mm-1034 === Subject: Re: Help with a sequence, please > I need help with this sequence: 1, 7, 25, 65, 140 .... > The sequence is > SUM (i to n) i * T(i) > where SUM is summatory and T(i) is the triangular number of i, that is > (i^2 + i) / 2 > I need a formula to calculate the n-th number of this sequence. Sandra Check Neil J. Sloane's Encyclopaedia of Integer Sequences. Your series is A001296, and a formula is listed. see http://www.research.att.com/~njas/sequences/index_b.html ID Number: A001296 (Formerly M4385 and N1845) URL: http://www.research.att.com/projects/OEIS?Anum=A001296 Sequence: 0,1,7,25,65,140,266,462,750,1155,1705,2431,3367,4550,6020, 7820,9996,12597,15675,19285,23485,28336,33902,40250,47450, 55575,64701,74907,86275,98890,112840,128216,145112,163625, 183855,205905,229881,255892 Name: 4-dimensional pyramidal numbers: (3n+1)*C(n+2,3)/4. Also Stirling2(n,n-2). References M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835. A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16. F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223. Sum(j=1,n,j*triangle(j)) - Jon Perry (perry(AT)globalnet.co.uk), Program: (PARI) t(n)=n*(n+1)/2 for(i=1,30,print1(,sum(j=1,i,j*t(j)))) See also: a(n)=f(n, 2) where f is given in A034261. Adjacent sequences: A001293 A001294 A001295 this_sequence A001297 A001298 A001299 Sequence in context: A033814 A001845 A056685 this_sequence A000970 A048477 A058481 Keywords: nonn,easy,nice Offset: 0 Author(s): njas === Subject: Re: JSH: Factor of x of x >I think MOST people with some mathematical understanding will accept >that if you have a function of x, f(x) and that function equals 0 when >x=0, then it makes sense that f(x) has some factor of x in common with People with little mathematical understanding might think that. >Now I've found myself thinking about that a lot, and hopefully here I >can make a case for why I keep at this when people like Dik Winter or >Arturo Magidin or all these other posters come after me and claim it's >all settled and I'm wrong. >What I think is not wrong is that with f(x), if f(0)=0, then there's >some factor of x in there!!! Let f(x) be the function yielding the count of rational primes p such that 0 <= p <= x. Then f(0) = 0, but f(3) = 2 and in any subring of the complex numbers containing 2 and 3 their only common divisors are the units of that ring. Since no leading expert in number theory could possibly commit the schoolboy howler of thinking that x and f(x) had any nontrivial common factor, this is all a joke, right? I'm really glad to see that you are recovering your sense of humour but perhaps you should consider changing your gag writer. It's not a very funny joke. John Roberts-Jones === Subject: Re: JSH: Factor of x >Yes. b_2(x) is a root of the polynomial: > (b-1)^2 - (x - 1)(b - 1) + 7(x^2 + x) > b^2 - 2b + 1 - (x - 1)b + x - 1 + 7(x^2 + x) > b^2 - (x + 1)b + (7x^2 + 2x) >So what? Both a's and both b's have a factor in common with x. >Nothing strange here. Oh, you are wondering how it is possible >that both a_2(x) and b_2(x) = a_2(x) + 1 can have a factor in >common with x. > >Nope. > > In that case I am wondering about the reason of your posting. > Ok, so you're wondering. He's right to wonder. It looks like an excursion which has little to do with your main difficulty with Rick Decker's polynomial. The Decker quadratic is 7*(25*x^2 + 30*x + 2). Note that in the Decker quadratic, as in your previous cubic, the polynomial variable is 5. Note that if you replace this with t, you have Q(x) = 7*(x^2*t^2 +6*x*t + 2). When 5 was there instead of t, Decker was able to change the form of the polynomial to look like: 7*((x^2 + x)*5^2 + (x - 1)*5 + 7), and to consider a factorization of the form (5*a_1(x) + 7)*(5*a_2(x) + 7). Note that this doesn't work when you substitute t for 5: Q(x) would become 7*((x^2 + x)*t^2 + (-x*t + 6*x)*t + 2), which essentially goes nowhere. Using 5 as a polynomial variable in either your cubic or Decker's quadratic makes essentially no sense. The derivation of the fact that a_1(x) and a_2(x) are roots of a^2 - (x - 1)*a + 7*(x^2 + x) goes back to a factorization in terms of a *polynomial* variable like t, not a *fixed constant* like 5. Perhaps you can escape the Decker example by simply and logically replacing 5 by t or some other polynomial variable. >And something similar is true in your example. The factor a_1(x) has in >common with x is the same as the factor b_2(x) has in common with x, and >is in general coprime to the factor a_2(x) has in common with x. > >What about at x=0? > > What about it? Oh, 0 is not a factor of anything, and anything is a factor > of 0. So the factor a_2(0) has in common with 0 is a_2(0). The factor > Go the other way, as the question is about how a_1(x) and b_2(x) have > factors in common with x such that they *both* equal 0 when x=0 > On the other hand a_2(0) = -1, which is as far from 0 as needed Dik > Winter. > The point is that *most* people with some mathematical knowledge would > accept that if you have a function f(x), where f(0) = 0, then hey, > maybe that means that it has some factor in common with, like you > could have > f(x) = x^{1/3} > where x^{1/3} is, of course, a factor in common with x. Let f(x) = sqrt(x^3 + 1) - sqrt(x^4 + 1). Note that f(0) = 0. Now what factor does f(x) have in common with x ??? > My point here is that posters like you are challenging what most > people accept as rather basic, Your thinking here is transparent, intuitive, and wrong. You think that if you have a function f(x) such that f(0) = 0, it must be the case that f(x) has a factor like, say, x^(1/43), or (x^2 - x)^67, or some such; that is, you can divide out something that looks like a power of x. The example above and many others show that this is not true. Not to mention functions with various kinds of discontinuity and non-differentiability. >and your coy answer here, I think, > indicates that you *know* what you're doing, but are doing it anyway. > b_2(0) has in common with 0 is b_2(0). Moreover, a_2(0) and b_2(0) are > coprime (by your definition of b_2(x)). But indeed, my statement that > the factor a_1(x) has in common with x is the same as the factor b_2(x) > has in common with x is too sweeping. It is true when x is coprime to all > other factors of the constant term of the quadratic. It may not be true > when that is false. > > There's one point here which is why it's in the subject line: Factor > of x What is the factor of x in a function like f(x) = (x^13 + 5)^(1/5) - (x^(1/3) + 5^4)^(1/20), where of course f(0) = 0 ??? Nora B. > I think MOST people with some mathematical understanding will accept > that if you have a function of x, f(x) and that function equals 0 when > x=0, then it makes sense that f(x) has some factor of x in common with > x. > Now I've found myself thinking about that a lot, and hopefully here I > can make a case for why I keep at this when people like Dik Winter or > Arturo Magidin or all these other posters come after me and claim it's > all settled and I'm wrong. You are almost invariably wrong every time you launch a new idea. You have an amazing instinct for wrong mathematics. Do you remember the episode a few days ago when you were trying to factor a product of two primes ? Immediate counterexamples, and a whole series of unsuccessful second and third and fourth guesses by you, every one of them wrong. And here, with this factor of x idea, you are wrong again! > What I think is not wrong is that with f(x), if f(0)=0, then there's > some factor of x in there!!! See above. >BTW, also something similar occurs in the integers. Both 15 and 16 have >a factor in common with 6. In the algebraic integers the situation is >a bit more complicated, because there are so many divisors and no primes. > >You're babbling. What you just said just doesn't relate to the issue at >hand. > > How would I know when you do not tell what the issue at hand is? > > The subject line *gives* the issue at hand. > I'm making a point where I hope to rely on what mathematically aware > readers know themselves happens around 0. >Yup, has already been shown. Given an irreducible, primitive, monic > polynomial with integer coefficients: > x^n + c_(n-1).x^(n-1) + ... + c1.x + c0 >each of the roots has a factor in common with each of the prime divisors >of c0. > >What about at x=0? > > Is x = 0 a root? Yes, it is when c0 = 0. In that case the polynomial is > not irreducible, primitive, monic. You may take the first word from that > list. > > Basically the functions a_1(x) and a_2(x) as defined simply travel > over an interesting path where the polynomial that defines them goes > from being in general irreducible over *integers* with an integer x, > and your position is that the *functions* themselves are aware of this > in some way. > My position is that functions are functions. Right. That explains a lot. > Functional behavior is not that intelligent as to worry about whether > or not some polynomial has integer roots are not! ??? > I mean for readers who don't get it yet, the distinction Dik Winter is > making here is the difference between > x^2 + 2x + 1, which is reducible, > and x^2 + 2x + 2, which, of course is not. > > It might seem too esoteric with x there, so let's put in a value for > > x, and let x=13. Then > > a^2 - 12a + 7(13)(14) > > and we already know that *one* of the a's, is coprime to 13, or wait, > > do we? > >No, we don't, *both* are not coprime to 13. > >So you say that *both* a_1(13) and a_2(13) have some factor in common >with 13. >Interesting. > > Why? The proof has been given a number of times. Only you do not believe > it. > Well, knowing that with *functions* a_1(x) and a_2(x) that a_1(0) = 0, > and a_2(0) = -1, it is NOT a great leap to at least consider the > possibility that a_1(0) has some factor in common with x, while a_2(0) > does not. a_1(x) = 0 when x = 0. What you may have intended to say here was that you think it is reasonable to expect that a_1(x) has some kind of factor in common with x: I believe you think it must be some kind of power of x. That is not true. There is not even good reason to think that a_1(x) in the examples considered here is continuous. Nora B. > James Harris === Subject: Re: JSH: Factor of x > The point is that *most* people with some mathematical knowledge would > accept that if you have a function f(x), where f(0) = 0, then hey, > maybe that means that it has some factor in common with, like you > could have > f(x) = x^{1/3} > where x^{1/3} is, of course, a factor in common with x. Some example of functions f(x) with f(0)=0 f(x) = sin(x) f(x) = 2 - sqrt(4+x) f(x) = 0, if x=0; 47 if x !=0 None of these has a factor in common with x. There are more things on heaven and earth than polynomials in powers of x. - William Hughes > My point here is that posters like you are challenging what most > people accept as rather basic, and your coy answer here, I think, > indicates that you *know* what you're doing, but are doing it anyway. > b_2(0) has in common with 0 is b_2(0). Moreover, a_2(0) and b_2(0) are > coprime (by your definition of b_2(x)). But indeed, my statement that > the factor a_1(x) has in common with x is the same as the factor b_2(x) > has in common with x is too sweeping. It is true when x is coprime to all > other factors of the constant term of the quadratic. It may not be true > when that is false. > > There's one point here which is why it's in the subject line: Factor > of x > I think MOST people with some mathematical understanding will accept > that if you have a function of x, f(x) and that function equals 0 when > x=0, then it makes sense that f(x) has some factor of x in common with > x. > Now I've found myself thinking about that a lot, and hopefully here I > can make a case for why I keep at this when people like Dik Winter or > Arturo Magidin or all these other posters come after me and claim it's > all settled and I'm wrong. > What I think is not wrong is that with f(x), if f(0)=0, then there's > some factor of x in there!!! >BTW, also something similar occurs in the integers. Both 15 and 16 have >a factor in common with 6. In the algebraic integers the situation is >a bit more complicated, because there are so many divisors and no primes. > >You're babbling. What you just said just doesn't relate to the issue at >hand. > > How would I know when you do not tell what the issue at hand is? > > The subject line *gives* the issue at hand. > I'm making a point where I hope to rely on what mathematically aware > readers know themselves happens around 0. >Yup, has already been shown. Given an irreducible, primitive, monic > polynomial with integer coefficients: > x^n + c_(n-1).x^(n-1) + ... + c1.x + c0 >each of the roots has a factor in common with each of the prime divisors >of c0. > >What about at x=0? > > Is x = 0 a root? Yes, it is when c0 = 0. In that case the polynomial is > not irreducible, primitive, monic. You may take the first word from that > list. > > Basically the functions a_1(x) and a_2(x) as defined simply travel > over an interesting path where the polynomial that defines them goes > from being in general irreducible over *integers* with an integer x, > and your position is that the *functions* themselves are aware of this > in some way. > My position is that functions are functions. > Functional behavior is not that intelligent as to worry about