative is described by a 2x2 matrix. The condition for contractivity is then that both eigenvalues have modulus < 1. For a holomorphic function, the two eigenvalues have the same moduulus, which is the modulus of the complex derivative. Similarly, consider a quaternionic transformation as a map from R^4 to R^4, with derivative given by a 4x4 matrix. The condition for contractivity is then that all four eivenvalues have modulus < 1. -- http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Quaternionic Derivative - There are right and left quaternionic derivatives (q-derivatives) of q-differentiable functions. Let f be a q-differentiable function and let at a certain point q0 N = f(q) - f(q0) and D = q - q0 be the numerator and the denominator of a would-be difference quotient. In quaternions one has two different difference quotients QL = N.D^(-1) and QR = D^(-1).N; they are conjugate by QR = D^(-1) . QL . D and therefore have the same eigenvalues when considered as operators on R4. This situation continues to exist in the limit when |q - q0| approaches zero. Therefore it does not matter with which quaternionic derivative the iterations are performed. Johan E. Mebius > >>In fixed point iteration, one uses the relation |f'(z_0)|<1, =1, > 1 to >>conclude that the fixed point z_0 is attracting, resp. indifferent, resp. >>repuslive. >>Does anyone know which version of the Quaternionic Derivative one should use >>when one is tries to determine the corresponding situation for a quaternion >>fixed point z_0 if f is a quaternion function? >> >> >> > >Consider a complex transformation as a map from R^2 to R^2. Its >derivative is described by a 2x2 matrix. >The condition for contractivity is then that both eigenvalues have >modulus < 1. For a holomorphic function, the two eigenvalues have >the same moduulus, which is the modulus of the complex derivative. > >Similarly, consider a quaternionic transformation as a map >from R^4 to R^4, with derivative given by a 4x4 matrix. >The condition for contractivity is then that all four eivenvalues >have modulus < 1. > === Subject: Re: Relative Cardinality Nntp-Posting-Host: hera.cwi.nl > ... > Sorry, you said comparability by magnitude is necessary and > sufficient. I claim that if that number exists, I can compare > sqrt(2) with it. Rather, I claim that I can compare sqrt(2) with > all existing numbers (using existing in your sense). > > Well, but that probably would require that all existing numbers can be > squared. But if all atoms in the universe are already taken up by the > number, there is no room for the square (besides, we know that perfect > squares don't exist in the real world, just like perfect circles and > triangles). So you have to throw away the original number when > squaring, and afterwards it does not exist, so claiming that it is > larger or smaller than sqrt(2) would be silly. Yes, I was also thinking along such lines. But also that if the number exists that takes up all atoms in the universe, that is the only number that exists. What happens when we create that number? At what point are we throwing 1 out of existence? > Something like that. Do I get the fool's cap now for a moment, or is > it welded to WM's head? Welded well and good. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Relative Cardinality > > > A fundamental set contains at least such a set as given by Cantor in > > order to explain this point. A fundamental set for the natural number n > > is a set which contains at least one set which contains exactly n > > elements. > > Ok. Take the set {1,2,3,4,5,...,n}, than {{1,2,3,4,5,...,n}} is a > fundamental set for n? In that case n exists. (And this is irrespective > of n.) > > Three points are the beginning of all abuse of mathematics. No, n may > mean a number but it is not a number. Oh. > > This is the umpteenth definition of exist I see. Earlier you gave > > as a definition that it should be comparable... > > > > Comparability by magitude is the necessary and sufficient condition for > > a real number to exist. It is impossible to satisfy however, if there > > is not either a fundamental set or an n-adic representation. > > Can you prove that? > > I take it for granted as long as nobody is able to compare numbers in > another way. Yes, that is the spirit. Something is necessary and sufficient because you take it for granted. There are other ways to compare apart from comparing digit by digit. > > It can be compared with other existing numbers as well. But it cannot > > be compared with sqrt(2) because that one is not existing yet if only > > its first digits exist. > > Sorry, you said comparability by magnitude is necessary and sufficient. > I claim that if that number exists, I can compare sqrt(2) with it. > Rather, I claim that I can compare sqrt(2) with all existing numbers > (using existing in your sense). > > You cannot compare sqrt(2) with all existing numbers (in my sense) > because sqrt(2) does not exist in my sense. So it does not exist because it does not exist. > That is irrelevant. I am trying to use *your* definition of exist, and > can not find it is very consistent. I would think that with *your* > definition and theorem (comparability is necessary and sufficient) sqrt(2) > would exist, because I can compare it with every existing number. > > Again: We now that sqrt(2) has no fundamental set (because it is not a > natural) and it has not a complete decimal expansion, because it is not > a rational. What do you mean with complete in this sentence? > But to shorten the discussion: You cannot compare sqrt(2) with a number > which in order to exist requires all the memory space of the universe > because in that case there is no possibility to store sqrt(2) > simultaneously. If you allow the existence of a number that requires all the memory space of the universe, you allow the possibility that at some time only one number does exist, that only can be compared with itself. > You see, my definition of exist circumvents also your sophisticated > attacks. I think hit has circular tendencies. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Relative Cardinality I take it for granted as long as nobody is able to compare numbers in > > another way. Yes, that is the spirit. Something is necessary and sufficient because you > take it for granted. There are other ways to compare apart from comparing > digit by digit. Well there is some incredible obsession with WM and n-ary digit representation. Actually lots of cranks are obsessed with n-ary digit representations of numbers by the looks of it. One thing that is bugging me is this: WM claims something doesn't exist if we don't yet know it. And furthermore he claims that numbers don't exist if we don't have their n-ary representation. Now what would be bugging me is that say the ancient greeks did not have the n-ary system. They studied ratios of lengths in euclidean geometry. Now if they couldn't find the digit sequence for a decimal (or any n-ary) representation for their ratios, does it mean that their numbers didn't exist? I suppose at this stage WM would argue that because they had a different system of writing numbers, then as long as they could write it in their system then the number existed, because apparently the rules of the system can be arbitrary (as per prior WM posts). If so, then how about we look at say Q(sqrt(2)), when wanting to work with sqrt(2). Any number there is represented by (n/m) + (p/q) sqrt(2) Surely taking n=0,m=1,p=1,q=1 would represent sqrt(2). The comparison thing and limitted memory was also bugging me. A number which takes up all memory to store cannot be compared with any other number. Though the number of numbers existing at that time is 1 since we took up all the memory for our single huge number (strangely then the number 1 doesn't exist at the same time, so the number of numbers existing at any single time is likely not a number). However we've there's no one to look at it or compare it. In fact since we have no memory to store HOW we encoded the number and thus it's just nonsense scattered accross the universe. Plus how did we, being part of the universe end up scattering ourselves to the correct positions. Also can't we just think of an encoding and just READ OFF the number being currently encoded by the universe? What if we use several different encodings at the same time? Do we get different numbers from the same we didn't use more information? In any case, I think we can safely rule out WM being a Platonist. Jiri === Subject: Re: Relative Cardinality Nntp-Posting-Host: apps.cwi.nl ... > Yes, that is the spirit. Something is necessary and sufficient because you > take it for granted. There are other ways to compare apart from comparing > digit by digit. ... > I suppose at this stage WM would argue that because they had a > different system of writing numbers, then as long as they could write > it in their system then the number existed, because apparently the > rules of the system can be arbitrary (as per prior WM posts). The Greek have had various notations for numbers. The oldest one was pretty similar to the Roman system, but that was abandoned already pretty early. The final system was a system with 10,000 as base (myriads). There are a few variants. Diophantus used a positional system, but as far as I have been able to ascertain he had no sure way to represent 0 (or was it just the absence of some digit). It was written d1.d2.d3, etc. where d1 to d3 were numbers in the range 1 to 999. Possibly a missing position was noted as such. The second variant has strange mirrors in the alephs. They used myriads of various orders where such a myriad was noted by the letter M with the order (in ordinary numerals) on top. A myriad of order n+1 was 10,000 times a myriad of order n. And you noted a number by writing down the number of the various myriads, followed by the notation for the myriad, and finally the units (if they were present). A priori this allowed the notation of all numbers upto 10,000^1000-1. I do not think they had any consideration for larger numbers. Anyhow, they had no specific notation for rationals, except as noting numerator and denominator. I think the first use of any n-ary system to consistently note fractions dates from the book De Thiende by Simon Stevin from 1585, where he proposed decimal notation of fractions. So I think that according to Mueckenheim, rational numbers did not exist before that time. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Relative Cardinality I take it for granted as long as nobody is able to compare numbers in > > another way. Yes, that is the spirit. Something is necessary and sufficient because you > take it for granted. There are other ways to compare apart from comparing > digit by digit. Well there is some incredible obsession with WM and n-ary digit > representation. Actually lots of cranks are obsessed with n-ary digit > representations of numbers by the looks of it. One thing that is bugging me is this: WM claims something doesn't > exist if we don't yet know it. And, if everyone forgets it, then it doesn't exist (again). > And furthermore he claims that numbers > don't exist if we don't have their n-ary representation. Now what > would be bugging me is that say the ancient greeks did not have the > n-ary system. They studied ratios of lengths in euclidean geometry. > Now if they couldn't find the digit sequence for a decimal (or any > n-ary) representation for their ratios, does it mean that their > numbers didn't exist? I suppose at this stage WM would argue that because they had a > different system of writing numbers, then as long as they could write > it in their system then the number existed, because apparently the > rules of the system can be arbitrary (as per prior WM posts). If so, then how about we look at say Q(sqrt(2)), when wanting to work > with sqrt(2). Any number there is represented by (n/m) + (p/q) sqrt(2) Surely taking n=0,m=1,p=1,q=1 would represent sqrt(2). The comparison thing and limitted memory was also bugging me. It's been a major theme in my posts. > A number > which takes up all memory to store cannot be compared with any other > number. And there's no way to do any computation. > Though the number of numbers existing at that time is 1 since > we took up all the memory for our single huge number (strangely then > the number 1 doesn't exist at the same time, so the number of numbers > existing at any single time is likely not a number). However we've > there's no one to look at it or compare it. This is not necessarily true; if consciousness is not limited to the 3-dimensional, physical universe, it _can_ look at the number. > In fact since we have no > memory to store HOW we encoded the number and thus it's just nonsense > scattered accross the universe. I threw him a bone and let him assume this was possible. It doesn't make his arguments any easier. > Plus how did we, being part of the > universe end up scattering ourselves to the correct positions. That's a bone I threw to him. I assumed that we had a fixed permutation > Also > can't we just think of an encoding and just READ OFF the number being > currently encoded by the universe? What if we use several different > encodings at the same time? Again, a major theme of my posts, in particular: If the universe reads 111...111, what number does this represent? Even if you have different rules for determining how to decode the number, you still need to write down an indication of what rule to use. But this all collapses back on my universal rule which is a function mapping the configurations of the universe to R. WM quit posting, right after I found a good example of this. For posterity's sake, I'll write it down. 2,3,4,5. Consider how to decipher: x0004 There are several possibilities: (1) As a decimal number (2) 10 raised to that number (3) Factorial of the given number (4) Prime of the given number (i.e., the 4th prime, which is 7) ... These are 4 possible ways to interpret 0004, so it looks like we have 4 rules: f1(0004) = 4 f2(0004) = 10000 f3(0004) = 24 f4(0004) = 7 But you need to indicate, in the input, which rule to use; that's what the x position is for: To get an unambiguous number, you have to include it. So, instead of determining which rule to use, you have a universal rule which states that f(10004) = 4 f(20004) = 10000 f(30004) = 24 f(40004) = 7 (etc), and once again, there are only a finite number of Muecken numbers (at most 10^5). > Do we get different numbers from the same > we didn't use more information? In any case, I think we can safely rule out WM being a Platonist. Definitely. He doesn't believe there's any difference between a proof existing and someone knowing the proof. === Subject: Re: Relative Cardinality we took up all the memory for our single huge number (strangely then > the number 1 doesn't exist at the same time, so the number of numbers > existing at any single time is likely not a number). However we've > there's no one to look at it or compare it. This is not necessarily true; if consciousness is not limited to the > 3-dimensional, physical universe, it _can_ look at the number. BUT, if there then is an outside consciousness that can look at the universe and think about what number it is, then couldn't the outside consciousness think about other abstract things? Such as other numbers? I think WM pretty much stated that thought is just a bunch of chemical/electrical activity in the brain and thus entierly physical. > In any case, I think we can safely rule out WM being a Platonist. Definitely. He doesn't believe there's any difference between a proof > existing and someone knowing the proof. So, wouldn't such thinking extend to this: If I find a proof of Riemann hypothesis in the morning, then in the evening I discover that it is wrong. Now I didn't KNOW it was wrong all day, so was the proof correct as long as I thought it was correct? I mean the property of being wrong was unknown to me and thus nonexistent. Jiri === Subject: Re: Relative Cardinality are available. A number exists if the axiom system in which one is working allows it to > exist. It is defined then. But it does not yet exist. I.e., there is not yet shown that this number can be put in order < with other numbers. > 3) A number cannot exist, if this is impossible. In > another post, he has insisted that there exist distinct real numbers > with no rationals between them. You are wrong. Ty to better understand simple written text. I said that there can *not* exist two irrationals without a rational between them (in normal order). This proves that there are not more irrationals than rationals (+1). > No, it cannot be compared with sqrt(2) but only with a rational which > consists of the first 10^20 (or even some more) digits of sqrt(2). Does WM claim that if x < y and y < z one cannot deduce that x < z? No. But sometimes x < y cannot be obtained, neither by decimal representation nor by fractions or continued fractions. > The same arguing is deluding the antidiagonal as > different from every line number. Which line number is it the same as? Always the next one in this never ending sequence. 1) We find the antidiagonal different from every line number *which > can be tested *. But since one test serves *all* *simulteneously*, that is no restriction. There is no consistent test of all simultaneously. (And a bijection is valid only for finite sets. Its validity for infinite sets is not supported by any axioms.) Regads, WM === Subject: Re: Relative Cardinality > > 1) A number exists, if a fundamental set or an n-adic representation > are available. A number exists if the axiom system in which one is working allows it to > exist. > > It is defined then. But it does not yet exist. I.e., there is not yet > shown that this number can be put in order < with other numbers. That may be WM's definition of a number existing, but whatever cannot be deduced deducable from the axioms is irrelevant to what exists in that axiomatic system. > > 3) A number cannot exist, if this is impossible. In > another post, he has insisted that there exist distinct real numbers > with no rationals between them. > > You are wrong. Ty to better understand simple written text. I said that > there can *not* exist two irrationals without a rational between them > (in normal order). This proves that there are not more irrationals than > rationals (+1). Wrong! Because between every two reals there are both infinitely many rationals and infinitely many irrationals. Mutually dense sets need not be anything like equinumerous. WM's mind is not only finite, it is miniscule. > > No, it cannot be compared with sqrt(2) but only with a rational which > consists of the first 10^20 (or even some more) digits of sqrt(2). Since any rational can easily be compared with sqrt(2), sqrt(2) is as actual as any rational, at least by WM's comparison test. Does WM claim that if x < y and y < z one cannot deduce that x < z? > > No. But sometimes x < y cannot be obtained, neither by decimal > representation nor by fractions or continued fractions. Does WM declare that there is any square root of a rational that cannot be compared with any rational as totally as any rational can be compared with any other rational? > > The same arguing is deluding the antidiagonal as > different from every line number. Which line number is it the same as? > > Always the next one in this never ending sequence. But the rule for constructing the antidiagonal forces it to be different from every possible line simultaneously. there cannot exist a line that the anti-diagonal is equal to unless all one-digit numbers are equal to each other. 1) We find the antidiagonal different from every line number *which > can be tested *. Since any line number CAN be tested, that means all line numbers. > There is no consistent test of all simultaneously. That WM is handicapped, does not impose that handicap on anyone else. (And a bijection is > valid only for finite sets. Its validity for infinite sets is not > supported by any axioms.) It is supported by the ZFC axioms, in particular. SO that WM is wrong, again. === Subject: Primes: ar all odd primes (as factors) listed by 2^k-1 where k=2..inf Hi ng, came across that three problems: a) if I list all prime-factors of all n_k = 2^k-1 for k=2..oo , do then all odd primes occur? (rows/columns for k=0 and k=1, not really needed ...) k n_k prime-factors ---------------------------------- 0 0 0 1 3 7 5 31 .... 1_____1_____1__________________.... 2 3 3 3 7 7 4 15 3 5 5 31 31 6 63 3^2 7 (1) 7 127 127 8 255 3 5 17 9 511 7 73 10 1023 3 31 11 11 2047 23*89 12 4095 3^2 7 5 13 ... with the principal diagonal listed this way: 3,7,5,31,(1),127,17,73,11,(23*89),13,... Do all odd primes occur in that list? I guess, the arguments in the Lucas-Lehmer-sequences-discussion point in this direction, so I wonder, whether I can apply them here. b) and as a strengthening: do all composites (means all odd natural numbers) occur as factors of that list? It is a strengthening, since all powers of primes also have to occur in all combinations. I think, I've a vague idea of proving b) by using the impression, that the cycles of occurence of powers of the primes are all coprime, but b) is obviously depending on a). c) I also assume (for other reasons) that each prime p must occur at any position k with 2 <= k < p , or more precisely log2(p+1) <= k < p So all primes smaller than, say 16 must occur before k=16 But I'm not much experienced in proving... Gottfried Helms === Subject: Re: Primes: ar all odd primes (as factors) listed by 2^k-1 where k=2..inf > came across that three problems: > a) if I list all prime-factors of all n_k = 2^k-1 for k=2..oo , > do then all odd primes occur? > (rows/columns for k=0 and k=1, not really needed ...) k n_k prime-factors > ---------------------------------- > 0 0 0 1 3 7 5 31 .... > 1_____1_____1__________________.... > 2 3 3 > 3 7 7 > 4 15 3 5 > 5 31 31 > 6 63 3^2 7 (1) > 7 127 127 > 8 255 3 5 17 > 9 511 7 73 > 10 1023 3 31 11 > 11 2047 23*89 > 12 4095 3^2 7 5 13 > ... with the principal diagonal listed this way: 3,7,5,31,(1),127,17,73,11,(23*89),13,... Do all odd primes occur in that list? I guess, the arguments > in the Lucas-Lehmer-sequences-discussion point in this > direction, so I wonder, whether I can apply them here. b) and as a strengthening: do all composites (means all odd natural > numbers) occur as factors of that list? > It is a strengthening, since all powers of primes also have > to occur in all combinations. > I think, I've a vague idea of proving b) by using the impression, > that the cycles of occurence of powers of the primes are all coprime, > but b) is obviously depending on a). Parts a and b are fairly simple. For an odd m, look at the sequence mod m: { 1, 2, 4, 8, ... 2^k, ... }. Since there are only m-1 non-zero integers mod m, this sequence must start to repeat at some point. This means that 2^n = 2^k mod m for some k < n < m. Since 2 is relatively prime to m, we can conclude that 2^{n-k} - 1 = 0 mod m This means that m is a factor of 2^{n-k} - 1. Therefore, yes, all odd numbers occur as a factor of some 2^k-1. > c) I also assume (for other reasons) that each prime p must occur > at any position k with 2 <= k < p , or more precisely log2(p+1) <= k < p So all primes smaller than, say 16 must occur before k=16 Since the sequence mod m above must start to repeat after at most m-1 terms, m must be a factor of 2^k-1 for some k < m. The lower limit is obvious since m <= 2^k-1 implies that log_2(m+1) <= k. Rob Johnson take out the trash before replying === Subject: Primes: are all odd primes (as factors) listed by 2^k-1 where k=2..inf Hi ng, came across that three problems: a) if I list all prime-factors of all n_k = 2^k-1 for k=2..oo , do then all odd primes occur? (rows/columns for k=0 and k=1, not really needed ...) k n_k prime-factors ---------------------------------- 0 0 0 1 3 7 5 31 .... 1_____1_____1__________________.... 2 3 3 3 7 7 4 15 3 5 5 31 31 6 63 3^2 7 (1) 7 127 127 8 255 3 5 17 9 511 7 73 10 1023 3 31 11 11 2047 23*89 12 4095 3^2 7 5 13 ... with the principal diagonal listed this way: 3,7,5,31,(1),127,17,73,11,(23*89),13,... Do all odd primes occur in that list? I guess, the arguments in the Lucas-Lehmer-sequences-discussion point in this direction, so I wonder, whether I can apply them here. b) and as a strengthening: do all composites (means all odd natural numbers) occur as factors of that list? It is a strengthening, since all powers of primes also have to occur in all combinations. I think, I've a vague idea of proving b) by using the impression, that the cycles of occurence of powers of the primes are all coprime, but b) is obviously depending on a). (Well, actually I remember the wieferich-prime-problem, that p1=1093 and p1=3511 could happen to occur only in quadratic manner; but let's deal with those exceptions separately) c) I also assume (for other reasons) that each prime p must occur at any position k with 2 <= k < p , or more precisely log2(p+1) <= k < p So all primes smaller than, say 16 must occur before k=16 But I'm not much experienced in proving... Gottfried Helms === Subject: Re: Primes: are all odd primes (as factors) listed by 2^k-1 where k=2..inf > Hi ng, came across that three problems: > a) if I list all prime-factors of all n_k = 2^k-1 for k=2..oo , > do then all odd primes occur? > (rows/columns for k=0 and k=1, not really needed ...) k n_k prime-factors > ---------------------------------- > 0 0 0 1 3 7 5 31 .... > 1_____1_____1__________________.... > 2 3 3 > 3 7 7 > 4 15 3 5 > 5 31 31 > 6 63 3^2 7 (1) > 7 127 127 > 8 255 3 5 17 > 9 511 7 73 > 10 1023 3 31 11 > 11 2047 23*89 > 12 4095 3^2 7 5 13 > ... with the principal diagonal listed this way: 3,7,5,31,(1),127,17,73,11,(23*89),13,... Do all odd primes occur in that list? I guess, the arguments > in the Lucas-Lehmer-sequences-discussion point in this > direction, so I wonder, whether I can apply them here. b) and as a strengthening: do all composites (means all odd natural > numbers) occur as factors of that list? > It is a strengthening, since all powers of primes also have > to occur in all combinations. > I think, I've a vague idea of proving b) by using the impression, > that the cycles of occurence of powers of the primes are all coprime, > but b) is obviously depending on a). (Well, actually I remember the wieferich-prime-problem, that > p1=1093 and p1=3511 could happen to occur only in quadratic manner; > but let's deal with those exceptions separately) c) I also assume (for other reasons) that each prime p must occur > at any position k with 2 <= k < p , or more precisely log2(p+1) <= k < p So all primes smaller than, say 16 must occur before k=16 But I'm not much experienced in proving... Gottfried Helms You appear to be asking for which d we can find a k such that d | (2^k - 1)and if there are any bounds on k given d. The answer is that for every odd d there is a k such that d | (2^k - 1) and k < d. To see this you have to look at the subset of the integers mod d which are represented by numbers that are relatively prime to d. All books on elementary number theory will describe what I am talking about and prove the results. Since d is odd, 2 is relatively prime to d and so 2 mod d is in this group. This group has order the number of integers between 1 and d which are relatively prime to d, knows as phi(d). Clearly phi(d) is less than d. Since every element of a group of order n yields 1 when raised to the n'th power, we see that 2^phi(d) is congruent to 1 modulo d which is the same as saying that d is a divisor of 2^phi(d) - 1. Achava The === Subject: Re: Primes: are all odd primes (as factors) listed by 2^k-1 where k=2..inf > Hi ng, came across that three problems: > a) if I list all prime-factors of all n_k = 2^k-1 for k=2..oo , > do then all odd primes occur? > (rows/columns for k=0 and k=1, not really needed ...) k n_k prime-factors > ---------------------------------- > 0 0 0 1 3 7 5 31 .... > 1_____1_____1__________________.... > 2 3 3 > 3 7 7 > 4 15 3 5 > 5 31 31 > 6 63 3^2 7 (1) > 7 127 127 > 8 255 3 5 17 > 9 511 7 73 > 10 1023 3 31 11 > 11 2047 23*89 > 12 4095 3^2 7 5 13 > ... with the principal diagonal listed this way: 3,7,5,31,(1),127,17,73,11,(23*89),13,... Do all odd primes occur in that list? I guess, the arguments > in the Lucas-Lehmer-sequences-discussion point in this > direction, so I wonder, whether I can apply them here. > I don't quite follow that last claim, but ... by Fermat's Little Theorem, if p is an odd prime, p occurs as a factor of 2^{p-1} - 1. So the answer to your first question is Yes. > b) and as a strengthening: do all composites (means all odd natural > numbers) occur as factors of that list? > It is a strengthening, since all powers of primes also have > to occur in all combinations. > I think, I've a vague idea of proving b) by using the impression, > that the cycles of occurence of powers of the primes are all coprime, > but b) is obviously depending on a). (Well, actually I remember the wieferich-prime-problem, that > p1=1093 and p1=3511 could happen to occur only in quadratic manner; > but let's deal with those exceptions separately) c) I also assume (for other reasons) that each prime p must occur > at any position k with 2 <= k < p , or more precisely log2(p+1) <= k < p So all primes smaller than, say 16 must occur before k=16 If we're still discussing (1), then ... yes ... p appears no later than k=p-1. > > But I'm not much experienced in proving... > > Gottfried Helms === Subject: Re: Primes: are all odd primes (as factors) listed by 2^k-1 where k=2..inf But that backs me for the next consideration: -------------------------------------------- Since all primes are in that list, then d) for each prime the smallest index k denotes the L2(p) meaning L2(p) = the length of the first cyclic multiplicative group (containing 1 and 2) mod (p) with powerbase 2 L2(p) = min(k_p) or more general ... mod(n) , where L2(n) is the smallest k, where n occurs as a composite. This seems to give confirmation for my formula for L2(p): d.1) L2(p) = min(k_p) d.2) L2(p^a) = L2(p) * p^(a-1) // a>1; p is not wieferich = L2(p) * p^(a-2) // a>1; p is wieferich d.3) with n = p^a * q^b * ... * s^d // primepowers of n L2(n) = lcm( L2(p^a), L2(q^b), ..., L2(s^d) ) = lcm( L2(p), L2(q), ..., L2(s), n/wrad(n) ) (where wrad(n) is returning the squarefree-product of n's primefactors, and for wieferich-primes uses their square) The latter is already a usefool tool to discuss things like are there consecutive prime-powers of 2^a and p^b and others. -------------------------------------------- But the next two seem more difficult to grasp, but likely of additional use: e) The prime-factor-composition of the number Mn = 2^n - 1 into Mn = q^b * r^c *...* s^d is found in row k (= n) All primefactors q,r,...,s in that row have lengthes L2(q), L2(r), ..., L2(s) which correspond to the prime-factorization of n, in that the lengthes are divisors of n. Example Let n=10 =2*5 Mn = 1023 so 3 must be a factor of Mn as L2( 3) = 2 so 31 must be a factor of Mn as L2(31) = 5 so 11 must be a factor of Mn as L2(11) = 10 But now: is the opposite view of things allowed, to say: f) each Mn collects *all* prime-factors of the appropriate lengthes, or Mn is the collection of *all* prime-factors/prime- powers, which divide n This seems so to me; for instance let n=6 =2*3 Mn = 63 = 7*9 2 is L2(3) --> 3 is a factor of Mn 2*3 is L2(3^2) --> 3^2 is a factor of Mn 3 is L2(7) --> 7 is a factor of Mn for another instance let n=11 Mn = 2047 = 23*89 11 is L2(23) --> 23 is a factor of Mn 11 is L2(89) --> 89 is a factor of Mn there are no more primes with the length L2()=11 and -put into the focus of f)-: there are *no more* primes in Mn with appropriate L2()-value than given by all possible combinations of factors of n A consequence of f) is then, that the primes can be separated in two classes: primes pu of unique length L2(pu) -- no other prime has this L2-value primes p of common length L2(p) -- other primes may have this L2-value too. then 2^pu - 1 is a mersenne-prime 2^p - 1 is not. ... and other consequences, for instance giving a rule for creating pseudoprimes by using the L2()-function. Could d),e) and f) been deduced in this way (always respecting the 2 irregularities of the wieferich primes) ? Gottfried Helms --------------- To Ed: concerning the Lucas-Lehmer-remark: I read a proof recently, where the completeness of all primes in the LL-sequence was mentioned, if I recall right. > >>Hi ng, >> >> came across that three problems: >> >> >> a) if I list all prime-factors of all n_k = 2^k-1 for k=2..oo , >> do then all odd primes occur? >> (rows/columns for k=0 and k=1, not really needed ...) >> >> k n_k prime-factors >> ---------------------------------- >> 0 0 0 1 3 7 5 31 .... >> 1_____1_____1__________________.... >> 2 3 3 >> 3 7 7 >> 4 15 3 5 >> 5 31 31 >> 6 63 3^2 7 (1) >> 7 127 127 >> 8 255 3 5 17 >> 9 511 7 73 >> 10 1023 3 31 11 >> 11 2047 23*89 >> 12 4095 3^2 7 5 13 >> ... >> >> with the principal diagonal listed this way: >> >> 3,7,5,31,(1),127,17,73,11,(23*89),13,... >> >> Do all odd primes occur in that list? I guess, the arguments >> in the Lucas-Lehmer-sequences-discussion point in this >> direction, so I wonder, whether I can apply them here. I don't quite follow that last claim, but ... by Fermat's Little > Theorem, if p is an odd prime, p occurs as a factor of 2^{p-1} - 1. > So the answer to your first question is Yes. > > > >> b) and as a strengthening: do all composites (means all odd natural >> numbers) occur as factors of that list? >> It is a strengthening, since all powers of primes also have >> to occur in all combinations. >> I think, I've a vague idea of proving b) by using the impression, >> that the cycles of occurence of powers of the primes are all coprime, >> but b) is obviously depending on a). >> >> (Well, actually I remember the wieferich-prime-problem, that >> p1=1093 and p1=3511 could happen to occur only in quadratic manner; >> but let's deal with those exceptions separately) >> >> c) I also assume (for other reasons) that each prime p must occur >> at any position k with >> >> 2 <= k < p , >> >> or more precisely >> >> log2(p+1) <= k < p >> >> So all primes smaller than, say 16 must occur before k=16 > > > If we're still discussing (1), then ... yes ... p appears > no later than k=p-1. > > >>But I'm not much experienced in proving... >> >>Gottfried Helms > > === Subject: Re: Primes: are all odd primes (as factors) listed by 2^k-1 where k=2..inf But that backs me for the next consideration: -------------------------------------------- Since all primes are in that list, then d) for each prime the smallest index k denotes the L2(p) meaning L2(p) = the length of the first cyclic multiplicative group (containing 1 and 2) mod (p) with powerbase 2 L2(p) = min(k_p) or more general ... mod(n) , where L2(n) is the smallest k, where n occurs as a composite. This seems to give confirmation for my formula for L2(p): d.1) L2(p) = min(k_p) d.2) L2(p^a) = L2(p) * p^(a-1) // a>1; p is not wieferich = L2(p) * p^(a-2) // a>1; p is wieferich d.3) with n = p^a * q^b * ... * s^d // primepowers of n L2(n) = lcm( L2(p^a), L2(q^b), ..., L2(s^d) ) = lcm( L2(p), L2(q), ..., L2(s), n/wrad(n) ) (where wrad(n) is returning the squarefree-product of n's primefactors, and for wieferich-primes uses their square) The latter is already a usefool tool to discuss things like are there consecutive prime-powers of 2^a and p^b and others. -------------------------------------------- But the next two seem more difficult to grasp, but likely of additional use: e) The prime-factor-composition of the number Mn = 2^n - 1 into Mn = q^b * r^c *...