mm-1599 === Subject: Re: Inflationary Theory ; I'm confused Thesis: Space can't expand. If space would be able to expand, it would be bigger than it is, a contradiction. Into what would space itself expand? Into more space. The limit of the incredible is reached with a theory that says that space expands faster than light. Imagine a light ray that goes from point A to point B. When the photon starts moving it can never reach point B or *any* point in space because even if it is traveling at 300 000 Km/sec, in one second much more than 300 000 Km of space separates where the photon was, from its actual place, one second later. SPEED IS KM/SEC. If in one second more than 300 000 Km appear (from nowhere) nothing at all in that universe can move and the temperature of the Universe is absolute zero. Temperature of a gas is the mean speed of its molecules. If the molecules do not move, the gas is at absolute zero. You can't reach point B from point A unless you go faster than light, for *ANY* point A and B. But what means that space is expanding at 300 000 Km sec. What is a kilometer ???? It is thousand times the distance of the stick in Paris, or whatever definition you use. IN ALL definitions will be a physical process that needs movement. You take the Paris stick out of the museum and measure a distance. Or you use some interferometry to make a stick or whatever. Any measurement supposes an *unchanging* measure yardstick. And if space is expanding this yardsticks aren't conceivable any more. After a while they all grow by a variable amount! This becomes even worst when you consider the argument that space expansion doesn't occur within galaxies or other bound objects. Space would expand between objects but the size of the objects would not change, diluting themselves in an ever greater space. The behavior of a yardstick would change if there is a galaxy nearby or not. Suppose a yardstick measuring the distance between galaxy A and B. It would detect an expansion, but if the galaxies were inside a galaxy cluster it would not. All this contradictions have a common base: Space is expanding. Space is not expanding. Light gets stretched when it goes through empty space. Why ? I do not know. I prefer knowing what I do not know than supposing that space is expanding or similar solutions. Astronomers have gotten carried away with this, and I have read some In the beginning was the big bang, etc books with some amusement. The pope is happy, and blesses BB as the confirmation of the bible's creation story albeit with a physics touch now. I remain a skeptic. All the physical explanations sound correct but the basic problem is untouched. How can space expand ??? I have a dent against Creation, Creators, big bangs that started everything. We should at least acknowledge that we haven't the foggiest idea what the Universe is. For starters we can speak only about the observed universe. this is the scientific side. Speaking about what lies beyond the horizon is meta-physics (a very amusing activity) but not science, not physics. And there is *no way* to know what lies beyond the horizon by definition. Any being has an HORIZON, i.e. the measure of the farthest point it can perceive. Our horizon is expanding and the only thing that really expands, is the observable Universe. Our scopes reach farther and farther, and the poor big bang starts a race against it, getting pushed farther and farther since the scopes reach already 13.5 billion years and no bang is in sight. The observations show similar galaxies, clusters, etc as we see around us. No bang is in sight. This was the news this year, when Chandra and all new scopes start reaching the big bang: No bang is in sight. We find galaxy clusters, huge black holes, and old galaxies with a lot of iron in them. Astronomy is not cosmology. Astronomy is about observations, i.e. it can only assert what we know about the observed Universe. What lies beyond is forever the realm of meta-physics. And a theory could be brought for, that tries to explain how the observed Universe evolved, (if we can find a common denominator for it) but that will tell nothing about the universe but about the observed one, the one that lies between our horizon and us. Science can't go beyond the horizon. The basis of science are observations, and the universe can't be observed. This is an intrinsic limitation of any finite being. And please, we *are* finite. Our life span is 100 years if we are lucky, and our speeds never go beyond a few Km/Sec at most. We do not have the foggiest idea what the universe is. We didn't know the existence of planets in other stars just a few years ago. We had never space scopes until shortly. Let's calm down, and start looking around, observing, collecting data, trying to figure out what is out there. And forget the universe. It will be *always* beyond our reach. Science can't offer