mm-2129 === Subject: Re: Has Perelman's proof remained solid? > Hey all, > I was just curious if anyone has any information regarding the current > validity of Perelman's Thurston's geometrization proof. I saw his > talk at Penn in April, but have not seen any information since then > regarding the current status of his work or if any problems have been > found. If you're interested in keeping up on the status of Perelman's work, a good place to look is: http://www.math.lsa.umich.edu/research/ricciflow/perelman.html There are various notes and commentaries by some experts on this stuff. Thus far, Kleiner and Lott has posted their notes on the first Perelman paper. One might interpret this as affirming the correctness of it, except that in big letters on the page is stated, We do not take responsibility for the mathematical accuracy of anything posted here. Nevertheless, I have heard through the grapevine that the first paper has basically been reviewed. The second paper is still under review. === Subject: Re: Thoughts on the Collatz conjecture === >Subject: Re: Thoughts on the Collatz conjecture >> >> I am still convinced that the conjecture is true, because after >> hastily throwing together a computer program that I believe makes the >> case. >> >> My argument here is, as each level of the Collatz tree grows new >> branches are born derived from a smaller integer start number and so >> does the density bands (see below) and their symmetry. This causes >> a symmetrical squeeze play so to speak for other possible counter >> examples start numbers and trees. >> >> I did a computer program where integer start numbers are entered in >> order where start number n = 1,2,3,4,5,6..n with there associated >> sequences. Each start # turns on a corresponding numbered pixel and >> all the path members turn on their respected pixels. These pixels stay >> on. >> Naturally some of these pixels are on when they get hit again for an >> on because of the branching tree affect and also returning back to >> 4,2,1 and terminating on 1. What happens as each new starting (n) the >> (5) density bands out ahead of the all white band become more apparent >> after about 32 y rows of all on (white) pixels. This would be an >> integer start number the size of n =32*640 = 20480. >> >> I set this thing up for reading across --- >> 640 pixels where x ( n) (start #)= 1 to 640 and y=1 >> Then where x = 1 to 640 and n=641 to 1280 and y=2 etc. >> Y is set for a max. of 350 pixels. It bypasses any on pixel that is >> not viewable on screen where y >350. >> >> It creates an interesting effect with 6 distinct bands with each of >> the 5 bands having distinct symmetrical density patterns out ahead of >> (n) the seed that is the all white band behind the seed. >> >> This density band effect could be do to certain delta factors out >> ahead of the seed or something to do with 2^n? >> >> The short basic program is listed below with plenty of documentation >> so someone can translate to Java, c++ or any other language. >> >> >> ? = Docs >> >> 4 ? A Collatz conjecture pixel evaluation >> 5 ' This program turns on the appropriate pixel for each starting >> integer and all its sequence members. >> Pixels once on, stay on. >> 10 CLS >> 12 Screen 9: ? Set graphics screen mode to 350 X 640 pixels >> 15 DEFDBL A: ?Double precision for any variable beginning with A >> 20 A=1:A3=A:A4=1:Y=1:A5=640:A6=640: PSET(A,Y): ? A is starting >> integer (seed) and turns on pixel x(A) = 1 and Y = 1: This line never >> used again. >> 30 A1$=STR$(A): ? Line 30-54 checks to see if integer is odd or even. >> 40 A2=LEN(A1$) >> 50 J$=MID$(A1$,A2,1): IF J$= 1 THEN GOTO 200 >> 51 IF J$= 3 THEN GOTO 200 >> 52 IF J$= 5 THEN GOTO 200 >> 53 IF J$= 7 THEN GOTO 200 >> 54 IF J$= 9 THEN GOTO 200 >> 60 A3=A3/2:A=A3:GOSUB 500: IF A =< 1 THEN 320 ELSE 30:' This line >> handles even integers and goes to subroutine that evaluates the >> correct x and y pixel to turn on. >> 200 A3=(A*3) +1:A=A3:GOSUB 500:GOTO 60: ' Handles odd integers of seed >> and its sequence. Ect. >> 320 A4=A4+1:A=A4:A3=A4:GOSUB 500:GOTO 30: 'Retrieves the next seed and >> repeats the whole process creating a new sequence from that seed. >> 500 IF A>A6 THEN A6=A6+A5:Y=Y+1:ELSE 530: ' Sets Up A for right row >> (Y) >> 510 If Y>350 THEN Y=1:GOTO 540: ' If integer value in any sequence is >> (350*640) then this line bypasses the pixel command (PSET) because >> pixel will not be in a viewable area of the screen. >> 515 IF A>A6 THEN 500: ' Go back to line 500 and add another 640 to >> variable A6 >> 520 IF A=< A6 AND Y>1 THEN >> Y=Y-1:A7=Y*A5:A8=A-A7:Y=Y+1:PSET(A8,Y):Y=1:GOTO 540: ' Sets up x(A8) >> value when y>1 and thus the correct x,y coordinates for any applicable >> integer with a value > 640. >> 530 If Y=1 THEN PSET(A,Y): GOTO 540:' A branch from 500 where Else >> means Y=1 >> 540 A6=A5:Y=1:RETURN: ' Resets variables and returns for next >> integer. >> 600 END >> >> Please excuse the hastily thrown together code. Should have done a >> renumber also! >> >> You have to think of each row of 640 pixels as rows cut off at that >> point and then stacked on each other where you can easily view how >> these density patterns out ahead of the seed number are formed. This >> would probably go unnoticed if the line stayed continuous as in the >> number line. >> >> This creates 5 distinct and fascinating density band patterns that >> grow in width as the first solid white band or seed band grows in >> width. >> Please note, when first starting out the bottom (last) density pattern >> starts to show a checkerboard pattern on an angle. >> >> If nothing more, its interesting! >> >> As always, any evaluations or comments are welcome. >> >> Dan >Replying to the last of this thread ( at my point of view at this time >). > I have new data. > Is anyone interested in forming a group? > Someone just started one. > http://www.grammabeautiful.com/ > I would have mentioned it to you but I assumed you saw the announcement in > sci.math. > It would be great to crack this problem. We should lay down the >pride and work together. > I think we can crack this problem. > If it is undecidable, you won't ever crack it. But that's not what I'm > interested in. If told tommorrow that the issue was resolved, I would still > continue to study sequence functions, Mersenne Hailstones, attractors and > loops. >Ernst What, you think I don't have a thought in my head that you can not control? Get real.. I have new data. As far as a web page goes.. It's a web page. A group is a group. I consider you , Mansanator a long time groupie. However you , Mansanator, have not come up with 1/2 the ideas I have. I have your Mersenne number pegged trust me. I suggest a serious group. If you are a real person Mansanator then you are welcome. I must stress I will Validate each and every person in my Group. If you are a false person you will Die!@!!!! So want to know the latest I have found??? Get real. Again to those who are interested in Cracking this problem I have new data. Can Mensanator say the same??? So in closing; If you are a duck-of-knowing Quack... Ernst (Quack) Berg === Subject: Re: Thoughts on the Collatz conjecture > What, you think I don't have a thought in my head that you can not > control? I have no idea what you're talking about. I'm not trying to control you. If you think you can solve the Collatz Conjecture, fine. I merely said that _I_ would not be disappointed if all the effort to solve it ends up in vain. > Get real.. > I have new data. Good for you. Do you want to share it with others interested in the Collatz Conjecture? Are you afraid that someone will steal your ideas and you won't get the credit? It's ok if you feel that way. Just work out your ideas on your own and publish a paper. But you are going to find it hard to collaborate with others if you're secretive. > As far as a web page goes.. It's a web page. Anybody can do that. You've got a web page, I've got a web page, lots of others have web pages. Easterly is trying to set up an interactive community that is dynamic, not just a static web page. In some of our discussions, I needed to illustrate some of my points with pictures. It would be nice to post them in a central repository. > A group is a group. That's what Easterly is setting up. > I consider you , Mansanator a long time groupie. If by that you mean I have a passioante interest in the Collatz Conjecture, then you are correct. > However you , Mansanator, have not come up with 1/2 the ideas I have. So? I never claimed to be a number theorist. > I have your Mersenne number pegged trust me. Are you upset because I pointed out that some of your ideas were wrong? Isn't that what a group is supposed to do, help each other? Would you prefer I let you persist in your folly? > I suggest a serious group. If you are a real person Mansanator then > you are welcome. I must stress I will Validate each and every person > in my Group. If you are a false person you will Die!