* s^d is found in row k (= n) All primefactors q,r,...,s in that row have lengthes L2(q), L2(r), ..., L2(s) which correspond to the prime-factorization of n, in that the lengthes are divisors of n. Example Let n=10 =2*5 Mn = 1023 so 3 must be a factor of Mn as L2( 3) = 2 so 31 must be a factor of Mn as L2(31) = 5 so 11 must be a factor of Mn as L2(11) = 10 But now: is the opposite view of things allowed, to say: f) each Mn collects *all* prime-factors of the appropriate lengthes, or Mn is the collection of *all* prime-factors/prime- powers, which divide n This seems so to me; for instance let n=6 =2*3 Mn = 63 = 7*9 2 is L2(3) --> 3 is a factor of Mn 2*3 is L2(3^2) --> 3^2 is a factor of Mn 3 is L2(7) --> 7 is a factor of Mn and -put into the focus of f)-: there are *no more* primes in Mn with appropriate L2()-value than given by all possible combinations of factors of n Could d),e) and f) been deduced in this way (always respecting the 2 irregularities of the wieferich primes) ? Gottfried Helms --------------- To Ed: concerning the Lucas-Lehmer-remark: I read a proof recently, where the completeness of all primes in the LL-sequence was mentioned, if I recall right. > >>Hi ng, >> >> came across that three problems: >> >> >> a) if I list all prime-factors of all n_k = 2^k-1 for k=2..oo , >> do then all odd primes occur? >> (rows/columns for k=0 and k=1, not really needed ...) >> >> k n_k prime-factors >> ---------------------------------- >> 0 0 0 1 3 7 5 31 .... >> 1_____1_____1__________________.... >> 2 3 3 >> 3 7 7 >> 4 15 3 5 >> 5 31 31 >> 6 63 3^2 7 (1) >> 7 127 127 >> 8 255 3 5 17 >> 9 511 7 73 >> 10 1023 3 31 11 >> 11 2047 23*89 >> 12 4095 3^2 7 5 13 >> ... >> >> with the principal diagonal listed this way: >> >> 3,7,5,31,(1),127,17,73,11,(23*89),13,... >> >> Do all odd primes occur in that list? I guess, the arguments >> in the Lucas-Lehmer-sequences-discussion point in this >> direction, so I wonder, whether I can apply them here. I don't quite follow that last claim, but ... by Fermat's Little > Theorem, if p is an odd prime, p occurs as a factor of 2^{p-1} - 1. > So the answer to your first question is Yes. > > > >> b) and as a strengthening: do all composites (means all odd natural >> numbers) occur as factors of that list? >> It is a strengthening, since all powers of primes also have >> to occur in all combinations. >> I think, I've a vague idea of proving b) by using the impression, >> that the cycles of occurence of powers of the primes are all coprime, >> but b) is obviously depending on a). >> >> (Well, actually I remember the wieferich-prime-problem, that >> p1=1093 and p1=3511 could happen to occur only in quadratic manner; >> but let's deal with those exceptions separately) >> >> c) I also assume (for other reasons) that each prime p must occur >> at any position k with >> >> 2 <= k < p , >> >> or more precisely >> >> log2(p+1) <= k < p >> >> So all primes smaller than, say 16 must occur before k=16 > > > If we're still discussing (1), then ... yes ... p appears > no later than k=p-1. > > >>But I'm not much experienced in proving... >> >>Gottfried Helms > > === Subject: Re: Square roots - decimal expansions -- Space no ? === Subject: Re: Fractional derivatives > >> >Hi I read someplace that fractional derivatives can be computed by >tinkering in the Fourier domain of that function. > >My question is, that not all functions have a Fourier transform ( >Dirichlet conditions)... >so how do we define their fractional derivatives? >> >>Not all functions have derivatives; why should every function >>have a fractional derivative? > >Does every C-infinity function have a fractional derivative? Yes. (See Neil's reply to yesterday's silly reply.) >If not, what can be said about the set of values of c such >that every/some/all-in-some-interesting-subset C-infinity >function has a derivative of order c? > >Presumably these questions can actually be answered, maybe >in an interesting way, if instead of C-infinity one >puts in real analytic on all of R or maybe even >defined by a single power series on all of R. > >Lee Rudolph ************************ C. Ullrich === Subject: Re: Fractional derivatives > >> I don't recall seeing any such definition that doesn't need >> some sort of global growth condition. And the definitions >> are not local, as has been pointed out. This would all tend >> to point towards a no. http://en.wikipedia.org/wiki/Differintegral http://mathworld.wolfram.com/FractionalDerivative.html Well duh. (I was misremembering the definition of fractional integrals as necessarily using integrals from -infinity to x. Of course an integral from 0 to x is fine, and doesn't require any growth condition at infinity. Duh.) ************************ C. Ullrich === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> >> >> The numbers that can be proved transcendental are countable. > >Aren't there uncountably many Liouville numbers? >> >>Of course there are, and they're all transcendental. >>Which in a sense contradicts what he said. Or maybe >>not: >> >>A proof is (or can be written as) a finite sequence of >>English words, and so there are only countably many >>proofs, hence only countably many provably transcendental >>numbers. Most Liouville numbers x are 'indescribable'; >>there exist only countably many Liouville numbers x >>such that there exists a proof of the fact that x is >>uncountable. > >I am not contradicting any of what you say, but it is very easy to >describe precisely an uncountable collection of transcendentals. Of course it is. But for most x in that collection it's impossible to give a precise description of x. >Take any enumeration of the algebraic numbers in [0,1] as decimal >expansions, and perform the Cantor diagonal construction, changing >every 1 or 2 to either a 3 or a 4, and every other digit to either >a 1 or a 2. > >Derek Holt. ************************ C. Ullrich === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> >> >> The numbers that can be proved transcendental are countable. > >Aren't there uncountably many Liouville numbers? >> >>Of course there are, and they're all transcendental. >>Which in a sense contradicts what he said. Or maybe >>not: >> >>A proof is (or can be written as) a finite sequence of >>English words, and so there are only countably many >>proofs, hence only countably many provably transcendental >>numbers. Most Liouville numbers x are 'indescribable'; >>there exist only countably many Liouville numbers x >>such that there exists a proof of the fact that x is >>uncountable. > I am not contradicting any of what you say, but it is very easy to > describe precisely an uncountable collection of transcendentals. > Take any enumeration of the algebraic numbers in [0,1] as decimal > expansions, and perform the Cantor diagonal construction, changing > every 1 or 2 to either a 3 or a 4, and every other digit to either > a 1 or a 2. > Derek Holt. Wouldn't the set of transcendental numbers be a shorter and simpler description of an uncountable collection of transcendentals? The membership criterion doesn't even depend on such arcane topics as decimal representations and therefore is much easier to visualize. My original point, as Ullrich suggests, was that the set { x in R : x is provably transcendental } is a countable set. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My original point, as Ullrich suggests, was that the set > { x in R : x is provably transcendental } is a countable set. Of course; so is { x in R : x is provably equal to x }. - Tim === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> My original point, as Ullrich suggests, was that the set >> { x in R : x is provably transcendental } is a countable set. > Of course; so is { x in R : x is provably equal to x }. Not so. (Ax)x=x is a theorem of ZF. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My original point, as Ullrich suggests, was that the set > { x in R : x is provably transcendental } is a countable set. > >> Of course; so is { x in R : x is provably equal to x }. > >Not so. So. Or at least _if_ we're interpreting things so as to make the set of provably transcendental numbers countable then the set of provably self-equal numbers is countable for the same reason - for most real x it's impossible to give a proof that x = x simply because it's impossible to specify x. >(Ax)x=x is a theorem of ZF. ************************ C. Ullrich === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My original point, as Ullrich suggests, was that the set > { x in R : x is provably transcendental } is a countable set. > >> Of course; so is { x in R : x is provably equal to x }. > >Not so. (Ax)x=x is a theorem of ZF. My uncountable set T of transcendentals is defined by T = sum_{n=1^{infinity} a_n 10^{-n} | a_n in {1,2} if f(n) > 2, a_n in {3,4} if f(n) <= 2 } where f : N -> {0,..,9} is a specific recursive function. It can be proved that (Ax)( x in T => x is transcendental ). Is there a difference between this and saying that (Ax)x=x is a theorem of ZF ? Perhaps this is a misconception on my part, but I find T a lot easier to visualize than the set of all transcendental numbers. Derek Holt. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> My original point, as Ullrich suggests, was that the set >> { x in R : x is provably transcendental } is a countable set. >Of course; so is { x in R : x is provably equal to x }. >> >>Not so. (Ax)x=x is a theorem of ZF. > My uncountable set T of transcendentals is defined by > T = sum_{n=1^{infinity} a_n 10^{-n} | a_n in {1,2} if f(n) > 2, > a_n in {3,4} if f(n) <= 2 } > where f : N -> {0,..,9} is a specific recursive function. > It can be proved that (Ax)( x in T => x is transcendental ). > Is there a difference between this and saying that (Ax)x=x is a theorem of ZF ? Yes. Deciding whether a specific x is a member of your set T, as you have presented it, looks something like an attempt to solve the halting problem. > Perhaps this is a misconception on my part, but I find T a lot > easier to visualize than the set of all transcendental numbers. Do you find it hard to visualize the set of all algebraic numbers? Do you find it hard to visualize the complement of a set? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My original point, as Ullrich suggests, was that the set > { x in R : x is provably transcendental } is a countable set. > >> Of course; so is { x in R : x is provably equal to x }. > >Not so. (Ax)x=x is a theorem of ZF. > >> My uncountable set T of transcendentals is defined by > >> T = sum_{n=1^{infinity} a_n 10^{-n} | a_n in {1,2} if f(n) > 2, >> a_n in {3,4} if f(n) <= 2 } > >> where f : N -> {0,..,9} is a specific recursive function. > >> It can be proved that (Ax)( x in T => x is transcendental ). > >> Is there a difference between this and saying that (Ax)x=x is a theorem of ZF ? > >Yes. Deciding whether a specific x is a member of your set T, as you >have presented it, looks something like an attempt to solve the halting >problem. What exactly do you mean by a specific x ? >> Perhaps this is a misconception on my part, but I find T a lot >> easier to visualize than the set of all transcendental numbers. > >Do you find it hard to visualize the set of all algebraic numbers? No >Do you find it hard to visualize the complement of a set? Possibly. For example, if a recursively enumerable subset of the integers has a complement that is not recursively enumerable, then it might be easier to visualize the set than its complement. Derek Holt. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My uncountable set T of transcendentals is defined by > T = sum_{n=1^{infinity} a_n 10^{-n} | a_n in {1,2} if f(n) > 2, > a_n in {3,4} if f(n) <= 2 } >where f : N -> {0,..,9} is a specific recursive function. >It can be proved that (Ax)( x in T => x is transcendental ). >Is there a difference between this and saying that (Ax)x=x is a theorem of ZF ? >> >>Yes. Deciding whether a specific x is a member of your set T, as you >>have presented it, looks something like an attempt to solve the halting >>problem. > What exactly do you mean by a specific x ? A computable Cauchy sequence of rationals will do. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> What exactly do you mean by a specific x ? >A computable Cauchy sequence of rationals will do. Fine. And since there are only countably many of those, there are only countably many specific x for which you can prove x=x. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > What exactly do you mean by a specific x ? >>A computable Cauchy sequence of rationals will do. > Fine. And since there are only countably many of those, there are > only countably many specific x for which you can prove x=x. But if you are given an x, then you know that there are denumerably many such x's that are algebraic and denumerably many that are transcendental. Therefore, the fact that x can be given provides no information as to whether it is transcendental. On the other hand, *every* x that can be given is equal to itself. No further information is needed. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > >> What exactly do you mean by a specific x ? > >A computable Cauchy sequence of rationals will do. > >> Fine. And since there are only countably many of those, there are >> only countably many specific x for which you can prove x=x. > >But if you are given an x, then you know that there are denumerably many >such x's that are algebraic and denumerably many that are transcendental. >Therefore, the fact that x can be given provides no information as to >whether it is transcendental. > >On the other hand, *every* x that can be given is equal to itself. No >further information is needed. Yes. What does this have to do with the fact that there are only countably many x for which a proof that x = x exists? ************************ C. Ullrich === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >What exactly do you mean by a specific x ? >> >>A computable Cauchy sequence of rationals will do. >Fine. And since there are only countably many of those, there are > only countably many specific x for which you can prove x=x. >> >>But if you are given an x, then you know that there are denumerably many >>such x's that are algebraic and denumerably many that are transcendental. >>Therefore, the fact that x can be given provides no information as to >>whether it is transcendental. >> >>On the other hand, *every* x that can be given is equal to itself. No >>further information is needed. > Yes. What does this have to do with the fact that there are only > countably many x for which a proof that x = x exists? What is the subject line of this thread? The situation we started with was that there are many more transcendentals than algebraics. However, if we restrict our attention to numbers that are capable of actually appearing in proofs, it turns out that there are equally many of each. This is related to the claim that proving a number transcendental is hard. I don't know what the rest of you have been talking about. Proving that x = x is not hard. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My original point, as Ullrich suggests, was that the set > { x in R : x is provably transcendental } is a countable set. > >> Of course; so is { x in R : x is provably equal to x }. > > Not so. (Ax)x=x is a theorem of ZF. That doesn't contradict what I said. That's not a proof of 0=0, for example, though it is a useful second-last sentence in such a proof. A proof that some predicate P holds for some particular x must end with a sentence that includes an expression for x. How many real numbers have a finite expression? - Tim === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> My original point, as Ullrich suggests, was that the set >> { x in R : x is provably transcendental } is a countable set. >> > Of course; so is { x in R : x is provably equal to x }. >> >> Not so. (Ax)x=x is a theorem of ZF. > That doesn't contradict what I said. That's not a proof of 0=0, for > example, though it is a useful second-last sentence in such a proof. > A proof that some predicate P holds for some particular x must end > with a sentence that includes an expression for x. How many real > numbers have a finite expression? See my response to Robert Israel. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My original point, as Ullrich suggests, was that the set > { x in R : x is provably transcendental } is a countable set. > >> Of course; so is { x in R : x is provably equal to x }. > >Not so. (Ax)x=x is a theorem of ZF. And so is (Ax)((x in S) implies (x is transcendental)) for certain values of S. The point is that in any formal language with at most countably many characters, there are only countably many numbers x that can be specified. What you want to prove about x is irrelevant to this. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >> My original point, as Ullrich suggests, was that the set >> { x in R : x is provably transcendental } is a countable set. >Of course; so is { x in R : x is provably equal to x }. >> >>Not so. (Ax)x=x is a theorem of ZF. > And so is (Ax)((x in S) implies (x is transcendental)) > for certain values of S. The point is that in any formal > language with at most countably many characters, > there are only countably many numbers x that can be specified. > What you want to prove about x is irrelevant to this. Irrelevant in what context? Not in the context of this thread, which was: >> Also I read somewhere that most of the numbers are transcendental,... > Algebraic numbers are countable and therefore transcendental numbers > have the same cardinal as the real numbers. >> it is funny then that it takes so much trouble to produce their >> examples.. > It's very easy to create transcendental numbers. What is hard is to > prove that a specific number is transcendental. It is not hard to prove that a specific number is equal to itself. The cardinality of the set of proofs has nothing to do with this. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions > My original point, as Ullrich suggests, was that the set > { x in R : x is provably transcendental } is a countable set. > >> Of course; so is { x in R : x is provably equal to x }. > >Not so. (Ax)x=x is a theorem of ZF. > >> And so is (Ax)((x in S) implies (x is transcendental)) >> for certain values of S. The point is that in any formal >> language with at most countably many characters, >> there are only countably many numbers x that can be specified. >> What you want to prove about x is irrelevant to this. > >Irrelevant in what context? Not in the context of this thread, which >was: > Also I read somewhere that most of the numbers are transcendental,... > >> Algebraic numbers are countable and therefore transcendental numbers >> have the same cardinal as the real numbers. it is funny then that it takes so much trouble to produce their > examples.. > >> It's very easy to create transcendental numbers. What is hard is to >> prove that a specific number is transcendental. > >It is not hard to prove that a specific number is equal to itself. The >cardinality of the set of proofs has nothing to do with this. Of course it does. It's exactly as relevant here as it is in the assertion that there are only countable many provably transcendental numbers. Unless you have a proof of that that's something entirely different from what I and it seems others have in mind. Exactly what proof do _you_ have in mind for your assertion about the fact that there are only countably many provably transcendentals? ************************ C. Ullrich === Subject: Re: proved transcendental numbers (was: Transcendental Dimensions >>It is not hard to prove that a specific number is equal to itself. The >>cardinality of the set of proofs has nothing to do with this. > Of course it does. It's exactly as relevant here as it is in the > assertion that there are only countable many provably transcendental > numbers. > Unless you have a proof of that that's something entirely different > from what I and it seems others have in mind. Exactly what proof > do _you_ have in mind for your assertion about the fact that there > are only countably many provably transcendentals? The same one you are thinking of. But if I have a number x (represented, say, as a computable Cauchy sequence of rationals) then it is considerably easier to conclude that x=x than it is to conclude that x is transcendental. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Two distribution theory questions On 21 Jul 2005 10:01:07 -0700, James >I'm reading up on distribution theory this summer and I became >interested in the following question (it is not a home work question, >I'm just interested): > >If Dk is the space of test functions f : R -> C with support in [-k,k], >is Dk separable? Where the topology is uniform convergence for every derivative, yes. >What would be a countably dense subset? > >I'm thinking that maybe if {Fn} are the standard bump functions and Pmn >is a polynomial with complex rational coefficients cut off in such a >way that fn * Pmn lies in Dk then these will do? I'm not sure exactly what you mean by that. Here's a set of functions that I'm pretty sure is dense: Let f >= 0 be a nontrivial test function. (Say the integral of f is 1, not that that matters, but it's needed for something I say below.) Consider all linear combinations with rational coefficients of the functions g = f(ax + b), where a, b are rational such that g is supported in [-k,k]. (If I wanted to prove that those were dense I'd start with the fact that if h is a test function then the convolution of h with dilates of f converges to f, then I'd approximate that convolution by a linear combination of translates of a dilate of f...) >James ************************ C. Ullrich === Subject: Separable Linear Order How does one proof a separable linear order is Lindelof? === Subject: Re: Fields as universal algebras? > > > >> In field theory, you often consider subrings which are not subfields, >> but in ring theory you seldom consider additive subgroups which are not >> subrings. > >What about ideals? If you do not require your rings to have a 1, then ideals are subrings. > Also, it is not really as substructures that ideals are of interest, so much as a representatives of congruences (compatible equivalence relations). > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > (nice quote; over the years more than one student has shown me a copy of the comic strip where Calvin declares himself to be a math atheist) -John Coleman > At the risk of appearing to give aid and comfort to the crackpots, let > me make the pedantic point that non-standard models of the (1st-order) > Peano axioms DO contain numbers which are infinite. To say that non-standard elements are infinite is only to say that they satisfy a certain infinite set of formulas. > Certainly infinite sets and power sets exist as absractions. But, > abstractions don't necessarily obey exactly that same laws of logic > as directly observable objects. Infinite sets exist as (improper) idealizations. Indeed, idealizations don't necessarily obey the same laws of logic as the directly observable objects. But, at least, there should exist a path from the abstractions back to the observable objects. Directly observable sets are idealized - and that's a good thing - but idealized sets must also be materialized again, for the sake of _applications_. The latter tends to be forgotten. That's basically what frustrates the Applied and causes anti-Cantorism. <87sly8axyo.fsf@phiwumbda.org> <87oe8was7v.fsf@phiwumbda.org> don't necessarily obey the same laws of logic as the directly observable > objects. But, at least, there should exist a path from the abstractions > back to the observable objects. Directly observable sets are idealized - > and that's a good thing - but idealized sets must also be materialized > again, for the sake of _applications_. The latter tends to be forgotten. For those who missed it, the key sentence is: But, at least, there should exist a path from the abstractions back to the observable objects. > >> Are you claiming that set theory is directly applied in General >> Relativity? Do you actually have evidence of that? > > I have seen a paper in which transfinite induction was > used. And in the existence of maximal solutions to the > initial value problem, appeal is (at least sometimes) > made to Zorn's lemma. I'm not surprised. Whether people like it or not, mathematics cannot be separated from its applications. > > >Why are you bringing physics into this? Whether black >holes exist or not has nothing to do with set theory. > >>Theoretically: no. In practice: yes. Because some consequences of set >>theory have invaded into physics by the fact that mathematics becomes >>somewhat _applied_ there, huh ! Geez ... > > Are you claiming that set theory is directly applied in General > Relativity? Do you actually have evidence of that? Ignoring > mathematical foundations, which physicists, and most everybody else, > typically do, it seems that the limit of 1/sqrt(1-v^2/c^2) as > v approaches c is infinite with or without set theory. This is the usual argument that foundational issues in mathematics can do no harm when it comes to applications outside. I do not agree with this argument, having gathered too much evidence of the contrary. Now I don't say that mathematicians should be blamed for this, if it happens. But the fact is that most physicists have a blind faith in mathematics and can be easily deluded by the fact that infinities actually exists, within mainstream mathematics, and think that they do exist in physics as well. Yes, I have evidence. Read this: http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#cz A naked singularity was found in a heat exchanger. But since a naked singularity cannot possibly exist there, it leads to quite a different conclusion: <858y03c4do.fsf@lola.goethe.zz> <85zmsic1xg.fsf@lola.goethe.zz> <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> On the other hand my original concern is what happens when the theory does not have (as Q does) the axiom All non-zero numbers have a predecessor. This is what I meant by PA - induction, since definitions of PA without reference to Q do not include this axiom (since it would be redundant, since it can be proven by induction). My apologies for misreplying to your query about what I meant by PA - induction. It's led to a lot of ink under the bridge! > >> >Why are you bringing physics into this? Whether black >holes exist or not has nothing to do with set theory. >> >>Theoretically: no. In practice: yes. Because some consequences of >>set theory have invaded into physics by the fact that mathematics >>becomes somewhat _applied_ there, huh ! Geez ... > > Physics can influence where mathematics is heading, but not what it is > finding there. Its verdict on mathematics can't be true/false, > but just interesting/irrelevant. That's the other way around. Mathematics _influences_ where physics is heading and what it is finding there. Its verdict on physics is true/ false. All physicists except one (: me) have no doubt that mathematics is quite reliable for this purpose. That's exactly what bothers me. Roger Penrose entered physics, together with his machinery called set theory, and he made predictions concerning the nature of black holes. Herewith the notion of 'infinity' is put into practice and it finds a physical interpretation, namely the singularity that resides behind the event horizon of a black hole. It is emphasized that the 'infinity' concept here is the one as understood by mainstream mathematics, since i.e. intiutionism hasn't any applications in physics and it is not interested in such. (As Brouwer has said: mathematics has nothing to do with reality. Intuitionism is extremely idealistic.) But suppose it had been otherwise. Suppose that intuitionism had become the mainstream in mathematics. Then we would have had quite a different concept of infinity, far more restrictive anyway. Now guess what would have happened to physics if Roger Penrose had been such an intuitionist or a constructivist ? Would we still have then those theories about the Cosmic Censor ? Personally, I don't think so. I find this disturbing. Our picture of the world should not depend upon the kind of mathematics accidentally used. Discussion, linux) > malbrain@yahoo.com said: >> Main Entry: fi=B7nite >> Pronunciation: 'fI-nIt >> Function: adjective >> Etymology: Middle English finit, from Latin finitus, past participle of >> finire >> 1 a : having definite or definable limits > possibilities> b : having a limited nature or existence >> >> This definition from webster should suffice. Binary numbers with ones >> in finite positions have a limited number of possibilities. >> >> karl m >> >> > have no problem with that definition. It seems that your definition (and also the dictionary definition above) conflate two distinct mathematical notions: finiteness and having a maximal element (or the closely related notion of boundedness). I will give an example, but I don't expect that Tony will get it. Suppose I take the set of natural numbers and adjoin a new element, let's call it top. Suppose I extend the ordering so that for all n in N, we have n < top. Then my new set looks something like this: 0 < 1 < 2 < .... < n < n+1 < .... < top. The first ellipsis has the numbers between 2 and n while the second ellipsis has *every* natural number greater than n+1 [1]. This is a perfectly well-defined ordered set (in fact, well-ordered). It even has a name: omega + 1. It clearly has a maximal element. Now, Tony is apparently incapable of defining having an end in purely mathematical terms, so we are left to guess what he means, but my best guess is having a maximal element. However, this would give an example of a finite set with an infinite subset. Surely, Tony doesn't want this. But then what the heck *does* having an end or having a limit mean? Footnotes: [1] If there actually were infinite natural numbers, then they are included in the second ellipsis. -- Jesse F. Hughes Time and again, history has shown that people who think their beliefs trump reality lose, and lose badly. Luckily, I don't have to listen to you. -- James Harris on reality avoidance <85d5pe8ujg.fsf@lola.goethe.zz> <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> >> It's usually something like Q (with the definition of <)... Andrew Boucher: > Sorry, I didn't see you sneak in the definition of < (in terms of > addition). This is a perfectly standard axiom of Q. a perfectly standard but is it the standard? I don't have the original definition of Robinson here, but for instance in Nelson Predicative Arithmetic (which is closer to the source, since it predates your references) Q is defined with these axioms (p. 8): (x)(Sx != 0) (x)(Sx = Sy => x = y) (x)(x + 0 = x) (x)(x + Sx = S(x+y)) (x)(x*0 = 0) (x)(Sy = x*y + x) (x)(!x = 0 => (there exists y)(Sy = x) The last axiom can be proven by induction, so is not included in the axioms of PA (at least when PA is defined straightaway, and not from Q). For instance, here's a standard defintion of PA (Mendelson), with S a total function: (x)(y)(Sx = Sy) (x)(!Sx = 0) (x)(xy)(Sx = Sy => x = y) (x)(x + 0 = x) (x)(y)(x = Sy => S(x + y)) (x)(x*0 = 0) (x)(y)(x*Sy = (x*y) + x) Induction Hence PA - induction (as I meant it anyway!) doesn't give you access to Every non-zero number has a predecessor. So my *comment* was that (x)(y)(Sx = Sy) (x)(!Sx = 0) (x)(xy)(Sx = Sy => x = y) (x)(x + 0 = x) (x)(y)(x = Sy => S(x + y)) (x)(x*0 = 0) (x)(y)(x*Sy = (x*y) + x) + schemes of WOP does not imply all schemes of induction. Indeed it will not imply the scheme B0 & (x)(Bx => B(Sx)) => (x)Bx, where Bx is !x = 0 & (there exists y)(Sy = x). This turns the problem into whether Well Ordering Principle > for this particular ordering implies induction, instead of (what I was > thinking) whether Well Ordering Principle for some generic < implies > induction. In your case the proof appears trivial. In order to prove > (Well Ordering Principle for < in terms of addition => Induction), > suppose WOP; you need to prove that every non-zero !!! > number has a predecessor. If > not then there is a least non-zero !!! > number without a predecessor, say L. L is > not 1, since 1 has 0 as a predecessor. Consider the set {L,1}. This > can be well-ordered, which implies that 1 < L. So K + 1 = L for some > K, so L has a predecessor. I don't get this (second-order) proof at all. Where is the proof second-order? The proof uses two schemes of WOP. The first instance uses the predicate (!x =0) & ! (there exists y)( Sy = x). Abbreviate this predicate as Px. The second instance uses (Px & (y)(Py => x <= y)) v x = 1. > Here is the answer to the original question you raised (in first-order > logic). Ey(x = Sy) > Q8 x < y <-> Ez(z=/=0 & y = x + z) Since it has both Q3 and Q8 as axioms, this is not actually what my original comment was meant to be about ! > Helene.Boucher@wanadoo.fr > >> It's usually something like Q (with the definition of <)... Andrew Boucher: > Sorry, I didn't see you sneak in the definition of < (in terms of > addition). This is a perfectly standard axiom of Q. > > a perfectly standard but is it the standard? > > I don't have the original definition of Robinson here, but for instance > in Nelson Predicative Arithmetic (which is closer to the source, since > it predates your references) Q is defined with these axioms (p. 8): > (x)(Sx != 0) > (x)(Sx = Sy => x = y) > (x)(x + 0 = x) > (x)(x + Sx = S(x+y)) > (x)(x*0 = 0) > (x)(Sy = x*y + x) Shouldn't the above read '(x)(x*Sy = x*y + x)' ? > (x)(!x = 0 => (there exists y)(Sy = x) Shouldn't the above read '(x)(x != 0 => (there exists y)(Sy = x)'? > > The last axiom can be proven by induction, so is not included in the > axioms of PA (at least when PA is defined straightaway, and not from > Q). For instance, here's a standard defintion of PA (Mendelson), with > S a total function: > (x)(y)(Sx = Sy) > (x)(!Sx = 0) > (x)(xy)(Sx = Sy => x = y) Shouldn't the above read '(x)(y)(Sx = Sy => x = y)' ? > (x)(x + 0 = x) > (x)(y)(x = Sy => S(x + y)) Shouldn't the above read '(x)(y)(x + Sy => S(x + y))' ? > (x)(x*0 = 0) > (x)(y)(x*Sy = (x*y) + x) > Induction > > Hence PA - induction (as I meant it anyway!) doesn't give you access to > Every non-zero number has a predecessor. So my *comment* was that > (x)(y)(Sx = Sy) > (x)(!Sx = 0) > (x)(xy)(Sx = Sy => x = y) > (x)(x + 0 = x) > (x)(y)(x = Sy => S(x + y)) > (x)(x*0 = 0) > (x)(y)(x*Sy = (x*y) + x) > + schemes of WOP > > does not imply all schemes of induction. Indeed it will not imply the > scheme > B0 & (x)(Bx => B(Sx)) => (x)Bx, > where Bx is !x = 0 & (there exists y)(Sy = x). > This turns the problem into whether Well Ordering Principle > for this particular ordering implies induction, instead of (what I was > thinking) whether Well Ordering Principle for some generic < implies > induction. In your case the proof appears trivial. In order to prove > (Well Ordering Principle for < in terms of addition => Induction), > suppose WOP; you need to prove that every > > non-zero !!! > > number has a predecessor. If > not then there is a least > > non-zero !!! > > number without a predecessor, say L. L is > not 1, since 1 has 0 as a predecessor. Consider the set {L,1}. This > can be well-ordered, which implies that 1 < L. So K + 1 = L for some > K, so L has a predecessor. I don't get this (second-order) proof at all. > > Where is the proof second-order? The proof uses two schemes of WOP. > The first instance uses the predicate > > (!x =0) & ! (there exists y)( Sy = x). > > Abbreviate this predicate as Px. The second instance uses > > (Px & (y)(Py => x <= y)) v x = 1. > > Here is the answer to the original question you raised (in first-order > logic). > > Q3 x =/= 0 -> Ey(x = Sy) > Q8 x < y <-> Ez(z=/=0 & y = x + z) > > Since it has both Q3 and Q8 as axioms, this is not actually what my > original comment was meant to be about ! <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> Good a proof reader! > (x)(Sy = x*y + x) Shouldn't the above read '(x)(x*Sy = x*y + x)' ? Yes! (x)(!x = 0 => (there exists y)(Sy = x) Shouldn't the above read '(x)(x != 0 => (there exists y)(Sy = x)'? Unless I'm missing something, ! x = 0 is the same as x != 0... The last axiom can be proven by induction, so is not included in the > axioms of PA (at least when PA is defined straightaway, and not from > Q). For instance, here's a standard defintion of PA (Mendelson), with > S a total function: > (x)(y)(Sx = Sy) > (x)(!Sx = 0) > (x)(xy)(Sx = Sy => x = y) Shouldn't the above read '(x)(y)(Sx = Sy => x = y)' ? Yes! > (x)(y)(x = Sy => S(x + y)) Shouldn't the above read '(x)(y)(x + Sy => S(x + y))' ? Yes! > > Good a proof reader! > > > (x)(Sy = x*y + x) Shouldn't the above read '(x)(x*Sy = x*y + x)' ? > > Yes! > (x)(!x = 0 => (there exists y)(Sy = x) Shouldn't the above read '(x)(x != 0 => (there exists y)(Sy = x)'? > > Unless I'm missing something, ! x = 0 is the same as x != 0... The last axiom can be proven by induction, so is not included in the > axioms of PA (at least when PA is defined straightaway, and not from > Q). For instance, here's a standard defintion of PA (Mendelson), with > S a total function: > (x)(y)(Sx = Sy) > (x)(!Sx = 0) > (x)(xy)(Sx = Sy => x = y) Shouldn't the above read '(x)(y)(Sx = Sy => x = y)' ? > > Yes! > > (x)(y)(x = Sy => S(x + y)) Shouldn't the above read '(x)(y)(x + Sy => S(x + y))' ? > > Yes! > I have made more grotesque typos with less excuse. <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> > (x)(y)(x = Sy => S(x + y)) Shouldn't the above read '(x)(y)(x + Sy => S(x + y))' ? Yes! > Well actually, no! It should read: (x)(y)(x + Sy = S(x + y)) Shame on you! Terrible proof reader! <3k7o8jFt4lfgU2@individual.net> <3k7tbfFtamaiU1@individual.net> Shame on you! Terrible proof reader! On reading that back, it may give the wrong impression. Of course I am joking (the joke being the shame is on me for writing so badly). Any trouble to point out my errors, and I am only surprised that anyone other than Jeffrey and I are still reading this part of the thread. >> Nothing of any importance about mathematics would change >> if we substituted different words for the basic concepts. Yes it would. Mathematicians would stop looking so foolish when they say things like: All natural numbers are finite but there are an infinite number of them. Not only do they seem foolish when they say such things, they also create enough confusion as to create a whole group of people who are known as anti-Cantorians. Furthermore, they appear to be quite ignorant about the problem with the above quote. > Mathematicians would stop looking so foolish when they say > things like: > > All natural numbers are finite but there are an infinite number > of them. Of course they look foolish. It should be: All natural numbers are finite but there is an infinite number of them. -- Alec McKenzie > >> Nothing of any importance about mathematics would change >> if we substituted different words for the basic concepts. > > Yes it would. > > Mathematicians would stop looking so foolish when they say > things like: > > All natural numbers are finite but there are an infinite number > of them. > > Not only do they seem foolish when they say such things, they > also create enough confusion as to create a whole group of > people who are known as anti-Cantorians. Furthermore, they > appear to be quite ignorant about the problem with the above > quote. All naturals are finite, and since adding 1 to any natural produces a larger natural, there is no end to them. Some people seem think that called this non-ending non-finite is non-ignorant. I mist this gem, so I am going piggyback... >> Somehow they >> think an all-encompassing definition of 'infinite' must >> be provided before someone can say what an infinite set is. >> I am not sure what they mental hangup is. I wonder >> how any of them would ever learn a foreign language. > >Ha, ha, ha. _This_ anti-Cantorian has learned six languages: Dutch, >German, French, English, Latin and Greek. We in the Netherlands are >privileged with our knowledge of foreign languages. Yeah, right. Try to ask directions anywhere on the streets in French and you will draw a blank face (unless you find someone who came from Morocco). Pray step off your high horse. Did you ever hear one of the people from government talk in English? Did you not cringe? > Yet I find that >an all-encompassing definition of 'infinite' must be provided. Why? Do you have an all-encompassing definition of the Dutch word pas? Offhand I know two meanings, and neither fits the use in the term bankpas. In mathematics (as in real life) the meaning of a word depends on context. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ > > > How do you disagree with an axiom? By assuming a contrary axiom, as is done in non-Euclidean geometry. The parallel postulate is denied in non-euclidean geometry. Axioms are posits, or assumptions. They are NOT self evidence truths. Bob Kolker !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> How do you disagree with an axiom? By assuming a contrary axiom, as is done in non-Euclidean > geometry. Oh, but that is not disagreeing with it. In fact, it is expressing faith that the axiom indeed _is_ an axiom. If I use a hammer for driving a nail, this does not mean that I disagree with a screwdriver. I'll still use the screwdriver when having to drive a screw. > The parallel postulate is denied in non-euclidean geometry. It is not as much denied as exchanged. > Axioms are posits, or assumptions. They are NOT self evidence > truths. They are not truths at all, they are tools. -- Kastrup, Kriemhildstr. 15, 44793 Bochum <42DC3F5F.30904@netscape.net> <85pste8vvp.fsf@lola.goethe.zz> <854qaq8u5i.fsf@lola.goethe.zz> <85mzoi5wgm.fsf@lola.goethe.zz> <851x5tx1w4.fsf@lola.goethe.zz> <42df8a07$0$2871$afc38c87@news.optusnet.com.au> <8564v3fegm.fsf@lola.goethe.zz> > >> How do you disagree with an axiom? By assuming a contrary axiom, as is done in non-Euclidean > geometry. Oh, but that is not disagreeing with it. In fact, it is expressing > faith that the axiom indeed _is_ an axiom. Axioms are agreements -- an shared expression of faith. > If I use a hammer for driving a nail, this does not mean that I > disagree with a screwdriver. I'll still use the screwdriver when > having to drive a screw. The parallel postulate is denied in non-euclidean geometry. It is not as much denied as exchanged. What does this mean, it seems backward? You deny one axiom in favor of agreement with another. > Axioms are posits, or assumptions. They are NOT self evidence > truths. They are not truths at all, they are tools. > Axioms are agreements to share pre-conceived truths. Yes, they are also tools. karl m !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> >>> How do you disagree with an axiom? >> >> By assuming a contrary axiom, as is done in non-Euclidean >> geometry. >> >> Oh, but that is not disagreeing with it. In fact, it is expressing >> faith that the axiom indeed _is_ an axiom. Axioms are agreements -- an shared expression of faith. Uh, no. Axioms have nothing to with faith at all. If you are playing chess, you don't have _faith_ that a knight moves always two squares and then one perpendicular. If it moves differently, that does not cause you to lose faith in the knight, but rather in your opponent's mental sanity. Axioms are the rules of the game. They are arbitrary, but it does not usually make sense to question them, since the purpose is to _play_ the game. Only when you are intend to design a new game does it make sense playing with the rules. >> If I use a hammer for driving a nail, this does not mean that I >> disagree with a screwdriver. I'll still use the screwdriver when >> having to drive a screw. >> >> The parallel postulate is denied in non-euclidean geometry. >> >> It is not as much denied as exchanged. What does this mean, it seems backward? You deny one axiom in favor > of agreement with another. I don't deny it. I just decide I want to play a game with different rules. That does not make the rules for the original game less valid. > Axioms are agreements to share pre-conceived truths. Yes, they are > also tools. Then the movements of pieces in chess are also pre-conceived truths. I find this an odd way of looking at them. -- Kastrup, Kriemhildstr. 15, 44793 Bochum <42DC3F5F.30904@netscape.net> <85pste8vvp.fsf@lola.goethe.zz> <854qaq8u5i.fsf@lola.goethe.zz> <85mzoi5wgm.fsf@lola.goethe.zz> <851x5tx1w4.fsf@lola.goethe.zz> <42df8a07$0$2871$afc38c87@news.optusnet.com.au> <8564v3fegm.fsf@lola.goethe.zz> <85ll3ya08b.fsf@lola.goethe.zz> > >> >>> How do you disagree with an axiom? >> >> By assuming a contrary axiom, as is done in non-Euclidean >> geometry. >> >> Oh, but that is not disagreeing with it. In fact, it is expressing >> faith that the axiom indeed _is_ an axiom. Axioms are agreements -- an shared expression of faith. Uh, no. Axioms have nothing to with faith at all. If you are playing > chess, you don't have _faith_ that a knight moves always two squares > and then one perpendicular. If it moves differently, that does not > cause you to lose faith in the knight, but rather in your opponent's > mental sanity. Faith (?), n. [OE. feith, fayth, fay, OF. feid, feit, fei, F. foi, fr. L. fides; akin to fidere to trust, Gr. to persuade. 1. Belief; the assent of the mind to the truth of what is declared by another, resting solely and implicitly on his authority and veracity; reliance on testimony. e.g. we agree on the basis of our experience with the axiom's veracity and viability. karl m <42DC3F5F.30904@netscape.net> <85pste8vvp.fsf@lola.goethe.zz> <854qaq8u5i.fsf@lola.goethe.zz> <85mzoi5wgm.fsf@lola.goethe.zz> <851x5tx1w4.fsf@lola.goethe.zz> <42df8a07$0$2871$afc38c87@news.optusnet.com.au> > How do you disagree with an axiom? By assuming a contrary axiom, as is done in non-Euclidean geometry. The > parallel postulate is denied in non-euclidean geometry. Axioms are > posits, or assumptions. They are NOT self evidence truths. Bob Kolker There are so many points possible to address, in a variety of ways, about the infinite, sets infinite, theories of infinite sets, and various perceptions of perspectives of infinity, increase without bound, untrammeled induction, universality, etcetera. I am telling you so! Here I want to ask Bob: are there any self-evident truths? The answer may be yes, but in a reflexive kind of way. It would be interesting to diagram the progression of various arguments and stated opinions about basically transfinite cardinals as the obviously to some, not contradictory, the word, uh, ... people argue about it, uh, contentionary, conflictory, confrontational, ah, controversial flashpoint of discussion of infinity. cardinals is: what good are they. What can they do. What can they show me. For many the answer is not much. There is a group of methods called measure theory that does have some of its generally accepted formulations expressed in terms of transfinite cardinals, in one dimension, as basically the difference between cardinality of the continuum and everywhere discontinuous sets, and it is plain to say that what results there may be in terms of the continuum between various every discontinuous sets can be formulated without recourse to the transfinite cardinals. Another possible notion of the utillity of the transfinite cardinals is explained in different ways of generally the discrete calculi or logarithms. One nice thing about most applied mathematics is that it's possible to construct a physical experiment using definitions of laws of nature, or statistical experiments, to exhibit observed results that agree with prediction made based upon those applicable mathematical methods. In a nonstandard measure theory, Vitali's result may be seen to not hold, and non-measurable sets don't exist, and transfinite cardinals are ineffective and evne misleading as a nomenclature for various quantities in those models, using what is called a kind of infinitesimal, which is a similar, yet different, thing, as an infinity. Infinite sets are equivalent. Bob: the universe is infinite. Now, where that is so, that leads to various notions including, where basically the powerset, diagonal, and nested interval results are held as the epitome of pure mathematics about the infinite, direct contradictions of those things. I have addressed many or the most of cardinals and post-Cantorian theory of the infinite. Thus, I claim some knowledge of pure mathematics, math. So, the existence of a self-evident truth, yes or no, and why. My answer you have. I can keep typing like this for days. Hell, I have. This took fifteen minutes. You must be able to strike like a hawk. So: what's new in this post? Ross -- Careful study of the complete thread will show that my statements are largely correct. > > For the simple reason that 'infinity' is not a concept that is limited > to mathematics alone. It spreads out i.e. into physics, and gives rise > there to singularities that exist but one can never perceive them, due > to a Cosmic Censorship that prevents us to take a look into the inside You are beating a dead horse. Mathematics as such as no empirical content, whatsoever. It is purely abstract. It may be the case that a mapping or correspondence can be established bewteen some mathematics systems and measurable quantities, but that is purely happenstantial. Bob Kolker >> >> For the simple reason that 'infinity' is not a concept that is limited >> to mathematics alone. It spreads out i.e. into physics, and gives rise >> there to singularities that exist but one can never perceive them, due >> to a Cosmic Censorship that prevents us to take a look into the inside > > You are beating a dead horse. Mathematics as such as no empirical > content, whatsoever. It is purely abstract. > > It may be the case that a mapping or correspondence can be established > between some mathematics systems and measurable quantities, but that is > purely happenstantial. So let's say to mathematicians who want to enter physics: access denied. > > >an all-encompassing definition of 'infinite' must be provided. >> >> Does a Christmas tree in certain engineering contexts actually have >> be a conifer? > >No. But it seems that you have deleted an essential add-on: > For the simple reason that 'infinity' is not a concept that is limited > to mathematics alone. It spreads out i.e. into physics, and gives rise > there to singularities that exist but one can never perceive them, due > to a Cosmic Censorship that prevents us to take a look into the inside It wasn't relevant. The colloquial definition of Christmas Tree spread out too, so why should it not have to have a single definition too? (And it's not like there is a single definition of inifinite within mathematics.) Martin > > It's the standard definition of the actual infinite, but it is not > perfectly good. Worse. It's not good at all. What is wrong with it? No one has shown it leads to a contradiction. Bob Kolker >> >> It's the standard definition of the actual infinite, but it is not >> perfectly good. Worse. It's not good at all. > > What is wrong with it? No one has shown it leads to a contradiction. No contradiction. That's perhaps the only good thing about it. If that is the only thing you care about, let me tell you that most of us care about other things, such as a physics that has a reliable mathematical machinery at its disposal. says... >No contradiction. That's perhaps the only good thing about it [set theory]. >If that is the only thing you care about, let me tell you that most of >us care about other things, such as a physics that has a reliable >mathematical machinery at its disposal. Give an example of a calculation in physics in which using modern (Cantorian) mathematics gives you the wrong answer, and using some other kind of mathematics gives you the right answer. -- Daryl McCullough Ithaca, NY > It's the standard definition of the actual infinite, but it is not > perfectly good. Worse. It's not good at all. >> >> What is wrong with it? No one has shown it leads to a contradiction. > >No contradiction. That's perhaps the only good thing about it. If that >is the only thing you care about, let me tell you that most of us care >about other things, such as a physics that has a reliable mathematical >machinery at its disposal. And physics does have reliable mathematical machinery at its disposal. You are blaming mathematics for the choices made by physicists. It's the physicist who are choosing to continue using the mathematics that involve infinity. Martin > >> > >>It's the standard definition of the actual infinite, but it is not >>perfectly good. Worse. It's not good at all. > >What is wrong with it? No one has shown it leads to a contradiction. >> >>No contradiction. That's perhaps the only good thing about it. If that >>is the only thing you care about, let me tell you that most of us care >>about other things, such as a physics that has a reliable mathematical >>machinery at its disposal. > > And physics does have reliable mathematical machinery at its disposal. > You are blaming mathematics for the choices made by physicists. It's > the physicist who are choosing to continue using the mathematics that > involve infinity. I am not blaming mathematics. This is what I said somewhere else in this thread: Now I ... > don't say that mathematicians should be blamed for this, if it happens. > But the fact is that most physicists have a blind faith in mathematics > and can be easily deluded by the fact that infinities actually exists, > within mainstream mathematics, and think that they do exist in physics > as well. === Subject: Re: Update: Objections to Cantor's Theory > Well, it might not be the way to *introduce* the concept to a beginner > -- one needs to start with an informal account of the underlying > intuitions. But that doesn't mean that it isn't, at the end of the day, > the right way to understand the concept's essence. I don't think the essence of a non-deterministic machine has anything to do with 7-tuples. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Update: Objections to Cantor's Theory On 21 Jul 2005 11:40:06 -0700, petry > >>If mose of the debate on the internet is junk, then >>it seems strange to use the internet as evidence >>for the prevalence of the anti-Cantorian view. > >For many of the anti-Cantorians, it is most obvious that >there's something absurd about Cantor's Theory, but they >really haven't a clue about how to argue the point to the >mathematicians. Yup. Similarly it's clear to most people (or to most people who've heard about the things relativity says) that the predictions of relativity are absurd - measuring rods don't contract just because they're moving, etc. But they haven't got a clue how to argue the point to the physicists. There's a reason for that: relativity actually _is_ how things are. The fact that something is obiously absurd really doesn't prove it's not so. ************************ C. Ullrich === Subject: Re: Update: Objections to Cantor's Theory > On 21 Jul 2005 11:40:06 -0700, petry > >If mose of the debate on the internet is junk, then >it seems strange to use the internet as evidence >for the prevalence of the anti-Cantorian view. >> >>For many of the anti-Cantorians, it is most obvious that >>there's something absurd about Cantor's Theory, but they >>really haven't a clue about how to argue the point to the >>mathematicians. > > Yup. Similarly it's clear to most people (or to most > people who've heard about the things relativity says) > that the predictions of relativity are absurd - > measuring rods don't contract just because they're > moving, etc. But they haven't got a clue how to > argue the point to the physicists. There's a reason > for that: relativity actually _is_ how things are. Any such comparison between mathematical and physical reality is completely besides the point. One of our problems with mathematics is that some of us (: me) find physics the leading discipline and that A little bit of Physics would be NO Idleness in Mathematics. BTW, mathematics has not such convincing arguments for its truth as relativity has with E = mc^2 and the atomic bomb. === Subject: Re: Update: Objections to Cantor's Theory > >I think that I would object to the use of the word anti-Cantorian > and I would prefer that you use the word intuitionist. >> >> Though I feel much sympathy for intuitionism, I would object to that. >> >> For the reason that intuitionism has already a well-defined meaning, >> which conflicts on many issues with Petry's description of an >> anti-Cantorian. > > Somebody else emailed me exactly that objection, and I agree with them > and you. They suggested the word constructivist. Closer, but not quite. Constructivist is already a reserved word as well: Google it up. The main problem with this terminology is that some constructivists - and most certainly all intuitionists - don't want to be involved with physical reality. The laboratory of mathematics is one of Petry's main issues in defining the anti-Cantorian, though. === Subject: Re: Update: Objections to Cantor's Theory anti-Cantorian > Read: mathematical crackpot There are no other anti-Cantorians. On the other hand there are constructivists, intuitionists and finitists; but they are certainly not the type of anti-Cantorians showing up in math newsgroups (i.e. unscholared crackpots). F. === Subject: Re: Update: Objections to Cantor's Theory > anti-Cantorian > > Read: mathematical crackpot > > There are no other anti-Cantorians. I do not know. I would say that Kronecker was as anti-Cantorian as you can get. On the other hand, he did not reject Cantorianism on the grounds that it was invalid, inconsistent, or whatever, but on the grounds that in his opinion mathematicians should only operate with finite numbers and with a finite numbers of operations. So in his opinion transcendental numbers did not exist. (Note the contrast with Mueckenheim who is of the opinion that irrational numbers do not exist. For Kronecker, algebraic numbers did exist, that is a field were his major contributions are.) So when Lindemann presented his proof that pi was transcendental, Kronecker complimented him for a beautiful proof, but nevertheless worthless, as transcendental numbers did not exist. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Update: Objections to Cantor's Theory > > There a r e no other anti-Cantorians. [...] I would say that Kronecker w a s as anti-Cantorian as > you can get. > Agree. (Same may be true for the late Poincare.) On the other hand, he did not reject Cantorianism on the > grounds that it was invalid, inconsistent, or whatever, but on the > grounds that in his opinion mathematicians should only operate with > finite numbers and with a finite numbers of operations. > Right. Actually, he was a forerunner of constructivism, that's well known. F. === Subject: Re: Update: Objections to Cantor's Theory Nntp-Posting-Host: apps.cwi.nl > There a r e no other anti-Cantorians. > > [...] I would say that Kronecker w a s as anti-Cantorian as > you can get. > > Agree. (Same may be true for the late Poincare.) ... > On the other hand, he did not reject Cantorianism on the > grounds that it was invalid, inconsistent, or whatever, but on the > grounds that in his opinion mathematicians should only operate with > finite numbers and with a finite numbers of operations. > > Right. > > Actually, he was a forerunner of constructivism, that's well known. Yes, so I doubt your statement that there are no other anti-Cantorians than cranks. I would not put Kronecker in that category. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Update: Objections to Cantor's Theory > > Yes, so I doubt your statement that there are no other anti-Cantorians > than cranks. I would not put Kronecker in that category. > Kronecker i s d e a d. (Actually, he died long ago.) And I wouldn't call modern intuitionists, konstructivists or finitist anti-Cantorians. They are non-Cantorians, but not anti-Cantorians. That's a certain difference, imho. F. P.S. I agree, Kronecker certainly w a s an anti-Cantorian, as is well known, but that's _history_! === Subject: Re: Update: Objections to Cantor's Theory !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > > anti-Cantorian > > > > Read: mathematical crackpot > > > > There are no other anti-Cantorians. I do not know. I would say that Kronecker was as anti-Cantorian as you > can get. On the other hand, he did not reject Cantorianism on the > grounds that it was invalid, inconsistent, or whatever, but on the > grounds that in his opinion mathematicians should only operate with > finite numbers and with a finite numbers of operations. So in his > opinion transcendental numbers did not exist. Did not exist does not jibe with an axiomatic approach. don't apply to reality could. > (Note the contrast with Mueckenheim who is of the opinion that > irrational numbers do not exist. For Kronecker, algebraic numbers > did exist, that is a field were his major contributions are.) So > when Lindemann presented his proof that pi was transcendental, > Kronecker complimented him for a beautiful proof, but nevertheless > worthless, as transcendental numbers did not exist. Weird. -- Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Update: Objections to Cantor's Theory ... > I do not know. I would say that Kronecker was as anti-Cantorian as you > can get. On the other hand, he did not reject Cantorianism on the > grounds that it was invalid, inconsistent, or whatever, but on the > grounds that in his opinion mathematicians should only operate with > finite numbers and with a finite numbers of operations. So in his > opinion transcendental numbers did not exist. > > Did not exist does not jibe with an axiomatic approach. don't > apply to reality could. Apparently (I have read a bit in his works) he thought that being roots of polynomials with integer coefficients was sufficient reason for a number to exist, and thought that was constructive. On the other hand, he clearly thought that the various constructions to get the reals from the rationals were not constructive. But even in his works he is himself not entirely consistent. See also the biography at . -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Update: Objections to Cantor's Theory > >> It is plausible that in the future, mathematics will be split >> into two disciplines - scientific mathematics (i.e. the science >> of phenomena observable in the world of computation), and >> philosophical mathematics, wherein Cantor's Theory is merely >> one of many possible formal theories of the infinite. > >Call this one relegious mathematics or matheology. The Cantorians close >their eyes in order to avoid obvious contradictions and to maintain >their useless pet, simply insisting that logic is not valid in the >infinite. You've got that completely backwards. The one thing that (classical) > mathematicians insist on is that logic works the same regardless > of whether the domain is naturals, reals, infinite sets, or whatever. Then they should agree that there cannot exist more irrationals than rationals in the real continuum, because, in normal order <, there does never exist a pair of irrational numbers without a rational number between them. There is no logic available to circumvent this fact. === Subject: Re: Update: Objections to Cantor's Theory > mueckenh@rz.fh-augsburg.de says... > > >> It is plausible that in the future, mathematics will be split >> into two disciplines - scientific mathematics (i.e. the science >> of phenomena observable in the world of computation), and >> philosophical mathematics, wherein Cantor's Theory is merely >> one of many possible formal theories of the infinite. > >Call this one relegious mathematics or matheology. The Cantorians close >their eyes in order to avoid obvious contradictions and to maintain >their useless pet, simply insisting that logic is not valid in the >infinite. You've got that completely backwards. The one thing that (classical) > mathematicians insist on is that logic works the same regardless > of whether the domain is naturals, reals, infinite sets, or whatever. > > Then they should agree that there cannot exist more irrationals than > rationals in the real continuum, because, in normal order <, there does > never exist a pair of irrational numbers without a rational number > between them. There is no logic available to circumvent this fact. There is nothing in standard logic or any standard axiom systemthat reqeuires assumption of anything like WM's relative cardinality, and unless it is assumed as an axiom or otherwise required, WM's non-standard conclusions are unsupported by anything except WM's non-standard assumptions. === Subject: Re: Update: Objections to Cantor's Theory mueckenh@rz.fh-augsburg.de says... >> You've got that completely backwards. The one thing that (classical) >> mathematicians insist on is that logic works the same regardless >> of whether the domain is naturals, reals, infinite sets, or whatever. > >Then they should agree that there cannot exist more irrationals than >rationals in the real continuum, because, in normal order <, there does >never exist a pair of irrational numbers without a rational number >between them. There is no logic available to circumvent this fact. What *logical* claim are you saying that mathematicians are violating? Betweenness, order, rational, irrational, are *mathematical* concepts, not logical concepts. Yes, of course, the *mathematics* of infinite sets is different from the mathematics of finite sets. But the *logic* is the same. The logical operators, to refresh your memory are: and or implies not exists forall What *logical* statement do you think mathematicians are saying is true for finite sets but not infinite sets? -- Daryl McCullough Ithaca, NY === Subject: Re: Update: Objections to Cantor's Theory > > >> mueckenh@rz.fh-augsburg.de says... >> >> >>> It is plausible that in the future, mathematics will be split >>> into two disciplines - scientific mathematics (i.e. the science >>> of phenomena observable in the world of computation), and >>> philosophical mathematics, wherein Cantor's Theory is merely >>> one of many possible formal theories of the infinite. >> >>Call this one relegious mathematics or matheology. The Cantorians close >>their eyes in order to avoid obvious contradictions and to maintain >>their useless pet, simply insisting that logic is not valid in the >>infinite. >> >> You've got that completely backwards. The one thing that (classical) >> mathematicians insist on is that logic works the same regardless >> of whether the domain is naturals, reals, infinite sets, or whatever. > >Then they should agree that there cannot exist more irrationals than >rationals in the real continuum, because, in normal order <, there does >never exist a pair of irrational numbers without a rational number >between them. There is no logic available to circumvent this fact. Sure there is. The fact that Q is dense in R does not entail that Card(R) <= Card(Q). There is no logic that can show that it does. (Note: we are using the standard definition of Card() here, not your attempted definition). Martin === Subject: Re: Update: Objections to Cantor's Theory a set and its powerset? The view is that one can be constructed. karl m But the view is not justified! The bijection as a meaningful tool to measure set-sizes is no even laid down in the axioms. It is nothing but a thoughtless extrapolation from the finite into the infinite. === Subject: Re: Update: Objections to Cantor's Theory > > What is absurd about there not existing a bijection between > a set and its powerset? The view is that one can be constructed. karl m > > But the view is not justified! The bijection as a meaningful tool to > measure set-sizes is no even laid down in the axioms. The C in ZFC specifically allows construction of injections/bijections between sets. So that WM is wrong! Again! Consistent, isn't he? === Subject: Re: Update: Objections to Cantor's Theory > The bijection as a meaningful tool to > measure set-sizes is no even laid down in the axioms. It is nothing but > a thoughtless extrapolation from the finite into the infinite. ^^^^^^^^^^^ You mis-spelled 'sensible'. Of course it's an extrapolation. We're trying to develop something beyond what our intuition readily handles. The only strategy we have is to take something that works where we know what the answer ought to be, and try to generalise it. This generalisation also has many properties that we'd like to see: the Schroeder-Bernstein theorem, for example. Now, you can say 'I don't like the axiom of infinity', and work in what's left. That's your choice. But including the axiom of infinity and then using the existence of a bijection between two sets to say that they have the same number of elements works well: for those of us who are happy with the axiom of infinity and wish to be able to use infinite sets, there is no competitor that I know of. At least, none that makes as much sense. === Subject: Re: Update: Objections to Cantor's Theory > Now, you can say 'I don't like the axiom of infinity', > and work in what's left. That's your choice. But including > the axiom of infinity and then using the existence of > a bijection between two sets to say that they have the > same number of elements works well: for those of us > who are happy with the axiom of infinity and wish to > be able to use infinite sets, there is no competitor > that I know of. At least, none that makes as much sense. Oh yeah ? Let's see. Theorem ------- The number of even naturals is half the number of all naturals. Proof ----- Consider the finite sequence of natural numbers from 1 to N: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... N Then the total number of naturals is N and, depending on whether N is odd or even, the number of even naturals in that sequence is (N-1)/2 or N/2 respectively. The quotient of the two counts is 1/2.