any further explanation. === Subject: Re: Inflationary Theory ; I'm confused > Thesis: > Space can't expand. Proof: > If space would be able to expand, it would be > bigger than it is, a contradiction. Wrong. Put a number in a spreadsheet cell. Allow iteration. Allow the cell to increase its value by a tiny bit on each pass. What is the value expanding into? > Into what would > space itself expand? An alternative view, is that all of *now* is contracting. > Into more space. Wrong. Since space is not contained, then it is free to be merely a relationship between all the matter in the Unvierse. > The limit of the incredible is reached with a > theory that says that space expands faster than > light. Depends on how you define incredible, doesn't it? What is incredible to me is how people don't use a search engine, and find good information sites like: URL:http://www.astro.ucla.edu/~wright/cosmolog.htm ... > Science can't offer any further explanation. Armchair philosophers run out of explanations a lot faster than science does. David A. Smith === Subject: Re: True = [ proven | provable ] >> One such means of constructing such a model is by the use of witnesses, as >> described above. > Right, this is Henkin's proof, now standard. The axiom of choice is >needed in the general case to show that every consistent set of >sentences has a consistent complete extension. I had thought that the formula of a language (and, in particular, the sentences) could be well-ordered, once you have a well-ordering of the constant symbols, function symbols, relation symbols, and variables. I had also thought that a well-ordering of the sentences would be sufficient to prove that every consistent set of sentences has a consistent complete extension. As a consequence, I had thought that the axiom of choice would only be needed to guarantee a well-ordering of constant symbols, function symbols and relation symbols. Also, I had thought that a well-ordering was needed for constant symbols, functions symbols and relation symbols so that one can well-order those formulae with one free variable, and therefore to introduce the witnesses for those formulae. In other words, I had thought that, in the general case, the axiom of choice was needed to introduce the witnesses (the axiom of choice being specifically needed to well-order the symbols of the (extended) language). David ----- === Subject: Re: scoring between 0 and 1 [Floortje] > I have a quiz where someone is supposed to guess a certain numer (4400). > For a correct awnser (difference=0) they get 1 point and the further away > they are they should lose more of that point but never reach zero. Is > there a forumula that will give (0)=1 and |(inf)|=0 ? There are any number of ways to do this, of course. Here's one nice family of ways: pick real constants A>1 and B>0, and use the formula: f(difference) = A^(-B*|difference|) where ^ denotes exponentiation. Note that the value is: 1 --------------- B*|difference| A As difference goes from 0 to +inf, B*|difference| goes from 0 to +inf too (since B>0), so the power goes from 1 to +inf (since A>1), so the reciprocal goes from 1 to 0. Note that all functions in this family have the property that the score is cut in half whenever the difference increases by log_base_A(2)/B. For example, pick A=2 and B=1. Then log_base_2(2)/1 = 1, and so the score gets cut in half each time the difference increases by 1: 2^(0) = 1 2^(-1) = 1/2 2^(-2) = 1/4 2^(-3) = 1/8 etc. You can pick A and B to make log_base_A(2)/B as little or big as you like. === Subject: Re: scoring between 0 and 1 >If their guess might be any real number and the number they are trying >to guess is N, the you might look at score = exp(- k * | x - N |) >where k adjusts the steepness of dropping off to zero. I'm guessing >you might want small k values like .01 or smaller. >If their guess is from [0, infinity) you might try something like: >score = (1/N) x exp(-(1/N) x + 1) >--Lynn Additionally, if you don't like the exponential shape you might consider: score = 1 / ( 1 + k (x - N)^2 ) for small values of k. --Lynn === Subject: Re: JSH: Benchmark challenge > Program to factor integers with benchmark timer > (COMPAQ Presario Laptop, 2GHz Pentium 4) > ----------------------------------------------- > > Input a positive integer: 1996465633 > Prime Factor: 35141 > Prime Factor: 56813 > Elapsed time: 0.090738 msec > < similar times snipped > > Your program isn't by any chance using Fermat's method? It seems to be a > bit too fast for trial division. > Hi Christian, > If there is some interest in comparing factoring methods with benchmark > timings, I'll post the source code. Yes, that would be welcome. What I would be curious about: There are simple methods that anyone without much mathematical background should easily understand (trial division and Fermat's method), and then there are more