@!!!! Go ahead. Set up your own group. If you insist on requiring personal information in order to join, then i won't join. Take a hint from Easterly's group. The only valid thing he asks for is an e-mail address, which is reasonable. Anything beyond that is an unreasonable intrusion into my privacy. > So want to know the latest I have found??? > Get real. I'm not going to lose any sleep over not knowing your new data. You've had new data before and it didn't work out. If you want to gloat, post a message here when your paper is published. > Again to those who are interested in Cracking this problem I have new > data. Let's sit back and see if anyone takes your bait. Wouldn't it be ironic if I'm the only one who'll talk to you? > Can Mensanator say the same??? Yes, as a matter of fact I have. But I didn't post it here on sci.math. You'll have to visit Easterly's group to see it. > So in closing; If you are a duck-of-knowing Quack... > Ernst (Quack) Berg === Subject: Is this a famous problem? Here's a puzzle which made the rounds in 1967. Is it still well known? To me it seems more elegant and interesting than the Goldbach or Collatz Conjecture, (and perhaps easier and ``more important'' as well). Part 1 may be a fun challenge but isn't overly difficult. Part 2, AFAIK is still unsolved. To state the problem, one definition is needed: Y(S) = max (n / gcd(n,m)) n,m in S Part 1: Find every finite subset S of the positive integers such that Y(S) = |S| or Y(S) < |S|. (|S| is the size of S.) Part 2: Prove your answer to Part 1, i.e. demonstrate that no satisfactory S has been overlooked. James === Subject: Re: Who here believes maths is all there is? > Perhaps God said: > > Let there be the empty set and the set inclusion operator. > Then God created Godel, and that just screwed everything up. Makes sense. My personal view is, all there is a proper subset of math. I should have written the proper is all caps. But this line puts it all together so brilliantly, my statement is vague and superfluous in comparison. === Subject: Re: Cartesian coordinates for verticies of a n-dimensional simplex? in message : > I am want to construct simplexes for n-dimensions. A Simplex is the > simplest shape that can define a dimension. For example, a 2d simplex is > an equilateral triangle. A 3d simplex is a 3-sided pyramid with a base, > etc. I am interested only in a regular, unit simplexes, i.e., all > vertices are equidistant from all other vertices, and each vertex is > exactly 1 unit away from the origin that the simplex is centered on. I > have found a broken link to David Anderson's paper on the n-dimensional > simplex, but that doesn't help me much... Again, I am looking for the The key to this is the fact, which can be proved by induction, that the angle subtended at the center of an n-d simplex by any two vertices is acos(-1/n). One convenient Cartesian coordinate system is to choose the first coordinate axis through an arbitrary vertex, the second through the projection of another arbitrarily chosen vertex onto the hyperplane perpendicular to the first axis, and so on. In this system, the coordinates of the vertices v_i are: v_1 = (1,0,0,...) v_2 = (-1/n,sqrt(1 - (1/n^2)),0,...) v_3 = (-1/n,sqrt(1 - (1/n^2))*(-1/(n-1)),...) . . . For example, in 2d it's: v_1 = (1,0) v_2 = (-1/2,sqrt(3/4)) v_3 = (-1/2,-sqrt(3/4)) In 3d it's: v_1 = (1,0,0) v_2 = (-1/3,sqrt(8/9),0) v_3 = (-1/3,-sqrt(2/9),sqrt(2/3)) v_4 = (-1/3,-sqrt(2/9),-sqrt(2/3)) In 4d: v_1 = (1,0,0,0) v_2 = (-1/4,sqrt(15/16),0,0) v_3 = (-1/4,-sqrt(5/48),sqrt(5/6),0) v_4 = (-1/4,-sqrt(5/48),-sqrt(5/24),sqrt(5/8)) v_5 = (-1/4,-sqrt(5/48),-sqrt(5/24),-sqrt(5/8)) -- Jim Heckman === Subject: Re: Cartesian coordinates for verticies of a n-dimensional simplex? > in message : > I am want to construct simplexes for n-dimensions. A Simplex is the > simplest shape that can define a dimension. For example, a 2d simplex is > an equilateral triangle. A 3d simplex is a 3-sided pyramid with a base, > etc. I am interested only in a regular, unit simplexes, i.e., all > vertices are equidistant from all other vertices, and each vertex is > exactly 1 unit away from the origin that the simplex is centered on. I > have found a broken link to David Anderson's paper on the n-dimensional > simplex, but that doesn't help me much... Again, I am looking for the > The key to this is the fact, which can be proved by induction, > that the angle subtended at the center of an n-d simplex by any > two vertices is acos(-1/n). One convenient Cartesian coordinate > system is to choose the first coordinate axis through an > arbitrary vertex, the second through the projection of another > arbitrarily chosen vertex onto the hyperplane perpendicular to > the first axis, and so on. In this system, the coordinates of > the vertices v_i are: > v_1 = (1,0,0,...) > v_2 = (-1/n,sqrt(1 - (1/n^2)),0,...) > v_3 = (-1/n,sqrt(1 - (1/n^2))*(-1/(n-1)),...) > For example, in 2d it's: > v_1 = (1,0) > v_2 = (-1/2,sqrt(3/4)) > v_3 = (-1/2,-sqrt(3/4)) > In 3d it's: > v_1 = (1,0,0) > v_2 = (-1/3,sqrt(8/9),0) > v_3 = (-1/3,-sqrt(2/9),sqrt(2/3)) > v_4 = (-1/3,-sqrt(2/9),-sqrt(2/3)) > In 4d: > v_1 = (1,0,0,0) > v_2 = (-1/4,sqrt(15/16),0,0) > v_3 = (-1/4,-sqrt(5/48),sqrt(5/6),0) > v_4 = (-1/4,-sqrt(5/48),-sqrt(5/24),sqrt(5/8)) > v_5 = (-1/4,-sqrt(5/48),-sqrt(5/24),-sqrt(5/8)) > -- > Jim Heckman Craig === Subject: Re: Cartesian coordinates for verticies of a n-dimensional simplex? Another way (although not quite what is asked for) is to