(1-1/N) or just 1/2, respectively. Now let the sequence grow to infinity, that is: take the limit of the above quotient for N -> oo . In both cases, the limit is 1/2. This proves the theorem. See? Nothing else is needed than the classical, pre-Cantorian concept of a limit, which is accepted by all kind of mathematicians I know of. Note: it is assumed here that the sequence of all naturals is the same as the _set_ of all naturals, which seems crucial for the argument. === Subject: Re: Update: Objections to Cantor's Theory > >> Now, you can say 'I don't like the axiom of infinity', >> and work in what's left. That's your choice. But including >> the axiom of infinity and then using the existence of >> a bijection between two sets to say that they have the >> same number of elements works well: for those of us >> who are happy with the axiom of infinity and wish to >> be able to use infinite sets, there is no competitor >> that I know of. At least, none that makes as much sense. > >Oh yeah ? Let's see. > >Theorem >------- >The number of even naturals is half the number of all naturals. > >Proof >----- >Consider the finite sequence of natural numbers from 1 to N: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... N > >Then the total number of naturals is N and, depending on whether N is >odd or even, the number of even naturals in that sequence is (N-1)/2 >or N/2 respectively. The quotient of the two counts is 1/2.(1-1/N) or >just 1/2, respectively. >Now let the sequence grow to infinity, that is: take the limit of the >above quotient for N -> oo . In both cases, the limit is 1/2. > >This proves the theorem. > >See? Nothing else is needed than the classical, pre-Cantorian concept >of a limit, which is accepted by all kind of mathematicians I know of. Sure. However, it leaves you completely unable to compare the cardinality of these two sets {1, 2, 3, 4, .... } and { one, two, three, four, ..... } or, for that matter {1, 2, 3, 4, .... } and { 1, 2, watermelon, 4, ..... } without adding more axioms. Alan -- Defendit numerus === Subject: Re: Update: Objections to Cantor's Theory >> Now, you can say 'I don't like the axiom of infinity', >> and work in what's left. That's your choice. But including >> the axiom of infinity and then using the existence of >> a bijection between two sets to say that they have the >> same number of elements works well: for those of us >> who are happy with the axiom of infinity and wish to >> be able to use infinite sets, there is no competitor >> that I know of. At least, none that makes as much sense. > Oh yeah ? Let's see. > Theorem > ------- > The number of even naturals is half the number of all naturals. Theorem The number of perfects squares is 0 times the number of all naturals. > Proof > ----- > Consider the finite sequence of natural numbers from 1 to N: > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... N Proof Consider the finite sequence of natural numbers from 1 to N: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... N > Then the total number of naturals is N and, depending on whether N is > odd or even, the number of even naturals in that sequence is (N-1)/2 > or N/2 respectively. The quotient of the two counts is 1/2.(1-1/N) or > just 1/2, respectively. Then the the total number of naturals is N and the number of perfect squares in that sequence is floor(sqrt(N)). The quotient of the two counts is floor(sqrt(N))/N. > Now let the sequence grow to infinity, that is: take the limit of the > above quotient for N -> oo . In both cases, the limit is 1/2. Now let the sequence grow to infinity, that is: take the limit of the above quotient for N -> oo. The limit is 0. > This proves the theorem. This proves the theorem. > See? Nothing else is needed than the classical, pre-Cantorian concept > of a limit, which is accepted by all kind of mathematicians I know of. See all the neat stuff you can prove? You can also prove that the number of perfect cubes and the number of primes both also equal 0 times the number of naturals, and so are presumably equal to each other. Of course this idea fails miserably when considering non numerical sets. Stephen === Subject: Re: Update: Objections to Cantor's Theory ^OX9W/.#XpUmm`>TD2zNE-t}emfPkFR.Z5`flY:3QYT$>dUwN^sm;MBV:F7aL9x*q!` ln!l}>Y6_45$%R|P7DSrBkEph@1-;P*s~F_28vO@e4p/'>}Pc?@rl8cz]d9RXOt > Set theory is a disease from which mathematics > will one day recover (Poincare) > > > Poincare never said that. You will not be able to give a valid > reference for this. > > I can't, but Keith Ramsay already has. :-) > If you consider Keith's reference to a different quote of Poincare's that says something quite different as supporting your alleged quote, then I guess I can't stop you. === Subject: Re: Update: Objections to Cantor's Theory > will one day recover (Poincare) > Poincare never said that. You will not be able to give a valid > reference for this. I can't, but Keith Ramsay already has. :-) > If you consider Keith's reference to a different quote of Poincare's > that says something quite different as supporting your alleged quote, > then I guess I can't stop you. Poincare's opinion is quite clear: There is no actual infinity. The Catorians have forgotten that and have fallen into contradictons. [H. Poincar.8e, Les math.8ematiques et la logique III, Rev. m.8etaphys. morale 14, p. 316, (1906).] Here is another opinion, somewhat newer: Abraham Robinson (1964): (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. (ii) Nevertheless, we should continue the business of Mathematics 'as usual', i.e., we should act as if infinite totalities really existed. (In: Formalism 64, auch abgedruckt in Robinson 1979, p. 507.) === Subject: Re: Update: Objections to Cantor's Theory > > Here is another opinion, somewhat newer: > Abraham Robinson (1964): .... Nevertheless, we should continue the business of > Mathematics 'as usual', i.e., we should act as if infinite totalities > really existed. (In: Formalism 64, auch abgedruckt in Robinson 1979, > p. 507.) > WM conveniently forgets, when he quotes Robinson, that Robinson say that even if WM were right, mathematicians should act as if WM were wrong. So that when we act like WM is wrong, we are only following advice posted by WM himself. What Robinson is saying is that even if we are wrong about existence, we are right to reject WM. === Subject: Re: Update: Objections to Cantor's Theory Discussion, linux) > >> It is plausible that in the future, mathematics will be split >> into two disciplines - scientific mathematics (i.e. the science >> of phenomena observable in the world of computation), and >> philosophical mathematics, wherein Cantor's Theory is merely >> one of many possible formal theories of the infinite. Call this one relegious mathematics or matheology. The Cantorians close > their eyes in order to avoid obvious contradictions and to maintain > their useless pet, simply insisting that logic is not valid in the > infinite. Wrong. An obvious contradiction would be a proof of some statement and its negation. That is what mathematicians ask for and that is what you self-proclaimed anti-Cantorians have failed to give. Mathematicians have been precise on what a proof is as well, giving an explicit set of axioms and rules of inference. ,---- | Certainly infinite sets and power sets exist as absractions. | But, abstractions don't necessarily obey exactly that same | laws of logic as directly observable objects. `---- That is a fairly explicit claim: logic is not valid in the infinite. But that claim is from your camp. -- I'm starting to absorb information [...] as I find myself more and more fascinated by my own prime counting function. The rate of information absorption is on an exponential scale, like it always has been for me for things I'm interested in. -- James S. Harris === Subject: Re: Update: Objections to Cantor's Theory Discussion, linux) > The anti-Cantorians claim that while infinite sets and > power sets of those infinite sets are undeniably useful > abstractions, [...] > But, abstractions don't necessarily obey exactly that same > laws of logic as directly observable objects. I wonder how any abstractions can be undeniably useful if we are incapable of reasoning about them with our standard deductive logic (and Petry doesn't specify any fragment of logic that abstractions obey). -- Jesse F. Hughes You see 300 of something, anything, and you go `[Man], that's a lot of stuff.' -- Jim Bigler, quoted in the Pittsburgh Post-Gazette. === Subject: Re: Update: Objections to Cantor's Theory |ideas about set theory introduced by Cantor in the latter |part of the nineteenth century. You might want to think a little about how to place this same thing as what you're describing as Cantor's Theory. [...] |in concrete mathematics, and wrecklessly apply it to recklessly. [...] |In science, truth |must have observable implications, and such a reality check |would reveal Cantor's Theory to be a pseudoscience; many of the |formal theorems in Cantor's Theory have no observable implications. I think an example would make it clearer to the reader what you have in mind. Keith Ramsay === Subject: Re: Update: Objections to Cantor's Theory [...] |> #... it has come to pass that we have encountered certain |> #paradoxes, certain apparent contradictions that would have |> #delighted Zeno the Eleatic and the school of Megara. And |> #then each must seek the remedy. For my part, I think, and |> #I am not the only one, that the important thing is never to |> #introduce entities not completely definable in a finite |> #number of words. Whatever be the cure adopted, we may |> #promise ourselves the joy of the doctor called in to follow |> #a beautiful pathologic case. |> |> So it seems that rather than describing set theory as a |> disease, he is describing set theory as a patient to be |> cured of a disease, i.e. paradoxes. His idea of a proper |> cure wasn't the same as Cantor's, nor was it the same as |> yours. | | |However, I don't think you are interpreting it correctly. Look at |his proposed cure: | | For my part, I think, and I am not the only one, |that the important thing is never to introduce entities not completely |definable in a finite number of words. | |That is almost exactly the anti-Cantorian view. There don't exist |more than a countable number of entities, given Poincare's cure. Poincare is famous for being in favor of predictive definitions. (Often this also means that one doesn't believe in the existence of a set containing all real numbers, although I don't remember whether he ever said he believed that. One can believe in a hierarchy of different kinds of real numbers, each kind being countable, the diagonal proof showing that there are reals of a higher type that aren't of a lower type.) There's a big difference between predicativism and believing in the kind of essential relationship between mathematics and computation that you do, however. |He is claiming that it is Cantor's ideas about set theory that is the |disease. That's not in the quotation. The quote doesn't even mention Cantor. |I don't think anyone is arguing that there is something |wrong with sets per se. Hence, he probably wouldn't actually describe set theory, as such, as a disease, would he? Just particular verions of it. Which is what I took him to be writing. |So I'm sticking with the quote. I hope you mean you're going to quote what he actually Please don't present this poor paraphrase as if it were a direct quotation. Keith Ramsay === Subject: Re: Update: Objections to Cantor's Theory > >[...] >|> #... it has come to pass that we have encountered certain >|> #paradoxes, certain apparent contradictions that would have >|> #delighted Zeno the Eleatic and the school of Megara. And >|> #then each must seek the remedy. For my part, I think, and >|> #I am not the only one, that the important thing is never to >|> #introduce entities not completely definable in a finite >|> #number of words. Whatever be the cure adopted, we may >|> #promise ourselves the joy of the doctor called in to follow >|> #a beautiful pathologic case. >|> >|> So it seems that rather than describing set theory as a >|> disease, he is describing set theory as a patient to be >|> cured of a disease, i.e. paradoxes. His idea of a proper >|> cure wasn't the same as Cantor's, nor was it the same as >|> yours. >| >| >|However, I don't think you are interpreting it correctly. Look at >|his proposed cure: >| >| For my part, I think, and I am not the only one, >|that the important thing is never to introduce entities not completely >|definable in a finite number of words. >| >|That is almost exactly the anti-Cantorian view. There don't exist >|more than a countable number of entities, given Poincare's cure. > >Poincare is famous for being in favor of predictive definitions. >(Often this also means that one doesn't believe in the existence >of a set containing all real numbers, although I don't remember >whether he ever said he believed that. One can believe in a >hierarchy of different kinds of real numbers, each kind being >countable, the diagonal proof showing that there are reals of >a higher type that aren't of a lower type.) What _I_ wonder about here is exactly what introduce means. The set of real numbers _can_ be defined in a finite number of words (the set of all sets of rationals such that...). I don't know what Poincare meant by the word, but given _my_ interpretation of the word introduce we can introduce the set of all reals without having introduced each element of that set. >There's a big difference between predicativism and believing >in the kind of essential relationship between mathematics and >computation that you do, however. > >|He is claiming that it is Cantor's ideas about set theory that is the >|disease. > >That's not in the quotation. The quote doesn't even >mention Cantor. > >|I don't think anyone is arguing that there is something >|wrong with sets per se. > >Hence, he probably wouldn't actually describe set theory, >as such, as a disease, would he? Just particular verions >of it. Which is what I took him to be writing. > >|So I'm sticking with the quote. > >I hope you mean you're going to quote what he actually >Please don't present this poor paraphrase as if it were >a direct quotation. > >Keith Ramsay ************************ C. Ullrich === Subject: Re: Update: Objections to Cantor's Theory |I also share with you a sense that mathematics |has recently tended to be a little overboard on the abstract. However, |I do believe that the trend is reversing. Could you be a little more specific about both of these? I don't know what sort of thing you have in mind. I don't even really know what time scale you're talking about. It's not clear to me that such a trend has existed in the past 30 years, say, nor a recent reversal. My impression is that a lot of these apparent trends are just cyclical, and also localized to particular fields. But I would also say that it's difficult to get a good sense of the large trends, since even very broadly knowledgable people tend to be aware of current research in only a minority of fields. Keith Ramsay === Subject: Re: Update: Objections to Cantor's Theory > > |I also share with you a sense that mathematics > |has recently tended to be a little overboard on the abstract. > However, > |I do believe that the trend is reversing. > > Could you be a little more specific about both of these? > I don't know what sort of thing you have in mind. I don't > even really know what time scale you're talking about. > It's not clear to me that such a trend has existed in the > past 30 years, say, nor a recent reversal. > > My impression is that a lot of these apparent trends are > just cyclical, and also localized to particular fields. > But I would also say that it's difficult to get a good > sense of the large trends, since even very broadly > knowledgable people tend to be aware of current research > in only a minority of fields. You could be right. On one of my other posts in this thread, I talked in more detail in terms of analysis. Stephen === Subject: Ellipse problems (or is it not an ellipse problem?) What is the equation of an ellipse when, in a right circular cone, the plane cuts through at 90 degrees to one element on the surface of the cone? The cone side lengths are infinite. Also what are the equations for the lengths of the major and minor axes with respect to the angle of the cone tip? Is this shape an ellipse or some other conic section? If so can you provide help on this? I've tried looking this up in Geometry textbooks but since I haven't done geometry in University I do not understand the language used in describing the geometry. The internet is of limited help on this problem, with the equations for the elipse being general enough, not taking into account the angle of the cone. Jonathan === Subject: Re: Ellipse problems (or is it not an ellipse problem?) > What is the equation of an ellipse when, in a right circular cone, the > plane cuts through at 90 degrees to one element on the surface of the > cone? By element, I assume you're talking about a generatrix G, one of the lines on the surface of the cone. > The cone side lengths are infinite. Also what are the equations for the lengths of the major and minor axes > with respect to the angle of the cone tip? Let t denote the angle between G and the cone's axis. (Then I suppose that what you called the angle of the cone tip would be 2*t.) And let d denote the distance along generatrix G from the tip of the cone to the cutting plane. Then the length of the major axis is d*tan(2*t), and the length of the minor axis is 2*d*sin(t)/Sqrt(cos(2*t)), assuming my calculations are correct. > Is this shape an ellipse or some other conic section? If so can you > provide help on this? Not necessarily. You started the post assuming that it was an ellipse, and so I proceeded under that same assumption. However, it will not be an ellipse if t is pi/4 (that is, 45 degrees -- in which case you'll get a parabola) or more (in which case you'll get a hyperbola). W. Cantrell === Subject: cool giant exactly solvable matrix equation! Let i and j be integers (indices) that range from {0,...,H}. Let M be an (H+1)x(H+1) matrix with the following matrix elements: M(i,j) = g_i^j = g_i to the power of j where {g_0, g_1, ..., g_H} are H+1 positive real numbers inside the range (0,R). Let v be an (H+1) vector with matrix elements v(i) = g_i / (R-g_i) Let z be the (H+1) vector determined by solving the matrix equation: M z = v Find the solution for the H+1 matrix elements of z for arbitrary H. (Hint: Using mathematica, I have examined the solutions for H=2, 3, 4, 5. It seems the solutions fall into a simple pattern that should allow one to write down a closed-form solution. Anybody recognize what that would be?) === Subject: Re: cool giant exactly solvable matrix equation! > Let i and j be integers (indices) that range from {0,...,H}. Let M be an (H+1)x(H+1) matrix with the following matrix elements: > M(i,j) = g_i^j = g_i to the power of j > where {g_0, g_1, ..., g_H} are H+1 positive real numbers inside the > range (0,R). Let v be an (H+1) vector with matrix elements > v(i) = g_i / (R-g_i) Let z be the (H+1) vector determined by solving the matrix equation: > M z = v Find the solution for the H+1 matrix elements of z for arbitrary H. (Hint: Using mathematica, I have examined the solutions for H=2, 3, 4, > 5. > It seems the solutions fall into a simple > pattern that should allow one to write down a closed-form solution. > Anybody recognize what that would be?) > the matrix in the first part appears to be a Vandemonde type (qv on the web) which has a known polynomial determinant. Not sure about explicit inverses or the second part. rusty === Subject: Re: cool giant exactly solvable matrix equation! > >> Let i and j be integers (indices) that range from {0,...,H}. >> >> Let M be an (H+1)x(H+1) matrix with the following matrix elements: >> M(i,j) = g_i^j = g_i to the power of j >> where {g_0, g_1, ..., g_H} are H+1 positive real numbers inside the >> range (0,R). >> Let v be an (H+1) vector with matrix elements >> v(i) = g_i / (R-g_i) >> Let z be the (H+1) vector determined by solving the matrix equation: >> M z = v >> Find the solution for the H+1 matrix elements of z for arbitrary H. >> (Hint: Using mathematica, I have examined the solutions for H=2, 3, 4, >> 5. >> It seems the solutions fall into a simple >> pattern that should allow one to write down a closed-form solution. >> Anybody recognize what that would be?) >the matrix in the first part appears to be a Vandemonde type (qv on the web) >which has a known polynomial determinant. Not sure about explicit inverses >or the second part. The inversion of a Vandermonde matrix amounts to polynomial interpolation, i.e. Mz = v says that the entries z_i are coefficients of a polynomial P(t) = sum_{j=0}^H z_j t^j such that P(g_i) = v_i for each i. The Lagrange interpolation formula says P(t) = sum_i v_i product_{j <> i} (t-g_j)/(g_i-g_j) Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Model Aircraft Design > If the ring diameters are 10 and 12 inches as you state, then as far as > I can see the lengths of the straight top and bottom edges of the vanes > will be approximately one inch (the difference between the radii) - but > not exactly one inch... they will in fact be slightly longer as these > edges are not exactly radial. Where does the 2 inches come from? > Ooops. Yes you are correct. should have been one inch as you state. And yes, I can see why the top and bottom of the vanes will not be exactly 1 but slightly longer. The width halfway up the vane will be exactly 1 wide correct ? Tim === Subject: Re: Model Aircraft Design <3kci55Ftpjr0U1@individual.netIf the ring diameters are 10 and 12 inches as you state, then as far as > I can see the lengths of the straight top and bottom edges of the vanes > will be approximately one inch (the difference between the radii) - but > not exactly one inch... they will in fact be slightly longer as these > edges are not exactly radial. Where does the 2 inches come from? > Ooops. Yes you are correct. should have been one inch as you state. And yes, I can see why the top and bottom of the vanes will not be exactly > 1 but slightly longer. The width halfway up the vane will be exactly 1 > wide correct ? Tim Yes, I think so. The way I figure it is that you draw two concentric ellipses: x^2 + y^2/2 = 6^2 x^2 + y^2/2 = 5^2 Then you draw two straight lines y = +sqr(2) y = -sqr(2) And the ellipses and lines then delineate the shape of the vane. I make the length of the straight edges of the vane about 1.0171 inches. Also, I had a go at the circle approximation thing. Let A and B respectively be the intersections of the two straight lines with one of the ellipses (+ve x). Let C be the intersection of the ellipse with the x-axis (+ve x). Draw a circle through A, B and C (centre lying on the x-axis). This is not the best circle approximation, but it's relatively easy to work out. For the outer ellipse I make the radius of the aproximating circle 11.9580 inches (to 4 d.p.) For the inner ellipse I make the radius of the approximating circle 9.9495 inches. Measuring the error in the circle approximation as the difference between the circle's radius and the length of a line through the circle's centre to the ellipse, I make the maximum error about 0.000074 inches for the outer ellipse, and about 0.00013 inches for the inner. Good luck with your flying machine... hope you get it to work! === Subject: Historical remarks: Irrationality of e and Liouville Transcendental Dimensions): > It's fairly easy to prove that e is transcendental, I think. > It might even have been proved before Liouville's theorem > was published? Charles Hermite (1822-1901) proved that e was transcendental in 1873, while Liouville's work on transcendental numbers occurred during the 1840's. The following remarks are from a March 4, 2005 sci.math post of mine: > For example, recently I've been working on some issues > concerning the size of the set of Liouville numbers > and its complement, and I can't begin to tell you how > many times I've read that in 1844 Liouville published > the well-known theorem concerning how well algebraic > numbers of degree n can be approximated by rational > numbers. No, no, no! Liouville might have known about > this in 1844, but the result itself didn't appear in > print until Liouville's 1851 paper on transcendental > numbers. Liouville's 1844 transcendental proofs > involved numbers given as continued fractions, > and it just so happens that continued fractions > provide very efficient rational approximations, > but at that time (at least in print) Liouville > had not yet identified the more general approximation > property in our present definition of the Liouville > numbers as the fundamental principle at work in > his proofs. Incidentally, L.9ftzen (see [9] below, p. 521) has this to say about Liouville's 1844 transcendence papers: Liouville did not prove [the result concerning approximations of algebraic numbers by rationals in his 1844 publications], but his notes [unpublished journal notes of Liouville's that L.9ftzen had access to] make a reconstruction of his proof possible ... Here are some Liouville papers about these issues that are on the internet. The 1844 transcendental papers are not among these, however. [His 1844 papers were mostly brief announcements with some proof sketches, and the complete details didn't appear until his 1851 paper.] [A] Joseph Liouville, Sur l'irrationnalit.8e du nombre e = 2,718..., Journal de Math.8ematiques Pures et Appliqu.8ees (= Liouville's Journal) (1) 5 (May 1840), 192. Liouville's paper is on-line at http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm This paper modifies Fourier's method of proving e is irrational (see below) to prove that e is not quadratically irrational. (This doesn't follow from the already known fact that e^r is irrational for every nonzero rational number, by the way.) [B] Joseph Liouville, Addition a la note sur l'irrationnalit.8e du nombre e, Journal de Math.8ematiques Pures et Appliqu.8ees (= Liouville's Journal) (1) 5 (June 1840), 193-194. Liouville's paper is on-line at http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm This paper extends the proof of Liouville [A] to prove that e^2 is not quadratically irrational. [C] Joseph Liouville, Sur la limite de (1 + 1/m)^m, m .8etant un entier positif qui croft ind.8efiniment, Journal de Math.8ematiques Pures et Appliqu.8ees (= Liouville's Journal) (1) 5 (1840), 280. Liouville's paper is on-line at http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm This paper gives a proof that (1 + 1/m)^m approaches e as the real variable m approaches infinity. Cauchy had 'proved' this in his 1821 book Cours d'Analyse, but Cauchy's proof involved an unsupported interchange of the operation limit as m --> infinity with the operation corresponding to the infinite summation associated with the binomial expansion of (1 + 1/m)^m (they were proving this for positive real number values of m, not just positive integer values of m). [D] Joseph Liouville, Sur des classes tr.8fs-.8etendues de quantit.8es dont la valeur n'est ni alg.8ebrique, ni meme r.8eductible .87 des irrationnelles alg.8ebriques, Journal de Math.8ematiques Pures et Appliqu.8ees (= Liouville's Journal) (1) 16 (1851), 133-142. Liouville's paper is on-line at http://gallica.bnf.fr/Catalogue/noticesInd/FRBNF34348784.htm Below is an excerpt from a much longer essay on types of numbers [rational, Pythagorean/Hilbert field, constructible, constructible with marked rulers, expressible in terms of real radicals, expressible in terms of radicals, algebraic, elementary in various ways (Joseph Ritt, Timothy Chow, etc.), provably recursive in various ways, recursive, various levels of montonically recursive classes, arithmetical hierarchy, hyper-arithmetical, etc.] that I've been working on, off and on, for the past year or so. I don't know when it'll eventually get finished (the Usenet version), but I'm simultaneously working on a much more extensive LaTeX manuscript version that eventually I'll make available either through publication or in the mathematics arXiv at http://front.math.ucdavis.edu/ In 1714 Roger Cotes obtained an infinite continued fraction expansion for e-1. Although this can be used to establish the irrationality of e-1, and hence the irrationality of e, it doesn't appear as if Cotes ever attempted to draw any irrationality conclusions from his result. In 1737 Leonhard Euler obtained the same infinite continued fraction expansion for e-1 that Cotes did, as well as an infinite continued fraction expansion for (e+1)/(e-1). In addition, Euler argued that these expansions showed they were irrational, and Euler was then able to argue fairly convincingly, although perhaps not entirely rigorously, that e and e^2 were irrational. Note that the irrationality of e^2 implies the irrationality of e, but not conversely (since there exist irrational numbers whose squares are rational). However, the irrationality of (e+1)/(e-1) is equivalent to the irrationality of e, since it is clear that e rational implies (e+1)/(e-1) is rational, and the identity (x+1)/(x-1) = e when x = (e+1)/(e-1) gives the other direction. My source for these remarks about Cotes and Euler is Kline [7], pp. 459-460. Kline [7] does not indicate how Euler proved e^2 is irrational, and so I do not know if this can be deduced purely from the irrationality of e and of (e+1)/(e-1) (which I haven't been able to do), or whether something about the specific nature of the continued fraction representations of these numbers (or something else) was used by Euler. However, according to Pringsheim [11] (p. 327), it appears that Euler also obtained continued fraction expansions for (e^2 - 1)/2 and [e^(1/3) - 1]/2. Thus, perhaps when Kline [7] commented that Euler essentially obtained the irrationality of e^2, Kline should have additionally mentioned that Euler had obtained an expansion for (e^2 - 1)/2. In 1761 Johann Heinrich Lambert continued the continued fraction investigations of Euler and proved that e^x and tan(x) (radian measure) were each irrational for every nonzero rational number x. Thus, Lambert proved: * e^x is irrational for all nonzero rational values of x. [But note this leaves unresolved the irrationality of numbers such as e^2 + e, 4e^5 + 3e^2 - e, etc.] * ln(x) is irrational for all positive rational numbers x. [If ln(x) were rational for x different from 1, then exp(ln x) would be the exponential of a nonzero rational number, and hence irrational.] * Pi is irrational. [Consider tan(Pi/4).] For Lambert's proof, see Hobson [5] (Sections #302-303, pp. 374-375), Laczkovich [8], Stevens [15], Struik [16] (Chapter V.17: Lambert. Irrationality of