complicated methods (Pollard's method is still relatively simple, but what I would call real mathematics). At which point do the more complicated methods overtake the simpler, slower ones in real life implementations? === Subject: Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity Hi Tim, I'm rather ignorant about the paraconsistent logic. When I first heard reading about it, it was in the context of multivalent logics, or multi-valued logics, in the thread titled Base, for it has besides the T and F an indeterminate third truth value, generally U, where that is often ascribed to Kleene, after Lucasiewicz. In terms of its intuitionistic slant and excluded middle, that is considered. In the null axiom or axiom-free set theory, the excluded middle on the ur-element, the ur-paradox of the assertion of existence, does and does not apply. It's considered five years ago in that beginner's post about multivalent logic, or many other posts on sci.math. I admit as well to not being as well-studied as I would like, ergo I study. We here actually basically know what we're discussing, or talking about. Being as well a voracious learner and enjoying knowledge for its own sake, you might understand why I'm very particular about my own personal logical theory of everything and demand my own input. This group is full of people who love to have mathematical knowledge and share it with others, each for their own reasons. Mathematical logic is somewhat vast, and is hopefully compressible to three or four pages. Well, enough of that business. When you say you talk about transfinite cardinals and functions, do you mean along the lines of the functions from reals to reals, or computational complexity and Turing, ie asymptotic finite combinatorics? Do you use that lambda calculus, theory of functions and types, Coquand, Luo, Pierce, Martin-Lof? I heard of it, type theory and lambda calculus, read a couple of those Pierce books. It's cumbersome to apply it in computer programming unless you're writing a compiler, where it can be useful. Basically I want to learn more about you. Do you have a math degree or is your learning self-directed? It's easy to see what words you write here, and the rapidity of assimilation of material. What's your philosophy about mathematics, or mathematical discussion? When you use the transfinite, as you say, is it transfinite induction you use? Ross Finlayson === Subject: Re: Basically a sieve method, relation to quantum by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0NNrAl06950; >So why is it a super sieve, why is it a sieve at all and how does it >relate to quantum methods for factoring? Feelings Nothing more than feelings Ad Hominem === Subject: Hausdorff Theorem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0NNrBg06983; I'm trying to find a proof of the following Hausdorff theorem: If f is a function on [a,b] and C(f),the set of points of continuity of f, is dense in [a,b], then C(f) is not a set of first category in [a,b]. As the only reference I have the highly inaccessible Hausdorff's Grundzuge der Mengenlehre from 1914. I will be most grateful for any help towards the proof of this important theorem. Martin === Subject: Re: Diofrantic by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0NNrBw06972; >I would be >interested if someone could select a simple example, (or >reference), to demonstrate the general idea of the method >of descent, for >a^3 +b^3 = K*c^3 I don't know about ``descent'' methods, but the first thing to note is that a^3 + b^3 = K*c^3 is an elliptic curve. J. W. S. Cassels, ``Lectures on Elliptic Curves,'' London Mathematical Society Student Texts, 24, Cambridge University Press, Chapter 8, Section (i), page solutions can be found from methods in John Cremona's book ``Algorithms for Modular Elliptic Curves'' and or using his programs (see http://www.maths.nott.ac.uk/personal/jec/ftp/data/INDEX.html), or using The APECS package for Maple by Ian Connell (http://www.math.mcgill.ca/connell/). See also, L. J. Mordell, ``Diophantine Equations,'' Academic Press, 1969, pages 124 to 130. John Robertson === Subject: Re: Homeogeneous Spaces by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0NNrCw07015; >Topological space S is homogeneous when for all x,y in S, > some auto-homeomorphism h:S -> S with h(x) = y. >Is a connected subspace of homogeneous space homogeneous? > No. [0,1] and [0,1) subset R are counterexamples. >Is an open subspace of homogeneous space homogeneous? >Is an open connected subspace of a homogeneous space homogeneous? >Counter examples, of course, are welcome. >---- No. S=[0,1] with the natural topology is homogeneous, [0,1) is open and connected _in S_ but it is not homogeneous. V. Anisiu === Subject: Re: fastest way to derive sum(k^3, for k=1, ...n)? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0NNrCA07048; >Hi all, >anybody recommends a fastest way to compute the >sum(k^3, k=1...n)? >I tried hard to remember this formula, but keep forgetting it... >Everytime it took me quite a while to derive it... >Any fastest way? In the first place, as already mentioned by others, the equation sum(k^3, k=1..n) = sum(k, k=1..n)^2 holds. As for the proof, imagine the LHS of this equation as the sum of cubes (in a geometrical sense) with sides 1,2,..,n, where each cube with side k is in turn composed by k^3 unit cubes. Now the RHS of the equation abovesays that we can rearrange all those samll cubes that they form a rectangular parallelepiped of height 1 such that the base is a square with side sum(k, k=1,..n). Assume that we have shown this for some n, say n=3 to make things easier. The question is how to decompose the cube with side 4 in order to get the appropriate bigger parallelepiped. Now this cube can be considered as composed of 4 quadratic layers. If you number the unit cubes of each layer in the following way 1 7 7 7 2 2 6 6 3 3 3 5 4 4 4 4 then it easy to see how to achieve this goal: 1 2 2 3 3 3 4 4 4 4 1 2 2 3 3 3 4 4 4 4 1 2 2 3 3 3 4 4 4 4 1 2 2 3 3 3 4 4 4 4 x x x x x x 7 7 7 7 x x x x x x 7 7 7 7 x x x x x x 7 7 7 7 x x x x x x 6 6 6 6 x x x x x x 6 6 6 6 x x x x x x 5 5 5 5 Again, this needs to be generalized to arbitrary n, but I leave this out here as it is rather obvious. Johann === Subject: Re: Convergent? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0NNrDQ07067; >Prove the convergence or otherwise of >1/1 + 1/2(1+log2) + 1/3(1+log3)(1+log(1+log3)) + >1/4(1+log4)(1+log(1+log4))(1+log(1+log(1+log4))) + ... You may wish to experiment with Mathematica: Define the function g[n] as: h[x_] := 1 + Log[x]; g[n_Integer] := Product[Nest[h, n , m - 1 ], {m, 2, n}]/n Then g[n] defines the n'th term in your series. After spending a short time on this, it seems to me that the series converges to + infty. This is because for each summand, the factor (1 + Log[n])/n converges to 0 at a slower rate than 1/n, whereas all the other factors converge to 1 from above. Best, Zaeem. === Subject: Re: Convergent? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0NNrC607055; >Prove the convergence or otherwise of >1/1 + 1/2(1+log2) + 1/3(1+log3)(1+log(1+log3)) + >1/4(1+log4)(1+log(1+log4))(1+log(1+log(1+log4))) + ... Interesting. You may wish to experiment with Mathematica: Define the function g[n] in Mathematica as follows: h[x_] := 1 + Log[x]; g[n_Integer] := Product[Nest[h, n , m - 1 ], {m, 2, n}]/n g[n] gives the n'th term of your series. In each of the terms g[n], the factor (1+Log[n])/n converges to zero at a slower rate than 1/n. All the other factors seem to converge to 1 from above. I suspect your series converges to + infty. === Subject: Re: ******* TRY THESE SCI.MATH ********** >> How FAR can you move the BAR? >> >> | >> <12 | 34567898765432> >> | >> <11 | 1111111111111> >> <12 | 1212121212121> >> <12 | 3123123123123> >> | >> >> To any finite position. >> >> THAT PORTION OF THE SEQUENCE IS ON THE LIST TOO. >> >> >> >> How far can you move the bar across a random real and >> still be covered in the computables list? >> >> To any finite position. >> > How many digits can you move the bar over? > To any finite position. (+oo is not finite.) The answer to HOW MANY is a quantity. <1 2 3 4 5 6 ..> How many digits are in N that (have a digit after them)? Remember 'where true' in SQL. Herc === Subject: Re: ******* TRY THESE SCI.MATH ********** > -> > -> If you have the list of computables, a random real number can be > on > -it to an infinite number > -> of digits, and yet not be on the list True / False / Other > -> ____ > -> > -A real number r can be such that for each natural number k, there is > -member of the list of computables such that r agrees with it to k > -digits, yet r is not on the list. > Its not in English. > Yes, it is. Show me a dictionary with the words 'r' and 'k'. Herc === Subject: Re: Surrogate factoring, theory versus implementation <35i6elF4locllU1@individual.net> > If that prototype had worked as well as I'd hoped, I wouldn't be > posting about it. > And if pigs had wings, they would fly. Are you sure? That prototype didn't work as well as I'd hoped. That's just a reality. But it does work. It does factor. The method I use is very original, while your reply is not. I have a theory, with a paper outlining the theory, and a prototype program showing that the equations in the paper are correct and that *something* is in fact happening. You are a babbling boob talking about flying pigs. See the difference? You produce nothing. I put the information out there: mathematics and a program. You talk saying nothing of value. I talk from a mathematical foundation. You are just some dumb-ass I feel like replying to, while I am the discoverer. You are nothing but someone I can use to vent upon. So I vent, and you exist. Your life has purpose. James