put the n-simplex in (n+1)-space. Then you can use a simple scheme like this (shown for n=4): (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) and now it is obvious that they all have the same distance from each other. The center is (1/4,1/4,1/4,1/4). === Subject: Re: Cartesian coordinates for verticies of a n-dimensional simplex? > Another way (although not quite what is asked for) is to put the > n-simplex in (n+1)-space. Then you can use a simple scheme like this > (shown for n=4): > (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) > and now it is obvious that they all have the same distance from each > other. The center is (1/4,1/4,1/4,1/4). This is a simplex that is a tetrahedron and while it could be described in 3 dimensions, it is actually moved out into 4d because the fourth point is in the 4th dimension, rather than being simply a 4th point in the 3rd dimension. Is that correct? So, this is actually a simplex for the case of n=3, right? Craig === Subject: Re: Cartesian coordinates for verticies of a n-dimensional simplex? > Another way (although not quite what is asked for) is to put the > n-simplex in (n+1)-space. Then you can use a simple scheme like this > (shown for n=4): > (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) > and now it is obvious that they all have the same distance from each > other. The center is (1/4,1/4,1/4,1/4). Yeah, that's the way it's usually done when calculating the roots of the Lie algebras of type A_n, or their Weyl groups. -- Jim Heckman === Subject: Re: How to transform spherical/polar coordinates to Cartesian for n-dimensions? in message : > Please don't top post. I've restored the correct order below. > in message : in message <2rh1b.31872$0u4.12349@news1.central.cox.net>: I am looking for the method to transform spherical coordinates of > arbitrarily high dimensions (n) to Cartesian coordinates (of the > same number > of dimensions of course). I have found how to go from polar (2d) to > Cartesian, and spherical (3d) to Cartesian, but I have not yet found > the > n-dimension solution. It is my intent to implement this in Java for > a program I am writing. Any help or links would be greatly > appreciated. The usual scheme, modulo renumbering of indices, is: x_1 = r sin(a_1) > x_2 = r cos(a_1) sin(a_2) > x_3 = r cos(a_1) cos(a_2) sin(a_3) > ... > x_n = r cos(a_1) cos(a_2) cos(a_3) ... cos(a_{n-1}) Of course, r >= 0, -pi/2 <= a_i <= pi/2 except -pi < a_{n-1} <= pi > Was that last line supposed to be: > x_n = r cos(a_1) cos(a_2) cos(a_3) ... cos(a_{n-1}) > or > x_n = r cos(a_1) cos(a_2) cos(a_3) ... sin(a_{n-1}) > or > x_n = r cos(a_1) cos(a_2) cos(a_3) ... cos(a_{n-1}) sin(a_n) > The first. More specifically: > x_n = r cos(a_1) cos(a_2) ... cos(a_{n-2}) cos(a_{n-1}) > ? It looked like in the x_1 through x_3 that the last factor was always > going to be sin of a_n... > Yes. That's true for all x_i except the last one. For each x_i > after the first, you change the sine at the end of the previous > x_{i-1} to a cosine, and for all x_i except the last one you add > the sine of a new angle a_i. > To answer your other question, x_1 = x, x_2 = y, x_3 = z, ... > However you want to label the Cartesian coordinates doesn't > really matter. For more than 3 or 4 dimensions, it's usually > more convenient to use numbered indices instead of different > letters. D'oh! I was thinking lat/lon instead of the theta/phi that mathematicians usually use. So change all the sines to cosines and vice versa, and take 0 <= a_i <= pi except for 0 <= a_{n-1} < 2pi. -- Jim Heckman === Subject: Re: How to transform spherical/polar coordinates to Cartesian for n-dimensions? > in message : > Please don't top post. I've restored the correct order below. > in message : in message <2rh1b.31872$0u4.12349@news1.central.cox.net>: I am looking for the method to transform spherical coordinates of > arbitrarily high dimensions (n) to Cartesian coordinates (of the > same number > of dimensions of course). I have found how to go from polar (2d) to > Cartesian, and spherical (3d) to Cartesian, but I have not yet found > the > n-dimension solution. It is my intent to implement this in Java for > a program I am writing. Any help or links would be greatly > appreciated. The usual scheme, modulo renumbering of indices, is: x_1 = r sin(a_1) > x_2 = r cos(a_1) sin(a_2) > x_3 = r cos(a_1) cos(a_2) sin(a_3) > ... > x_n = r cos(a_1) cos(a_2) cos(a_3) ... cos(a_{n-1}) Of course, r >= 0, -pi/2 <= a_i <= pi/2 except -pi < a_{n-1} <= pi Was that last line supposed to be: x_n = r cos(a_1) cos(a_2) cos(a_3) ... cos(a_{n-1}) > or > x_n = r cos(a_1) cos(a_2) cos(a_3) ... sin(a_{n-1}) > or > x_n = r cos(a_1) cos(a_2) cos(a_3) ... cos(a_{n-1}) sin(a_n) > The first. More specifically: > x_n = r cos(a_1) cos(a_2) ... cos(a_{n-2}) cos(a_{n-1}) > ? It