Pi, pp. 369-374), and Wallisser [17]. In 1794 Adrien-Marie Legendre proved that Pi^2 is irrational, which also implies the irrationality of Pi. Legendre conjectured in Note 4 of his 1794 paper that Pi is not the root of any polynomial with rational coefficients (i.e. that Pi is a transcendental number). It is very easy to prove x irrational ==> 1/x irrational and x irrational ==> x^(1/n) irrational for each n = 1, 2, 3, .... Thus, by 1794 it was known that Pi^r is irrational for each nonzero rational number r such that, when expressed in lowest terms m/n, m is one of -2, -1, 1, 2 and n is any positive integer. Incidentally, the ideas above can be used to give a simple proof of the following theorem. This result is probably well known but I don't recall having seen it in print anywhere. Theorem: If x^n is irrational for each n = 1, 2, 3, ..., then x^r is irrational for each nonzero rational number r. As an application, it is easy to prove that every positive integer power of 1 + sqrt(2) is irrational. Hence, it follows that every nonzero rational power of 1 + sqrt(2) is irrational. It is instructive to note that irrationality results about e are much easier to prove than corresponding irrationality results about Pi. This is due to the fact that the continued fraction expansion for e is much simpler than the continued fraction expansion for Pi and the nice differentiability properties of e^x (which imply, among other things, that all the coefficients in the series expansion of e^x about x=0 are rational). Indeed, the transcendence of Pi was proved as a consequence of having sufficiently strong results involving e so that the identity e^(Pi*i) = -1 could be used to draw conclusions about Pi. It is sometimes claimed that Lambert's proofs were not entirely rigorous. Thus, besides the additional results that Legendre proved, Legendre is sometimes credited with giving the first rigorous proofs that e and Pi are irrational. I said this in Renfro [12], for instance, having seen it mentioned on p. 401 of Smith [14], among other places. However, these claims that Lambert's proofs were incomplete (in a footnote on p. 521, Wallisser [17] lists 8 such references besides the two I mention in this paragraph) appear to have been based on an incorrect assessment of Lambert made on p. 67 of Rudio's influential historical survey [13]. According to Archibald (see Archibald [1], pp. 253-254, or what is essentially the same thing, Archibald's notes to Part II of Chapter II in Klein [7], pp. 88-90), Klein's comments on p. 59 of [7], which seem to imply that the irrationality of Pi was not rigorously established until Legendre's work, were likely based on Rudio's [13] comments on the matter. Incidentally, Lambert's rigor was supported in at least one paper before Rudio [13], which adds to the mystery of Rudio's criticism of Lambert's proofs. Although Legendre's method is quite as rigorous as that on which it is founded, still, on the whole, the demonstration of Lambert seems to afford a more striking and convincing proof of the truth of the proposition; his investigation, however, is given in such detail, and so many properties of continued fractions, no well known, are proved, that it is not very easy to follow his reasoning, which extends over more than thirty pages. The object of the present paper is to exhibit Lambert's demonstration of this important theorem concisely, and in a form free from unnecessary details, and to apply his method to deduce some results with regard to the irrationality of certain circular and other functions. (p. 12 of Glaisher [4]) That Lambert's proof is perfectly rigorous and places the fact of the irrationality of Pi beyond all doubt, is evident to every one who examines it carefully; and considering the small attention that had been paid to continued fractions previously to the time at which it was written, it cannot but be regarded as a very admirable work. (p. 14 of Glaisher [4]) Pringsheim [11], who does not mention Glaisher's paper [4], investigated Lambert's proof and found it to be perfectly adequate, and then Pringsheim [11] went on to speculate as to why Rudio might have claimed Lambert's proofs were lacking. In 1892 F. Rudio stated that Lambert's proof was not sufficient and that Legendre had supplied the deficiency. This statement is an error, as has been shown by Pringsheim's careful study. He found Lambert's work was more rigorous than Legendre's. (p. 494 of Mitchell/Strain [10]) At the turn of the century the University of Munich had several professors, such as Pringsheim and Tietze, whom were interested in continued fractions. In 1898 Pringsheim e and Pi. A large part of the paper is devoted to the question of how Rudio had arrived at his conclusion about the gap in Lambert's proof. Pringsheim states: Lambert has written two papers on the quadrature of the circle; a popular one entitled: Vorl.8aufige Kenntnisse f.9fr die, so die Quadratur und die Rectification des Circuls suchen; and a scientific one entitled: M.8emoire sur quelques propriet.8es remarquables des quantit.8es transcendantes circulaires et logarithmiques. The former serves more as an orientation to the problem and gives a good but very general description of his results. Obviously Rudio only considered this paper, which is completely reprinted in his monograph on Archimedes, Huygens, Lambert, Legendre. (p. 522 of Wallisser [17]) Brezinski [2] (pp. 110-111) also discusses some of these issues. [1] Raymond Clare Archibald, Remarks on Klein's 'Famous Problems of Elementary Geometry', The American Mathematical Monthly 21 (1914), 247-259. [JFM 45.0742.03] http://www.emis.de/cgi-bin/JFM-item?45.0742.03 [2] Claude Brezinski, HISTORY OF CONTINUED FRACTIONS AND PAD.83 APPROXIMANTS, Springer-Verlag, 1991. [MR 92c:01002; Zbl 714.01001] http://www.emis.de/cgi-bin/MATH-item?0714.01001 Reviewed by Wolfgang J. Thron in Mathematics of Computation 62 (1994), 432-433. [3] George Chrystal, ALGEBRA: AN ELEMENTARY TEXTBOOK FOR THE HIGHER CLASSES OF SECONDARY SCHOOLS AND FOR COLLEGES, 7'th Edition, two parts, American Mathematical Society, 1999. [MR 22 #12066; Zbl 91.01402; JFM 18.0051.01; JFM 21.0073.01] http://www.emis.de/cgi-bin/Zarchive?an=0091.01402 http://www.emis.de/cgi-bin/JFM-item?18.0051.01 http://www.emis.de/cgi-bin/JFM-item?21.0073.01 Chapter XXXII, Section 17 (pp. 512-514) gives a couple of irrationality results on continued fractions that he attributes to Legendre. Chrystal doesn't say whether the proofs are Legendre's original proofs. [4] James Whitbread Lee Glaisher, On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities, Report of the British Association for the Advancement of Sciences, 1871, 12-16. [JFM 3.0198.04] http://www.emis.de/cgi-bin/JFM-item?03.0198.04 [5] Ernest W. Hobson, A TREATISE ON PLANE AND ADVANCED TRIGONOMETRY, 7'th edition, Dover Publications, 1928/1957. [MR 19,876h; Zbl 78.13205] http://www.emis.de/cgi-bin/Zarchive?an=0078.13205 [6] Ernest W. Hobson, Squaring the circle: A history of the problem, pp. 1-57 in SQUARING THE CIRCLE AND OTHER MONOGRAPHS, Chelsea Publishing Company, 1913/1957. [JFM 44.0050.02; Zbl 52.16301; MR 14,1114a] http://www.emis.de/cgi-bin/JFM-item?44.0050.02 http://www.emis.de/cgi-bin/Zarchive?an=0052.16301 Hobson's essay is on-line at http://name.umdl.umich.edu/ABN2635 appeared in 1895 and the first English translation in 1897. This is a reprint of the 1930 English translation which included some notes by Raymond Archibald.] [JFM 28.0438.05; Zbl 70.38003; MR 17,883e] http://www.emis.de/cgi-bin/JFM-item?28.0438.05 http://www.emis.de/cgi-bin/Zarchive?an=0070.38003 The 1897 English translation of Klien's book is on-line at http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABN2381 [8] Mikl.97s Laczkovich, On Lambert's proof of the irrationality of Pi, The American Mathematical Monthly 104 (1997), 439-444. [MR 98a:11090; Zbl 876.11032] http://www.emis.de/cgi-bin/MATH-item?0876.11032 http://www.maa.org/pubs/monthly may97 toc.html [9] Jesper L.9ftzen, JOSEPH LIOUVILLE (1809-1882): MASTER OF PURE AND APPLIED MATHEMATICS, Springer-Verlag, 1990. [MR 91h:01067; Zbl 701.01015] http://www.emis.de/cgi-bin/MATH-item?0701.01015 Reviewed by E. Zitarelli in Historia Mathematica 20 (1993), 205-210. [10] Ulysses Grant Mitchell and Mary Strain, The number e, Osiris 1 (1936), 476-496. [Zbl 14.24401; JFM 62.0003.06] http://www.emis.de/cgi-bin/Zarchive?an=0014.24401 http://www.emis.de/cgi-bin/JFM-item?62.0003.06 [11] Alfred Pringsheim, Ueber die ersten Beweise der Irrationalit.8at von e und Pi, Sitzungsberichte der Bayerischen Akademie der Wissenschaften Mathematisch-Physikalische Klasse 28 (1898), 325-337. [JFM 29.0373.08] http://www.emis.de/cgi-bin/JFM-item?29.0373.08 [12] Dave L. Renfro, sci.math posts, 25 March 2001. Extensive list of references (print and internet) for proofs that (1) e is irrational, (2) Pi is irrational, (3) e is transcendental, and (4) Pi is transcendental. ABHANDLUNGEN .86BER DIE KREISMESSUNG, versehen, Leipzig. B. G. Teubner (Leipzig), 1892. [JFM 24.0050.02] http://www.emis.de/cgi-bin/JFM-item?24.0050.02 [14] Eugene Smith, The history and transcendence of Pi, pp. 388-416 in Jacob W. Young (editor), MONOGRAPHS ON TOPICS OF MODERN MATHEMATICS, Dover Publications, 1911/1955. [Zbl 67.29202] http://www.emis.de/cgi-bin/Zarchive?an=0067.29202 The book this essay is in can be found on-line by searching for 'Young' using Identify works by author and title at http://www.hti.umich.edu/u/umhistmath/ Mathematischen Gesellschaft in Hamburg 18 (1999), 151-158. [MR 2000i:11114; Zbl 1045.11049] http://www.emis.de/cgi-bin/MATH-item?1045.11049 Harvard University Press, 1969. [MR 39 #11; Zbl 205.29202] http://www.emis.de/cgi-bin/Zarchive?an=0205.29202 [17] Rolf Wallisser, On Lambert's proof of the irrationality of Pi, pp. 521-530 in Franz Halter-Koch and Robert F. Tichy (editors), ALGEBRAIC NUMBER THEORY AND DIOPHANTINE ANALYSIS, Walter de Gruyter & Co., 2000. [MR 2001h:01022; Zbl 973.11005] http://www.emis.de/cgi-bin/MATH-item?0973.11005 ************** 3. FOURIER'S PROOF THAT e IS IRRATIONAL *************** Although Euler is credited with proving the irrationality of e in 1737, Euler's proof is not the well known proof (see [18]) that makes use of the series expansion e = 1/0! + 1/1! + 1/2! + 1/3! + ... The well known proof of the irrationality of e that appears in most textbooks is due to Joseph Fourier, although this doesn't seem to be very widely known. For example, there is no mention of Fourier in this regard in Maor's semi-historical book [21] about e. At least Maor doesn't claim that this proof is due to Euler, which Gourdon/Sebah [20] do: Euler gave in 1737 a very elementary proof of the irrationality of e based on the sequence ... ([20], accessed 22 July 2005). Maor [21] simply says on p. 192 that Euler first proved the irrationality of e in 1737, and then Maor [21] cites Courant/Robbins's book WHAT IS MATHEMATICS for the well known series proof that Maor [21] gives on pp. 202-203. Brabenec [19] (Problem 23, p. 82) also incorrectly attributed Fourier's proof to Euler: Euler used a Maclaurin series as the key element in his proof, ... What follows is all that I've managed to uncover so far about the historical background of Fourier's proof that e is irrational. Hobson [6] (p. 44) and Ribenboim [22] (pp. 285 & 301) say this proof is due to Joseph Fourier (1768-1830). Both cite Stainville's 1815 book M.83LANGES D'ANALYSE ALG.83BRIQUE ET DE G.83OMETRIE, possibly implying that the proof first appeared here, although neither explicitly says this. L.9ftzen [9] (p. 516) also mentions that this proof is due to Fourier, and L.9ftzen cites pp. 57-58 of Rudio [13], a book that I have not yet seen a copy of. However, Rudio [13] says the usual textbook proof for the irrationality of e appears on p. 339 of Stainville's book and that Rudio says this proof is due to Fourier. Finally, Mitchell/Strain [10] (p. 495) and Ross [23] (p. 72) state that this proof was given by Fourier in 1815. I don't know whether Fourier came up with this proof prior to 1815, nor does L.9ftzen know. [18] The Math Forum: Ask Dr. Math: The Irrationality of e. http://mathforum.org/library/drmath/view/53910.html [19] Robert L. Brabenec, RESOURCES FOR THE STUDY OF REAL ANALYSIS, Classroom Resource Materials, Mathematical [Zbl 1059.26001] http://www.emis.de/cgi-bin/MATH-item?1059.26001 [20] Xavier Gourdon and Pascal Sebah, Irrationality proofs, Numbers, Constants and Computation web pages. http://tinyurl.com/4c62w [21] Eli Maor, e: THE STORY OF A NUMBER, Princeton University Press, 1994. [MR 95a:01002; Zbl 805.01001] http://www.emis.de/cgi-bin/MATH-item?0805.01001 [22] Paulo Ribenboim, MY NUMBERS, MY FRIENDS: POPULAR LECTURES ON NUMBER THEORY, Springer--Verlag, 2000, xii + 375 pages. [MR 2002d:11001; Zbl 947.11001] http://www.emis.de/cgi-bin/MATH-item?0947.11001 [23] Marty Ross, Irrational thoughts, The Mathematical Gazette ***************************************************************** Dave L. Renfro === Subject: Re: Historical remarks: Irrationality of e and Liouville > > ... I read Irrationality of (e and Liouville) before I read (Irrationality of e) and Liouville. -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: Pulsar density to aid in alien-count Re: Computing how much life exists in the Cosmos Re: Some relief as to the question of whether a individual thought comprises 10^20 photons or whether a single individual thought can occur from that of 1 single photon. The answer to that will differentiate whether life is cosmically abundant or whether life is sparse. I suppose a method of checking into which of those is correct is to ask for the cosmic density of neutrinos. A given point in space, how many neutrinos pass through that given point per time. Let us make that point the size of a human brain. So how many neutrinos pass through a given brain per second? Is it on the order of 10^20 or is it on the order of 1. The answer may even surprize myself. In that the basis of thought and thinking and intelligent life is ultimately based on neutrino density or neutrino cross section. Forget the analogy that a thought is a lightbulb of 10^20 photons per second but rather that neutrino cross section is a thought. So what is the neutrino density of a human brain per second? How many neutrinos traverse a human brain per second? Is it much larger than 10^20? Perhaps neutrino cross section is the key to intelligence in the universe and thus key to the equation of amount of life in the Universe. Equation for Amount of Life in Universe: 10^148 = 10^78 multiply 10^70 where the amount of life is contained in the 10^70 term. So that if neutrino cross section is much larger than 10^20 per instant of time--(second of time??) then life is rare in the cosmos and we maybe alone. If neutrino cross section equals about 10^20 then life in the cosmos is patterned and sparse. But if neutrino cross section is small then life is very abundant and that perhaps every solar system contains some life therein. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Pulsar density to aid in alien-count Re: Computing how much life exists in the Cosmos Re: Come to think of it. Is the total number of neutrinos in the Universe equal to a number that of 10^148?? The number 10^148 is the number of total Coulomb Interactions in a atom of plutonium and since the Universe is 231Pu Atom Totality then there are 10^148 photons in the Cosmos. But what about the total number of neutrinos in the cosmos? Is the total number of neutrinos that of 10^70?? Or is it that of 10^148. If my equation of life is accurate: Equation for Amount of Life in Universe: 10^148 = 10^78 multiply 10^70 where life is contained in the 10^70 term then it must be linked with the total number of neutrinos in the cosmos. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? > But what are the postulates? What are the laws? > Laws are simply very good theories, for instance; The law of gravity is a theory. CARLSON: Ms. Scott -- hold on. That's not -- in some ways, that's not really the question. I mean, the question is: Shall we admit the truth that evolution is a theory? It's the theory of evolution, not the law of evolution. And what's wrong with admitting that? SCOTT: Well, in science, a theory is an explanation. Of course evolution is a theory, just like gravitation. But what we should be... CARLSON: Wait, I thought gravity was a law. The law of gravity, right... SCOTT: No, gravity... CARLSON: ... or is this so far over my head I don't know what you're talking about? I thought it was a law. SCOTT: Well, I'll tell you what, if you drop something, it's going to fall. That's an observation: unsupported things fall. But you explain that observation with the theory of gravity, which is that the mass of what whatever it is you dropped, a pencil or a pen or something, is attracted by the mass... CARLSON: Well you are blowing my mind... SCOTT: That's not an observation. CARLSON: ... law of gravity. Honestly, is it not the law, it's really a theory of gravity? SCOTT: It's a theory of gravity. But remember, a theory is an explanation. SPRIGG: ... should point out, Scott, though, that theories of origins and theories that are testable in terms of current experimentation are somewhat different in a scientific perspective. We can't experimentally confirm evolution. SCOTT: Sure we can... CNN Crossfire: Secret Court Stymies Justice Department; Creationists Square off with Evolutionists; Should Bush Be Telling Americans to Exercise? http://www.cnn.com/TRANSCRIPTS/0208/24/cf.00.html > We can all name Newton's Laws and Einstein's Postulates, which lie at > the base of classical and relativistic physics. But what are String Theory's postulates and laws? Perhaps there are none? Moving Dimensions Thoery has a postulate: The General Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions. The Specific Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions at the rate of c in quantized units of the Planck length. Classical physics, quantum mechanics, and relativity descend from this > simple postulate. Light, and thus all energy, is quantized as the > dimension which transports it expands in a quantized manner. Light > travels at a constant velocity in all frames because velocity is > measured relative to time which is measured relative to the light that > is transported by the fourth expanding dimension. Thus both > fundamental constants h and c emerge from the fundamental nature of the > expansion of the fourth dimension relative to the three spatial > dimensions. And thus MDT provides a simple, unifying postulate > accounting for the classical, relativistic, and quantum mechanical > properties of this universe. > > http://physicsmathforums.com === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? > strings resolves the incompatibility between quantum mechanics and > general relativity (which, as currently formulated, cannot both be > right). What exactly is this incompatibility? Incidentally, if string theory can resolve the incompatibility then _it_ must be incompatible with quantum mechanics or general relativity. > properties-that is, the different masses and other properties of both > four forces of nature (the strong and weak nuclear forces, > electromagnetism, and gravity)-are a reflection of the various ways > in which a string can vibrate. How does this resolve the incompatibility? -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? <5G3Ee.2794$R5.627@news.indigo.iestrings resolves the incompatibility between quantum mechanics and > general relativity (which, as currently formulated, cannot both be > right). > What exactly is this incompatibility? A group of human ideas that have been translated into the abstract code called human language. > Incidentally, if string theory can resolve the incompatibility > then _it_ must be incompatible with quantum mechanics or general relativity. Are you claiming that ALL aspects of string theory are incompatible with ALL aspects of quantum mechanics or general relativity therefore SOME aspects of string theory cannot be compatible with SOME aspects of quantum mechanics or general relativity? http://www.philosophypages.com/lg/e07a.htm > properties-that is, the different masses and other properties of both > four forces of nature (the strong and weak nuclear forces, > electromagnetism, and gravity)-are a reflection of the various ways > in which a string can vibrate. > How does this resolve the incompatibility? It was not intended to resolve the incompatibility but was merely the provision of basic information that could be used to create some smooth sounding laws, as requested. > -- > Timothy Murphy > e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie > tel: +353-86-2336090, +353-1-2842366 > s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? I've read those books, ``Quantum Fields and Strings.'' So far I have not found any postulates nor laws of String Theory. Here they are Einstein's postulates of relativity: 1. First postulate (principle of relativity) The laws of electrodynamics and optics will be valid for all frames in which the laws of mechanics hold good. Every physical theory should look the same mathematically to every inertial observer. The laws of physics are independent of location space or time. 2. Second postulate (invariance of c) The speed of light in vacuum, commonly denoted c, is the same to all inertial observers, is the same in all directions, and does not depend on the velocity of the object emitting the light. When combined with the First Postulate, this Second Postulate is equivalent to stating that light does not require any medium (such as aether) in which to propagate. Here're Newton's 3 Laws of Motion: I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. II. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Acceleration and force are vectors (as indicated by their symbols being displayed in slant bold font); in this law the direction of the force vector is the same as the direction of the acceleration vector. III. For every action there is an equal and opposite reaction. And here's the postulate of Moving Dimensions Theory that unifies them all, along with the phenomena in quantum mechanics: The General Postulate of Moving Dimensions Theory: The fourth dimension is expanding relative to the three spatial dimensions. The Specific Postulate of Moving Dimensions Theory: The fourth dimension is expanding relative to the three spatial dimensions at the rate of c in quantized units of the Planck length. Classical physics, quantum mechanics, and relativity descend from this simple postulate. Light, and thus all energy, is quantized as the dimension which transports it expands in a quantized manner. Light travels at a constant velocity in all frames because velocity is measured relative to time which is measured relative to the light that is transported by the fourth expanding dimension. Thus both fundamental constants h and c emerge from the fundamental nature of the expansion of the fourth dimension relative to the three spatial dimensions. And thus MDT provides a simple, unifying postulate accounting for the classical, relativistic, and quantum mechanical properties of this universe. But where are String Theory's postulates? http://physicsmathforums.com === Subject: Re: What are String Theories Laws and Postulates. The General Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions. The Specific Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions at the rate of c in quantized units of the Planck length. > [EL] Dimensions do not move. In an attempt to educate you, here are some examples of base dimensions: [L] = Length. [M] = Mass. [T] = Time. [F] = Force. [Q] = Charge. There are also many derived dimensions such as: Action = [LFT] = [L^2.M.T^-1] Work = [LF] = [L^2.M.T^-2] As you can see, if you can see, dimensions are arbitrated abstracts representing phenomenal continua. Each continuum then, must contain observable changes of state we call variance, which we need to quantify by measuring such variance or variable quantities against an invariant scale that we arbitrate as a standard. If you are not mentally ill, you should realise by now that you were confused, and that you have been confusing the being measured with the invariant scale with which it is being measured. How can we measure the length of your fingers if we were constantly hitting them with the ruler that should stand still against them to take a reading! Forget about the universe and come down to earth with me for a while. We shall make a shirt of you and we need to measure the perimeter of your chest while you are breathing. When you inhale, your chest expands and the measuring strip end slides against the scale in my hand to a maximal, indicating the longest perimeter of your chest being measured against an invariant scale printed on the measuring strip. That strip did not stretch, or else the quantification would be unreliable and meaningless. All what we wanted was to map your dimensions to cut the cloth to fit you, and such mapping demands a reliable scale that does not change between measuring you and measuring cloth. Did you understand, OR NOT YET! EL === Subject: Re: What are String Theories Laws? What Are String Thoery's Postulates? Has String Theory Accomplished Anything? How Much Has String Theory Cost Us? > I've read those books, ``Quantum Fields and Strings.'' So far I have not found any postulates nor laws of String Theory. Here they are Einstein's postulates of relativity: 1. First postulate (principle of relativity) > The laws of electrodynamics and optics will be valid for all frames in > which the laws of mechanics hold good. The POR as stated by Einstein. However these days it is changed to all the laws of nature. > Every physical theory should look the same mathematically to every > inertial observer. Follows from the POR. > The laws of physics are independent of location space or time. Follows from the definition of inertial frame so is not a postulate. 2. Second postulate (invariance of c) > The speed of light in vacuum, commonly denoted c, is the same to all > inertial observers, is the same in all directions, and does not depend > on the velocity of the object emitting the light. When combined with > the First Postulate, this Second Postulate is equivalent to stating > that light does not require any medium (such as aether) in which to > propagate. It is not equivalent to stating an aether does not exist. For example nothing in our current laws forbids EM to be governed by the Proca lagrangain which does not require an aether, gives light a very small but non zero mass, and means the speed of light is not the same for all inertial observers. Here're Newton's 3 Laws of Motion: I. Every object in a state of uniform motion tends to remain in that > state of motion unless an external force is applied to it. > II. The relationship between an object's mass m, its acceleration a, > and the applied force F is F = ma. Acceleration and force are vectors > (as indicated by their symbols being displayed in slant bold font); in > this law the direction of the force vector is the same as the direction > of the acceleration vector. III. For every action there is an equal and opposite reaction. And here's the postulate of Moving Dimensions Theory that unifies them > all, along with the phenomena in quantum mechanics: The General Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions. The Specific Postulate of Moving Dimensions Theory: > The fourth dimension is expanding relative to the three spatial > dimensions at the rate of c in quantized units of the Planck length. Classical physics, quantum mechanics, and relativity descend from this > simple postulate. Light, and thus all energy, is quantized as the > dimension which transports it expands in a quantized manner. Light > travels at a constant velocity in all frames because velocity is > measured relative to time which is measured relative to the light that > is transported by the fourth expanding dimension. Thus both > fundamental constants h and c emerge from the fundamental nature of the > expansion of the fourth dimension relative to the three spatial > dimensions. And thus MDT provides a simple, unifying postulate > accounting for the classical, relativistic, and quantum mechanical > properties of this universe. Moving dimension theory has been examined on this newsgroup many times and is obvious crap of the first order. But where are String Theory's postulates? > Idiot. Bill > http://physicsmathforums.com > > === Subject: Re: A Priority Queueing Question with Bulk Arrival and Renein === Subject: Re: Motivation for matrix algebra <17243100.1121936074263.JavaMail.jakarta@nitrogen.mathforum.orgmatrix D representing the linear map d with respect to > the basis (1,x) is the 2x2 matrix 0 0 > 0 1 I apppreciate your reponse. I'm spending time to understand the origin of all of the requirements, such as why the axioms are defined as such, multiplication as such, etc. That is, going right down to the fundamental thought processes that must have occured within mathematicians minds when developing the various parts of the theory. At the present time that has me figuring out why linear functions are of such seeming importance and applicable to so much in the world. I will write back once I have learned more. === Subject: Proof that f(x) = ax + b is not a linear function! :) A linear function satisfies: 1) f(x + y) = f(x) + f(y) 2) f(nx) = nf(x) for all scalars, n. See http://mathworld.wolfram.com/LinearFunction.html Suppose f(x) = ax + b defines a function where a and b are scalars. Then f(x + y) = a[x + y] + b = ax + ay + b and f(x) + f(x) = [ax + b] + [ay + b] = ax + ay + 2b and so f is not a linear function for all scalars a and b. We show that 2) fails as well for good measure. f(nx) = a[nx] + b but nf(x) = n[ax + b] = a[nx] + nb therefore f is not a linear function for all scalars a and b. Please inform the mathematical community of this fundamental discovery! We are sorry that so many texts will have to be updated. :) === Subject: Re: Proof that f(x) = ax + b is not a linear function! :) :-) > A linear function satisfies: > 1) f(x + y) = f(x) + f(y) > 2) f(nx) = nf(x) for all scalars, n. > See http://mathworld.wolfram.com/LinearFunction.html > > > Suppose f(x) = ax + b defines a function where a and b are scalars. > > Then f(x + y) = a[x + y] + b = ax + ay + b > and f(x) + f(x) = [ax + b] + [ay + b] = ax + ay + 2b > and so f is not a linear function for all scalars a and b. > > > We show that 2) fails as well for good measure. > > f(nx) = a[nx] + b > but nf(x) = n[ax + b] = a[nx] + nb > therefore f is not a linear function for all scalars a and b. > > > Please inform the mathematical community of this fundamental discovery! > We are sorry that so many texts will have to be updated. :) > === Subject: Re: Proof that f(x) = ax + b is not a linear function! :) >A linear function satisfies: > 1) f(x + y) = f(x) + f(y) > 2) f(nx) = nf(x) for all scalars, n. > See http://mathworld.wolfram.com/LinearFunction.html > Suppose f(x) = ax + b defines a function where a and b are scalars. Then f(x + y) = a[x + y] + b = ax + ay + b > and f(x) + f(x) = [ax + b] + [ay + b] = ax + ay + 2b > and so f is not a linear function for all scalars a and b. > We show that 2) fails as well for good measure. f(nx) = a[nx] + b > but nf(x) = n[ax + b] = a[nx] + nb > therefore f is not a linear function for all scalars a and b. > Please inform the mathematical community of this fundamental discovery! > We are sorry that so many texts will have to be updated. :) congratulations, you have shown that there is a difference between linear functions and vector space homomorphisms R->R. if b=0, a linear function is a vector space homomorphism for all a. === Subject: Re: Proof that f(x) = ax + b is not a linear function! :) > A linear function satisfies: > 1) f(x + y) = f(x) + f(y) > 2) f(nx) = nf(x) for all scalars, n. > See http://mathworld.wolfram.com/LinearFunction.html > Suppose f(x) = ax + b defines a function where a and b are scalars. Then f(x + y) = a[x + y] + b = ax + ay + b > and f(x) + f(x) = [ax + b] + [ay + b] = ax + ay + 2b > and so f is not a linear function for all scalars a and b. > We show that 2) fails as well for good measure. f(nx) = a[nx] + b > but nf(x) = n[ax + b] = a[nx] + nb > therefore f is not a linear function for all scalars a and b. > Please inform the mathematical community of this fundamental discovery! > We are sorry that so many texts will have to be updated. :) Can you cite an example of such a text ?? I don't know of any that would make the mistake of claiming that your 'f' is linear -- it's affine, but _not_ linear ... (Don't be confused by the fact that f's graph is a straight line -- that's not at all the same as being linear ... ) === Subject: Re: Proof that f(x) = ax + b is not a linear function! :) It seems calculus texts do so, as well as many other algebra ones. It seems to be normal for ax + b to be termed a linear function. Check out many of the links here: Some are from universities. :) === Subject: Re: Proof that f(x) = ax + b is not a linear function! :) Technically, ax+b is known as an affine function in linear algebra. But the term linear function (like most other terms) has meaning that varies from one field of learning to another. === Subject: Re: Proof that f(x) = ax + b is not a linear function! :) > Technically, ax+b is known as an affine function in > linear algebra. > > But the term linear function (like most other > terms) has meaning that > varies from one field of learning to another. Quite right, I think that historicaly the term linear for functions f(x)=ax+b comes in the sense that the graph of f is a straight line, nothing to do with the fact of being f linear or not. Fernando. === Subject: Re: Proof that f(x) = ax + b is not a linear function! :) <220720051320354382%anniel@nym.alias.net.invalid> Is this not a foundation-shattering discovery then? :( Much research showed that the origin of the term linear function was first applied to f(x) = ax + b because its graph is a straight line. It seems a misnomer for the definition given to be for a linear function then, since clearly, it does not apply to all functions whose plots are straight lines. === Subject: Uncle Al's ROAST --- was [Re: Reliving Chem 1120 aka GenChem2] to uncle Al. But Richard missed that he started a great post for an outright hilarious ROAST for uncle Al....... ahahaha... AHAHA... Let's never forget that Al's nemesis, Archibald Plutonium, made it into Discovery Magazine but smart Al Schwartz never did. ahahaha... AHAHAHA.... ahahaha: > :> : We're three weeks into the three month full parity > :> : Eotvos experiment run, Schultzy. > ::> You said that the results would be obtained by August. > :> What happened? > [Al] > : Science. Learn how to do a clean experiment - > : and what to do if it isn't clean when you try it. > [Richard] > Interesting how your story keeps changing, as it always does. > So when are we going to see those kg-sized diamonds? > [Al] > :> : Here's a challenge for you, inorganiker. Two Co(3+) chelated > :> : by three1,2-ethylenebis-5,5-(2,2'-bipyridyl) ligands form a nice > :> : threefold helicate, total charge +6. Propose a reasonably > :> : isomorphous helicate-forming ligand that gives me a > :> : corresponding -6 charge as counterion. > [Richard] > :> First you can give us a reference for your claim that the Wisconsin > :> cryptosporidium outbreak was due to EPA regulations about > :> chlorination of water. > [Al] > : Fine, you can't do it. Uncle Al will propose what he has. > [Richard] > I take it that you now admit that you cannot provide any references for > any of the statements to which I referred (reasonably enough, since > all of them were manifestly untrue). Too bad that you (like most bullies) > are essentially a coward and lack the guts to admit that you were wrong. It's really too bad that you avoided taking the opportunity to go to > Vietnam (using your time much more productively by flunking out of > grad school). It would have given you such a great chance to beat up > on people who had the great disadvantage of not being Caucasian. > And your well-known disdain for authority would have come in handy > when your platoon voted on whether or not to frag the lieutenant. > [hanson] B.Sc. in chemistry from Moo U and then he became an industrial chemist, apparently never getting into the upper echelons, which may be the reason why he attacks and denigrates management on every occasion. But, may be that was not his fault because of the industrial accident that he suffered which could have turned him damaged goods with some peculiar personality disorders. So, Rich, be not too hard onto uncle Al for his particular cross that he bears. It may be much heavier then normal ones.... and it may be chirally twisted... as he himself complains (see below). [Richard] > It really bothers you [Al] that you can't bully me, doesn't it. Perhaps it > should bother you more that not only are you incapable of calculating > time zone differences, you don't even know the difference between an > inorganic and a physical chemist. > Richard Schultz, schultr@mail.biu.ac.il, Chemistry, Bar-Ilan University, > Israel > ----- [Richard quotes Al] > A mensch takes personal responsiblity for his actions. > When demonstrated to be empirically wrong a mensch > admits error and becomes a better mensch. Uncle Al > [hanson] But Richard,.... ahahaha... you must understand that Al never makes mistakes, lies or changes his stories... Al is **evolving**, which means that he and his opinions do change...... AHAHAHA.... for reasons beyond his control, of course... ahahaha... AHAHAHA... ahaha... So, since you, Richard, talked about us let me say that I actually like uncle Al's less abridged CV much better. It provides more laughs... of piss & HAc: Hey, you!... UncleanAl... UncleAn0... yo!, UrinAl.... == If you are as smart as you say you are and claim to have the right answer to everything, then how come that you all you can afford is a 1989 Volks wagon Golf GL(1) ... ahahaha .... smart, really smart, urinAl....... ahahaha... AHAHAHA.... == If you are as smart as you say you are then how come that your wife made you file for divorce (2) ... ahahaha... ... smart, really smart, urinAl....... ahahaha... AHAHAHA.... == If you are as smart as you say you are then how come that a laugh like : AHAHAHA..... ahaha... AHAHAHA..... (3) does crank you more that it cranks lesser idiots?... ahahaha... ... smart, really smart, urinAl....... ahahaha... AHAHAHA.... 