whether > or not some polynomial has integer roots are not! > I mean for readers who don't get it yet, the distinction Dik Winter is > making here is the difference between > x^2 + 2x + 1, which is reducible, > and x^2 + 2x + 2, which, of course is not. > > It might seem too esoteric with x there, so let's put in a value for > > x, and let x=13. Then > > a^2 - 12a + 7(13)(14) > > and we already know that *one* of the a's, is coprime to 13, or wait, > > do we? > >No, we don't, *both* are not coprime to 13. > >So you say that *both* a_1(13) and a_2(13) have some factor in common >with 13. >Interesting. > > Why? The proof has been given a number of times. Only you do not believe > it. > Well, knowing that with *functions* a_1(x) and a_2(x) that a_1(0) = 0, > and a_2(0) = -1, it is NOT a great leap to at least consider the > possibility that a_1(0) has some factor in common with x, while a_2(0) > does not. > James Harris === Subject: Re: Final Rout of Synchronization Clocks in Relativity Expires: 28 days >Sorry Henri, that's wrong. Only _inertial_ instruments >measure the speed as c according to SR. >> Yes. OK. You have to include the great SRian escape route. inertial. >>Try this experiment: stand up, turn round then sit down. >>Now consider a flash of light moving away from a recent >>supernova in the Andromeda galaxy. It is 2 million light >>years away so in the frame that co-rotated with you as you >>turned, that light moved 2*pi*R or a distance of 13 million >>light years in just a few seconds. Obviously the speed of >>light in a rotating frame cannot possibly be c. >> I doubt if the one-way speed of light is ever c. >> But anyway, you have put forward a good case for scrapping rotating frames >> altogether. I certainly agree with that. >How does this look in a rotating frame? is a perfectly >valid question one can ask of any experiment so we have to >be able to answer it. After all, the Earth rotates so in the >limit all actual experiments have to take that into account. >The point is that it is obvious from the above that SR does >not say that the speed of light is c in a rotating frame. > ... You were the one who first introduced such to this > argument remember. >>No, you replied to Paul Anderson: >> ... > SR says that the speed of light is c relative to its source. Therefore >if a > ring laser is analysed in the co-rotating frame, no fringe shifts >should be > expected according to SR. > Sagnac clearly refutes SR. >>SR does NOT imply that the speed of light is c relative to >>its source in the co-rotating frame, only that it is isotropic >>in the inertial frame as Paul stated. >> The point I made to Paul was that if the 'co-rotaing frame' argument was >used >> against source dependency then it would also apply to SR since SR is based >> source dependency. >The point I make is that you introduced SR, not me. >In addition SR is based on source independence, specifically >the opposite of what you say. George, if you are in a remote inertial spaceship and send a ray of light out in front of the ship, what do you expect its speed will be relative to the ship. If you reflect the light back with a mirror on a long pole, how long will it take to return? 2D/c. If you send it around a square path using three fixed mirrors, how long will it take to return? 4D/c. The speed of the light is obviously c relative to its source. If you rotate the spaceship in the plane of the mirrors, the time taken will NOT be 4D/c for reasons whicj I have explained. SR doesn't know what it says. > That means that there will also be one according to SR which > clearly states that light moves at c relative to its source. >>You are right, there will be a shift according to SR but >>you have been arguing the opposite. >> I was being a little facetious. >OK, but you cannot then say I am confused when you are >contradicting yourself, glass houses and all that, what? >> The point is, the SR explanation is the same as source dependency. >Nope, they are opposites. Ballistic theory says the light >is emitted at c in the rotating frame and you have to >calculate its speed in the inertial frame. SR says it >will be c in the inertial frame and you have to calculate >what it will be in the rotating frame. You also have to calculate its direction. > Source and observer are the same in a ring gyro. >>Both are rotating and accelerating, hardly inertial. >> Well if the whole thing is not inertial, SR cannot apply at all - so how >the >> hell can you even suggest that SR can provide any kind of explanation for >> sagnac effect when it is entirely beyond its scope? >You only know that the speed is c in the inertial frame >so you calculate it based on that. Since the speed is >isotropic but the mirrors move, there is a predicted >fringe shift. In Ritzian theory you only know the speed >is c relative to the source at emission so have to >calculate in the co-rotating frame. In that the mirrors >are not moving or rotating so there is no fringe shift. That is an oversimplification and incorrect. >>SR says the speed is c in any inertial frame, whether there >>is a 'lab' handy to define that or not. You have removed the >>lab but not the inertial frame. >> And what if the ring is also moving inertially in a direction >perpendicular to >> its plane? >in _any_ inertial frame >>See my other post where I have shown that there is no speed >>change on the reflections and the path lengths vary as (v/c)^2 >>for v<> You cannot assume v<c as well. >It does but we know the shift is linear in (v/c) for small v >and it is easier to calculate. In practice at high v, the >mirror will have moved too far and the light beam will miss >it so you get a crude variation of the toothed wheel method >for measuring the speed of light. Would the cross beam in the MMX do the same? And it does that whether or not the light speed is source dependent. >>No, remember SR says the light travels at c ONLY in an inertial >>frame. The co-rotating frame is not inertial. Ballistic theory >>on the other hand says the light is emitted a c relative to the >>source and each mirror on subsequent reflections. >> No it doesn't. You cannot assume that it even arrives at each mirror at c. >I don't assume it, this is the calculation that demonstrates >it http://www.briar.demon.co.uk/Henri/speed.gif Sorry George. Your diagram doesn't make any sense. > I really don't see where SR comes into this at all. >>It didn't until you introduced it in your reply to Paul. You >>said SR says that the speed of light is c relative to its >>source. which it appears you know is untrue, SR says it is c >>in any inertial frame but the rotating table is not inertial. >>When you bear that in mind you will find SR >> Frankly, I don't want to 'find' SR. It is completely illogical from start >> finish and there is absolutely no evidence to back it up. >Oops, that should have said .. you will find SR predicts >a fringe shift. No doubt you can accuse me of treating SR >as a religion now ;-) Earth centricism acurately predicts that the moon rotates around the sun, too, George. >George Henri Wilson. www.users.bigpond.com/hewn/index.htm === Subject: Re: Final Rout of Synchronization Clocks in Relativity Expires: 28 days >> No Paul. George led me to believe that that was the SRian view. >> What is YOUR explanation? >Explanation of what? >You have been told what SR say about the Sacnac many times, >so why the hell do you keep asking over and over again? >I have certainly explained it MANY times to you. >George Disman explained it a couple of postings ago in this thread. >Are you so senile that you don't remember from one posting >to the next? If that the case, why don't you go back an read >the previous postings again? >Print the following out, and pin it up over your computer: >REMEMBER - I HAVE BEEN TOLD: > According to SR, the speed of light is isotropic c > when the Sagnac ring is rotating, the light will > use different times to go around the ring in the two > opposite directions, because the light path obviously > is shorter in one direction than the other when measured > in the inertial frame where the light moves at the speed c. >You have obviously no rational argument against this explanation, >so you try to ridcilule the expression 'inertial' by vague statements >like this: HaHa!!! But all you have given is the aether solution. You don't even accept an aether. (or so you say Hahahah!) Besides, you don't have any evidence that the speed of light is isotropic in an inertial frame so your explanation is no more than circular speculation. It is no more credible than claiming that the fairies cause the fringes to shift. >| That is a perfect example of why SR is just a disguised aether theory. >| An aether with an 'arbitrary inertial frame'. >| I'm not impressed! >| Yes. OK. You have to include the great SRian escape route. inertial. >What the hell are you implying with these remarks, if it isn't >that there is no such thing as an inertial frame, because >that implies the existence of an ether, which you claim >doesn't exist? Of course inertial frames exist - but in an instantaneous universe they are all effectively the same so we only need one. >Note this. YOU state something which cannot - if it means >anything at all - be interpreted otherwise than a claim that >there is no such thing as an inertial frame. Bull. I'm simply pointing out that there is no evidence that light speed is isotropic. Of course it is not. >And then YOU states that: > George led me to believe that that was the SRian view. >Say - how confised can you get? >I am challenging you to answer my questions below: >---------------------------------- >Do you really insist that inertial frames does not exist? >In that case - how do gyros work? >What is the reference for the rotation they measure? Gyros work for the same reason that a three mirror sagnac works. Light beams moving on opposite directions are deflected by different amounts when reflecting from the moving mirrors. >The fact is of course that the concept inertial frame >a system of co-ordinates in which the equations >of Newtonian mechanics hold good.? Because he knew NM was correct - if not complete. >Paul Henri Wilson. www.users.bigpond.com/hewn/index.htm === Subject: Re: Final Rout of Synchronization Clocks in Relativity Expires: 28 days > I am pretty convinced now that source dependency would make virtually no > difference to the operation of a sagnac. >>Pretty convinced indeed. Of course you are. >>No experimental evidence can shatter your blind irrational >>faith in source dependency. >>You can twist anything if you have enough faith. >>And you do. >>Paul >> Are you suggesting that light DOES NOT move at c relative to its source? > No. What everybody has been suggesting to moron > quantumoids for the last 110 years, is that > *photons* move at C relative to their source. > You freaking moronic, born-again followers > of the true eight-fold path to New York stupid. Why Zed? Henri Wilson. www.users.bigpond.com/hewn/index.htm === Subject: analysis problem........ show that f(x) = x sin(x) is not uniformly continuous -------------------- i want to choose x_1,x_2,epsilon. thus i want to show |x_1 - x_2| |f(x_1)-f(x_2)|>=e_0 but i can't.......... === Subject: Re: analysis problem........ >show that f(x) = x sin(x) is not uniformly continuous >-------------------- >i want to choose x_1,x_2,epsilon. >thus >i want to show >|x_1 - x_2| |f(x_1)-f(x_2)|>=e_0 >but i can't.......... Whenever you talk about continuity (or uniform continuity), specify the open set on which the function is defined. Let's assume this is R, the real numbers. Suppose I claim f is uniform continuous but you are challenging my claim. The definition of uniform continuity says that for any epsilon > 0, there exists an epsilon > 0 such that |f(x1) - f(x2)| < epsilon whenever x1, x2 in R and |x1 - x2| < delta. If f is not uniform continuous, then some epsilon > 0 exists such that no suitable delta can be found. You, the challenger, are supposed to produce an epsilon which will fail. You might pick epsilon = 1 or epsilon = 10^(-100) or epsilon = pi/sqrt(2). but must make your choice known. Your disproof should similarly start with something like Let epsilon = 0.5. Having heard your choice of epsilon, it is my job to produce a corresponding delta which will work for your epsilon. Perhaps I pick delta = MIN(1, epsilon/100). This will be positive whenever epsilon > 0 Observe that my choice of delta is allowed to depend on your epsilon but not on x1, x2. You, the challenger, now show I am lying. Find a way to choose x1 and x2, using my delta, such that |x1 - x2| < delta but |f(x1) - f(x2)| is large. You are showing that, no matter how small my delta is, you can find two close points x1, x2 such that f(x1) and f(x2) are at least epsilon apart. It may help to graph f. -- John Adams served two terms as Vice President and one as President, but lost reelection. Later his son became President despite losing the popular vote. That son lost his reelection attempt badly. Now history is repeating itself. pmontgom@cwi.nl Microsoft Research and CWI Home: San Rafael, California === Subject: Re: analysis problem........ very difficult....... any d>0 |x1-x2| < d = min(1, e/100)=> |f(x1)-f(x2)| = |x1*sin(x1) - x2*sin(x2)| >= 100|x1-x2| = 100*(e/100) =e um.............. i can't find x1, x2.... very very hot difficult.......um............. === Subject: Re: analysis problem........ > show that f(x) = x sin(x) is not uniformly continuous > -------------------- > i want to choose x_1,x_2,epsilon. > thus > i want to show > |x_1 - x_2| |f(x_1)-f(x_2)|>=e_0 > but i can't.......... That isn't what you need to show. Given d, e_0 there exists x_1,x_2 such that ... -- 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: Constructions of a Heptagon See the construction of a Heptagon: http://hptgn.tripod.com/ === Subject: Re: need help in understanding Torkel's ZFC comment > > 5. Is ZFC an attempt to formalize Set Theory (which also avoids the > paradoxes)? Where have authors been able to construct formal proofs > in ZFC? > > Get a grip, idiot: You know, you'd make a really lousy teacher. (In case - or rather since - you haven't figured it out already, I do already know the answers to my 6 questions. And, in spite of what you might have heard elsewhere, it IS a good idea to ask a question that you know the answer to, to see where the other person's head is at - or to smoke 'em out.) > Inspired by Whitehead and Russell's monumental Principia Mathematica, > the Metamath Proof Explorer has over 3,000 completely worked out proofs > in logic and [ZFC] set theory. > Essentially everything that is possible to know in mathematics can be > derived from a handful of axioms known as Zermelo-Fraenkel set theory. If you believe that, then I have a bridge that I'd like to show you (on sale). > which is the culmination of many years of effort to isolate the > essential nature of mathematics and is one of the most profound > achievements of mankind. Great! Because my system is a lot better. Wow - one of the most profound achievements of mankind (and outdone.) *blush* Charlie Volkstorf PS Where can I find quotes about how great ZFC is? I want to include system fixes its mistakes and is a lot better. > Taken from the > Metamath Proof Explorer Home Page > http://metamath.planetmirror.com/mpegif/mmset.html > F. === Subject: Re: need help in understanding Torkel's ZFC comment > How about if you point me to the best example or two of proofs using > ZFC that is given - preferably not just proving part of ZFC with > another part? >The square root of 2 is irrational: > http://au.metamath.org/mpegif/sqr2irr.html > Q: What does that have to do with ZFC? There are 42 lines and > offhand it doesn't look like any of them are from ZFC. > A: This theorem was proved from axioms: ax-1 ax-2 ax-3 ax-mp ax-4 > ax-5 ax-6 ax-7 ax-gen ax-8 ax-9 ax-10 ax-11 ax-12 ax-13 ax-14 > ax-15 ax-16 ax-17 ax-ext ax-rep ax-un ax-pow ax-reg ax-inf. > > (http://au.metamath.org/mpegif/sqr2irr.html) As I said elsewhere, 19 of these are axioms outside of ZFC, and I don't think you really need ZFC to prove that sqrt(2) is irrational anyway. This is NOT a good example of a formal use of ZFC, IMEO, and if anything shows that ZFC alone doesn't suffice. (I will look more closely now that I can concentrate on the best example.) Charlie Volkstorf > F. === Subject: Re: need help in understanding Torkel's ZFC comment |> Charlie-Boo says... |>1. Would ZFC be better if it included its rules of inference? |> For mathematical purposes, the theory *is* its set of theorems. How |> you describe those theorems as axioms/axiom schemas/rules of inference |> doesn't really matter much. | |But if different authors use different rules of inference, the |theorems (and thus the theory) may vary. The logic associated with ZFC is classical first-order logic. There are multiple formalizations of classical first-order logic. Since it doesn't matter which one is used, one doesn't assume that any one of them is the set of inference rules for ZFC. |And if some are later |discovered to be inconsistent - oops. So you're worried that first-order logic might turn out not to have been correctly formalized? Or that it might turn out to be incoherent? |Why reinvent the wheel? Why indeed? First-order logic is an off the shelf component, and restating it each time you want to present an axiom system would be a case of reinventing the wheel. So would taking your rules of inference to include extra stuff besides first-order logic (when you could just as well simply use a standard set of rules of inference). |Those |are just some of the problems caused by using an incomplete system and |each author having to fill in the gaps. I've read Cohen's book on the continuum hypothesis and the axiom of choice. I've seen various papers referring to ZFC in journals. I don't remember any of them having to fill in gaps. I think you're still thinking that ZFC should include some stuff that simply isn't considered part of it. |> The standard view is to try to separate |> the rules of *logic*, which are the same for any first-order theory, |> from the *axioms*, which are different for different theories. The |> advantage is that you can prove facts about all first-order theories |> (such as compactness) and then they automatically apply to ZFC, or |> to PA, or to GNB, etc. | |Yes, if the rules of inference are given. But Zermelo et. al. never |gave rules of inference (although unintentionally including some |misconstrued as axioms.) For most purposes, giving rules of inference for first-order logic is unnecessary. Zermelo wasn't trying to implement ZFC on a computer or anything like that, so he didn't need formal rules. He wasn't trying to do syntactic proof-theory. He just needed an ordinary mathematician's understanding of the quantifiers and connectives, just like the way we would reason about any other first-order theory. Axiom schemes like the axiom scheme of replacement are counted as axioms because they have content. They restrict which structures can count as models. Ordinary rules of inference in first-order logic, in order to be valid, need to be valid for all possible models, whether they are models of ZFC or not. |> But for most purposes, it doesn't really matter how you describe the |> theory. | |There's also the question of efficiency - minimizing the number of |axioms (dropping redundant ones). There are plenty of ramifications |to using a poorly designed system! Efficiency for what purpose? I can't think of many purposes for which having fewer axioms would always help in efficiency. More compact storage of the axiom set is not a very important consideration. Keith Ramsay === Subject: want to improve my math skills I was wondering if anyone could point me to a website, or book (with the answers) that would test me on problem solving. Like word problems improve my problem solving skills but for some weird reason I can't find anything on the web. PLEASE HELP!!!! please feel free to email me === Subject: Re: Graph Theory: Adjacency Matrices for Graphs without 3-cycles > One possible matrix for a graph with no 3-cycles is > > 0 1 0 1 0 1 0 1 ... > 1 0 1 0 1 0 1 0 ... > 0 1 0 1 0 1 0 1 ... > 1 0 1 0 1 0 1 0 ... > etc. > > This appears to be a sufficient condition for no 3-cycles. Is it also > a necessary condition? > > A necessary and sufficient condition for a matrix M = (m_ij) to be the > adjacency matrix of some (simple) graph G is that it be a symmetric 0-1 > matrix with 0's on the diagonal. > > A necessary and sufficient condition that G not contain 3-cycles is > that m_ij m_jk m_ki be 0 for all i,j,k, which is equivalent to saying > that M^3 has all 0's on the diagonal, which is in turn equivalent to > saying that the sum of the cubes of the eigenvalues of M is 0. (When > M is a 3x3 matrix, this is in turn equivalent to det M = 0, but I don't > think there is a similarly simple characterization in general.) > > -Jim Ferry > I don't think that the matrix of a graph is a true matrix. The 1's > and 0's > do not represent values; but merely the presence or absence of an > edge. The matrix proposed represents the no 3-cycle graphs with the > most edges for a given |V|. > I do not understand matrix nomenclature and 'eigenvalues well > enough to comment on your observations. Sorry; Bill J. I don't know what's not true about it. The adjacency matrix of a graph with n vertices (numbered 1,2,...,n) is the matrix with entries M_{ij} = 1 if is an edge, 0 if it is not. Yes, 1 and 0 represent presence or absence of an edge. So what? You can still treat this matrix as you would any other n x n symmetric matrix. Yes, your graph has the most edges of any |V|-vertex graph with no 3-cycles. But not every graph without 3-cycles is a subgraph of it. Consider e.g. the 5-cycle graph 1--2 | | 3 | / 5--4 (view in fixed-width font) This has the adjacency matrix [ 0 1 0 0 1 ] [ 1 0 1 0 0 ] [ 0 1 0 1 0 ] [ 0 0 1 0 1 ] [ 1 0 0 1 0 ] It is not a subgraph of any of your graphs (which don't have any 5-cycles). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: re:The Lost Proof of Fermat claiming that the self evident equation: x^3 + y^3 = (x+y)*[(x+y)^2 - 3xy ] is incorrect, to put it mildly. Troll = Moron. What is the world coming to??? I refuted their moronic babble: 3^3 + 4^3 = 91 = (3+4)*[ 49 - 36 ] = 7*13 = 91 (x+y)*[(x+y)^2 - 3xy ] = (x+y)^3 - 3xy(x+y) = (x+y)^3 - 3x^2 y - 3x y^2 = x^3 + 3x^2 y + 3x y^2 - 3x^2 y - 3x y^2 + y^3 = x^3 + y^3 http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: Re: The Lost Proof of Fermat Egads! I am under TROLL ATTACK!!! Calling all trolls Calling all trolls Wake up and smell the logic! x^3 + y^3 = (x+y)*[(x+y)^2 - 3*x*y ] 3^3 + 4^3 = (3+4)*[(3+4)^2 - 3*3*4 ] 3^3 + 4^3 = 91 = 7*[49 - 36 ] = 7*13 = 91 Wake up ya @#$%% **&&^ TROLLS!!!! Are there any REAL mathematicians here??? === Subject: Re: The Lost Proof of Fermat Utilizing the generalized equation (x+y)*[(x+y)^(p-1) + [{(x^p+y^p)/(x+y)} - (x+y)^(p-1)] ] becomes: x^3 + y^3 = (x+y)*[(x+y)^2 - 3xy] x^5 + y^5 = (x+y)*[(x+y)^4 -5x^3 y -5(xy)^2 -5x y^3 ] x^7 + y^7 = (x+y)*[(x+y)^6 -7yx^5 -14x^4 y^2 -21(xy)^3 -14x^2 y^4 - 7xy^5 ] In general, x^p + y^p = (x+y)*[(x+y)^(p-1) - p*f(x,y) ] === Subject: Re: The Lost Proof of Fermat > Utilizing the generalized equation > (x+y)*[(x+y)^(p-1) + [{(x^p+y^p)/(x+y)} - (x+y)^(p-1)] ] > becomes: > x^3 + y^3 = (x+y)*[(x+y)^2 - 3xy] Let's see, for x=2 and y=3, left side is 8 + 27 = 35 and the right side is (5)*[5^2 - 12] = 5*[13] = 65. I don't think it works. > In general, x^p + y^p = (x+y)*[(x+y)^(p-1) - p*f(x,y) ] Multiply through your right side by your (x+y) factor. You get (x+y)^p - p*(x+y)f(x,y). Obviously, if you try to expand (x+y)^p, you will get an x^p term and a y^p term, which is what you have on the left side... then you magically subtract off an f(x,y) which you don't really describe. Perhaps you want to know the factorization x^p + y^p = (x+y)( x^p - x^(p-1)y + x^(p-2)y^2 - x^(p-3)y^3 + ... + y^p ) (prime p only). It's clean, easy to prove, no messy coefficients... and fairly well-known. J === Subject: Re: The Lost Proof of Fermat >>Ok but the above representation of L'Hopitals rule is over-simplified. >>It would be more accurate to say that IF lim f'(x)/g'(x) x->a = c in R >>then lim f(x)/g(x) x->a = c. > Not sure whether you're deliberately trying to mess with people, or if > you're just stupid, but you can *EASILY* look this stuff up. > Doug I agree. However even though everyone can easily look it up, it is also very easy to misinterpret what it says. The most common mistakes (besides from forgetting to verify that the preconditions are satisfied) seems to be people thinking that lim f(x)/g(x) = lim f'(x)/g'(x) (for === Subject: Re: It doesn't Re: How Does the Fibonacci Sequence Start from Zero > I thought Fibonacci obtained the sequence as part of tackling a numerical > problem about rabbit breeding. So it doesn't start with 0. It could if you believe in creationist mathematics. :-) BTW, A: Because it messes up the order in which people normally read text. Q: Why is top-posting such a bad thing? A: Top-posting. Q: What is the most annoying thing on usenet and in e-mail? -- --Tim Smith === Subject: Re: Ask Marilyn > Marilyn vos Savant seems to be defending the indefensible this week. A > reader takes her to task over her answer to this problem: Two people race > over a course of 100 miles, the first 50 of which must be covered at 55 > mph, and the second 50 must be covered at a rate of 65 mph. The difference > is that one racer must stop for 15 minutes in the 55 mph leg and the > second must stop for 15 minutes in the 65 mph leg. > Marilyn claims the race is a tie. There are many special cases that make > the problem work out Marilyn's way, but not every scenario works. ... > As one example, suppose the first racer stops after 3/4 hour for 1/4 hour. I don't even need to read your example to know that you must have made an arithmetic mistake. Here's how to see this: Let's consider a different problem. You and I each have a video tape of a single car's run on a test track. It went 50 miles at 55 mph, then went 50 miles at 65 mph. If we both start watching our tapes at the same time, we will finish at the same time. Suppose we each take a 15 minute break, where we pause the tape for 15 minutes, then resume it. We obviously will still finish at the same time, no matter when we each take our 15 minute break. You can clearly construct an isomorphism between the original problem and the tape watching situation, with the 15 minute break in tape watching mapping to the 15 minute break in the race. -- --Tim Smith === Subject: Re: Ask Marilyn > In sci.math, Baldin Pramer > : > Marilyn vos Savant seems