looked like in the x_1 through x_3 that the last factor was always > going to be sin of a_n... > Yes. That's true for all x_i except the last one. For each x_i > after the first, you change the sine at the end of the previous > x_{i-1} to a cosine, and for all x_i except the last one you add > the sine of a new angle a_i. > To answer your other question, x_1 = x, x_2 = y, x_3 = z, ... > However you want to label the Cartesian coordinates doesn't > really matter. For more than 3 or 4 dimensions, it's usually > more convenient to use numbered indices instead of different > letters. > D'oh! I was thinking lat/lon instead of the theta/phi that > mathematicians usually use. So change all the sines to cosines > and vice versa, and take 0 <= a_i <= pi except for > 0 <= a_{n-1} < 2pi. > -- > Jim Heckman Craig === Subject: Re: is there any Gram-Schimt techique for integer matrix? > For some reason, I want to restrict the construction of an orthogonal > matrix within the domain of integer matrix. > Is there a Gram-Schimt kind of technique which can deal with integer > matrix and get the result as near as orthogonal matrix as > possible?(even better if the result can be orthonormal after a scaling > factor for the whole matrix) > I don't know if the following makes things easier... let's say there > is originally an real matrix which is orthogonal/orthonormal. After > some scaling and round off into integer matrix, it is no longer > orthogonal. How to make some modification to restore its orthogonal as > much as possible(within the integer domain) > Please give me some advice on how to do this problem? If an earlier > mathematician had worked on this problem, please point me to the > resources or the similar places... maybe some mathematician had > already worked it out... For a matrix M, it is common to denote its transpose by M^t. I would start with an arbitrary integer matrix A such that det(A) =/= 0, then apply a modified Gram-Schmidt algorithm where all vector normalization operations are omitted, that is the steps where a vector v is replaced by v/||v|| . The result will be a rational entry matrix N such that N. N^t is a diagonal matrix. Now (N.N^t). (N.N^t)^t will be a diagonal matrix whose entries are squares of rational numbers q_1, ... q_n, n being the order of the matrices. Dividing the rows of N.N^t by the rationals q_1, ... q_n, we get a rational entry matrix P such that P.P^t = I . Then, multiplying P by some appropriately-chosen positive integer m, we get that M:= m*P is an integer entry matrix such that M. M^t = k*I, for some k>0. Maybe what you really want to do is to approximate a real-valued orthogonal matrix by a rational-valued orthogonal matrix? That seems harder. The easiest case should be 2x2 matrices. David Bernier === Subject: Re: is there any Gram-Schimt techique for integer matrix? > Maybe what you really want to do is to approximate > a real-valued orthogonal matrix by a rational-valued > orthogonal matrix? Yeah, you said it that I have a real-valued orthogonal matrix and I scaled and round-off it becomes no longer orthogonal I want to change/modify the integer matrix a little bit to make it as orthogonal as possible, without changing the whole matrix structure by using normal G-S procedure... Maybe some mathematicians had already worked on this many years ago? I can also do a computer search, but that's too time consuming... Can you give me more information/pointers on this issue? -Walala === Subject: Re: is there any Gram-Schimt techique for integer matrix? > Maybe what you really want to do is to approximate > a real-valued orthogonal matrix by a rational-valued > orthogonal matrix? > Yeah, you said it that I have a real-valued orthogonal matrix and I scaled > and round-off it becomes no longer orthogonal I want to change/modify the > integer matrix a little bit to make it as orthogonal as possible, without > changing the whole matrix structure by using normal G-S procedure... > Maybe some mathematicians had already worked on this many years ago? I can > also do a computer search, but that's too time consuming... > Can you give me more information/pointers on this issue? > -Walala Actually the Gram-Schmidt procedure preserves rationality if we leave out normalizing the output vectors to unit length. Consider two rational vectors u, v with the residue r when u is projected onto v: r = u - (u,v)/||v||^2 Note that r is again a rational vector. I regard this as a widely known fact of uncertain provenance. It underlies such specific developments as the root-free Cholesky decomposition, root-free QR, etc. === Subject: Re: is there any Gram-Schimt techique for integer matrix? Maybe what you really want to do is to approximate > a real-valued orthogonal matrix by a rational-valued > orthogonal matrix? Yeah, you said it that I have a