1989 VW Golf GL.... for at least another 11 years. You cannot imagine what it is like for the Gifted to be immersed in a society of people like yourself. .... urinAl continued in his same post with: Orange County Mensa (top 2% for entry) has at least two members with IQs in excess of 190. One is a slumbunny elementary school teacher, the other is an unemployed programmer....and then there is you, urinAl. urinAl at poster Schoenfeld: >Al: I, am the right hand of vengeance. >Al: I, am the last living thing you are ever going to remember. >Al: I am truth incarnate. GOD SENT ME. .. ahaha... so, God sent an avenging loudmouth, the size of an urinAl to state: Uncle Al was a professional sperm donor.... I did 6 cm^3 and got applause. .... Al never stated how many dozens of shots it took him to get the 6 cc's... ahahaha..... AHAHAHA... but, after said fun and applause was over he announced in his message Uncle Al votes no. No kids. ..... Kinda sad, as this made him bitter in his old age now and he is showing that in his posts, saying: the poor, stupid, and crippled. Invest in the future you want. and so Al did: a 1989 VW Golf GL for at least another 11 years. ....... ahahaha... AHAHAHA.... More & to your credit, Al, you've invented a cockroach repellant but never sold any of it ..... and then you announced that you are going to produce gem quality kilogram sized diamonds, from about 1996 to 2005, to be grown in your Devil Solvent, apparently a very, very slow growth medium/matrix, which could explain why none of your diamonds have reached neither the karat and much less the kg size state ... yet...... ahahaha..... AHAHAHA..... So, no $$ from cockroaches nor from the devil, but like a true searcher, and as an autodidact, you, Al, decided to take on Einstein, in a chiral fashion, to prove him wrong..... after you saw that Wei Tou-Ni group of the NTHua Uni in Taiwan[?] or (PR China [?]. Stray thought 7/12/05) were conducting an Eotvoes XP. The results will be in, per your own prognosis, mid September 2005. ----- Then a stroke of true geniuos 2005, when you stated: I, [Al] looked to sci.physics as a resource to massage heterodox but empirically allowable ideas. I have no imaginable use for the toolbox past mid-September ... ahahaha.... I must give credit where credit due, Al. That is a brilliant exit strategy! ahahaha... AHAHAHA.... Now, tell me why I shouldn't ROTFLMAO....ahaha... AHAHAHAHA.... over all this. You are ing funny, Al. Hilarious actually. But, not much else. ---- And, for all those laughs that you have provided you are entitled to be Al Schwartz, the one stooopid idiot, .... ahahaha.... .....which will bring great distress to your chief admirer & disciple Herbie Glaser aka G=EMC^2 Glazier AHAHAHAHA...... ahahahaha.... AHAHAHA....... carry on, Al.... There must be laughter in the world, or we turn out like you, Al, Al, write ing book, a distillation of your posts... and make some money, finally . If I am in your neighborhood, I'll be there at the book signing and make propaganda for you...... ahahahaha.... Richard, see: Al is a very great humanitarian, to boot. Al gives advice to anybody whether they like it or not, and he becomes excruciatingly grief-stricken and mad when your laugh at him. All these faults you see in Al empirically, do not matter because he compensates for them in full when he mentors his grateful NG disciples who are mostly lazy and stooopid students for whom Al does their homework, only to turn around and to bitch loudly and complain that the educational system is bad.... ahahaha... ahahahanson ahahaha.... ahahanson === Subject: Re: Uncle Al's ROAST --- was [Re: Reliving Chem 1120 aka GenChem2] ....... ahahaha... AHAHA... > ahahaha... AHAHAHA.... ahahaha: > :> : We're three weeks into the three month full parity > :> : Eotvos experiment run, Schultzy. > ::> You said that the results would be obtained by August. > :> What happened? [Al] > : Science. Learn how to do a clean experiment - > : and what to do if it isn't clean when you try it. [Richard] > Interesting how your story keeps changing, as it always does. > So when are we going to see those kg-sized diamonds? [Al] > :> : Here's a challenge for you, inorganiker. Two Co(3+) chelated > :> : by three1,2-ethylenebis-5,5-(2,2'-bipyridyl) ligands form a nice > :> : threefold helicate, total charge +6. Propose a reasonably > :> : isomorphous helicate-forming ligand that gives me a > :> : corresponding -6 charge as counterion. [Richard] > :> First you can give us a reference for your claim that the Wisconsin > :> cryptosporidium outbreak was due to EPA regulations about > :> chlorination of water. [Al] > : Fine, you can't do it. Uncle Al will propose what he has. > ,.... ahahaha... >...... AHAHAHA.... > ... ahahaha... AHAHAHA... ahaha... ... ahahaha > ....... ahahaha... AHAHAHA.... < ... ahahaha... > ....... ahahaha... AHAHAHA.... > AHAHAHA..... ahaha... AHAHAHA..... > ... ahahaha... >....... ahahaha... AHAHAHA.... > .... AHAHAHA... ahaha... > ... ahahaha..... AHAHAHA... > ....... ahahaha... AHAHAHA.... >...... ahahaha..... AHAHAHA..... > ... ahahaha.... > ahahaha... AHAHAHA.... > ....ahaha... AHAHAHAHA.... > .... ahahaha.... > AHAHAHAHA...... ahahahaha.... AHAHAHA....... > .....ahahaha.... AHAHAHA.... > A...... ahahahaha.... >.... ahahaha... ahahahanson > ...... ahahaha.... > ahahaha.... ahahanson Evidence suggests the idiot actually retypes it every time. http://www.mazepath.com/uncleal/xheli.png Do it as a reasonably isomorphous helicate (6-) anion containing two Co(3+) ions. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: Uncle Al's ROAST --- was [Re: Reliving Chem 1120 aka GenChem2] Al, write ing book, a distillation of your posts... and make some money, finally. If I am in your neighborhood, I'll be there at the book signing and make propaganda for you...... ahahahaha.... AHAHAHAH..... ahahaha........ Here's a good start for you: > ....... ahahaha... AHAHA... >> ahahaha... AHAHAHA.... ahahaha: >> ,.... ahahaha... >>...... AHAHAHA.... >> ... ahahaha... AHAHAHA... ahaha... > ... ahahaha >> ....... ahahaha... AHAHAHA.... > < ... ahahaha... >> ....... ahahaha... AHAHAHA.... >> AHAHAHA..... ahaha... AHAHAHA..... >> ... ahahaha... >>....... ahahaha... AHAHAHA.... >> .... AHAHAHA... ahaha... >> ... ahahaha..... AHAHAHA... >> ....... ahahaha... AHAHAHA.... >>...... ahahaha..... AHAHAHA..... >> ... ahahaha.... >> ahahaha... AHAHAHA.... >> ....ahaha... AHAHAHAHA.... >> .... ahahaha.... >> AHAHAHAHA...... ahahahaha.... AHAHAHA....... >> .....ahahaha.... AHAHAHA.... >> A...... ahahahaha.... >>.... ahahaha... ahahahanson >> ...... ahahaha.... >> ahahaha.... ahahanson > [unrinAl] > Evidence suggests the idiot actually retypes it every time. > [hanson] ahahaha.... and urinAl obviously got really pissed and cranked, empirically and big time, about HIS own posting contents in my roast for him, but he left standing only what really impressed him, the: .. AHAHAHA.... ahahahaha... Good job, Al... ahahaha.. As a further honor for urinAl, the Gifted pissed one, I present an encore of the roast: to uncle Al. But Richard missed that he started a great post for an outright hilarious ROAST for uncle Al....... ahahaha... AHAHA... Let's never forget that Al's nemesis, Archibald Plutonium, made it into Discovery Magazine but smart Al Schwartz never did. ahahaha... AHAHAHA.... ahahaha: > :> : We're three weeks into the three month full parity > :> : Eotvos experiment run, Schultzy. > ::> You said that the results would be obtained by August. > :> What happened? > [Al] > : Science. Learn how to do a clean experiment - > : and what to do if it isn't clean when you try it. > [Richard] > Interesting how your story keeps changing, as it always does. > So when are we going to see those kg-sized diamonds? > I take it that you now admit that you cannot provide any references for > any of the statements to which I referred (reasonably enough, since > all of them were manifestly untrue). Too bad that you (like most bullies) > are essentially a coward and lack the guts to admit that you were wrong. It's really too bad that you avoided taking the opportunity to go to > Vietnam (using your time much more productively by flunking out of > grad school). It would have given you such a great chance to beat up > on people who had the great disadvantage of not being Caucasian. > And your well-known disdain for authority would have come in handy > when your platoon voted on whether or not to frag the lieutenant. > [hanson] B.Sc. in chemistry from Moo U and then he became an industrial chemist, apparently never getting into the upper echelons, which may be the reason why he attacks and denigrates management on every occasion. But, may be that was not his fault because of the industrial accident that he suffered which could have turned him damaged goods with some peculiar personality disorders. So, Rich, be not too hard onto uncle Al for his particular cross that he bears. It may be much heavier then normal ones.... and it may be chirally twisted... as he himself complains (see below). [Richard] > It really bothers you [Al] that you can't bully me, doesn't it. Perhaps it > should bother you more that not only are you incapable of calculating > time zone differences, you don't even know the difference between an > inorganic and a physical chemist. > Richard Schultz, schultr@mail.biu.ac.il, Chemistry, Bar-Ilan University, > Israel > ----- [Richard quotes Al] > A mensch takes personal responsiblity for his actions. > When demonstrated to be empirically wrong a mensch > admits error and becomes a better mensch. Uncle Al > [hanson] But Richard,.... ahahaha... you must understand that Al never makes mistakes, lies or changes his stories... Al is **evolving**, which means that he and his opinions do change...... AHAHAHA.... for reasons beyond his control, of course... ahahaha... AHAHAHA... ahaha... So, since you, Richard, talked about us let me say that I actually like uncle Al's less abridged CV much better. It provides more laughs... of piss & HAc: Hey, you!... UncleanAl... UncleAn0... yo!, UrinAl.... == If you are as smart as you say you are and claim to have the right answer to everything, then how come that you all you can afford is a 1989 Volks wagon Golf GL(1) ... ahahaha .... smart, really smart, urinAl....... ahahaha... AHAHAHA.... == If you are as smart as you say you are then how come that your wife made you file for divorce (2) ... ahahaha... ... smart, really smart, urinAl....... ahahaha... AHAHAHA.... == If you are as smart as you say you are then how come that a laugh like : AHAHAHA..... ahaha... AHAHAHA..... (3) does crank you more that it cranks lesser idiots?... ahahaha... ... smart, really smart, urinAl....... ahahaha... AHAHAHA.... 1989 VW Golf GL.... for at least another 11 years. You cannot imagine what it is like for the Gifted to be immersed in a society of people like yourself. .... urinAl continued in his same post with: Orange County Mensa (top 2% for entry) has at least two members with IQs in excess of 190. One is a slumbunny elementary school teacher, the other is an unemployed programmer....and then there is you, urinAl. urinAl at poster Schoenfeld: >Al: I, am the right hand of vengeance. >Al: I, am the last living thing you are ever going to remember. >Al: I am truth incarnate. GOD SENT ME. .. ahaha... so, God sent an avenging loudmouth, the size of an urinAl to state: Uncle Al was a professional sperm donor.... I did 6 cm^3 and got applause. .... Al never stated how many dozens of shots it took him to get the 6 cc's... ahahaha..... AHAHAHA... but, after said fun and applause was over he announced in his message Uncle Al votes no. No kids. ..... Kinda sad, as this made him bitter in his old age now and he is showing that in his posts, saying: the poor, stupid, and crippled. Invest in the future you want. and so Al did: a 1989 VW Golf GL for at least another 11 years. ....... ahahaha... AHAHAHA.... More & to your credit, Al, you've invented a cockroach repellant but never sold any of it ..... and then you announced that you are going to produce gem quality kilogram sized diamonds, from about 1996 to 2005, to be grown in your Devil Solvent, apparently a very, very slow growth medium/matrix, which could explain why none of your diamonds have reached neither the karat and much less the kg size state ... yet...... ahahaha..... AHAHAHA..... So, no $$ from cockroaches nor from the devil, but like a true searcher, and as an autodidact, you, Al, decided to take on Einstein, in a chiral fashion, to prove him wrong..... after you saw that Wei Tou-Ni group of the NTHua Uni in Taiwan[?] or (PR China [?]. Stray thought 7/12/05) were conducting an Eotvoes XP. The results will be in, per your own prognosis, mid September 2005. ----- Then a stroke of true geniuos 2005, when you stated: I, [Al] looked to sci.physics as a resource to massage heterodox but empirically allowable ideas. I have no imaginable use for the toolbox past mid-September ... ahahaha.... I must give credit where credit due, Al. That is a brilliant exit strategy! ahahaha... AHAHAHA.... Now, tell me why I shouldn't ROTFLMAO....ahaha... AHAHAHAHA.... over all this. You are ing funny, Al. Hilarious actually. But, not much else. ---- And, for all those laughs that you have provided you are entitled to be Al Schwartz, the one stooopid idiot, .... ahahaha.... .....which will bring great distress to your chief admirer & disciple Herbie Glaser aka G=EMC^2 Glazier AHAHAHAHA...... ahahahaha.... AHAHAHA....... carry on, Al.... There must be laughter in the world, or we turn out like you, Al, Al, write ing book, a distillation of your posts... and make some money, finally . If I am in your neighborhood, I'll be there at the book signing and make propaganda for you...... ahahahaha.... Richard, see: Al is a very great humanitarian, to boot. Al gives advice to anybody whether they like it or not, and he becomes excruciatingly grief-stricken and mad when your laugh at him. All these faults you see in Al empirically, do not matter because he compensates for them in full when he mentors his grateful NG disciples who are mostly lazy and stooopid students for whom Al does their homework, only to turn around and to bitch loudly and complain that the educational system is bad.... ahahaha... ahahahanson ahahaha.... ahahanson === Subject: Re: Uncle Al's ROAST --- was [Re: Reliving Chem 1120 aka GenChem2] > >.... ahahaha... ahahahanson > ...... ahahaha.... > ahahaha.... ahahanson Evidence suggests the idiot actually retypes it every time. Hand-made Old-World craftsmanship -- you just don't see that anymore. === Subject: Re: Uncle Al's ROAST --- was [Re: Reliving Chem 1120 aka GenChem2] 6 cc ? I can abstain for a couple of weeks and produce two times that. Even sperm-wize whole humans outweigh Jews :) Hallelujah. -- My Ten Commandments: 1. Recognize the relationships between things and the laws which govern men's actions, so that you know what you are doing. 2. Direct your deeds to a worthy goal, but do not ask if they will achieve the goal; let them be models and examples rather than means to an end. 3. Speak to all others as you do to yourself, without regard to the effect you make, so that you do not expel them from your world and in your isolation lose sight of the meaning of life and the perfection of the creation. 4. Do not destroy what you cannot create. 5. Touch no dish unless you are hungry. (A pun that could read - Do not turn to the court of law unless you are hungry). 6. Do not desire what you cannot have. 7. Do not lie without need. 8. Honor children. Listen to their words with reverence and speak to them with endless love. 9. Do your work for six years; but in the seventh, go into solitude or among strangers, so that the memory of your friends does not prevent you from being what you have become. 10. Lead your life with a gentle hand and be ready to depart whenever you are called. - Leo Szilard === Subject: Re: Uncle Al's ROAST --- was [Re: Reliving Chem 1120 aka GenChem2] > to uncle Al. But Richard missed that he started a great post for > an outright hilarious ROAST for uncle Al....... ahahaha... AHAHA... > Let's never forget that Al's nemesis, Archibald Plutonium, made > it into Discovery Magazine but smart Al Schwartz never did. > ahahaha... AHAHAHA.... ahahaha: Hanson, stop laughing, enough already. You will get a sezure. After clearing your tears and drinking a glas of water, check out the thread Re: Is Life governed by simple principles just like growth of crystals?. No, it has nothing to do with the oncle. Evgenij === Subject: Re: Plus/minus square roots? > > >> > >> >Only the adherents of the magic circle can subscribe to > >(-3)x (-3) = (+9) ALWAYS......... > >......but that a^2 = (-9) gives value for a of (-3) OR (+3) >> >>Durr, what? > >You don't think that a square root of a number, is a number which >multiplied by itself, gives that result! >> >>Since when is -3 squared equal to -9 ? >> >> >Then again, you obviously believe that (-3) = (+3), >after all, they give the same result when squared (for fairy lovers) >> >>Seeing you in any upper division mathematics course would be a ing >>laugh riot. Or really sad, one or the other. >> >>In the real line, a squared number is always positive. I don't think >>you are ready for complex numbers yet, but hey - what do you have to >>lose at this point? >> >>Which of these complex numbers is the smallest? I am sure you would >>enjoy the notion of ordering in the complex plane. >> >>1 + i >>1 - i >>1/sqrt(2) + i/sqrt(2) >>1/sqrt(2) - i/sqrt(2) >> >> >Jim G >c'=c+v > >You have all had two years to come up with a less than zero >occuring in nature! You've tried it with temperature (Schwartz), length >(distance/velocity) van D, this money crap. >Try again?????????????? >You just don't get it! Math relating to fairies (imaginary) is >just that; fairy tales..........just like the Relativity theories it >supports. > >Jim G >c'=c+v > > > Since the imaginary numbers fall into the same category as the fairies > they represent, who can tell (reads gives a ) which is the > male????????? > (or is it the male which is negative, and the female positive, like the > pathetic references to electrons as example of less than zero > Stay in school till you're a 100 Eric; you'll never understand that > those i's > and various other math mumbo are necessary ONLY until the theories they > were developed to support are falsified (done and dusted) > > Jim G > c'=c+v > What do you mean falsified? Imaginary numbers are abstract constructions that serve several purposes. Since you are using a computer and therefore are using electricity (specifically alternating current) you are using devices that are designed using complex numbers. you can wish them away, but they exist whether you believe in them or not. === Subject: Re: Plus/minus square roots? constructions that serve several purposes. Since you are using a > computer and therefore are using electricity (specifically alternating > current) you are using devices that are designed using complex numbers. > you can wish them away, but they exist whether you believe in them or not. Don't take him seriously, he isn't worth the effort. It is mildly ironic to see him whine about the utility of abstract concepts that had they not existed, the device he is using to whine would also not exist. === Subject: u^2 - v^2 & u^2 + 2 v^2 both squares? Can u^2 - v^2 and u^2 + 2 v^2 be squares simultaneously? This is related to the polynomial a^4 + 10 a^2 b^2 + b^4. Can this be a square? === Subject: Re: u^2 - v^2 & u^2 + 2 v^2 both squares? The simultaneous equations u^2 - v^2 = x^2 and u^2 + 2 v^2 = y^2 define a curve of genus 1 in projective 3-space. This curve contains the point [u,v,x,y] = [1,0,1,1]. That gives the trivial solution to your problem, since u=1 (or -1) and v=0 obviously makes both of your quantities equal to squares. I assume you want solutions in positive integers. Anyway, since your genus 1 curve has a rational point, it can be put into Weierstrass form. Since the coefficients are so small, almost certainly the curve you get will appear in Cremona's tables, so you can look up the rank there. If the rank is 0, then you just need to check the torsion points (which are probably just the trivial solutions). You say it's related to the polynomial a^4 + 10 a^2 b^2 + b^4. Do you mean that your problem has a nontrivial solution if and only if there is a nontrivial solution to c^2 = a^4 + 10 a^2 b^2 + b^4 ? If so, it's even easier to put this curve into Weierstrass form, moving one of the obvious solutions [a,b,c] = [1,0,+/-1] or [0,1,+/-1] to be the inflection point at infinity. (My guess is that the other three of these points then become points of order 2.) Anyway, moral of the story is that if you're interested in problems of this sort, you need to learn about elliptic curves. === Subject: Re: u^2 - v^2 & u^2 + 2 v^2 both squares? > Can u^2 - v^2 and u^2 + 2 v^2 be squares simultaneously? Adding the equations u^2 - v^2 = A^2, and u^2 + 2 v^2 = B^2, we get v^2 = A^2 + B^2. The solutions to this equation are well-known (look up Pythagorean triples for the parameterization in terms of integers). I don't have enough paper with me to do any more. === Subject: Re: u^2 - v^2 & u^2 + 2 v^2 both squares? > >>Can u^2 - v^2 and u^2 + 2 v^2 be squares simultaneously? > > > Adding the equations > > u^2 - v^2 = A^2, and > u^2 + 2 v^2 = B^2, > > we get > > v^2 = A^2 + B^2. Actually, 2 u^2 + v^2 = A^2 + B^2 Rick === Subject: Re: u^2 - v^2 & u^2 + 2 v^2 both squares? > >>Can u^2 - v^2 and u^2 + 2 v^2 be squares simultaneously? > Adding the equations u^2 - v^2 = A^2, and > u^2 + 2 v^2 = B^2, we get v^2 = A^2 + B^2. Actually, 2 u^2 + v^2 = A^2 + B^2 I guess I didn't have enough paper to get even _that part_ right. 8-) === Subject: Legendre Transform I'm trying to get a better understanding of the Legendre Transform. The Wikipedia entry[1] for Legendre Transform mentions that it is related to integration by parts--what is the relationship? Also, I've heard that there is a relation to the Laplace transform--what is it? I was introduced to the Legendre transform in thermodynamics as a transform from convex functions of the form y = f(x) to 'tangent lines' of the form b = g(m), where b is the intercept of a tangent line through (x,y) and m is its slope. But apparently the Legendre transform is also equivalent to taking a derivative, computing a functional inverse, and then integrating. How do you show the equivalence? The Wikipedia page also says that the Legendre Transform of (1/2) x^t A x is (1/2) y^t A^-1 y (where x and y are vectors, x^t represents the transpose of x, and A is an invertible matrix). How is this result obtained and what does it mean? Tobin [1] http://en.wikipedia.org/wiki/Legendre_transform === Subject: Re: Legendre Transform > I'm trying to get a better understanding of the Legendre Transform. The > Wikipedia entry[1] for Legendre Transform mentions that it is related to > integration by parts--what is the relationship? Also, I've heard that > there is a relation to the Laplace transform--what is it? I would guess that it is 'related' to integration by parts in that you can undo integration by parts by swapping the derivative and anti-derivative and applying the integration by parts rule. I have no idea how it relates to the Laplace transform, I don't see a direct relationship. Transforms in general, and specifically integral transforms, oddly fascinate me...I wish I knew what makes them work so well but all the books I read are more about method and existance theorems, rather than the theory on what makes them tick. === Subject: Re: The simplest proof that the brain is at least a universal computer >> He was careful to distinguish infinite, something that >> is certainly irrelevant to machinekind, from unbounded >> which is a model of physical extensibility of thought, >> e.g. mathematical work. > A computing machine is IMAGINED as a mechanical device capable of > a finite number of states or internal configurations (I.C.) q_1, > q_2, ..., q_r, working on a (both ways) INFINITE TAPE divided into > squares on which symbols are printed. (emphasis mine) Well, regardless of his or anyone else's idea of the difference between an > unbounded and an infinite tape, the important thing is surely that the > number of allowed states of the tape (and hence of the whole TM) is > countably infinite. Uncountability is introduced in a controlled way via > oracles. The easiest way to guarantee this is to require that at any given > time all but finitely many squares on the tape are blank. I tend to use the > word unbounded to describe a tape that's so constrained. Forget K(x), it's not relevant to the discussion, just len(x) will do :) -- Eray === Subject: Re: The simplest proof that the brain is at least a universal computer >> He was careful to distinguish infinite, something that >> is certainly irrelevant to machinekind, from unbounded >> which is a model of physical extensibility of thought, >> e.g. mathematical work. > A computing machine is IMAGINED as a mechanical device capable of > a finite number of states or internal configurations (I.C.) q_1, > q_2, ..., q_r, working on a (both ways) INFINITE TAPE divided into > squares on which symbols are printed. (emphasis mine) Well, regardless of his or anyone else's idea of the difference between an > unbounded and an infinite tape, the important thing is surely that the > number of allowed states of the tape (and hence of the whole TM) is > countably infinite. Uncountability is introduced in a controlled way via > oracles. The easiest way to guarantee this is to require that at any given > time all but finitely many squares on the tape are blank. I tend to use the > word unbounded to describe a tape that's so constrained. This is exactly how the Turing Machine was originally formulated, no? At least that is the description in the latest edition of the Cinderella book, i.e. Turing Machine Instantenous Description. Given an ID, for some x in Z+, K(ID) >> He was careful to distinguish infinite, something that >> is certainly irrelevant to machinekind, from unbounded >> which is a model of physical extensibility of thought, >> e.g. mathematical work. > A computing machine is IMAGINED as a mechanical device capable of > a finite number of states or internal configurations (I.C.) q_1, > q_2, ..., q_r, working on a (both ways) INFINITE TAPE divided into > squares on which symbols are printed. (emphasis mine) Well, regardless of his or anyone else's idea of the difference between an > unbounded and an infinite tape, the important thing is surely that the > number of allowed states of the tape (and hence of the whole TM) is > countably infinite. Uncountability is introduced in a controlled way via > oracles. The easiest way to guarantee this is to require that at any given > time all but finitely many squares on the tape are blank. I tend to use the > word unbounded to describe a tape that's so constrained. This is exactly how the Turing Machine was originally formulated, no? > At least that is the description in the latest edition of the > Cinderella book, i.e. Turing Machine Instantenous Description. Given an > ID, for some x in Z+, K(ID) string s. Or just that len(ID) in Z, ahahah :) -- Eray === Subject: Re: Cardinality of N and P(N) > BACKGROUND Cantor defined the cardinality of N, the set of natural numbers, as > aleph-0, which is the first transfinite number. Thus card(N) = > aleph-0. The cardinality of the power set of N (i.e., the set of all possible > subsets of N) is 2^aleph-0. Thus card(P(N)) = 2^aleph-0. This implies that N (the infinite set of natural numbers) cannot be put > in one-to-one correspondence with the members of P(N). In other words, > N is a countably infinite set, but P(N) is an uncountably infinite set. QUESTION My question is, what are the ramifications if it could be demonstrated > that P(N) is, in fact, countable; i.e., what if the members of P(N) > could be ordered in such a way as to be countable, and thus mappable to > members of N? Would this impact Cantor's hierarchy of transfinites (aleph-1, aleph-2, > etc.)