mm-2369 === Subject: Re: scale transformation > Maybe it's not a scale transformation. I want the rectangle > coordinates to move, but the rectangle sizes stay the same. Imagine > that the rectangles represent labels, and that I wanted to zoom in, > with the actual label sizes staying the same. What kind of > transformation is that? transform the middlepoints M of the rectangle with a scale transformation and rebuild the points of the rectangle P starting from its transformed middlepoint M' with the original scale M' = A*M P' = M' + P - M === Subject: Re: Relative Cardinality > As this would lead to strange results like Card(N) = >> Card({Primes}), >> Of course, Card(N) does equal Card(Primes). >> Does WM think there is a natural number n such that the >> n-th prime does not exist? >> >> Yes, it is so. I am not sure, whether sequences like 111...111 with n >> 1's or like 10^2n - 10^n + 1 do ever cease to supply primes now and >> then. In principle such numbers with 10^10000 digits do exist and >> perhaps could be prime. The prime number 10^100 does not exist, >> however, for the simple reason that we cannot count up to that number >> step by step. Counting up in cents to the U.S. national debt is impossible, so WM >declares that it does not exist? Isn't this basically the position of the ultrafinitists? I'm not sure how seriously they are taken in the field and I'm not sure if WM's position is as well thought out as theirs, but is it fundamentally irrational or unmathematical? OTOH, we should could have saved a lot of time debating this if he'd just said up front what his objection was. Alan -- Defendit numerus === Subject: Re: Please help with a diophant equation <110720051126515540%anniel@nym.alias.net.invalid> Whe the discriminant is negative for y>=2 (y<=-1)? === Subject: Re: Fields of characteristic 2 > I'm starting to study the abstract fields. > I am very curious to know some example of field of characteristic 2. > I know only the classical example of Z_2 (the integers modulo 2). You can get further when you take an irreducible polynomial over Z_2, >say x^2 + x + 1, and start calculating in the polynomials mod that >polynomial. (This is the standard way to extend fields that are not >algebraically complete.) Does there exist an infinite field of characteristic 2? Yes. There is even an algebraically complete field of characteristic 2. >For that you have to look in On Numbers and Games by John Conway. That's a strange assertion! I would agree that On Numbers and Games is a nice book, but I am sure that there must be a few other possibilities for learning about things like forming the algebraic closure of a field! Derek Holt. === Subject: Abstract Representations More on abstractions. This time I am focusing merely on abstract representations, a somewhat contrived term by which I mean sensory abstraction as opposed to linguistic abstraction. I spent some time in trying to define the conditions for an abstract representation. My first attempt was to define it in terms of lossy compression. This met a good deal of objection, because the purpose of lossy compression is more-or-less decreasing the bit error rate of a reproduction. On the other hand, what I have in mind is suppression of details, e.g. simply reducing the Kolmogorov complexity, without really caring about the bit error rate. (At least this is how it goes in this version). Let B be an image, and A be its abstract representation (sketch). The conditions I outlined were: 1) K(A) < c1. K(B) (entropy reduction) 2) K(A/B) < c2. K(B) (noise exclusion) 3) K(A:B) > c3. K(B) (similarity) With these conditions, I want to restrict A to sketches that are: 1) Less complex than the original, e.g. some details suppressed 2) Doesn't contain much information not in the original 3) Is similar enough to the original (e.g. information distance) The constant factors are supposed to make the definitions scale-free, but I cannot give any conditions on what the good constants are. A problem I have with these definitions is, whether condition 3 is redundant, or is something like it needed? It could also be defined as NID(A,B) < c (which turns out to be the same thing!) where NID is normalized information distance of Vitanyi and Cilibrasi. Another problem is the wishful thinking about scale-free definitions. For 1 and 3, there seems to be no problem, but for condition 2 I am not certain. Do my conditions look sensible to you? I'm looking for some references. A friend who reviewed my work said, you are asserting a good deal about abstraction, but there aren't many references to back it up. I based my ideas on Marvin Minsky's discussion of abstraction in his books and my own observations but I couldn't find many references from Kolmogorov complexity researchers. I did receive feedback from valuable colleagues, but more could not hurt. The purpose of these definitions: once you have these it turns out that you can have interesting pattern search