real-valued orthogonal matrix and I scaled > and round-off it becomes no longer orthogonal I want to change/modify the > integer matrix a little bit to make it as orthogonal as possible, without > changing the whole matrix structure by using normal G-S procedure... > Maybe some mathematicians had already worked on this many years ago? I can > also do a computer search, but that's too time consuming... > Can you give me more information/pointers on this issue? > -Walala > Actually the Gram-Schmidt procedure preserves rationality if we leave out > normalizing the output vectors to unit length. > Consider two rational vectors u, v with the residue r when u is projected > onto v: > r = u - (u,v)/||v||^2 > Note that r is again a rational vector. I regard this as a widely known > fact > of uncertain provenance. It underlies such specific developments as the > root-free Cholesky decomposition, root-free QR, etc. Ooops... I left out something in the residue equation, should be: r = u - (u,v)/||v||^2 v In other words we subtract a rational multiple of v from u... -- chip === Subject: Re: is there any Gram-Schimt techique for integer matrix? >> For some reason, I want to restrict the construction of an orthogonal >> matrix within the domain of integer matrix. >> Is there a Gram-Schimt kind of technique which can deal with integer >> matrix and get the result as near as orthogonal matrix as >> possible?(even better if the result can be orthonormal after a scaling >> factor for the whole matrix) >> I don't know if the following makes things easier... let's say there >> is originally an real matrix which is orthogonal/orthonormal. After >> some scaling and round off into integer matrix, it is no longer >> orthogonal. How to make some modification to restore its orthogonal as >> much as possible(within the integer domain) >> Please give me some advice on how to do this problem? If an earlier >> mathematician had worked on this problem, please point me to the >> resources or the similar places... maybe some mathematician had >> already worked it out... > For a matrix M, it is common to denote its transpose by M^t. > I would start with an arbitrary integer matrix A > such that det(A) =/= 0, then apply a modified Gram-Schmidt > algorithm where all vector normalization operations are > omitted, that is the steps where a vector v is > replaced by v/||v|| . The result will be a rational entry > matrix N such that N. N^t is a diagonal matrix. > Now (N.N^t). (N.N^t)^t will be > a diagonal matrix whose entries are squares > of rational numbers q_1, ... q_n, > n being the order of the matrices. > Dividing the rows of N.N^t by the rationals > q_1, ... q_n, we get a > rational entry matrix P such that > P.P^t = I . Then, multiplying P by Unfortunately, P = I. > some appropriately-chosen positive integer m, > we get that M:= m*P is an integer entry matrix > such that > M. M^t = k*I, for some k>0. > Maybe what you really want to do is to approximate > a real-valued orthogonal matrix by a rational-valued > orthogonal matrix? > That seems harder. The easiest case should be > 2x2 matrices. > David Bernier === Subject: Re: Does Gaussian Random Walk Have Maximum in Interval? >>For an instance of a Gaussian random walk over the time interval >>[t0,t1], let x(t) be the position of the walk at any time t in [t0,t1]. >>Then (tm,x(tm)) is a maximum iff x(tm) >= x(t) for all t in [t0,t1]. >>In general, does such a point exist? >If by Gaussian random walk you mean Brownian motion (or any other >suitably well-defined stochastic process), sure. X(t) is axiomatically >continuous, X(t) is certainly continuous, but I don't see how that's axiomatic - the fact that a process with independent gaussian increments with the appropriate variances has as continuous sample paths is something that needs to be proved. >so M = max{X(t): t0 <= t <= t1} is a well-defined random >varable. For standard Brownian motion, M is not bounded (i.e., P{M > >c} > 0 for all c), and M - X(t0) has density x |-> 2/sqrt(2 pi >(t1-t0)) exp(-x^2/[2(t1-t0)]) for x > 0. >Let T = min{t: X(t) = M}. Off hand, I do not know the joint >distribution of T and M. ************************ David C. Ullrich === Subject: Re: Which came first: negative integers or 0? >I believe that negative numbers actually came first. Zero and infinity were >denied existence on religious and philosophical grounds. One implies the >other. The existence of zero implies the existence of infinity? I hope you're not using the word implies in the sense in which it's used in mathematics - if you are, could you show us how the proof goes? >It goes back to the theory of creation. If the Greeks, and later >the Catholic church, admitted the existence of zero, then that would throw a >wrench into their theory of creation, as they refused to believe that the >universe