to be defending the indefensible this week. A > reader takes her to task over her answer to this problem: Two people > race over a course of 100 miles, the first 50 of which must be covered > at 55 mph, and the second 50 must be covered at a rate of 65 mph. The > difference is that one racer must stop for 15 minutes in the 55 mph leg > and the second must stop for 15 minutes in the 65 mph leg. > > Marilyn claims the race is a tie. There are many special cases that make > the problem work out Marilyn's way, but not every scenario works. > Unfortunately, her exact original problem statement is not included in > either the reader's response, or in her response to the reader. > > As one example, suppose the first racer stops after 3/4 hour for 1/4 > hour. It then takes him 7/44 hour to finish the remaining 55 mph part of > the course, then 50/65 hour to finish the 65 mph part of the course, for > a total time of 3/4 + 1/4 + 7/44 + 50/65 hour. If the second racer > drives straight through the 55 mph part then drives 3/4 hour in the 65 > mph part then stops 15 minutes, he has only 5/4 mile left to cover at > the end. His time will be 50/55 + 3/4 + 1/4 + 1/52 hour. > > Well, lessee, what scenarios are we covering here? Ghost In The Machine's analysis looks fine, but the problem probably wanted to assume instantaneous stops and acceleration. Baldin Pramer seems to have been arguing that the race would not be a tie, and he gives an example... I'm not sure if he says anything else after what was copied there, but maybe he missed the fact that 50/55 + 1/52 = 7/44 + 50/65 and so the race ends in a tie. J === Subject: (-2/3)^(-2/3) = (3/2)^(2/3)? According to GraphEq (-2/3)^(-2/3) > 0 -> http://peda.com/grafeq/gallery/rogue/xx_exponential.html. In fact (-2/3)^(-2/3) have 3 distinct values (2 complex and 1 real). One of them is ~ -0.655185-1.13481i. Right? If so, is it mathematically incorrect the statement: For negative x = even / odd, x^x is well-defined positive real number? Why y^b = x^a <==> y = (x^a)^(1/b) is not true for all x and b <> 0 in R, i.e. why operation exponentiation is a single-valued for positive base? For example: y + b = x + a <==> y = (x + a) - b is true for all values of x, y, a, b in R y * b = x * a <==> y = (x * a) / b is true for all values of x, y, a, (b <> 0) in R Calvin === Subject: Re: Group > If H and K are both given as subgroups of an overgroup G, then HK > is defined to be { hk | h in H, k in K }. This is not always a > subgroup of G (although it is sometimes). Actually, if H, K are subgroups of G, then HK is a subgroup of G if and only if HK=KH. It's basic to show that :) Philipp === Subject: Re: Circle geometry >Given: > a circle centre (x, y); > 2 points on the circumference Point1 at (a, b) and Point2 at (c, d). >The 3 sets of coordinates above are all known, thus the angle between >the 2 points can be calculated easily. >How can I mathematically/programatically determine whether Point1 is >clockwise or anti-clockwise to Point2 along the shortest arc. Bearing >in mind that the 2 points may fall anywhere on the circumference of >the circle. >Martin > Calculate the determinant: > |a-x b-y| > |c-x d-y| > If it is positive (c,d) is counterclockwise from (a,b) relative to the > center (x,y), if negative then clockwise. If the determinant is zero > then the points are either identical or opposite ends of a diameter. > --Lynn Wow, that's awesomely simple. Martin === Subject: Re: ?? categoricity or naive felicity ?? >|This is what I meant about the male alpha >|problem The interesting thing about the female omega phenomenon (which I would never be so tendentious as to label a problem) is that the lower-case and upper-case variants have such different iconicities. Lee Rudolph === Subject: Re: ?? categoricity or naive felicity ?? : >|This is what I meant about the male alpha : >|problem : : The interesting thing about the female omega phenomenon (which : I would never be so tendentious as to label a problem) is that : the lower-case and upper-case variants have such different : iconicities. I intend problem as in a puzzle or a mathematical exercise. One natural solution is hierarchies, though it is not always the most healthy solution. -- -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: ?? categoricity or naive felicity ?? >|This is what I meant about the male alpha >|problem > The interesting thing about the female omega phenomenon (which > I would never be so tendentious as to label a problem) is that > the lower-case and upper-case variants have such different > iconicities. On the other hand, it is unlikely that female participation on newsgroups is responsible for compositions like the one a friend of mine just sent me: ----- Lessons in Group Posting: (and why groups, generally, should be avoided) How many group posters does it take to change a light bulb? 1 to change the light bulb and to post that the light bulb has been changed 14 to share similar experiences of changing light bulbs and how the light bulb could have been changed differently 7 to caution about the dangers of changing light bulbs 27 to point out spelling/grammar errors in posts about changing light bulbs 53 to flame the spell checkers 41 to correct spelling/grammar flames 6 to argue over whether it's lightbulb or light bulb ... another 6 to condemn those 6 as anal-retentive 2 industry professionals to inform the group that the proper term is lamp 15 know-it-alls who claim *they* were in the industry, and that light bulb is perfectly correct 156 to email the participant's ISPs complaining that they are in violation of their acceptable use policy 109 to post that this group is not about light bulbs and to please take this discussion to a lightbulb group 203 to demand that cross posting to hardware forum, off-topic forum, and lightbulb group about changing light bulbs be stopped 111 to defend the posting to this group saying that we all use light bulbs and therefore the posts *are* relevant to this group 306 to debate which method of changing light bulbs is superior, where to buy the best light bulbs, what brand of light bulbs work best for this technique and what brands are faulty 27 to post URL's where one can see examples of different light bulbs 14 to post that the URL's were posted incorrectly and then post the corrected URL's 3 to post about links they found from the URL's that are relevant to this group which makes light bulbs relevant to this group 33 to link all posts to date, quote them in their entirety including all headers and signatures, and add Me too 12 to post to the group that they will no longer post because they cannot handle the light bulb controversy 19 to quote the Me too's to say Me three 4 to suggest that posters request the light bulb FAQ 44 to ask what is a FAQ 4 to say didn't we go through this already a short time ago? 143 to say do a Google search on light bulbs before posting questions about light bulbs 1 forum lurker to respond to the original post 6 months from now and start it all over again.... _________________________________________________________________ Ahhhhhhhhh - life in the fast lane. :-) It makes me think of this series of classes I was hired to teach on the Unix OS - and each of the 26 students barely spoke a common language. Worse, their knowledge bases were mutually disjunct. I never did *that* again. Once in a while, experience is a good teacher. :-) ----- :-) mitch === Subject: 0 and natural numbers I noticed in the sci.math FAQ that it states that informally the inclusion or exclusion of 0 from N is a subject of religious argument (or words to that effect). It then continues with the formal set theory definition, ie 0={}, 1={0}, 2={0,1}, 3={0,1,2}, etc. This is probably a naive question, but can one reject this definition and still accept Set theory as the fundamental basis of mathematics? Is there an alternative definition of N where 0 is not included using set theory? eg 1={1}, 2={1,1}, 3={1,1,2}, etc. [But what to do with the empty set as it is a subset of all sets?] I have also come across another inductive definition IIRC of N, defining the number 1 and the operation '+' and thus deriving all N. It is certainly simpler/achievable than trying to start with 0 and '+' because you then have to define 1 anyway to get anywhere. Google searches list (mis)quoted Peano's axioms as 0 or 1 being the lowest element of N. I assume this is according to the authors own world-view. What did Peano really say with respect to 0 in N? I note the religious reference in the FAQ and suspect I am just covering tired contentious ground, but I would appreciate any attempts at clarifying this issue. Anthony David === Subject: Re: 0 and natural numbers > I noticed in the sci.math FAQ that it states that informally the inclusion > or exclusion of 0 from N is a subject of religious argument (or words to > that effect). > It then continues with the formal set theory definition, ie 0={}, > 1={0}, 2={0,1}, 3={0,1,2}, etc. > This is probably a naive question, but can one reject this definition > and still accept Set theory as the fundamental basis of mathematics? Yes, there is the Zermelo definition. > Is there an alternative definition of N where 0 is not included > using set theory? > eg 1={1}, 2={1,1}, 3={1,1,2}, etc. 1={1} as a definition is circular. > [But what to do with the empty set as it is a subset of all sets?] > I have also come across another inductive definition IIRC of N, defining > the number 1 and the operation '+' and thus deriving all N. It is > certainly simpler/achievable than trying to start with 0 and '+' because > you then have to define 1 anyway to get anywhere. > Google searches list (mis)quoted Peano's axioms as 0 or 1 being the > lowest element of N. I assume this is according to the authors own > world-view. What did Peano really say with respect to 0 in N? > I note the religious reference in the FAQ and suspect I am just covering > tired contentious ground, but I would appreciate any attempts at clarifying > this issue. > Anthony David -- G.C. === Subject: Re: 0 and natural numbers > I noticed in the sci.math FAQ that it states that informally the inclusion > or exclusion of 0 from N is a subject of religious argument (or words to > that effect). > It then continues with the formal set theory definition, ie 0={}, > 1={0}, 2={0,1}, 3={0,1,2}, etc. > This is probably a naive question, but can one reject this definition > and still accept Set theory as the fundamental basis of mathematics? > Is there an alternative definition of N where 0 is not included > using set theory? > eg 1={1}, 2={1,1}, 3={1,1,2}, etc. 1 = {1} violates the axiom of regularity or foundations 2 = {1,1} = {1} = 1 gets you nowhere 3 = {1,1,2) = {1,1,1} = {1} = 1 treads water > [But what to do with the empty set as it is a subset of all sets?] Be amazed how the lowest of the low acheives emptyness of a Zen master. For lo, as above, from emptyness comes all: 0 = {}, 1 = {0}, 2 = {0,1} === Subject: Re: 0 and natural numbers > For lo, as above, from emptyness comes all: 0 = {}, 1 = {0}, 2 = {0,1} 0 = {|} 1 = {0|} -1 = {|0} 1/2 = {0|1} surreal numbers also come from nothing. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: 0 and natural numbers > For lo, as above, from emptyness comes all: 0 = {}, 1 = {0}, 2 = {0,1} > 0 = {|} > 1 = {0|} > -1 = {|0} > 1/2 = {0|1} > surreal numbers also come from nothing. They came out of a bar. ;-) === Subject: Re: 0 and natural numbers > For lo, as above, from emptyness comes all: 0 = {}, 1 = {0}, 2 = {0,1} > 0 = {|} > 1 = {0|} > -1 = {|0} > 1/2 = {0|1} > surreal numbers also come from nothing. > They came out of a bar. ;-) no, no, ... it's true! no need to mock the name! why, you don't oppose the 'builiding-up' of the ordinals, do you? then why this abhorence to surreals? i just wanted to know if there is any serious mathematical theorems out there about surreal numbers. or is it just beautiful, but of no consequence, sort of like alicia silverstone? === Subject: Re: 0 and natural numbers > Be amazed how the lowest of the low acheives emptyness of a Zen master. > For lo, as above, from emptyness comes all: 0 = {}, 1 = {0}, 2 = {0,1} and not just natural numbers: For a set x, define Cx to be the power set of x union x. Then the sequence {} C{} CC{} ... (continued transfinitely) is all the sets :-^) === Subject: Re: 0 and natural numbers > I noticed in the sci.math FAQ that it states that informally the inclusion > or exclusion of 0 from N is a subject of religious argument (or words to > that effect). > It then continues with the formal set theory definition, ie 0={}, > 1={0}, 2={0,1}, 3={0,1,2}, etc. > This is probably a naive question, but can one reject this definition > and still accept Set theory as the fundamental basis of mathematics? The fact that there is a particular way to define the natural numbers in terms of set theory does not mean that other definitions can't exist, or that those other definitions can't also be based on set theory. > Is there an alternative definition of N where 0 is not included > using set theory? > eg 1={1}, 2={1,1}, 3={1,1,2}, etc. How about 1 = {}, 2 = {1}, 3 = {1,2}, ...? > [But what to do with the empty set as it is a subset of all sets?] Did I say it wasn't? > I have also come across another inductive definition IIRC of N, defining > the number 1 and the operation '+' and thus deriving all N. It is > certainly simpler/achievable than trying to start with 0 and '+' because > you then have to define 1 anyway to get anywhere. We could make it even simpler and start with googolplex. > Google searches list (mis)quoted Peano's axioms as 0 or 1 being the > lowest element of N. I assume this is according to the authors own > world-view. What did Peano really say with respect to 0 in N? Peano began with 1. > I note the religious reference in the FAQ and suspect I am just covering > tired contentious ground, but I would appreciate any attempts at clarifying > this issue. To an analyst, the natural numbers are for building sequences. Sequences just naturally start with 1. To a set theorist, on the other hand, the natural numbers are the finite cardinals, i.e., the sizes of finite sets, and {} is a finite set. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: 0 and natural numbers > I noticed in the sci.math FAQ that it states that informally the inclusion > or exclusion of 0 from N is a subject of religious argument (or words to > that effect). > It then continues with the formal set theory definition, ie 0={}, > 1={0}, 2={0,1}, 3={0,1,2}, etc. > This is probably a naive question, but can one reject this definition > and still accept Set theory as the fundamental basis of mathematics? > The fact that there is a particular way to define the natural numbers in > terms of set theory does not mean that other definitions can't exist, or > that those other definitions can't also be based on set theory. > Is there an alternative definition of N where 0 is not included > using set theory? > eg 1={1}, 2={1,1}, 3={1,1,2}, etc. > How about 1 = {}, 2 = {1}, 3 = {1,2}, ...? > [But what to do with the empty set as it is a subset of all sets?] > Did I say it wasn't? > I have also come across another inductive definition IIRC of N, defining > the number 1 and the operation '+' and thus deriving all N. It is > certainly simpler/achievable than trying to start with 0 and '+' because > you then have to define 1 anyway to get anywhere. > We could make it even simpler and start with googolplex. But how do we 'fill in' the set to get to the googolplex? > Google searches list (mis)quoted Peano's axioms as 0 or 1 being the > lowest element of N. I assume this is according to the authors own > world-view. What did Peano really say with respect to 0 in N? > Peano began with 1. > I note the religious reference in the FAQ and suspect I am just covering > tired contentious ground, but I would appreciate any attempts at clarifying > this issue. > To an analyst, the natural numbers are for building sequences. Sequences > just naturally start with 1. To a set theorist, on the other hand, the > natural numbers are the finite cardinals, i.e., the sizes of finite sets, > and {} is a finite set. Anthony === Subject: Re: 0 and natural numbers >> Is there an alternative definition of N where 0 is not included >> using set theory? >> eg 1={1}, 2={1,1}, 3={1,1,2}, etc. >> How about 1 = {}, 2 = {1}, 3 = {1,2}, ...? >> [But what to do with the empty set as it is a subset of all sets?] >> Did I say it wasn't? >> I have also come across another inductive definition IIRC of N, defining >> the number 1 and the operation '+' and thus deriving all N. It is >> certainly simpler/achievable than trying to start with 0 and '+' because >> you then have to define 1 anyway to get anywhere. >> We could make it even simpler and start with googolplex. > But how do we 'fill in' the set to get to the googolplex? It's no more necessary with googolplex than it is with 1. If we choose to start with googolplex, then numbers before googolplex do not exist. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: 0 and natural numbers > How about 1 = {}, 2 = {1}, 3 = {1,2}, ...? The thing about the usual approach (due to von Neumann, I believe) is that, for example, 7 is a certain set with seven elements. And similarly for the other natural numbers. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: 0 and natural numbers Discussion, linux) >> How about 1 = {}, 2 = {1}, 3 = {1,2}, ...? > The thing about the usual approach (due to von Neumann, I believe) > is that, for example, 7 is a certain set with seven elements. > And similarly for the other natural numbers. That's a cute fact, but it's obviously not essential. -- So, at this time, I'd like to assure you that I am not interested in making sure mathematicians worldwide get fired. I've rethought my desire to go to Congress and try to get funding for mathematicians cut. -- James Harris is a reasonable man. Whew! === Subject: Re: 0 and natural numbers >> >How about 1 = {}, 2 = {1}, 3 = {1,2}, ...? > >>The thing about the usual approach (due to von Neumann, I believe) >>is that, for example, 7 is a certain set with seven elements. >>And similarly for the other natural numbers. >> >That's a cute fact, but it's obviously not essential. Literally not essential, perhaps, but don't you think it is desirable to have a bijection between a cardinal number and any set with it as its cardinality? I think that property is more than cute. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: 0 and natural numbers <873c8nyot2.fsf@phiwumbda.org> Discussion, linux) >The thing about the usual approach (due to von Neumann, I believe) >is that, for example, 7 is a certain set with seven elements. >And similarly for the other natural numbers. > >>That's a cute fact, but it's obviously not essential. > Literally not essential, perhaps, but don't you think it is desirable to > have a bijection between a cardinal number and any set with it as its > cardinality? I think that property is more than cute. I dunno. It's not a property found in Frege's definition nor in the mumble-mumble representation (not von Neumann's, but the s(x)={x} representation -- Zermelo?). But yeah, I guess it's handy the way you put it. Certainly, it's intuitively nice, but in the formulation you give, it has practical benefit too. -- Jesse Hughes There's a thrill that's gone that I'll probably not have in quite the same way again. After all, FLT was a unique animal, and we had a great dance. -J.S. Harris on proving Fermat's last theorem === Subject: Re: 0 and natural numbers > To an analyst, the natural numbers are for building sequences. Sequences > just naturally start with 1. But what if you're an analyst and a C programmer? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: 0 and natural numbers Discussion, linux) >> To an analyst, the natural numbers are for building sequences. Sequences >> just naturally start with 1. > But what if you're an analyst and a C programmer? You can only be one at a time. Duh. -- Jesse F. Hughes A factor is simply something that multiplies against another factor to produce a 'product'. -- James Harris offers a definition. === Subject: Re: 0 and natural numbers > > To an analyst, the natural numbers are for building sequences. Sequences > just naturally start with 1. > But what if you're an analyst and a C programmer? Then your in deep do-do. === Subject: Re: 0 and natural numbers >> To an analyst, the natural numbers are for building sequences. Sequences >> just naturally start with 1. > But what if you're an analyst and a C programmer? Or an assembly language programmer? You just have to learn to ignore divide by zero errors when dealing with the harmonic series. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: 0 and natural numbers > Google searches list (mis)quoted Peano's axioms as 0 or 1 being the > lowest element of N. I assume this is according to the authors own > world-view. What did Peano really say with respect to 0 in N? Peano himself used 1. But since it can be done equally well with 0, any author chooses the one he likes best. The difference between these two views is unimportant, but it is something that should be noted whenever natural numbers are first mentioned. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: 0 and natural numbers >> Google searches list (mis)quoted Peano's axioms as 0 or 1 being the >> lowest element of N. I assume this is according to the authors own >> world-view. What did Peano really say with respect to 0 in N? > Peano himself used 1. The de.sci.mathematik FAQ (http://www.informatik.uni-oldenburg.de/~tjark/dsm/faq/faq.html#SECTION00540 000000000000000) contains two versions, both by Peano: The later version Peano.G.: Formulaire de mathí©matiques 5 Bde. Turin, Bocca 1895-1908 starts with Zero, the earlier version G. Peano, Arithmetices principia Novo methodo exposita Turin (1889) starts with One, as does also R. Dedekind, Was sind und was sollen die Zahlen? (1887) > But since it can be done equally well with 0, > any author chooses the one he likes best. > The difference between these two views is unimportant, but it is > something that should be noted whenever natural numbers are first > mentioned. It is indeed unimportant. The Peano axioms say only that it is a sequence that starts at some point which is never reached again. The Peano axioms say nothing about whether the elements of the sequence have anything to do with numbers as we know them from counting or calculation. Helmut Richter === Subject: Re: 0 and natural numbers >> Google searches list (mis)quoted Peano's axioms as 0 or 1 being the >> lowest element of N. I assume this is according to the authors own >> world-view. What did Peano really say with respect to 0 in N? > Peano himself used 1. > The de.sci.mathematik FAQ > (http://www.informatik.uni-oldenburg.de/~tjark/dsm/faq/faq.html#SECTION00540 0 00000000000000) > contains two versions, both by Peano: > The later version > Peano.G.: Formulaire de mathí©matiques 5 Bde. Turin, Bocca 1895-1908 > starts with Zero, the earlier version > G. Peano, Arithmetices principia Novo methodo exposita Turin (1889) > starts with One, as does also > R. Dedekind, Was sind und was sollen die Zahlen? (1887) through French, Italian or German and at least get the gist of what is being said. > But since it can be done equally well with 0, > any author chooses the one he likes best. > > The difference between these two views is unimportant, but it is > something that should be noted whenever natural numbers are first > mentioned. > It is indeed unimportant. The Peano axioms say only that it is a > sequence that starts at some point which is never reached again. The > Peano axioms say nothing about whether the elements of the sequence > have anything to do with numbers as we know them from counting or > calculation. Interesting observation. Is this a common interpretation? The authors I referred to suggest the numbers follow from the axioms and are indeed the Natural set. Anthony === Subject: Re: 0 and natural numbers <874qt3rmsc.fsf@adavid.com.au> Discussion, linux) >> The de.sci.mathematik FAQ >> (http://www.informatik.uni-oldenburg.de/~tjark/dsm/faq/faq.html#SECTION00540 0 00000000000000) >> contains two versions, both by Peano: >> The later version >> Peano.G.: Formulaire de mathí©matiques 5 Bde. Turin, Bocca 1895-1908 >> starts with Zero, the earlier version >> G. Peano, Arithmetices principia Novo methodo exposita Turin (1889) >> starts with One, as does also >> R. Dedekind, Was sind und was sollen die Zahlen? (1887) > through French, Italian or German and at least get the gist of what is > being said. Dedekind is certainly available in English. I'd be very surprised if the other two weren't also available. Check Dover for a cheap copy of Dedekind and perhaps also the others. -- [I]t's good for the economy to charge for intellectual property, so open source software cannot be good, while Microsoft is the most far-thinking company around and is doing it all for the good of the public. -- Linus Torvalds paraphrases Microsoft VP Craig Mundie === Subject: Re: Sound levels >> It seems to me that if there are x machines each with a sound level >> of L(1) then the level of x machines should be >> L(x)=x{10log[I(1)/I(0)]} >> rather than what was given to me as above. >> I don't even know if I'm missing the concept or the maths! > The concept. A level *is* the logarithm of a ratio of whatever. If > multiply something with 10, you could as well call it a 10 dB increase, > if you multiply something with 100, you call it a 20 dB increase, > multiplying with 1000 gives 30 dB and so on. > And when something is twice as much than something else, you could say > it's got 3 dB more, because > 10 lg 2 = 3 (well, more or less) >> Please explain as if to a 5 year old *smiles* ... this one has really >> got me confused! > I like this. :-) > You can now inmprss your friends in the kindergarten by telling them you > have 3 dB more marbles than they have. > Steffen Gee Mr Steffen, muummy was ever so happy with me until I tried to count the baked beans using logarithms ... it seems that there are always some beans left over that end up on the floor :( impressed as I am for finally understanding some part of what is going on! Ivan. === Subject: Re: Finite dimensional normed space When you finish this, prove something more general: Any finite-dimensional Hausdorff topological real linear space is linearly homeomorphic to R^n for some n. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: How many algebraical operations for Matrix inverse and multiplication? > Sorry to bother you for a stupid question. > Assume there is an nxn matrix A and nxm matrix B, > to calculate inverse matrix of A, i.e. inv(A), how many > algebraical operations (product, addition, subtraction, etc.) are needed? > For A*B, the algebraical operations number is n*n*m, right? I take it you assume that both A and B are dense matrices; if they contain lots of zeroes you can streamline the computations considerably. Actually, if B is square, you can do matrix multiplication in gave 973 hits. Strassen's algorithm is O(n^2.81) and can be improved further. A paper by Pan (1978) Strassen's algorithm is not optimal, Proceedings of the 19th Annual Symposium on the Foundations of Computer Science, 166-176, describes an O(n^2.78) algorithm for matrix multiplication. While the only algorithms I know for matrix inversion are also O(n^3), I would not be surprised if Strassen's or Pan's algorithm can be tweaked to give a faster algorithm for inversion as well. === Subject: Re: How many algebraical operations for Matrix inverse and > For A*B, the algebraical operations number is n*n*m, right? ... > Actually, if B is square, you can do matrix multiplication in > gave 973 hits. Strassen's algorithm is O(n^2.81) and can be improved > further. A paper by Pan (1978) Strassen's algorithm is not optimal, > Proceedings of the 19th Annual Symposium on the Foundations > of Computer Science, 166-176, describes an O(n^2.78) algorithm for > matrix multiplication. Yup. Again a clash between different complexities. A long time ago I did look at Strassen's algorithm. It exchanges multiplications for a larger number of additions/subtractions. Pretty good if a multiplication is more time-consuming than an addition. When that is not the case it is bad. So, when you are working with floating point numbers, do not think about Strassen's algorithm (on most computers multiplication is just as fast as addition), it will be slower. On the other hand, you can formulate the matrix multiplication in terms of 2x2 matrix elements. In that case you can use Strassen to advantage. But to get a real time benefit, your matrices have to be large. Strassen's idea is similar to the multiplication of two complex numbers, which can be done with only three multiplications, but at the expense of five additions rather than two. You need a situation where an addition is three times faster then a multiplication to get any advantage (and, yes, they do exist, but not with floating-point). > While the only algorithms I know for matrix inversion are also O(n^3), > I would not be surprised if Strassen's or Pan's algorithm can be tweaked > to give a faster algorithm for inversion as well. I do not think so. To get benefits from the algorithms when using floating point you need to shift to (at least) 2x2 submatrices... -- 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: How many algebraical operations for Matrix inverse and multiplication? >> Sorry to bother you for a stupid question. >> Assume there is an nxn matrix A and nxm matrix B, >> to calculate inverse matrix of A, i.e. inv(A), how many >> algebraical operations (product, addition, subtraction, etc.) are needed? >> For A*B, the algebraical operations number is n*n*m, right? >I take it you assume that both A and B are dense matrices; if they >contain lots of zeroes you can streamline the computations considerably. >Actually, if B is square, you can do matrix multiplication in >gave 973 hits. Strassen's algorithm is O(n^2.81) and can be improved >further. A paper by Pan (1978) Strassen's algorithm is not optimal, >Proceedings of the 19th Annual Symposium on the Foundations >of Computer Science, 166-176, describes an O(n^2.78) algorithm for >matrix multiplication. >While the only algorithms I know for matrix inversion are also O(n^3), >I would not be surprised if Strassen's or Pan's algorithm can be tweaked >to give a faster algorithm for inversion as well. The order is the same. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: minimum plolynome i look for a free program to evaluate the minimum plolynome of cos(2pi/29)+cos(24pi/29). i know only do it for sqrt(3+sqrt(12+sqrt(13))) for example === Subject: Re: minimum plolynome >i look for a free program to evaluate the minimum plolynome of >cos(2pi/29)+cos(24pi/29). I don't know about free, but using Maple I get 1-18*t+56*t^2+224*t^3-112*t^4-384*t^5+64*t^6+128*t^7 Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: minimum plolynome >i look for a free program to evaluate the minimum plolynome of >cos(2pi/29)+cos(24pi/29). > I don't know about free, but using Maple I get > 1-18*t+56*t^2+224*t^3-112*t^4-384*t^5+64*t^6+128*t^7 > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2 The free program PARI can be used to solve problems of this sort. It can probably do it symbolically, but here's a numerical solution using PARI. /* set precision to 100 digits */ ? default(realprecision,100) /* set u = quantity to be studied */ ? u=cos(2*Pi/29)+cos(24*Pi/29) %11 = 0.1197633795424974386851514109724889057... /* the function algdep(u,n) gives a polynomial of deg at most n that has u as an approximate root */ ? algdep(u,5) %17 = 10063494516*x^5 - 36293375755*x^4 - 49092508314*x^3 + 11227524909*x^2 + 53146532101*x - 6434498006 ? algdep(u,6) %16 = 341393705*x^6 + 21576719*x^5 + 515616367*x^4 + 333974409*x^3 + 154330454*x^2 - 620440190*x + 71411094 ? algdep(u,7) %12 = 128*x^7 + 64*x^6 - 384*x^5 - 112*x^4 + 224*x^3 + 56*x^2 - 18*x + 1 ? algdep(u,8) %13 = 128*x^7 + 64*x^6 - 384*x^5 - 112*x^4 + 224*x^3 + 56*x^2 - 18*x + 1 ? algdep(u,9) %14 = 128*x^9 + 64*x^8 - 256*x^7 - 48*x^6 - 160*x^5 - 56*x^4 + 206*x^3 + 57*x^2 - 18*x + 1 /* If one takes huge coeffs, one can always make u an approximate root. The key is to find a poly whose coefficients don't get very above data, it seems clear that the degree 7 polynomial is the one that we want. And although the deg 9 poly has even smaller coeffs, you can check that it is x^2+1 times the deg 7 poly. */ You can get PARI source code and executables at http://www.parigp-home.de/ JoeS === Subject: Re: pre-calc question (rotation and pulleys) > Im not sure if this is the right place to ask this, but i really hope > someone can help me with this. i dont have much information though. > all i have is this: > a motor turns a rod at 1780rpm. that rod is connected to a shaft which > turns at 3800rpm. so in other words there is to circles connected by a > pulley or belt. as one cicle spins, the pulley turns, and then turns > the other circle. i have to find the relationship between the two in > terms of their diameter, like a ratio of 1:?. Assuming the belt is rigid, inextensible and does not slip on either pulley, the speed of the belt is everywhere equal. The speed of a point on the edge of a circle of radius a with rotation rate w is k*a*w, where k is either 2pi if you measure you angles in radians or 360 if you measure Given the above, can you find an expression for the radius of the pulley wheel in terms of the radius of the shaft and the rotation rates? -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. === Subject: Re: Perplexing Patterns of Square Numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i25DsXo24254; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i257RMi17742 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i257RMB00846; >... >> I tried to follow these messages about perplex patterns but I am a little lost. >> Could someone summarize what has already been done about this subject, >> that is how many patterns have been found for n from 1 to 20 (or >> something like that) and also how many one can expect to discover. >... >sci.math thread with details of the above. Briefly, throughout last >year B.S. Rangaswamy reported several perplex patterns (found without >using a computer, if I'm not mistaken) for n=5, 6, 7, and 8; in December >posted some programs, while Ralph Furmaniak posted a more general >but slightly slower program. (At n=10, his takes 8 minutes on his >machine and mine takes 1 minute on a 750 MHz Pentium.) Below are >the number of solutions I find for n=1 to 12. I think no one knows >how many or if any exist for n>12. >-jiw > n sols(n) Examples (unsquared) > 1 3 {1} {2} {3} > 2 4 {9 4} {8 7} {4 8} {6 8} > 3 7 {29 22 12} {21 22 12} {11 17 14} ... > 4 1 {46 35 36 81} > 5 16 {304 167 221 136 263} ... > 6 17 {865 668 932 476 472 738} ... > 7 13 {1913 2636 2333 3134 2343 2643 3114} ... > 8 12 {7918 5141 8034 9615 7042 8839 5341 6454} ... > 9 11 {30746 20596 23361 17818 13924 25616 22873 10596 25719} >10 9 {72927 57786 36719 94789 56828 65462 84523 61608 54432 98038} >11 8 {290369 218896 198022 105629 212771 121426 234925 261263 > 135884 248686 128108} >12 2 {957511 358784 827807 934038 475632 868708 628471 391023 > 767854 403042 504407 411636} {683281 798836 829893 903962 > 855838 531773 996627 509885 664623 992182 833332 411643} There are TEN Perplex-3(n=3)patterns,square roots of which are furnished below: 1. 21 22 12 2. 29 22 12 3. 11 16 13 4. 19 26 13 5. 31 26 13 6. 11 17 14 7. 12 22 21 8. 13 26 31 9. 23 16 31 10. 27 16 31 The task on hand is to form perplex-13 patterns out of zero-free 13-digit square numbers. With around half a million 13-digit squares and 368 numbers(ranging from 1054254^2 to 3161581^2)eligible for placement in last row/column, the complexity and vastitude of discovery gets multiplied in this case. Wishing all the best. - BSR === Subject: Re: Perplexing Patterns of Square Numbers >... >> Could someone summarize what has already been done about this subject, >... > for a sci.math thread with details [...] > Below are the number of solutions I find for n=1 to 12. > I think no one knows how many or if any exist for n>12. >-jiw > n sols(n) Examples (unsquared) > 1 3 {1} {2} {3} > 2 4 {9 4} {8 7} {4 8} {6 8} > 3 7 {29 22 12} {21 22 12} {11 17 14} ... > 4 1 {46 35 36 81} > 5 16 {304 167 221 136 263} ... > 6 17 {865 668 932 476 472 738} ... > 7 13 {1913 2636 2333 3134 2343 2643 3114} ... ... > There are TEN Perplex-3(n=3)patterns,square roots of which are furnished below: > 1. 21 22 12 > 2. 29 22 12 > 3. 11 16 13 > 4. 19 26 13 > 5. 31 26 13 > 6. 11 17 14 > 7. 12 22 21 > 8. 13 26 31 > 9. 23 16 31 > 10. 27 16 31 ... > 3 10 {29 22 12} {21 22 12} {11 17 14} ... rather than as above. I misread my program's output when creating the table. -jiw === Subject: Re: Sequence: (m-k) divides a(m)*a(k) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i25DsXu24228; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i25DM2i20227 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i25DM1b13592; ... >Let a(1) = 1; >Let a(m) = lowest positive unpicked integers such that: >(m-k) divides evenly into (a(m) * a(k)) >for EACH k, 1 <= k <= m-1. >I get (again, figured by hand, so not completely believable): >a(m) : 1, 2, 4, 3, 12, 30,... >Is it a permutation of the positive integers? No ,it isn't a permutation of the positive integers. In fact I prove that for m > 5 , 10 divides a(m). Proof: We know m > 5 and a(1)=1, a(2)=2, a(3)=4, a(4)=3 & a(5)=12 . There exist 10 cases; Case 1. m == 0 mod 10 , since m-5 divides a(5)*a(m) ,we deduce that 5 divides a(m) (I-1) since m-4 divides a(4)*a(m) ,we deduce that 2 divides a(m) (I-2) from (I-1) and (I-2) we conclude that 10 divides a(m). Case 2. m == 1 mod 10 , since m-1 divides a(1)*a(m) ,we deduce that 10 divides a(m). Case 3. m == 2 mod 10 , since m - 2 divides a(2)*a(m) ,we deduce that 5 divides a(m) (III-1) since m - 4 divides a(4)*a(m) ,we deduce that 2 divides a(m) (III-2) so from (III-1) and (III-2) we conclude that 10 divides a(m). Case 4. m == 3 mod 10 , since m - 1 divides a(1)*a(m) ,we deduce that 2 divides a(m) (IV-1) since m - 3 divides a(3)*a(m) ,we deduce that 5 divides a(m) (IV-2) so from (IV-1) and (IV-2) we conclude that 10 divides a(m). Case 5. m == 4 mod 10 , since m - 4 divides a(4)*a(m) ,we deduce that 10 divides a(m) . Case 6. m == 5 mod 10 , since m - 1 divides a(1)*a(m) ,we deduce that 2 divides a(m) (VI-1) since m - 5 divides a(5)*a(m) ,we deduce that 5 divides a(m) (VI-2) so from (VI-1) and (VI-2) we conclude that 10 divides a(m). Case 7. m == 6 mod 10 , since m - 1 divides a(1)*a(m) ,we deduce that 5 divides a(m) (VII-1) since m - 4 divides a(4)*a(m) ,we deduce that 2 divides a(m) (VII-2) so from (VII-1) and (VII-2) we conclude that 10 divides a(m). Case 8. m == 7 mod 10 , since m - 1 divides a(1)*a(m) ,we deduce that 2 divides a(m) (VIII-1) since m - 2 divides a(2)*a(m) ,we deduce that 5 divides a(m) (VIII-2) so from (VIII-1) and (VIII-2) we conclude that 10 divides a(m). Case 9. m == 8 mod 10 , since m - 3 divides a(3)*a(m) ,we deduce that 5 divides a(m) (IX-1) since m - 4 divides a(4)*a(m) ,we deduce that 2 divides a(m) (IX-2) so from (IX-1) and (IX-2) we conclude that 10 divides a(m). Case 10. m == 9 mod 10 , since m - 1 divides a(1)*a(m) ,we deduce that 2 divides a(m) (X-1) since m - 4 divides a(4)*a(m) ,we deduce that 5 divides a(m) (X-2) so from (X-1) and (X-2) we conclude that 10 divides a(m). Hence in all cases we see 10 divides a(m) and the proof is complete. a(m) for m=1,2,...,20 are: 1,2,4,3,12,30,60,420,840,1260,630,27720,4620,60060, 30030,120120,240240,2042040,3063060,232792560 Farideh === Subject: Re: How many algebraical operations for Matrix inverse and multiplication? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i25DsXl24235; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i25DbHi21564 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i25DbHL14182; >>Sorry to bother you for a stupid question. >>Assume there is an nxn matrix A and nxm matrix B, >>to calculate inverse matrix of A, i.e. inv(A), how many >>algebraical operations (product, addition, subtraction, etc.) are needed? If m=n, Strassen's algorithm uses only n^{2.807} operations. Later this has been improved to n^{2.376} (Coppersmith&Winograd). This is true for both matrix product and matrix inverse. For rectangular matrices, the cost of computing the product decreases as m decreases, and in fact if m0. === Subject: Re: How many algebraical operations for Matrix inverse and multiplication? >>Sorry to bother you for a stupid question. >>Assume there is an nxn matrix A and nxm matrix B, >>to calculate inverse matrix of A, i.e. inv(A), how many >>algebraical operations (product, addition, subtraction, etc.) are needed? > If m=n, Strassen's algorithm uses only n^{2.807} operations. > Later this has been improved to n^{2.376} (Coppersmith&Winograd). For what range of n is this method superior to the straight-forward n^3 algorithms? -Michael. === Subject: Re: Zorn's Lemma Question > Conversely, I think I read somewhere that the assertion that every > ring-with-identity has a maximal ideal is equivalent to AC (and hence > equivalent to Zorn's lemma.) But I don't know how to show this. You are right, see AUTHOR = {Hodges, Wilfrid}, TITLE = {Krull implies {Z}orn}, JOURNAL = {J. London Math. Soc. (2)}, FJOURNAL = {The Journal of the London Mathematical Society. Second Series}, VOLUME = {19}, YEAR = {1979}, NUMBER = {2}, PAGES = {285--287}, ISSN = {0024-6107}, CODEN = {JLMSAK}, MRCLASS = {04A25}, MRNUMBER = {80f:04004}, MRREVIEWER = {J. C. Robson}, } KP -- E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI PHONE: +31-15-2784572 TU Delft FAX: +31-15-2786178 Postbus 5031 URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft the Netherlands === Subject: Re: What is a Completed System? > 1) A family (a_i) of vectors in H such that for every h in H the > equalities > = 0 for every i in I(the index set) implies that h = 0. That one is the common definition; or at least, the one I learned. However, you said that this means the set {a_i} is a basis of H; this is true in general only if H is a Hilbert space. I'm not sure about the second definition, but it looks relatively easy to check. > What > does it mean for a vector space to be dense in another vector space? Do > use the definition of denseness as the one that deals with closure > (limits > of sequences)? Yes; or, use the norm as a metric and regard H as a metric space, then a subset S is dense in H iff for every y in H and e>0, there is an x in D such that d(x,y) So therefore any basis of a finite > dimensional hilbert space is a completed system? Any Hilbert space's basis is complete, because every element in the space has a unique representation as a sum of the basis elements, so if =0 for all a_i, h=0. -- The only math in the movie, The Matrix, is in the title. (paraphrased from a posting to alt.math.recreational) === Subject: Re: What is a Completed System? > 1) A family (a_i) of vectors in H such that for every h in H the > equalities > = 0 for every i in I(the index set) implies that h = 0. > That one is the common definition; or at least, the one I learned. > However, you said that this means the set {a_i} is a basis of H; this is > true in general only if H is a Hilbert space. I'm not sure about the > second definition, but it looks relatively easy to check. > What > does it mean for a vector space to be dense in another vector space? Do > I > use the definition of denseness as the one that deals with closure > (limits > of sequences)? > Yes; or, use the norm as a metric and regard H as a metric space, then a > subset S is dense in H iff for every y in H and e>0, there is an x in D > such that d(x,y) So therefore any basis of a finite > dimensional hilbert space is a completed system? > Any Hilbert space's basis is complete, because every element in the space > has a unique representation as a sum of the basis elements, so if a_i>=0 for all a_i, h=0. > -- > The only math in the movie, The Matrix, is in the title. > (paraphrased from a posting to alt.math.recreational) === Subject: easy l'hospital question~ i know that l'hospital rule can apply form 00 / 00 and i know that l'hospital rule can apply form ~ / 00 (regardless of numerator) but if lim (sin x) / x = 0 (trivial), but if l'hospital apply this, cos / 1 => oscillation why not same?? my thinking is wrong?? advice...please.......thank you. === Subject: Re: easy l'hospital question~ > i know that l'hospital rule can apply form ~ / 00 (regardless of numerator) You don't use l'hopital to show finite/infinity = 0 Since sin(x) is always finite [-1, 1], the answer falls out. -Tralfaz === Subject: Re: easy l'hospital question~ In the hypothesis, the numerator can not be Anything It must also be zero or infinity. > i know that l'hospital rule can apply form 00 / 00 > and > i know that l'hospital rule can apply form ~ / 00 (regardless of numerator) > but if > lim (sin x) / x = 0 (trivial), > x->00 > but if l'hospital apply this, cos / 1 => oscillation > why not same?? > my thinking is wrong?? > advice...please.......thank you. === Subject: Re: easy l'hospital question~ [[ This message was both posted and mailed: see the To, Cc, and Newsgroups headers for details. ]] Yours is a common misconception. But it's just that--a misconception. L'Hospital's Rule says that, IF lim_{x to a} f'(x)/g'(x) = L exists, and if either (a) lim f(x) = 0 and lim g(x) = 0, or (b) lim g(x) = infinity, then lim f(x)/g(x) exists and is L. In case (b) there's no hypothesis on lim f(x) at all. See Rudin, Principles of Mathematical Analysis, page 109. hot-girl's problem is that capitalized IF. The hypothesis isn't satisfied for (sin x)/x. --Ron Bruck PS. I just noticed that Rudin calls it L'Hospital's Rule. > In the hypothesis, the numerator can not be Anything It must also be zero > or infinity. > i know that l'hospital rule can apply form 00 / 00 > and > i know that l'hospital rule can apply form ~ / 00 (regardless of > numerator) > but if > lim (sin x) / x = 0 (trivial), > x->00 > but if l'hospital apply this, cos / 1 => oscillation > why not same?? > my thinking is wrong?? > advice...please.......thank you. === Subject: Re: easy l'hospital question~ > i know that l'hospital rule can apply form 00 / 00 > and > i know that l'hospital rule can apply form ~ / 00 (regardless of numerator) > but if > lim (sin x) / x = 0 (trivial), > x->00 > but if l'hospital apply this, cos / 1 => oscillation > why not same?? > my thinking is wrong?? > advice...please.......thank you. L'Hopital's rule may ONLY apply when the original limit form is indeterminant, .i.e., leads to 0/0 or to oo/oo situations. Lim_{x -> oo} sin(x)/x is not indeterminant. In fact, l'Hopital applications can be restricted to 0/0 cases with no loss of generality, since oo/oo can always be recast as (1/oo)/(1/oo) cases. === Subject: Re: easy l'hospital question~ >i know that l'hospital rule can apply form 00 / 00 >and >i know that l'hospital rule can apply form ~ / 00 (regardless of numerator) >but if >lim (sin x) / x = 0 (trivial), >x->00 sin(.0001)/.0001 = 0.999999998 It looks like the limit is 1. >but if l'hospital apply this, cos / 1 => oscillation cos(0)/1 = 1/1 = 1 --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: easy l'hospital question~ >>i know that l'hospital rule can apply form 00 / 00 >>and >>i know that l'hospital rule can apply form ~ / 00 (regardless of numerator) >>but if >>lim (sin x) / x = 0 (trivial), >>x->00 > sin(.0001)/.0001 = 0.999999998 > It looks like the limit is 1. We're talking about the limit at +oo here, not about the limit at 0. Jose Carlos Santos === Subject: Re: easy l'hospital question~ >We're talking about the limit at +oo here, not about the limit at 0. My apologies, I'm not used to ASCII math. I misinterpreted 00 as 0. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: easy l'hospital question~ > i know that l'hospital rule can apply form 00 / 00 > and > i know that l'hospital rule can apply form ~ / 00 (regardless of numerator) I have some doubts here. > but if > lim (sin x) / x = 0 (trivial), > x->00 > but if l'hospital apply this, cos / 1 => oscillation > why not same?? > my thinking is wrong?? Yes, you are thinking wrong. L'hopital's rule says that, under certain conditions, if lim f'(x)/g'(x) = a, then lim f(x)/g(x) = a. But if the limit lim f'(x)/g'(x) does not exist, the l'Hopital's rule says nothing at all. Jose Carlos Santos === Subject: Re: Help! Need a book recommendation... > I reckon you didn't understand the question. >>Following this thread from the outside, it's pretty obvious to me that >>Gerry understood the question. >>The fact that you didn't understand his answer explains something, >though. >> Yeah Doug, I can tell you're a real important guy. >Kudos! For my part, I can tell that you like others to do your homework >for you. >Doug Yeah, ok. === Subject: Re: Help! Need a book recommendation... >>But I'm getting stuck on things like not being able to solve a third order >>polynomial, or being able to integrate e^f(x), where f(x) is a second >>order polynomial. >[...] >>I don't want to read loads of words when a short deduction will do. >[...] >>I don't know what closed form means. >[...] >>This paper is part of an application for a master's. >Oh dear. === Subject: Re: Help! Need a book recommendation... >>But I'm getting stuck on things like not being able to solve a third order >>polynomial, or being able to integrate e^f(x), where f(x) is a second >>order polynomial. >[...] >>I don't want to read loads of words when a short deduction will do. >[...] >>I don't know what closed form means. >[...] >>This paper is part of an application for a master's. >Oh dear. Maybe next time you post here, you should specify that only useful, helpful people are allowed to respond. Doug === Subject: Re: (-2/3)^(-2/3) = (3/2)^(2/3)? > If Daniel Johnson will permit me to jump in here, his point was, I > believe, that while you CAN reduce (-2/3)^(-2/3) or (-8)^(1/3) to > real values, you cannot define (-2/3)^x or (-8)^x in a consistent > way for all real x and therefore the FUNCTION (-9)^x is not defined. > f(x) = (-9)^x could be defined for x real with infinity many > discontinuities. > More specifically, for rational x of the form m/(2n+1) ... > For example the software GraphEq uses this fact to > draw visually continuous graph of y = (-9)^x. > It's name is GrafEq, actually. The graph it gives looks like the union of > the graphs of y = 9^x and y = -(9^x). Is it possible y = (-2)^(x^x) to have real root for -1 < x < 0, x real? For example GrafEq plots visually continuous graphs (in 2nd and 3rd quadrant) of the system: |y = (-2)^u |u = x^x |x < 0 === Subject: Re: Difficult Analysis Problem >>I am facing the following strange problem : >>Given f : I -> R , with the property that f multiplied with f so f *f >>and f*f*f are infinity many times differentiable, is f infinitely time >>differentiable or not? >>If not, can you give me a counter example? You can find a proof in Joris, Henri Une C^oo-application non-immersive qui poss.8fde la propri.8et.8e universelle des immersions. (French) [A nonimmersive C^oo mapping having the universal property of immersions] Arch. Math. (Basel) 39 (1982), no. 3, 269-277. and a generalization in Duncan, John; Krantz, Steven G.; Parks, Harold R. Nonlinear conditions for differentiability of functions. J. Analyse Math. 45 (1985), 46-68. === Subject: Re: Interesting problem > ... >Let {Ua} be a Hamel Basis for R+ over Q+ >(Then {-Ua} is a Hamel Basis for R- over Q+) >Now define >f: R -> R>=0 by > f(x) = [SUM I](qi e^Uai) where x>0 and x = [SUM I](qi Uai) some >finite I > f(x) = [SUM I] (qi e^(-Uai)) where x<0 and x = -[SUM I](qi Uai) >some finite I > f(0) = 0 (not needed, just thrown in to be complete) > >Define A = f(R+) and B = f(R-) >Then A and B obviously partition R+, since e^x:R->R+ is a bijection > ... > Why if f injective? > Why can't we have q1 e^Ua1 + q2 e^Ua2 = q3 e^-Ua3 ? > Don Coppersmith That's one of the details I mentioned in the first line :-( (The other is that it's surjective) I'll look at this over the weekend. Unless someone else kindly deals with this before I get around to it :) Rick === Subject: JSH ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i25FoXs07789; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i25FhKi06709 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i25FhKS20712; The JSHow must go on ? Maurice === Subject: Re: JSH ? > The JSHow must go on ? > Maurice dont know what I an more curious about, JSH's next rant or how long he can keep himself from posting! === Subject: Re: JSH ? > The JSHow must go on ? > > Maurice > dont know what I an more curious about, JSH's next rant or how long he > can keep himself from posting! Maybe he finally got a job and is doing something more productive. === Subject: Re: JSH ? > The JSHow must go on ? > Maurice Going on past form, I'm sure he'll be back with the latest amazing breakthrough soon. === Subject: Re: quadratic inequalities good morning, thank you for your answer. > You mean you're given the matrix P and you want to find some X such > that X^T P X > 0? I assume the entries are all real. yes. And more generally find all the solutions X of this inequality, i.e. find a domain D such that all x in D satisfy this inequality (may be something like a cone). > If P + P^T has a positive eigenvalue lambda, an eigenvector X for > that eigenvalue will do. So will any nonzero linear combination of > eigenvectors for the positive eigenvalues of P + P^T. > On the other hand, if P + P^T has no positive eigenvalues, such an > X can't exist. Can you give me a reference ? And what about the case where X is a matrix, i.e. find X such that X^T P X is positive definite ? (should we just write [v_1, v_2,...v_r] with r the dimension of X and v_i are eigenvectors of P+P^T ?) === Subject: Beal proof now available Proof of the Beal Conjecture is available from the Open Directory Project under science>math>number theory>diophantine equations>fermat's last theorem>On the Beal Conjecture === Subject: Re: Beal proof now available > Proof of the Beal Conjecture is available from the Open Directory > Project under > science>math>number theory>diophantine equations>fermat's last > theorem>On the Beal Conjecture Have you won the dosh yet? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Randomness or 'Shuffledness' of a finite set. How does one measure the Randomness or 'Shuffledness' of a finite set? For example, if we have a set of 52 elements and say the value of each element is its index (of course a deck of cards comes to mind, but I do NOT want to obscure this question with having to deal with the suit of the card(s)). So, of course, when we have the set in the order: {1,2,3,...51,52} The (lets say) Shuffledness Index is 0 (i.e. it is NOT shuffled at all). And when the set is in some random (i.e. shuffled) state (for instance: {5,8,34,28,...32,41}, the 'Shuffledness Index is somwhere closer (or equal) to 1. Perhaps 1 (the ideal value) cannot be achieved, but I would like to know how to measure the state of the set and yield a value that gives one some indication of the state of the set. Thomas === Subject: Re: Randomness or 'Shuffledness' of a finite set. >How does one measure the Randomness or 'Shuffledness' of a finite set? >For example, if we have a set of 52 elements and say the value of each >element is its index (of course a deck of cards comes to mind, but I >do NOT want to obscure this question with having to deal with the suit >of the card(s)). >So, of course, when we have the set in the order: >{1,2,3,...51,52} >The (lets say) Shuffledness Index is 0 (i.e. it is NOT shuffled at >all). >And when the set is in some random (i.e. shuffled) state >(for instance: {5,8,34,28,...32,41}, the 'Shuffledness Index is >somwhere closer (or equal) to 1. >Perhaps 1 (the ideal value) cannot be achieved, but I would like to >know how to measure the state of the set and yield a value that gives >one some indication of the state of the set. I'd hold off on designing the index until you know how to calculate it. You may find that it makes more sense reversed, where 0 is random and positive and negative values are less random. Similar to a correlation coefficient. Or some other way. One factor you'd want to take into account is neighbors. There are 51 neighbor pairs. How many of them are original neigbors? You'd expect just under 2, or if you only count original-order neighbors (the card on the right is still on the right), just under 1. But that alone doesn't really tell you they're shuffled well. Consider the perfect riffle shuffle -- cut the deck in half and alternate. [1,28,2,29,3,30,4,31,...52] None of these cards is next to any original neighbor, and only 2 of them are in their original positions, but a person looking at a short sequence would easily be able to predict the next one. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Randomness or 'Shuffledness' of a finite set. > How does one measure the Randomness or 'Shuffledness' of a finite set? Perhaps one measure would be what is the complexity of going from a non-random state to the current state. So there are 2 points to solve : 1. What is a non-random state ? I assume [1,2,...,n] (the identy permutation) is the initial non random state. 2. What is going from ... to. I assume you have a set of building blocks (e.g. s1...sk are some permutations, e.g. transposition). So you are searching for the minimimum 't' such that your current state 's' is written as the product of building blocks : s = s_{i1} * s_{i2} * ... * s_{it} so s_{i1} ... s_{it} are the steps you have to do to suffle your deck. 't' measure the complexity of the suffling, in a sense it's distance to identy shuffle. I hope this makes sense. I think these question are strongly connected to the question of how many steps does it takes to suffle perfectly a deck. This question is much more well posed, and the answer is given by Diaconis (See Group representations in probability and statistics, a wonderfull book). G. === Subject: Re: Randomness or 'Shuffledness' of a finite set. > So you are searching > for the minimimum 't' such that your current state 's' > is written as the product of building blocks : btw, in a sense, this is connected to the distance on the Cayley graph on the symmetric group Sn, using some generating set (for instance the transpositions, but you could choose other building blocks, for instance transpositions [1,k] if you only allow to swap the 1st card with another). There is a book from Audrey Terras (fourier analysis on finite groups) that deals with distance and fourier on groups. Gabriel === Subject: Re: re:Pythagorean triples via trig half and double angle formulas >> [quoted text muted] > So what did you conclude? I've had the same idea, but not done much work on > it. Has anyone else considered such an approach? What does it sound like? > -Michael. I thought the web pages I cited covered what I thought on the topic. I cannot tell exactly which part you were asking about since the list mutes quotations. The pythagorean triple generater, based upon side angles: http://homework.jhax.net/math/Pythagorean_Triples.jsp Enter an arbitrary angle, tell it how accurate you want, and it computes the pythagorean triple and describes how it was computed. 120-degree triple generator (otherwise same as above): http://homework.jhax.net/math/Natural_Triples.jsp I also described the way I derived the technique. To give it more explicitly, start with the well-known Pythagorean generator: m^2-n^2 2mn m^2+n^2 So my approach when I was looking at it a few years ago was, construct your triangle using the above measures, and pick a side angle: cos(A) = (m^2-n^2)/(m^2+n^2) sin(A) = 2mn/(m^2+n^2) Divide the numerator and denominator by m^2 and it gives, in terms of a new rational factor f = n / m: cos(a) = (1-f^2)/(1+f^2) sin(a) = 2f/(1+f^2) You have rational sin and cos, whenever you plug in a rational factor. That rational factor f turns out to be tan(A/2), which you can discover by substituting or by graphing. So if you want to find a pythagorean triple with angle A as a side angle, you take the tangent of the half-angle, get a rational approximation that is as accurate as you like, and compute sin and cos as given above, and there you have a triangle with rational sides that can be scaled by least common multiple of denominators to become a pythagorean triple. It is not very complex. The formulae for 120-degree triples were derived later in a similar fashion. Was that what you were asking about, or something else? Ray Whitmer === Subject: Re: re:Pythagorean triples via trig half and double angle formulas >> [quoted text muted] > So what did you conclude? I've had the same idea, but not done much work on > it. Has anyone else considered such an approach? What does it sound like? > -Michael. > I thought the web pages I cited covered what I thought on the topic. I > cannot tell exactly which part you were asking about since the list mutes > quotations. > The formulae for 120-degree triples were derived later in a similar > fashion. Was that what you were asking about, or something else? I was specifically asking about the equal-tempered 53-step musical scale. Did you reach any conclusion? Did you actually listen to the sounds? -Michael. === Subject: Re: Sum(1/|cos(n)|,n>=1) >>Can you prove that, for a real a>1, the series >>Sum(1/(|cos(n)|*n^a),n>=1) is divergent or convergent? >> (1) There is a constant r such that there are only finitely many >> pairs (n,k) with |n - k pi/2| < 1/n^r. >Can you give me where this is proved and an estimation about the best Hata's result is in M. Hata, Acta Arithmetica 63 (1993) 335-349. This improved on a result of K. Mahler, Indag. Math. 15 (1953) 30-42 >> If |n - k pi/2| > 1/n^r for all k, >You mean : there is an r and an N such that, for n>N and all k in N, >we have (2) |n - k pi/2| > 1/n^r ? OK. >If yes, what is the relation between this r for (2) and the one for >(1)? Same one. >> we must have |cos(n)| > 1/(2 n^r). Thus the tail of your >> series is bounded by that of sum_n 2 n^(r-a), and converges if >> a > r+1. I believe the best estimate that has been proven (due to M. >> Hata) will tell you it will converge for a > 8.016045... >M.Hata gave for (2) r=7.016045...? Can you give me where this is >proved? M. Hata, Acta Arithmetica 63 (1993) 335-349 >> But it's probably true for all a > 2. In fact, for any a > 2, >> sum_n 1/(cos(nx) n^a) converges for almost every x in the sense >> of Lebesgue measure. >Maple gives me (and I agree) int(1/abs(cos(n*x)),x=0..2*Pi)=infinite, >so how do you prove this? Yes, of course that integral is infinite, but that's a different question. See e.g. Khinchin, Continued Fractions, Theorem 32: a consequence of this is that for any a > 2 and almost every x, there is N such that for all n > N there is no k with |n - k x| < n^(1-a). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: e^i(pi) = -1 revisited with Doubly Infinites Re: infinite rightward strings tacked-on to p-adics serves as Orthogonality and makes Doubly-Infinites the points of Lobachevskian Geometry >> >> Fields are those things which have >> two forms of addition. One called >> the fast adder for the intelligent >> people, the other called the >> slow adder for scientists and >> monkeys. >> They have something to do with snakes then? > No, they have something to do with worms. Worms? Have you got worms? >> And since it's Policy is Isomorphic to 0/2 >> There seem to be two main verbs in this sentence :-( > That's just because Americans are twice as > intelligent as Europeans, rather than > just Haiku times as intelligent as the > Japanese are. > The to is not a main verb, since it's > part of a dependent clause, if you can't > read, which you obviously can't. to isn't a verb. > The apostrophe in it's was put in their > on purpose. A nice ambiguity: it may abbreviate it is or it has --- either way it gives the sentence two main verbs. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: e^i(pi) = -1 revisited with Doubly Infinites Re: infinite rightward strings tacked-on to p-adics serves as Orthogonality and makes Doubly-Infinites the points of Lobachevskian Geometry >> >> >> Fields are those things which have >> two forms of addition. One called >> the fast adder for the intelligent >> people, the other called the >> slow adder for scientists and >> monkeys. >> >> They have something to do with snakes then? > > No, they have something to do with worms. > Worms? Have you got worms? >> And since it's Policy is Isomorphic to 0/2 >> >> There seem to be two main verbs in this sentence :-( > > That's just because Americans are twice as > intelligent as Europeans, rather than > just Haiku times as intelligent as the > Japanese are. > > The to is not a main verb, since it's > part of a dependent clause, if you can't > read, which you obviously can't. > to isn't a verb. The last time I heard that to wasn't an intransitive verb was the last time that math wasn't spelled in the moronic fashion maths. And since the last time I listened to moronic math stories was the last time I listened to moronic King of France stories, the only relevent question is: to be or not to be. Spelled for Plato tarts: (2B)^~(2B) = 2[(B)^~(B)] = 2[0] = 0. Hence QED. To is a verb. > The apostrophe in it's was put in their > on purpose. > A nice ambiguity: it may abbreviate it is or it has > --- either way it gives the sentence two main verbs. Either way it also doesn't matter. It's only British tards that use one main verb. === Subject: Summation problem abuse@newcastle.ac.uk Can anyone help me with the following problem? Let sum_{i=1}^{infty} q_i = 1 and that q_i > 0 Now for an postive integer r how do we find a set of q_i's such that sum_{i=1}^{infty} i^r q_i < infty but sum_{i=1}^{infty} i^{r+1} q_i > infty Colin === Subject: Re: Summation problem @ucsnew1.ncl.ac.uk: > Can anyone help me with the following problem? > Let sum_{i=1}^{infty} q_i = 1 and that q_i > 0 > Now for an postive integer r how do we find a set of q_i's such that > sum_{i=1}^{infty} i^r q_i < infty > but > sum_{i=1}^{infty} i^{r+1} q_i > infty I wonder how you'd do it if instead of sums you had integrals. Bart === Subject: Re: Bell curve distribution kicked back with a beer, ruminated at length, fell asleep, woke up, lit up a joint, then fell asleep again after thoughtfully blurting out: > A question: In a group like sci.math, why do you feel the need to > proclaim yourself as an atheist, in order to ask a math question. > Which is your priority: To get an answer to the question, or to post