? Would this impact the Continuum Hypothesis? -drt There is a fine point: too fine to be grasped by a casual reader of the popular presentations of set theory. If there is a small model of set theory (whose own universe is a set in a bigger universe) then you can have a countable model. The secret is: it looks countable from the outside (where you have plenty of bijections) but it can contain sets which are uncountable when evaluated from the inside (where many bijections are missing). Unless I am missing something, it comes from The Skolem-Loewenheim Theorem says: If a countable set S of sentences has an infinite model then it has a countable one. Of course, if one stays within one and the same model, the statement asserting the countability of P(N) would trigger the collapse of the standard set theory. === Subject: Re: Cardinality of N and P(N) > what if the members of P(N) > could be ordered in such a way as to be countable, and thus mappable to > members of N? I would say: Check to make sure your enumeration includes all subsets of N, not just the finite subsets. And conversely, check that the alleged members of N are finite numbers, and not some sort of infinite numbers. -- http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Cardinality of N and P(N) > BACKGROUND > > Cantor defined the cardinality of N, the set of natural numbers, as > aleph-0, which is the first transfinite number. Thus card(N) = > aleph-0. > > The cardinality of the power set of N (i.e., the set of all possible > subsets of N) is 2^aleph-0. Thus card(P(N)) = 2^aleph-0. > > This implies that N (the infinite set of natural numbers) cannot be put > in one-to-one correspondence with the members of P(N). In other words, > N is a countably infinite set, but P(N) is an uncountably infinite set. > > QUESTION > > My question is, what are the ramifications if it could be demonstrated > that P(N) is, in fact, countable; i.e., what if the members of P(N) > could be ordered in such a way as to be countable, and thus mappable to > members of N? Since it is already provable that P(N) is NOT countable, being able to prove that it also IS countable would mean the inconsistency of any set theory which allows both proofs. === Subject: Re: Cardinality of N and P(N) > > > Wouldn't it just prove that the notion of counting infinite > sets could be incorrect, at least for power sets? Assuming that > it only applied to the countability of power sets, and could not > be extended to other non-countable sets, would this really affect > the rest of Cantor's (et al) theorems? Any set is counted by setting up a one - one onto mapping with a set of known cardinality. Your problem is that you do not believe infinite sets exist. Bob Kolker === Subject: Re: Cardinality of N and P(N) > > QUESTION > > My question is, what are the ramifications if it could be demonstrated > that P(N) is, in fact, countable; Anything would follow since it is not the case the N and P(N) have the The assumption of a one to one onto map of P(N) to N leads to a contradiction, so we know no such mapping exists. If my grandmother had balls, not only would she be my grandpa, but it would follow that angels fly out of your arse. Bob Kolker === Subject: Re: Cardinality of N and P(N) <85mzofg57k.fsf@lola.goethe.zz> >> My question is, what are the ramifications if it could be >> demonstrated that P(N) is, in fact, countable; i.e., what if the >> members of P(N) could be ordered in such a way as to be countable, >> and thus mappable to members of N? Anyway, assume a mapping f(n) that maps n to a subset of N. > The mapping will not cover {n|n not in f(n)}. And what if {n | n not in f(n)} does not (or cannot) exist? Hypothetical, I know, but what if? -drt === Subject: Re: Cardinality of N and P(N) !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > My question is, what are the ramifications if it could be > demonstrated that P(N) is, in fact, countable; i.e., what if the > members of P(N) could be ordered in such a way as to be countable, > and thus mappable to members of N? > >> >> Anyway, assume a mapping f(n) that maps n to a subset of N. >> The mapping will not cover {n|n not in f(n)}. > And what if {n | n not in f(n)} does not (or cannot) exist? Hypothetical, I know, but what if? Look, you are being absurd. If a mapping n->f(n) is assumed to exist, that means that for every n, f(n) is defined. f(n) is a _set_. A set is defined as an entity that determinably contains or not contains each of its elements. So n not in f(n) is always determinable. And this determines, for every n, whether n is a member of the specified set. The where do you even want to start with does not exist? How can something not exist that is constructed in the most basic terms whatsoever? -- Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cardinality of N and P(N) > >> My question is, what are the ramifications if it could be >> demonstrated that P(N) is, in fact, countable; i.e., what if the >> members of P(N) could be ordered in such a way as to be countable, >> and thus mappable to members of N? >> Anyway, assume a mapping f(n) that maps n to a subset of N. > The mapping will not cover {n|n not in f(n)}. >> >> >> And what if {n | n not in f(n)} does not (or cannot) exist? >> >> Hypothetical, I know, but what if? > >Look, you are being absurd. If a mapping n->f(n) is assumed to exist, >that means that for every n, f(n) is defined. f(n) is a _set_. A set >is defined as an entity that determinably contains or not contains >each of its elements. So n not in f(n) is always determinable. And >this determines, for every n, whether n is a member of the specified >set. > >The where do you even want to start with does not exist? How can >something not exist that is constructed in the most basic terms >whatsoever? Actually it seems to me that you're being unduly dismissive. If it's clear that {n in N | n not in f(n)} has to exist, just because we've said what it is, then it's also perfectly clear that the set S = {sets x : x not in x} has to exist. But the existence of S leads to contradictions. ************************ C. Ullrich === Subject: Re: Cardinality of N and P(N) !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > >My question is, what are the ramifications if it could be > demonstrated that P(N) is, in fact, countable; i.e., what if the > members of P(N) could be ordered in such a way as to be countable, > and thus mappable to members of N? > >> >> Anyway, assume a mapping f(n) that maps n to a subset of N. >> The mapping will not cover {n|n not in f(n)}. > And what if {n | n not in f(n)} does not (or cannot) exist? Hypothetical, I know, but what if? >> >>Look, you are being absurd. If a mapping n->f(n) is assumed to exist, >>that means that for every n, f(n) is defined. f(n) is a _set_. A set >>is defined as an entity that determinably contains or not contains >>each of its elements. So n not in f(n) is always determinable. And >>this determines, for every n, whether n is a member of the specified >>set. >> >>The where do you even want to start with does not exist? How can >>something not exist that is constructed in the most basic terms >>whatsoever? Actually it seems to me that you're being unduly dismissive. > If it's clear that {n in N | n not in f(n)} has to exist, > just because we've said what it is, then it's also perfectly > clear that the set S = {sets x : x not in x} has to exist. sets x is not a well-defined set to start with, and it presumably is supposed to include S as well. But N is quite well defined outside of set theory. > But the existence of S leads to contradictions. Sure, but that is because sets x is something that presumably refers to what is being defined as well. But the construct {n in N | n not in f(n)} assumes only things that are defined quite in basic terms and _without_ _any_ internal _reference_ to what is being defined here. -- Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cardinality of N and P(N) > >> > >> My question is, what are the ramifications if it could be >> demonstrated that P(N) is, in fact, countable; i.e., what if the >> members of P(N) could be ordered in such a way as to be countable, >> and thus mappable to members of N? >> Anyway, assume a mapping f(n) that maps n to a subset of N. > The mapping will not cover {n|n not in f(n)}. >> >> >> And what if {n | n not in f(n)} does not (or cannot) exist? >> >> Hypothetical, I know, but what if? > >Look, you are being absurd. If a mapping n->f(n) is assumed to exist, >that means that for every n, f(n) is defined. f(n) is a _set_. A set >is defined as an entity that determinably contains or not contains >each of its elements. So n not in f(n) is always determinable. And >this determines, for every n, whether n is a member of the specified >set. > >The where do you even want to start with does not exist? How can >something not exist that is constructed in the most basic terms >whatsoever? >> >> Actually it seems to me that you're being unduly dismissive. >> If it's clear that {n in N | n not in f(n)} has to exist, >> just because we've said what it is, then it's also perfectly >> clear that the set S = {sets x : x not in x} has to exist. > >sets x is not a well-defined set to start with, and it presumably is >supposed to include S as well. > >But N is quite well defined outside of set theory. > >> But the existence of S leads to contradictions. > >Sure, but that is because sets x is something that presumably refers >to what is being defined as well. But the construct >{n in N | n not in f(n)} assumes only things that are defined >quite in basic terms and _without_ _any_ internal _reference_ to what >is being defined here. Fine. But you can't _prove_ that the axiom that allows us to say that given a set A and a property P then there is such a set as {x in A : P(x)} does not lead to contradictions. ************************ C. Ullrich === Subject: Re: Cardinality of N and P(N) On 21 Jul 2005 17:08:55 -0700, R Tribble <@tribble.com > And what if {n e N | n !e f(n)} does not (or cannot) exist? > But it d o e s exist: this is ensured by an axiom. You might also ask: but what if 0 = 1. Well... Hypothetical, I know, but what if? > In set theory this can't happen; or otherwise it wouldn't be set theory. F. === Subject: Re: Cardinality of N and P(N) > >My question is, what are the ramifications if it could be >demonstrated that P(N) is, in fact, countable; i.e., what if the >members of P(N) could be ordered in such a way as to be countable, >and thus mappable to members of N? > > > >> >>Anyway, assume a mapping f(n) that maps n to a subset of N. >>The mapping will not cover {n|n not in f(n)}. > > > > And what if {n | n not in f(n)} does not (or cannot) exist? > > Hypothetical, I know, but what if? One of the axioms of set theory is that if A is a set, and P(x) is a proposition, then {a in A | P(a)} is a set. Thus Kastrup's set {n in N | n not in f(n)} has to exist. But you are sort of on the right track in thinking that this axiom might be the one to mess things up. Older versions of set theory had a broader axiom that {a | P(a)} is a set, resulting in the famous contradiction by Russel by considering X = {a | a not in a} and seeing that X is in X and X is not in X are both true. === Subject: Re: Cardinality of N and P(N) > > > > And what if {n | n not in f(n)} does not (or cannot) exist? But the set does exist. It depends on what f is. It could be the empty set. Bob Kolker === Subject: Re: Cardinality of N and P(N) !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> And what if {n | n not in f(n)} does not (or cannot) exist? But the set does exist. It depends on what f is. It could be the empty set. But then the empty set could not be represented by any f(n). -- Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cardinality of N and P(N) > My question is, what are the ramifications if it could be > demonstrated that P(N) is, in fact, countable Very far-reaching: it would demonstrate that ZF is inconsistent, and require (another) reformulation of set theory. > i.e., what if the members of P(N) could be ordered in such a way as > to be countable, and thus mappable to members of N? Order has nothing to do with countability. If there's a surjection, there's a surjection regardless of ordering. > Would this impact Cantor's hierarchy of transfinites (aleph-1, > aleph-2, etc.)? Would this impact the Continuum Hypothesis? Yes. It would imply that the Continuum Hypothesis is true. And false. It would be possible to prove that aleph-alpha does not exist for any ordinal alpha. (And that they do exist, and all inaccessible) It would also be possible to derive 0=1, or any other mathematical absurdity expressible in ZF that you can think of. - Tim === Subject: Re: Cardinality of N and P(N) > i.e., what if the members of P(N) could be ordered in such a way as > to be countable, and thus mappable to members of N? > > Order has nothing to do with countability. Really? I was under the impression that the rationals had been proved to be countable by doing just that -- showing they could be ordered in such a way as to be countable. Am I wrong? -- Alec McKenzie === Subject: Re: Cardinality of N and P(N) Mail-To-News-Contact: abuse@dizum.com > >> i.e., what if the members of P(N) could be ordered in such a way as >> to be countable, and thus mappable to members of N? >> >> Order has nothing to do with countability. > >Really? I was under the impression that the rationals had been >proved to be countable by doing just that -- showing they could >be ordered in such a way as to be countable. Am I wrong? No ordering is necessary. I can show an injection from the rationals to the integers without any ordering needed: 1. Given a rational number, r, eliminate all common factors from the numerator and denominator. 2. Call the resulting denominator q. 3. If r is nonnegative, call the resulting numerator p. Calculate (2^p)*(3^q) 4. If r is negative, call the resulting numerator -p. Calculate (2^p)*(5^q) A few examples: 1.4 = 14/10 = 7/5 yields (2^7)*(3^5) = (128)*(243) = 31104 -3/6 = -1/2 yields (2^1)*(5^2) = 50 Any rational that you give me will turn into a positive integer. But, no matter what rationals you give me, many positive integers will never get hit, such as { 7, 11, 13, 14, 17, 19, 21, 22, 23, 26, 28, ...} So, without any ordering, I can show that the set of rationals is no larger than the set of positive integers. (The mapping will define an ordering on the rationals, but the ordering wasn't used to define the mapping.) -- Michael F. Stemper #include Life's too important to take seriously. === Subject: Re: Cardinality of N and P(N) > >> i.e., what if the members of P(N) could be ordered in such a way as >> to be countable, and thus mappable to members of N? >> >> Order has nothing to do with countability. > >Really? I was under the impression that the rationals had been >proved to be countable by doing just that -- showing they could >be ordered in such a way as to be countable. Am I wrong? > > No ordering is necessary. I can show an injection from the rationals to > the integers without any ordering needed: No doubt you can -- I made no suggestion that ordering was necessary. I was pointing out that rationals had been proved to be countable by ordering them in such a way that they could be counted. That would not be possible if order has nothing to do with countability. -- Alec McKenzie === Subject: Re: Cardinality of N and P(N) ... > No ordering is necessary. I can show an injection from the rationals to > the integers without any ordering needed: > > No doubt you can -- I made no suggestion that ordering was > necessary. I was pointing out that rationals had been proved to > be countable by ordering them in such a way that they could be > counted. That would not be possible if order has nothing to do > with countability. Sorry, I can not follow this. You are stating: 1. Ordering is possibly not necessary 2. There has been a proof that did order the rationals such that they can be counted. 3. That is not possible if order has nothing to do with countability. rationals to the integers. There is also an obvious injection from the integers to the rationals. And there is a theorem, that if there is an injection from A to B, and also an injection from B to A, that there is a bijection between the two. (This should be pretty intuitive, but in mathematics even the intuitive things are conjectures unless proven.) In all, there are other ways to prove that the rationals are countable than by imposing an order on them. Order has simply nothing to do with countability. -- 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: Cardinality of N and P(N) > Really? I was under the impression that the rationals had been > proved to be countable by doing just that -- showing they could be > ordered in such a way as to be countable. Am I wrong? The other way around: there was shown to be a surjection from the naturals onto the rationals. That surjection can then be used to define an ordering on the rationals with the same order-type as the naturals. - Tim === Subject: Observations On Fermat's Last Theorem And Conclusion OBSERVATIONS ON FERMAT'S LAST THEOREM AND CONCLUSION In 1994, English born Andrew Wiles of Princeton University, after eight years of tireless effort -spending one year in complete seclusion - proved and presented his proof in a 200 page brilliant treatise, that Fermat's Last Theorem is true. He was awarded the Wolfskehl Prize, which had eroded to $50,000.00 because of political events in Europe. The Wiles' proof, however, was not simple and was based on mathematical theories discovered hundreds of years after Fermat's time. Mathematicians therefore agree, that it could not possibly be the proof Scientific American, stated: Either Fermat was mistaken, and his proof, if existed, was flawed, or a simple cunning proof awaits discovery. For a full description of a simple conjecture on this problem, visit http://www.alexandertarics.com for your time. === Subject: Identical people in the same town I can't phrase this very well, but I'm sure you'll get my meaning: What is the probability of finding two identical looking people, of the same sex, in the same town of a population of 200,000? Please provide a complete proof. === Subject: Re: Identical people in the same town > I can't phrase this very well, but I'm sure you'll get my meaning: > > > What is the probability of finding two identical looking people, of the same > sex, in the same town of a population of 200,000? > > > Please provide a complete proof. > > 0. Proof: No two people are identical. Or any number from 0 to 1, depending on how you define identical looking. Your question is similar to How long is a piece of string? === Subject: Re: Identical people in the same town > I can't phrase this very well, but I'm sure you'll get my meaning: > What is the probability of finding two identical looking people, of the same > sex, in the same town of a population of 200,000? > Please provide a complete proof. > > 0. Proof: No two people are identical. Or any number from 0 to 1, depending on how you define identical looking. Your question is similar to How long is a piece of string? ...I'm sure you'll get my meaning. === Subject: Re: Identical people in the same town > I can't phrase this very well, but I'm sure you'll get my meaning: > > > What is the probability of finding two identical looking people, of the same > sex, in the same town of a population of 200,000? Quite high. I would expect a fair number of identical twins in a town that size. -- Alec McKenzie === Subject: Re: Identical people in the same town What is the probability of finding two identical looking people, of the same > sex, in the same town of a population of 200,000? Quite high. I would expect a fair number of identical twins in a > town that size. They are not of the same family. Identical looking was the operative phrase. === Subject: Re: Identical people in the same town > > I can't phrase this very well, but I'm sure you'll get my meaning: > > What is the probability of finding two identical looking people, of the same > sex, in the same town of a population of 200,000? This is not a mathematical problem. It is an empirical one. One would need to examine all people in a number of towns of that size to answer the question. > Please provide a complete proof. May I ask where the problem came from? -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: Identical people in the same town > This is not a mathematical problem. It is an empirical one. One would > need to examine all people in a number of towns of that size to answer > the question. Please provide a complete proof. May I ask where the problem came from? > It came from me - I saw two people who looked so alike it was uncanny - same hairstyle same glasses, same height, very close facial features, same color of hair, same body build. === Subject: Re: Identical people in the same town > > This is not a mathematical problem. It is an empirical one. One would > need to examine all people in a number of towns of that size to answer > the question. Please provide a complete proof. May I ask where the problem came from? > It came from me - I saw two people who looked so alike it was uncanny - same > hairstyle > same glasses, same height, very close facial features, same color of hair, > same body build. Maybe they were identical twins? The question What is the probability of identical twins in a town of size 200,000? is still difficult; but What is the probability of identical twins in a random sample of 200,000 people (in a certain country)? can probably be answered. -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: Identical people in the same town It came from me - I saw two people who looked so alike it was uncanny - same > hairstyle > same glasses, same height, very close facial features, same color of hair, > same body build. Maybe they were identical twins? The question What is the probability > of identical twins in a town of size 200,000? is still difficult; but > What is the probability of identical twins in a random sample of > 200,000 people (in a certain country)? can probably be answered. No, they were definitely not related. === Subject: Re: probability theory applied to theology > > > >>Serious, what is the probability that a mathematical theorem is >>valid, statistically speaking ? Isn't that probability just zero ? Under what model? What is your initial probability space? What > metrics is it governed by? > > I expect answers to a question, no further questions. Examine lots of instances of the theorem. Let's say there are M of them. Count the number of valid ones. Let's say there are N of them. Then, if you're lucky, the probability of the theorem being valid is approximately N/M. -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: probability theory applied to theology > >> >> >> >>Serious, what is the probability that a mathematical theorem is >>valid, statistically speaking ? Isn't that probability just zero ? > >Under what model? What is your initial probability space? What >metrics is it governed by? >> >>I expect answers to a question, no further questions. > > > Examine lots of instances of the theorem. Let's say there are M of > them. Count the number of valid ones. Let's say there are N of them. > Then, if you're lucky, the probability of the theorem being valid is > approximately N/M. There are more sophisticated interpretations of probability as some kind of strength of belief than this one. You might like to check out a book by E.T. Jaynes Probability Theory: The Logic of Science. === Subject: Re: probability theory applied to theology > > http://www.smh.com.au/news/national/faith-in-lap-of-the-odds/2005/07/18/1 > 121538922281.html > begins, > > It is 97 per cent certain God raised Jesus from the dead - based on > logic and mathematics, not faith - says an Oxford professor, Richard > Swinburne. It can't be based on logic, mathematics or faith. One would need to observe a long sequence of (God, Jesus) pairs and count how many of the Gods raised the Jesus's from the dead. If the ratio of raiser to total number of Gods was about 97/100 then one could express that, rather sloppily I think, as It is 97 per cent certain God raised Jesus from the dead. Can any other meaning be given to the claim? -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: probability theory applied to theology > >>http://www.smh.com.au/news/national/faith-in-lap-of-the-odds/2005/07/18/1 >>121538922281.html >>begins, >> >> It is 97 per cent certain God raised Jesus from the dead - based on >> logic and mathematics, not faith - says an Oxford professor, Richard >> Swinburne. > > > It can't be based on logic, mathematics or faith. One would need to > observe a long sequence of (God, Jesus) pairs and count how many of the > Gods raised the Jesus's from the dead. If the ratio of raiser to total > number of Gods was about 97/100 then one could express that, rather > sloppily I think, as It is 97 per cent certain God raised Jesus from > the dead. > > Can any other meaning be given to the claim? See my reply to your other post - look up E.T. Jaynes. === Subject: Re: probability theory applied to theology <8cc61$42dcbf1a$82a1e3ad$23107@news2.tudelft.nl> <87d5pfrwyw.fsf@phiwumbda.org> <8a7fa$42dcc6f1$82a1e3ad$23594@news2.tudelft.nl> <85hderdu1t.fsf@lola.goethe.zz> <1bec0$42dcc96e$82a1e3ad$23780@news2.tudelft.nl> <858y03dsed.fsf@lola.goethe.zz> <1a3e6$42dcddfa$82a1e3ad$25998@news2.tudelft.nl> (snip) > What I think you meant is what is the probability that a random > mathematical statement is true. If we specify a natural way of > constructing such statements, then it seems clear that the set of such > statements will countably infinite, but for a countably infinite set > there is no uniform distribution, hence the phrase random > mathematical statement is meaningless unless a distribution for > selecting from the defined set of mathematical statements is > specified. (snip) For a fixed formal language and first order theory it makes sense to ask about the probability that a randomly chosen wff (well formed formula) of length n is a theorem. Furthermore, if this probability has a limit as n -> 00, it makes sense to talk about the density of theorems among wffs. It seems plausible but by no means obvious that this density exists and is 0 for things like Peano Arithmetic. Surely this must have been studied. Does anyone know of any references? === Subject: Re: probability theory applied to theology > >... there is this person called Traveler who has some crazy > anti-physics web site. In a part of it he has quotes from many of > the great modern scientists, basically trying to make them all look > silly. >> >> >> >> Do you happen to have the url? Sorry, no, but he has some recent posts on sci.math, and probably more > recent stuff on the sci.physics groups. > Is it this one? http://users.adelphia.net/~lilavois/Crackpots/notorious.htm#Quotes -- Herman Jurjus === Subject: Re: probability theory applied to theology > >> > >> ... there is this person called Traveler who has some crazy >> anti-physics web site. In a part of it he has quotes from many of >> the great modern scientists, basically trying to make them all look >> silly. > > > Do you happen to have the url? > >> >> Sorry, no, but he has some recent posts on sci.math, and probably more >> recent stuff on the sci.physics groups. >Is it this one? > http://users.adelphia.net/~lilavois/Crackpots/notorious.htm#Quotes > Yes === Subject: combinatorics problem Hello! Suppose I have a set of objects which is partitioned into equivalence classes by an equivalence relation ~. I have a machine which can count how many objects there are, and how many pairs (x,y) there are such that x~y. I want to determine how many total classes there are. Is this possible? Snis Pilbor (it isn't a homework problem) === Subject: Re: combinatorics problem > Hello! Suppose I have a set of objects which is partitioned into > equivalence classes by an equivalence relation ~. I have a machine > which can count how many objects there are, and how many pairs (x,y) > there are such that x~y. I want to determine how many total classes > there are. Is this possible? Yes, but your answer won't necessarily be unique. If your machine prints out 9 and 41, there's more than one answer, because 41 = 5^2 + 4^2 and 41 = 6^2 + 2^2 + 1^2, so you could have two equivalence classes (with size 5 and 4), or three (with sizes 6, 2, 1). Example found using brute force. === Subject: Re: combinatorics problem > Hello! > > Suppose I have a set of objects which is partitioned into > equivalence classes by an equivalence relation ~. I have a machine > which can count how many objects there are, and how many pairs (x,y) > there are such that x~y. I want to determine how many total classes > there are. Is this possible? It isn't possible. Draw a bunch of points on a piece of paper, divide them into equivalence classes that are of unequal size, and then count up the number of such pairs. You will quickly see that it just cannot be done. On the other hand, I do some statistical methods to get the kinds of answers you might be looking for. If that would interest you, send me a message by email. Stephen === Subject: Re: combinatorics problem > Hello! Suppose I have a set of objects which is partitioned into > equivalence classes by an equivalence relation ~. I have a machine > which can count how many objects there are, and how many pairs (x,y) > there are such that x~y. I want to determine how many total classes > there are. Is this possible? > I don't _think_ so ... Suppose that C is one of the equivalence classes, containing k objects. How many pairs will your second machine count for C ?? Well, there are k pairs (x,x) with x in C and then there are k(k-1) pairs (x,y) with x, y in C and x != y, So C contributes k^2 to the second machine's count. Clearly, the second machine's output is just the sum over all such classes C of the counts it obtains so it follows that, if your *general* problem were solvable, then you could conclude that any n which can be expressed as a sum of squares has exactly _one_ such representation. But that's well-known to be *false* ... > Snis Pilbor > (it isn't a homework problem) === Subject: Re: combinatorics problem > Hello! Suppose I have a set of objects which is partitioned into > equivalence classes by an equivalence relation ~. I have a machine > which can count how many objects there are, and how many pairs (x,y) > there are such that x~y. I want to determine how many total classes > there are. Is this possible? > I don't _think_ so ... Suppose that C is one of the equivalence classes, containing > k objects. How many pairs will your second machine count for > C ?? Well, there are k pairs (x,x) with x in C and then there > are k(k-1) pairs (x,y) with x, y in C and x != y, So C contributes > k^2 to the second machine's count. Clearly, the second machine's > output is just the sum over all such classes C of the counts it > obtains so it follows that, if your *general* problem were > solvable, then you could conclude that any n which can be expressed > as a sum of squares has exactly _one_ such representation. But that's well-known to be *false* ... I'm not sure that's quite what's going on ... As I read the problem, you're trying to solve a system of Diophantine equations with a variable number of variables. You're trying to determine whether there's a solution to the system: x_1^2 + x_2^2 + ... + x_m^2 = A (1) x_1 + x_2 + ... + x_m = B (2) y_1^2 + y_2^2 + ... + y_n^2 = A (3) y_1 + y_2 + ... + y_n = B, (4) where x_i, y_j are positive integers, and m =/= n. (x_i and y_i are the number of elements in the ith equivalence class for each of two equivalence relations, of course.) If there is a solution, the problem is hopeless, since the answer isn't unique. If there are no solutions, then you want to solve (1) and (2). For instance, if A = 25, there are solutions to the subsystem of (1) and (3) with m =/= n (namely m=1, n=2, x_1 = 5, y_1 = 3, y_2 = 4) in spite of 25 being able to be written as the sum of two squares in more than one way. (Note that A = 50, which has the solutions x = (7,1) and y = (5,5) doesn't help, since m = 2 = n here.) Maybe writing a computer program to generate the ordered pairs (A,B) satisfying (1) and (2) with a fixed value of m will help. > Snis Pilbor > (it isn't a homework problem) I would hope not. 8-) If it's got a nice solution, then it would be worthy of the Putnam Exam. === Subject: Chains and anitchains. I'm reading Davey and Priestley's Lattices and Order, and I'm having some trouble with this sentence: Any set S may be converted into an ordered set S-bar by giving S the antichain order. which appears in Section 1.3. I'm not clear on how defining the antichain for S defines a chain on S-bar? I'm thinking there are n! possible chains, and I don't see how the antichain is used to pick one of them. (I'm also assuming the antichain for S is the total one: { (x, x) | x in S) }.) === Subject: Re: Chains and anitchains. === Subject: Re: Chains and anitchains. > I'm reading Davey and Priestley's Lattices and Order, and I'm having some > trouble with this sentence: > Nice book. > Any set S may be converted into an ordered set S-bar by giving S the > antichain order. > The antichain order for S is =, the order, call it <<== is a <<== b when a = b. This is an order, a _partial_ order. > which appears in Section 1.3. I'm not clear on how defining the > antichain for S defines a chain on S-bar? I'm thinking there are n! It doesn't. He didn't say S-bar is a chain, ie a total order, he said it's an (partial) order. > possible chains, and I don't see how the antichain is used to pick one > of them. (I'm also assuming the antichain for S is the total one: n! doesn't apply for infinite S > { (x,x) | x in S) }.) > This isn't a total order, it's a reflexive relation, it fact it's the antichain order. Most orders aren't total orders or chains. Indeed most of them are partial orders. { a,b,c } has six chains and 1 + 6 + 6 other orders. What are they? === Subject: Re: Chains and anitchains. > I'm reading Davey and Priestley's Lattices and Order, and I'm having some > trouble with this sentence: > Nice book. > Any set S may be converted into an ordered set S-bar by giving S the > antichain order. > The antichain order for S is =, the order, call it <<== is a <<== b when a = b. This is an order, a _partial_ order. > which appears in Section 1.3. I'm not clear on how defining the > antichain for S defines a chain on S-bar? I'm thinking there are n! It doesn't. He didn't say S-bar is a chain, ie a total order, he said it's an (partial) order. > possible chains, and I don't see how the antichain is used to pick one > of them. (I'm also assuming the antichain for S is the total one: n! doesn't apply for infinite S > { (x,x) | x in S) }.) > This isn't a total order, it's a reflexive relation, if fact it's the antichain order. Most orders aren't total orders or chains. Indeed most of them are partial orders. { a,b,c } has six chains and 1 + 6 + 6 other orders. What are they? === Subject: Re: Chains and anitchains. >I'm reading Davey and Priestley's Lattices and Order, and I'm having some >trouble with this sentence: Any set S may be converted into an ordered set S-bar by giving S the > antichain order. What a particularly opaque way to say that you can (partially) order any set S by declaring that no two distinct elements are comparable. >I'm not clear on how defining the antichain for >S defines a chain on S-bar? What makes you think it does? Nothing, in what you quoted, should. I don't possess the book, but I presume that their definition of ordered set is