was created ex nihilo. >Lurch >> A history question: which came first: negative integers or nought (= >> zero)? >> -- >> G.C. ************************ David C. Ullrich === Subject: Re: Which came first: negative integers or 0? I am getting my information from a book titled Zero: a biography of a dangerous idea. I think I may have incorrectly stated the title in a previous post. In this book, it was posited that if one were to take, in modern parlance, 1/n as n--> oo you would approach nothing. And for the case of oo, take 1/n as n--> 0 you approach oo. The author, however, described it in words. This is not a direct quote. In the book, there was no rigourous proof. I suppose the implication is something like: If oo, then 1/n -->0. But, hey, your the expert David, not me. I am still an undergrad. Lurch >I believe that negative numbers actually came first. Zero and infinity were >denied existence on religious and philosophical grounds. One implies the >other. > The existence of zero implies the existence of infinity? I hope you're > not using the word implies in the sense in which it's used in > mathematics - if you are, could you show us how the proof goes? >It goes back to the theory of creation. If the Greeks, and later >the Catholic church, admitted the existence of zero, then that would throw a >wrench into their theory of creation, as they refused to believe that the >universe was created ex nihilo. >Lurch >> A history question: which came first: negative integers or nought (= >> zero)? >> -- >> G.C. > ************************ > David C. Ullrich === Subject: Re: Which came first: negative integers or 0? > I am getting my information from a book titled Zero: a biography of a > dangerous idea. I think I may have incorrectly stated the title in a > previous post. In this book, it was posited that if one were to take, in > modern parlance, 1/n as n--> oo you would approach nothing. In this case you approach 0 but don't reach it. > And for the > case of oo, take 1/n as n--> 0 you approach oo. The author, however, In this case it's not clear to me that you approach anything. Certainly one can work with 0 and have no commitment to infinity. But to work with (actual) infinity, al la Cantor/Robinson/Conway, requires a degree of sophistication that one would expect only to come after one is at home with 0. In the case of lim n --> oo there is no actual infinity. > described it in words. This is not a direct quote. In the book, there was > no rigourous proof. I suppose the implication is something like: If oo, > then 1/n -->0. But, hey, your the expert David, not me. I am still an > undergrad. What kind of excuse is that :-) -- G.C. === Subject: Re: chaos > f: R -> R. orbit{x, f(x), (f^2)(x), ...} > Let {(f^i)(x)} be the rbit of x. > {(f^i)(x)} is chaotic if > (1) {(f^i)(x)} is not asymptotic periodic. > (2) Lipunov exponent > 0. > What does (1) mean? > What is the definition of asymptotic periodic? The orbit is asymptotically periodic if there is k (the period) such that f^(i+k)(x)-f^i(x) -> 0 > ex. Let n be a positive integer and f(x) = nx (mod 1) on [0,1]. > Which orbits of f are chaotic? > Please give me a hint. Hint. If x is rational, what happens? === Subject: Re: probability and/or logic puzzle > Yes, this came out of a textbook . . . > but I passed that class about 25 years ago, > so I'm not cheating on my homework. > Start with a normal deck of playing cards. > Discard all but the aces and kings. Now you > have eight cards left. Your friend draws > two cards and hides them from you. He truthfully > tells you that at least one of his cards is an ace. > What is the probability that he holds two aces? > > He replaces the cards and you shuffle them well. > Your friend again draws two cards, and truthfully > tells you that one of them is the ace of spades. > What is the probability that he now holds two aces? > (Hint: this isn't the same answer as above!) > Okay . . . I don't get it. Why isn't it the same > answer both times? We have to assume what rule that guy uses in making his statements... the Monty Hall problem... Let's say it is a simple conditional probability question: choose 2 of the 8 cards. How many possible choices? (8 choose 2) = 28. How many of these have no aces? Same as 2 kings, (4 choose 2) = 6. How many have at least one ace? 28 - (no aces) = 28 - 6 = 22. How many have 2 aces? (4 choose 2) = 6. How many have ace of spades? 