algorithms. I can send those who are interested the paper which contains a bit more discussion giving some examples (but excluding condition 3!) -- Eray Ozkural === Subject: Re: weight-density vs density > Isn't the density of a mass of matter just its weight-density, divided > by the acceleration (g) at which it will free fall? No. Weight-density is ill-defined since the gravitational > field will vary over the extent of any object. Furthermore, > at most places on earth the acceleration of free fall > will not be g. Of course you must mean that g varies with location; but at any > particular location, the mass of any particular body of matter is equal > to the weight it exerts on any support, divided by g at that location. Don Let's see... g varies with location, but over the various locations > that make up the volume of any real object, it doesn't vary? Hmmm... > Hmmm; you wouldn't be attributing that nonsense to me would you? Don > So if I take a one foot cube and move it in 11.99 inch steps from the > pole to the equator, there's no change in g from the pole to the > equator? === Subject: Re: weight-density vs density <42d29eed$0$27222$ed2619ec@ptn-nntp-reader01.plus.net> Is anyone familiar with Specific Gravity? I'm wondering why Tables show both weight-density, and density. Isn't the specific gravity of a substance the same using either one? Don === Subject: Re: weight-density vs density <42d29eed$0$27222$ed2619ec@ptn-nntp-reader01.plus.net> Is anyone familiar with Specific Gravity? I'm wondering why Tables show both weight-density, and density. Isn't the specific gravity of a substance the same using either one? Don === Subject: Re: weight-density vs density > Isn't the density of a mass of matter just its weight-density, divided > by the acceleration (g) at which it will free fall? No. Weight-density is ill-defined since the gravitational > field will vary over the extent of any object. Furthermore, > at most places on earth the acceleration of free fall > will not be g. Of course you must mean that g varies with location; but at any > particular location, the mass of any particular body of matter is equal > to the weight it exerts on any support, divided by g at that location. Don Let's see... g varies with location, but over the various locations > that make up the volume of any real object, it doesn't vary? Hmmm... > Hmmm; you wouldn't be attributing that nonsense to me would you? Don > So if I take a one foot cube and move it in 11.99 inch steps from the > pole to the equator, there's no change in g from the pole to the > equator? === Subject: Re: weight-density vs density > Isn't the density of a mass of matter just its weight-density, divided > by the acceleration (g) at which it will free fall? > How many ways are of asking the same question again and again before it starts to look a bit obsessive? === Subject: Re: weight-density vs density <42d2aaf6$0$7908$fa0fcedb@news.zen.co.uk Isn't the density of a mass of matter just its weight-density, divided > by the acceleration (g) at which it will free fall? How many ways are of asking the same question again and again before it > starts to look a bit obsessive? To whom? Don === Subject: Re: weight-density vs density <42d2aaf6$0$7908$fa0fcedb@news.zen.co.uk Isn't the density of a mass of matter just its weight-density, divided > by the acceleration (g) at which it will free fall? How many ways are of asking the same question again and again before it > starts to look a bit obsessive? To whom? Don === Subject: Re: weight-density vs density <42d2aaf6$0$7908$fa0fcedb@news.zen.co.uk Isn't the density of a mass of matter just its weight-density, divided > by the acceleration (g) at which it will free fall? How many ways are of asking the same question again and again before it > starts to look a bit obsessive? To whom? Don === Subject: Re: weight-density vs density Of course you must mean that g varies with location; but at any > particular location, the mass of any particular body of matter is equal > to the weight it exerts on any support, divided by g at that location. Don > I am glad you don't get bored of being proven wrong over this issue. It is fun to see you try to re-couch your terms to get your (false) point across. What is the mass of the sun? What is the mass of Pluto? How are the two masses related by your formula. Is this going to lead to f=wa/g again? I like that one. === Subject: Re: weight-density vs density > Isn't the density of a mass of matter just its weight-density, divided by the acceleration (g) at which it will free fall? >>No. >>Weight-density is ill-defined since the gravitational >>field will vary over the extent of any object. Furthermore, >>at most places on earth the acceleration of free fall >>will not be g. >> Of course you must mean that g varies with location; Do you have learning difficulties? Are you just thick? You've been told and told, g is defined, therefore g does not vary with location.