Harris === Subject: Re: Surrogate factoring, theory versus implementation > There are actually two discussions that can take place about my > surrogate factoring theory: > The theory itself, and implementations of the theory. > Not surprisingly some posters have seized on the efficacy of my > prototype program, which primarily is a proof-of-concept, which I have > put out there so you can get some sense of how things are going without > having to first read through the paper. > If that prototype had worked as well as I'd hoped, I wouldn't be > posting about it. > I'm talking this out now on Usenet *because* I'm seeing a failures in > that program that puzzle me, and having hit a wall, I'm talking the > problem out. > First thing with a young theory with an implementation that doesn't > behave as expected is to check the theory!!! > Maybe it's just wrong. > I've checked the theory, and it's not wrong. Well then, maybe you are an incompetent programmer -- in spite of your claims to the contrary. A solution to a problem would consist of a correct theory and a working implementation. You have contributed neither. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Surrogate factoring, theory versus implementation <41F435D4.7ED44863@ix.netcom.com> > I've checked the theory, and it's not wrong. > Well then, maybe you are an incompetent programmer -- in spite of your > claims to the contrary. > A solution to a problem would consist of a correct theory and a working > implementation. You have contributed neither. See, I am open to possibilities, you are not. You are single-mindedly devoted to simply arguing with me, and denying the value of my work. That makes you predictable, and is not an indication of intelligence. I MAY VERY WELL BE WRONG but the theory says otherwise. That an idea that solves the factoring problem--kind of supposedly a hard and significant problem--that was discovered last December still isn't best implemented in, well, in January, may be such an astounding failure to you as to be proof of mathematical failure, but you're clearly not very bright. I say, yes, I may be wrong, but PROVE I am wrong. Repeatedly some of you make these stupid replies or make dumb requests that don't touch the theory, which show your almost total lack understanding of how real discoveries are made. Heard of the Number Field Sieve? Did it pop up overnight? Do you think they went from theory to a full implementation in less than a month? Do you even have a clue of its history? Here's my reality check of you. Show that you have even rudimentary knowledge in the field of factoring. I dare you. It's a challenge. I say you are a babbling Usenet boob who talks way out of your league who has not a clue how ANY discovery gets done. Reply showing you have *some* understanding of the history of how something famous in the field of factoring was found, and how long it took to develop it. Prove you have even rudimentary knowledge. And I say you will dodge this challenge, run away from it, but still come back later posting as if people should take you seriously as that's what people like you do. You know nothing, demonstrate nothing, can produce nothing, but you talk nonetheless, as if you believe that merely talking proves you are of some value. I say you are of no value except as foils for me to vent upon when I get in the mood, like now. There you actually do have some value. James Harris === Subject: Re: the rain in Titan falls mainly in... > They found methane rain on Titan. I think that > is pretty cool (freezing cold, actually). > > WOW! You should try the A+ A- text adjustment on that site, works > great! > It's a bit easier to read that way. I've been reading the Herald > Tribune for years, and it's good to have it online too. The online > version updates with news more often than the print version. I saw that. Made me think of the Buffy episode last week where the librarian is arguing with the computer lab teacher that the information doesn't come from anywhere, its there one minute and gone the next, that books have smell! Lots of people are like that, if you showed them a good interface like that it'd sway the argument, especially for a teacher. Herc === Subject: Re: Homeogeneous Spaces by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id j0O09di09013; >>Topological space S is homogeneous when for all x,y in S, >> some auto-homeomorphism h:S -> S with h(x) = y. >>Is a connected subspace of homogeneous space homogeneous? >> No. [0,1] and [0,1) subset R are counterexamples. >>Is an open subspace of homogeneous space homogeneous? >>Is an open connected subspace of a homogeneous space homogeneous? >>Counter examples, of course, are welcome. >>---- >No. >S=[0,1] with the natural topology is homogeneous, >[0,1) is open and connected _in S_ but it is not homogeneous. >V. Anisiu Sorry, S is not of course homogenous. V.