something like An ordered set is a pair (S,R), where S is a set and R is a relation on S such that blah, blah. So the antichain for S, namely S-bar, is the pair (S,D) where D is the diagonal; while a chain on S-bar doesn't mean much of anything, insofar as S-bar is an ordered pair, not a set (yes, yes, I know that it's standard to *model* an ordered pair as a set, and that for that matter it's standard to act as though everything in sight is a set; but ignore that). >I'm thinking there are n! possible chains, and I >don't see how the antichain is used to pick one of them. (I'm also assuming >the antichain for S is the total one: { (x, x) | x in S) }.) If there's more context to the quoted sentence, that makes your statements following it more easily understood, please quote that too and we will all be much relieved. Lee Rudolph === Subject: post-office problem on three vars., closed form for 2 rel. prime pos. integers p,q, the problem is as follows: given p,q rel. prime, find the largest integer that _cannot_ be written as a linear comb: px+qy, x,y non-negative integers. or, equiv. find the least number n so that n, n+1,n+2,..... can all be written in the form px+qy, x,y non-neg. integers This problem has a nice closed form for the solution: it is (p-1)*(q-1). My question now, is for three variables, p,q,r all pos. integers and rel. prime, the goal is to find , as above, the largest number not expressible as: px+qy+rz , x,y,z non-neg. integers. Now, I think there are some algorithms out there to solve this problem. Is there any argument to show that this problem on three variables _does not_ have a closed form solution, or that this closed for is somehow unwieldy or something? CarlosR. === Subject: Coherence in the Big Bang Begin forwarded message: === Subject: More is different in the BIG BANG simulation experiment at the very lowest energy end of the spectrum ... superconductivity and superfluidity and now in the BIG BANG simulation http://physorg.com/news5340.html we have been provided with a series of very direct demonstrations which tangible, macroscopic realm ... In these phenomena I now include Einstein's gravity here since, in the geodesic local frame representation, the Einstein-Cartan tetrad field of gravity is the vacuum ODLRO field eu^a = Iu^a + bu(LpP^a/ih)(Goldstone Phase of post-inflation Higgs Ocean) to the roles, with which we are familiar, of the classical fields, the electromagnetic and gravitational fields ... This has been shown by a series of experiments of various kinds on coherence in quantum fluids. P.W. Anderson Coherent Matter Field Phenomena in Superfluids === Subject: ENERGY? MOMENTUM? in GRAVITY? Hell no! They must go! Math is not Physics, Emmy Noether's theorem and all that. Let us look at some of the banners on display. The one reading MOMENTUM has been up for a long time. When the mathematician sees it, he scribbles down 'mv'. When the physicist sees it, he reaches for his gun and fires a bullet into a block of wood. ... Newtonian mechanics is invariant under translation, and that is what makes MOMENTUM meaningful. EINSTEIN'S THEORY DOES NOT ENJOY THAT INVARIANCE and the word cannot be fitted into the new mathematical scheme. It must go, and ENERGY must go, that word sanctified by the industrial revolution. This tearing down of banners is naturally resented, and some people have gone to great pains to try to find in Einstein's theory things that are not there at all. J. L. Synge, Dublin Institute Advanced Studies in What is Einstein's Theory of Gravitation? First, exactly how do you define the group T4? Which operations does it comprise? Simple group of space-time displacements. x^u -> x'^u = x^u + L^u L^u constants independent of x^u So this kind of transformation implies an actual change of position of a physical object in physical spacetime? Yes of course. This is all done in detail in Wigner's papers. Is this instance of the abstract mathematical structure T4 simply a group of passive coordinate transformations in spacetime, or is it something else? Define passive coordinate transformations? A mere re-assignment of 4-tuple addresses to spacetime points (i.e. diffeomorphic re-mapping of the spacetime manifold to the address space R^4). Well there is no physics there. That's purely formal. You only get physics when you say something else on how such a mapping can be measured in some way. Math is not physics. I mean it as a ACTIVE PHYSICAL TRANSFORMATION. OK. Then T4 is a true physical symmetry group. Last I heard. Bob displaces himself by a L^u from Alice. Let both be inertial geodesic observers in globally flat space-time. If the action S of some physical theory is invariant under T4 then, by Noether's theorem, there exists a total 4-momentum P^u for that action S that is conserved in time (forget quantum gravity issues for now). That is total energy and the field with action S. OK. This all breaks down in curved spacetime of course where now we must locally gauge T4 down to x^u -> x'u = x^u + L^u(x^u') Xu'^u = L^u,u' OK. === Subject: Re: ENERGY? MOMENTUM? in GRAVITY? Hell no! They must go! > Math is not Physics, Emmy Noether's theorem and all that. Math is not physics but physics is math. Physics is the mathematical study of all conceivable universes. A universe is a mathematical model that describes spacetime, matter, energy and their interactions. Think of each model universe as filling one page in the atlas of all possible universes. Philosophy is written in this grand book, the universe, ... But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics. - Galileo Galilei. http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Finding the expected length to a pattern? >Suppose I have a coin with probability of having a head H p, and tail T with >probability 1-p. > >What is the expected waited tosses before I get a pattern >HTTHTTH, >HTHTHTHT, If I remember correctly, the quick way when to do that when p=1/2 is to compare substrings at the left and the right ends: So for HTTHTTH H matches H (count 2^1) HT does not match TH HTT does not match TTH HTTH matches HTTH (count 2^4) HTTHT does not match THTTH HTTHTT does not match TTHTTH HTTHTTH matches HTTHTTH (count 2^7) so the expected time is 2+16+128=146 Similarly for HTHTHTHT it is 4+16+64+256=340 A curiosity (from Barry Wolk): HTHT has an expected time of 4+16=20 while THTT has an expected time of 2+16=18. But in a race to see which appears first, HTHT will beat THTT with probability 9/14. In answer to your question, my guess (based the fact that mathematics often follows patterns) is that with p not 1/2 the expected time would be for HTTHTTH p^-1 + p^-2 * (1-p)^-2 + p^-3 * (1-p)^-4 and for HTHTHTHT p^-1 * (1-p)^-1 + p^-2 * (1-p)^-2 + p^-3 * (1-p)^-3 + p^-4 * (1-p)^-4 and a little experimentation makes this seem plausible. === Subject: Re: Finding the expected length to a pattern? > >> >>> I am interested in the following problem: >>> >>> Suppose I have a coin with probability of having a head H p, and tail >>> T >>> with >>> probability 1-p. >>> >>> What is the expected waited tosses before I get a pattern >>> >>> HTTHTTH, >>> HTHTHTHT, >>> >>> etc. >>> >>> This is not a HW problem. I am interested in the fastest approach. >>> Because I >>> already know how to do it using Markov chain modeling, but that takes >>> a >>> long >>> time. I am trying to understand Ross' approach(but his approach is >>> hard >>> to >>> understand...) so I want to see if anybody on the Internet has a >>> better/faster/easier-understanding approach. >> >> My apologies. I just posted a solution method using Markov chains, >> since I responded to a response which didn't include this paragraph. >> Are you interested in the fastest solution using pencil and paper? >> >> Scott >> -- >> Scott Hemphill hemphill@alumni.caltech.edu >> This isn't flying. This is falling, with style. -- Buzz Lightyear >> >> >> Sure. I know there is a solution that only needs a few steps of pencil >> and >> paper... >> >> It is in Ross 8th Edition of Introduction to Probability Models... >> >> But I don't understand it so I don't know how to generalize that >> approach. >> >> It needs to first decompose the string HTHTHTT to maximal substrings, >> etc... >> >> Does anybody know how to do it? Here are the mechanics. (See your reference for a proof as to why this > works.) I'll use your HTTHTTH as an example. For the first toss, bettor number one joins the game with a total wealth > of 1 unit. He places a bet on H. If he loses, he leaves the game with > zero wealth. If he wins he receives a payout of 1/p units. The game is > fair, because he has a 1-p chance of zero and a p chance of 1/p units, > for an expected value of 1 unit. If he loses, he retires from the game. > If we wins, he bets all his wealth on T for the second toss. Again, > if he loses, he leaves with nothing. If he wins, his total wealth will > now be 1/p * 1/(1-p) to make it fair. He will continue in this manner, > betting successively on the the remaining letters ..THTTH to finish > the pattern in the successive tosses. If he wins, he leaves the game > with 1/p^3 * 1/(1-p)^4 units, reflecting 3 bets on heads and 4 bets on > tails. Also, for the second toss, a second bettor joins the game, bringing with > him his total wealth of 1 unit. He places his bet on the first letter > of the pattern, H. He follows the same procedure that the first bettor > does, just one toss later. Each new toss brings one new bettor to the game. When a bettor finally wins, the total wealth in the game will be owned > by three bettors: the one who just won, having bet on HTTHTTH, with a total wealth > of 1/p^3 * 1/(1-p)^4. the one who came in three tosses later, having been successful so far > with HTTH, and having a total wealth of 1/p^2 * 1/(1-p)^2. the one who came in six tosses later, having just bet on the first > H in the pattern, and having a total wealth of 1/p. So the total wealth (no matter how many tosses have occurred) is 1/p^3 * 1/(1-p)^4 + 1/p^2 * 1/(1-p)^2 + 1/p But since each new bettor increases the expected total wealth of the > whole group by 1 unit when he joins the game, the expected number of > betters (which is the same as the expected number of tosses) is the > above number: 1/p^3 * 1/(1-p)^4 + 1/p^2 * 1/(1-p)^2 + 1/p Scott > -- > Scott Hemphill hemphill@alumni.caltech.edu > This isn't flying. This is falling, with style. -- Buzz Lightyear Hi Scott, This method works like magic. Do you have elobaration in plain language explaining why a string pattern should be decomposed in such a way and the final expected waiting time is the sum of all these probabilities for each decomposed substring pattern? It was just this blockade in mind that I could not go through Ross' approach. Why does the sum of probabilities become a expected length? === Subject: Re: Finding the expected length to a pattern? > >> >>> I am interested in the following problem: >>> >>> Suppose I have a coin with probability of having a head H p, and tail >>> T >>> with >>> probability 1-p. >>> >>> What is the expected waited tosses before I get a pattern >>> >>> HTTHTTH, >>> HTHTHTHT, >>> >>> etc. >>> >>> This is not a HW problem. I am interested in the fastest approach. >>> Because I >>> already know how to do it using Markov chain modeling, but that takes >>> a >>> long >>> time. I am trying to understand Ross' approach(but his approach is >>> hard >>> to >>> understand...) so I want to see if anybody on the Internet has a >>> better/faster/easier-understanding approach. >> >> My apologies. I just posted a solution method using Markov chains, >> since I responded to a response which didn't include this paragraph. >> Are you interested in the fastest solution using pencil and paper? >> >> Scott >> -- >> Scott Hemphill hemphill@alumni.caltech.edu >> This isn't flying. This is falling, with style. -- Buzz Lightyear >> >> >> Sure. I know there is a solution that only needs a few steps of pencil >> and >> paper... >> >> It is in Ross 8th Edition of Introduction to Probability Models... >> >> But I don't understand it so I don't know how to generalize that >> approach. >> >> It needs to first decompose the string HTHTHTT to maximal substrings, >> etc... >> >> Does anybody know how to do it? Here are the mechanics. (See your reference for a proof as to why this > works.) I'll use your HTTHTTH as an example. For the first toss, bettor number one joins the game with a total wealth > of 1 unit. He places a bet on H. If he loses, he leaves the game with > zero wealth. If he wins he receives a payout of 1/p units. The game is > fair, because he has a 1-p chance of zero and a p chance of 1/p units, > for an expected value of 1 unit. If he loses, he retires from the game. > If we wins, he bets all his wealth on T for the second toss. Again, > if he loses, he leaves with nothing. If he wins, his total wealth will > now be 1/p * 1/(1-p) to make it fair. He will continue in this manner, > betting successively on the the remaining letters ..THTTH to finish > the pattern in the successive tosses. If he wins, he leaves the game > with 1/p^3 * 1/(1-p)^4 units, reflecting 3 bets on heads and 4 bets on > tails. Also, for the second toss, a second bettor joins the game, bringing with > him his total wealth of 1 unit. He places his bet on the first letter > of the pattern, H. He follows the same procedure that the first bettor > does, just one toss later. Each new toss brings one new bettor to the game. When a bettor finally wins, the total wealth in the game will be owned > by three bettors: the one who just won, having bet on HTTHTTH, with a total wealth > of 1/p^3 * 1/(1-p)^4. the one who came in three tosses later, having been successful so far > with HTTH, and having a total wealth of 1/p^2 * 1/(1-p)^2. the one who came in six tosses later, having just bet on the first > H in the pattern, and having a total wealth of 1/p. So the total wealth (no matter how many tosses have occurred) is 1/p^3 * 1/(1-p)^4 + 1/p^2 * 1/(1-p)^2 + 1/p But since each new bettor increases the expected total wealth of the > whole group by 1 unit when he joins the game, the expected number of > betters (which is the same as the expected number of tosses) is the > above number: 1/p^3 * 1/(1-p)^4 + 1/p^2 * 1/(1-p)^2 + 1/p Scott > -- > Scott Hemphill hemphill@alumni.caltech.edu > This isn't flying. This is falling, with style. -- Buzz Lightyear Hi Scott, Ross' approach in his Introduction to Probability Models 8th Edition. But your bettor and casino analogy is a brand new one. Do you have a reference for this analogy along this line? Can you also show more example about how to decompose string pattern into max sub string pattern? In your above derivation, I also don't understand why at the end of the game, there are 3 bettors... There should be 7 bettors, am I right? === Subject: 2^n -1 hello....doctor~ M_n = 2^n -1 (n>=1) show that (M_m, M_n) = M_(m,n) namely, (2^m -1, 2^n -1) = 2^(m,n) -1 ---------------------------------------------------------------- i have a solution. (2^m -1, 2^n -1) = ((2^m -1)-(2^n -1), 2^n -1) = (2^m - 2^n, 2^n -1) = (2^n(2^(m-n) -1), 2^n -1) since (2^n, 2^n -1) =1, = (2^(m-n) -1, 2^n -1) thus, (M_m, M_n) = (M_(m-n), M_n) let be t = m+n. by mathematical induction, 1) if t=2, (M_1, M_1) = M_1 = M_(1,1) 2) if t The new problems (High School, Advanced, and Challenge) have been posted. > Please visit us at http://math.smsu.edu/~les/POTW.html > -- Web Site High School Problem Southwest Missouri State University's High School Problem Page In a cryptarithm, numbers are represented by replacing their digits by letters; a given letter consistently represents the same digit and different letters represent different digits. Solve the following cryptarithm: TRIED + DRIVE = RIVET -- comments Is this a plug for mass transit? ;-) 0 < t,d < 9 3 < r Oh ho hum, I'll cross post this on to alt.math.recreational. === Subject: Recovering from a computational mathematics education I have read messages from Dr. Herman Rubin and others discussing the sorry state of our mathematics curriculum. I tend to agree that learning computations without theory is not particular useful. I have felt all along that I was not really getting anything out of math class, but I did not know what to do about it. This was particularly evident to me when my first-year Calculus professor effortlessly summarized all the required mathematical knowledge from high-school on a single sheet of paper. Now that the damage has been done, what can I do about? I'm now in my third year of university and have taken math courses up to University Calculus II and Linear Algebra II without ever having to do a proof and without having been taught anything conceptual. Unfortunately, switching to a math degree is not an option at this point. I'd like to be able to upgrade myself from a human calculator to a thinker. Is there any way I can do this through self-study? Due to my lack of experience with real math, I'm not yet sure where my interests lie. Though, I do have a feeling that I have some interest in probability/statistics. In addition, a basic understanding of the fundamentals of analysis, algebra, and combinatorics would probably not hurt. Would it be advisable to just pick up a book like Apostol's Calculus and start reading, should I try to get some experience with proofs first, or should I do something entirely different? What would you recommend as a logical leaning sequence? === Subject: Re: Recovering from a computational mathematics education > > ... What would you recommend as a > logical leaning sequence? The logical order and the pedagogical order are not necessarily the same. Read what interests you. If it interests you, you will be more likely to stick with it and learn something. You mention probability and statistics; I like (in no particular order): Jacod, Jean & Protter, Philip Probability Essentials Springer Kingman, J F C & Taylor, S J Measure & Probability CUP Cramer, Harald Mathematical Methods of Statistics Princeton -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: automorphisms of subspaces of the reals > Q. Does there exist an infinite subspace X of the reals (with the > relative topology on X), such that the only automorphism of X is the > identity. > > A. I don't believe so. Here is a proof using the Cantor--Bendixson > analysis of subsets of the reals. > > Let U be an infinite subset of the reals. By Cantor--Bendixson, we can > write U as the union of a perfect closed set and a countable set. If > the countable set is infinite then it contains infinitely many isolated > points, and we are done (there is a homeomorphism that swaps two of the > isolated points and fixes the rest of the space). Otherwise, the > countable set is finite. In fact, there is at most one point in the > countable set. > I must be missing something here. Suppose U=Q (the rationals). It is an infinite countable set and contains no isolated points. Of course there are many nontrivial autohomeomorphisms of Q, but there are many countably infinite subsets of R with no isolated points. > Now if the set U has no nontrivial autohomeomorphisms then U contains > no intervals and thus U is totally disconnected in the subspace > topology. Write U as the union of a perfect set P together with at > most one isolated point. Then P is also totally disconnected. > > Let x < y < z be three points in P. There are points w,w' not in P > such that x < w < y < w' < z. Thus we can write P as the disjoint > union of (P intersect [w,w']) and the rest of P, and each of these sets > is clopen in P, and thus closed in the usual topology on R. If (P > intersect [w,w']) has a nontrivial automorphism then so does P; the > autohomeomorphism extends because the components are clopen. Thus (P > intersect [w,w']) has at most one isolated point. Discarding this > isolated point, we have a bounded perfect closed subset of U which is > clopen in U. > > Thus we assume P is bounded and has no isolated points. Since P is also > closed, P is compact. Thus P is a compact totally disconnected > infinite set with no isolated points. This means that P is a > homeomorphic image of the Cantor space. The Cantor space has infinitely > many autohomeomorphisms, so P has infinitely many homeomorphisms. > Since P is clopen in U, these extend to infinitely many distinct > autohomeomorphisms of U. This completes the proof. > > The proof actually shows that every infinite set of reals has > infinitely many autohomeomorphisms. > Thus there are no finite autohomeomorphism groups of infinite subsets > of the reals. > === Subject: Re: automorphisms of subspaces of the reals > Maybe I'm just dense, but I don't see how a function that's only defined > on Q can be continuous. > The generalization of continuity to arbitrary topological spaces is based on the observation (predating topology) that for a function f:X-->Y where X,Y are spaces like R^n, or for that matter any metric space, continuity has a number of equivalent versions: (1) epsilon-delta definition of continuity (2) f is continuous iff f is limit preserving (3) f is continuous iff the inverse image of any closed set is closed (4) f is continuous iff the inverse image of any open set is open In a sense this is the starting point for topology. The idea is to characterize the concept of open sets (alternatively closed sets) by axioms. A topological space is defined to be a set X together with a specified collection T of subsets of X, regarded as the open sets (for X), such that the open sets satisfy: (1) the empty set is open (2) X is open (3) an arbitrary union of open sets is open (4) a finite intersection of open sets is open Continuity is then defined purely based on open sets, no need for epsilon delta, in fact, the space is not even required to have a distance function. However keep in mind that the new topological version of the concept of continuity completely agrees with the older concepts for spaces where the older concepts make sense. For subspaces of a topological space, the implied topology inherited from the big space is called the relative topology. So for example, viewing the rationals as a subspace of the reals, the open sets of the rationals are defined as open subsets of the reals intersected with the rationals. They are open relative to the rationals, thus the term relative topology. So the concept of a continuous function on the rationals can now be defined based on the relative topology. If these concepts are new to you, then don't worry about the topological viewpoint for now -- you have alternatives. For subspaces of the reals, you can still use either the epsilon-delta version or the limit preserving version, whichever you prefer. I recommend the limit preserving version (it's more conceptual in my opinion) rather than epsilon-delta. The limit preserving version is this: A function f is continuous iff whenever a sequence x_n in the domain converges to a point x in the domain, then the image sequence f(x_n) converges to f(x). Based on this, the concept of a continuous function on the rationals has a definite meaning. quasi === Subject: Re: automorphisms of subspaces of the reals >On 20 Jul 2005 21:44:37 -0700, Butch Malahide > >> >Q. Does there exist an infinite subspace X of the reals (with the > relative topology on X), such that the only automorphism of X is the > identity. A. I don't believe so. Here is a proof using the Cantor--Bendixson > analysis of subsets of the reals. Let U be an infinite subset of the reals. By Cantor--Bendixson, we can > write U as the union of a perfect closed set and a countable set. >> >>The Cantor-Bendixson theorem applies to *closed* sets. How do you write >>the set of all irrational numbers as the union of a perfect closed set >>and a countable set? >[. . .] The proof actually shows that every infinite set of reals has > infinitely many autohomeomorphisms. >> >>Actually, assuming the axiom of choice, there is a dense subset U of >>the real line such that the only continuous mapping f:U-->U is the >>identity. Hmm - as you point out this is nonsense. For a second I assumed that the problem was with the detail I said I couldn't do below, but no, that detail is clear, there's an error elsewhere, that gets fixed if we assume that f is injective: >>Hint: Use transfinite induction. To start with, you can assume that U >>contains the set Q of all rational numbers. Then, any continuous >>mapping f:U-->R will be determined by its restriction to Q. There are >>just continuum many continuous (or otherwise) mappings from Q into R. >>In constructing the set U, make sure that none of those functions >>(except the identity) extends to a continuous mapping of U into U. > >Ah. I thought there should be a counterexample, but I couldn't see >how to construct one explicitly. What you say here seems probably >right - having thought about it for about a minute I'm stuck on >one detail (maybe you can fill in the detail or say how the thing >should be approached differently so as to avoid the problem): > >If f : Q -> R is continuous say B(f) is the set of all real x >such that either f has no limit at x or f has a limit at x >but this limit does not equal x. In general say _f(x) is the >limit of f at x if there is such a limit, or 0 if the limit >does not exist. > >It seems clear that if f : Q -> R is continuous but not the >identity then B(f) has cardinality c; this is the detail I >don't quite see how to prove, not that I've tried very hard. As you point out, this is very easy to see - there's an interval I such that f(Q intersect I) is disjoint from I, and hence I is contained in B(f). >Assuming this: > >Say the continuous functions f : Q -> R other than the >identity are enumerated as f_a for ordinals a < c. >We construct two increasing families of sets U_a and >T_a for ordinals a < c, such that U_a intersect T_a is >empty for all a, as follows: > >Let U_0 = Q and T_0 = {}. > >For each a, choose x in B(f) such that x is not in >U_b for any b < a, and such that _if_ f has a limit at >x then _f(x) is not in T_b for any b < a. The error is here - if, say, f is constant then there's only one possible _f(x), so we can't choose x so that _f(x) is not in T_b for b < a. But if f is continuous and injective and I is as above then f(Q intersect I) has no isolated points, so its closure has cardinality c, and we're set. >Let U_a be >the union of the previous U_b's with x added, and >let T_a be the union of the previous T_b's, with >_f(x) added if f has a limit at x. > >The fact that B(f) has cardinality c says that it's >always possible to find such an x. Now let U be the >union of the U_a for a < c and let T be the union of >the T_a. > >Now if f : U -> U is continuous and not the identity >then there exists a such that the restriction of f to >Q is f_a. Say x = x_a is the real that was added to U_a >above. Now the fact that f is continuous and x is in >U shows that f_a has a limit y at x, and that f(x) = y. >But this is a contradiction, since the construction >shows that y is in T, hence not in U. > > >************************ > > C. Ullrich ************************ C. Ullrich === Subject: Re: automorphisms of subspaces of the reals On 21 Jul 2005 13:10:43 -0700, Butch Malahide > > >> >> Actually, assuming the axiom of choice, there is a dense subset U of >> the real line such that the only continuous mapping f:U-->U is the >> identity. > >Nonsense! For one thing, there are the *constant* functions f:U-->U. Well, duh. (So the fact that I was unable to complete one of the details in working out what you suggested was actually a good thing, although the fact that I didn't notice your assertion was patently false was a little dumb.) >Maybe I meant to say that the only continuous *injection* from U into U >is the identity. Let me think this over. ************************ C. Ullrich === Subject: Re: automorphisms of subspaces of the reals Actually, assuming the axiom of choice, there is a dense subset U of > the real line such that the only continuous mapping f:U-->U is the > identity. Nonsense! For one thing, there are the *constant* functions f:U-->U. > Maybe I meant to say that the only continuous *injection* from U into U > is the identity. Let me think this over. Let's see if I can get it right this time. Let R be the real line, let c = |R| be the cardinal of the continuum, and let Q be the set of all rational numbers. PROPOSITION. Assuming the axiom of choice, there is a dense subset U of R such that the only continuous injection from U to U is the identity. We construct U by transfinite induction. At a typical stage in the construction, we have defined two disjoint subsets X,Y of R, each of cardinality less than c; X is the set of points we have decided to put in U, Y the set of points we have decided not to put in U; and X contains Q. Now consider a continuous injection f:Q-->R, not the identity. We want to spoil f by adding a few points to X and Y so as to guarantee that f can not be the restriction of a continuous injection from U to U. Define g:R-->R so that g(x) is equal to the limit of f at x whenever the limit exists, otherwise we don't care. Since f is continuous but not the identity, there is an open interval I of R such that, for all rational q in I, f(q) is not in I. Let A = {x in IY: g(x) is in X}, and B = {x in IY: g(x) is not in X}. If B is nonempty, choose x in B; add x to X, and add g(x) to Y. If B is empty, then |A| = c > |X|; choose distinct points x_1 and x_2 in A with g(x_1) = g(x_2); add x_1 and x_2 to X. === Subject: Re: automorphisms of subspaces of the reals Mail-To-News-Contact: abuse@dizum.com >> >> Actually, assuming the axiom of choice, there is a dense subset U of >> the real line such that the only continuous mapping f:U-->U is the >> identity. >Let's see if I can get it right this time. Let R be the real line, let >c = |R| be the cardinal of the continuum, and let Q be the set of all >rational numbers. >Now consider a continuous injection f:Q-->R, not the identity. Maybe I'm just dense, but I don't see how a function that's only defined on Q can be continuous. -- Michael F. Stemper #include Build a man a fire, and you warm him for a day. Set him on fire, and you warm him for a lifetime. === Subject: Re: automorphisms of subspaces of the reals >Maybe I'm just dense, but I don't see how a function that's only defined >on Q can be continuous. The generalization of continuity to arbitrary topological spaces is based on the observation (predating topology) that for a function f:X-->Y where X,Y are spaces like R^n, or for that matter any metric space, continuity has a number of equivalent versions: (1) epsilon-delta definition of continuity (2) f is continuous iff f is limit preserving (3) f is continuous iff the inverse image of any closed set is closed (4) f is continuous iff the inverse image of any open set is open In a sense this is the starting point for topology. The idea is to characterize the concept of open sets (alternatively closed sets) by axioms. A topological space is defined to be a set X together with a specified collection T of subsets of X, regarded as the open sets (for X), such that the open sets satisfy: (1) the empty set is open (2) X is open (3) an arbitrary union of open sets is open (4) a finite intersection of open sets is open Continuity is then defined purely based on open sets, no need for epsilon delta, in fact, the space is not even required to have a distance function. However keep in mind that the new topological version of the concept of continuity completely agrees with the older concepts for spaces where the older concepts make sense. For subspaces of a topological space, the implied topology inherited from the big space is called the relative topology. So for example, viewing the rationals as a subspace of the reals, the open sets of the rationals are defined as open subsets of the reals intersected with the rationals. They are open relative to the rationals, thus the term relative topology. So the concept of a continuous function on the rationals can now be defined based on the relative topology. If these concepts are new to you, then don't worry about the topological viewpoint for now -- you have alternatives. For subspaces of the reals, you can still use either the epsilon-delta version or the limit preserving version, whichever you prefer. I recommend the limit preserving version (it's more conceptual in my opinion) rather than epsilon-delta. The limit preserving version is this: A function f is continuous iff whenever a sequence x_n in the domain converges to a point x in the domain, then the image sequence f(x_n) converges to f(x). Based on this, the concept of a continuous function on the rationals has a definite meaning. quasi === Subject: Re: automorphisms of subspaces of the reals Actually, assuming the axiom of choice, there is a dense subset U of > the real line such that the only continuous mapping f:U-->U is the > identity. > >>Let's see if I can get it right this time. Let R be the real line, let >>c = |R| be the cardinal of the continuum, and let Q be the set of all >>rational numbers. > >>Now consider a continuous injection f:Q-->R, not the identity. > >Maybe I'm just dense, but I don't see how a function that's only defined >on Q can be continuous. What's the definition of the word continuous? ************************ C. Ullrich === Subject: Re: automorphisms of subspaces of the reals Actually, assuming the axiom of choice, there is a dense subset U of > the real line such that the only continuous mapping f:U-->U is the > identity. >>Let's see if I can get it right this time. Let R be the real line, let >>c = |R| be the cardinal of the continuum, and let Q be the set of all >>rational numbers. >>Now consider a continuous injection f:Q-->R, not the identity. > Maybe I'm just dense, but I don't see how a function that's only defined > on Q can be continuous. Q is a topological space. It's a subspace of R and therefore has a relative topology. The function f: Q -> R is continuous if the inverse image of each open set in R is an open set in Q. The restriction to Q of any continuous map from R to R is clearly continuous in this sense. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.