7. How many have ace of spades and another ace? 3. First answer: (two aces)/(at least one ace) = 6/22 = 3/11 Second answer: (ace of spades and another ace)/(ace of spades) = 3/7 -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Measure question > Say S is an abstract space, G is a subset of S and a Borel field, and > u(X) is a measure on G. > For each n, Q_n is in G, and the union of all Q_n=S. Also for each n, > u(Q_n) is finite. How does one prove that for each A in G: > limsup u(Q_n intersection A)= u(A) > All help appreciated. One cannot prove that. You should add some hypothesis. Say: Q_n is an increasing sequence. Or perhaps: limsup u(Q_n) = u(S). Then you have a chance to prove limsup u(Q_n intersection A)= u(A). -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Measure question >> Say S is an abstract space, G is a subset of S and a Borel field, and >> u(X) is a measure on G. >> For each n, Q_n is in G, and the union of all Q_n=S. Also for each n, >> u(Q_n) is finite. How does one prove that for each A in G: >> limsup u(Q_n intersection A)= u(A) >> All help appreciated. >One cannot prove that. >You should add some hypothesis. >Say: Q_n is an increasing sequence. >Or perhaps: limsup u(Q_n) = u(S). >Then you have a chance to prove >limsup u(Q_n intersection A)= u(A). If he adds the assumption that limsup u(Q_n) = u(S) his chance of proving this is still pretty small... ************************ David C. Ullrich === Subject: Re: Measure question >Say S is an abstract space, G is a subset of S and a Borel field, and >u(X) is a measure on G. >For each n, Q_n is in G, and the union of all Q_n=S. Things are very confused already. First, if G is a subset of S and a Borel field then S cannot be an abstract space, it must be a set of subsets of some set. Now when you say that the union of the Q_n is S it becomes pretty clear that you didn't mean that G was a subset of S at all. I _think_ that what you mean here is that S is a topological space, G is the field of Borel subsets of S, and u is a measure on G (I'm also very confused by what you mean by saying u(X) is a measure...) >Also for each n, >u(Q_n) is finite. How does one prove that for each A in G: >limsup u(Q_n intersection A)= u(A) You don't prove this because it's not true. (Simple counterexample: A = S = [1, infinity), u = Lebesgue measure, and Q+n = [n, n+1).) In addition to garbling the hypotheses that you stated you _omitted_ a crucial hypothesis! I could tell you what the hypothesis you omitted was, but that seems like a bad idea - you're trying to learn this stuff, and you're never going to learn to _solve_ these problems until you can at least _read_ and _state_ them correctly. So. Exactly what was the problem again? >All help appreciated. ************************ David C. Ullrich === Subject: Re: UFO Warp Drive Metric Engineering > Picture of the non-spherically symmetric non-static dipole exotic > vacuum zero point energy density /zpf field for the case of > Alcubierre's toy model UFO vacuum propeller metric is at Mr. Sofruity, I know you think your verbal diarrhea impresses the rubes, but can you please write in English next time? Stringing together a random assortment of buzzwords does not mean you've produced a readable sentence. === Subject: Connecting the Dots: Overview of my work Since I can go from talking about FLT, to discussing an esoteric error in the ring of algebraic integers, to my prime counting function, to partial differential equations, to what happens when you add 1/2 to the ring of integers, some might get a little lost, or not realize that all of it is connected by a rigid, logical framework based primarily on modern ideas. And in fact the modern ideas I use have a lot to do with your reading this post as object-oriented thinking is quite important in computer programming. What I want to do in this post is give a roadmap, connect all the dots, so that the big picture people understand how it all relates. First off, while object-oriented thinking permeates modern computer programming, and has always been in the sciences, mathematicians seem to have missed the train. If you've picked up a textbook on abstract algebra, you might notice there are all these little, dry rules. Being someone who focuses on the concrete, when I picked up a text on abstract algebra, back when I was fourteen at Duke University, I put it back down after a few minutes, and instead went back to playing with calculus. To me it's like when Ptolemy was a big wheel. For those who forgot their science history, Ptolemy worked out a system for figuring out where objects in the sky would be at various times using spheres and circles. Those were used because the heavens were considered the abode of God, and spheres and circles were considered perfection, so people figured God would only use perfect stuff. The problem is that left little errors, which were fixed with, guess what, little circles called epicycles. It was a kludgy system that involved a lot of calculation and still would be off. Well along comes Copernicus and Kepler, and Kepler drops the circle business and uses ellipses. Being himself religious Kepler came up with his own reasons for why God would use the supposedly less perfect ellipse, rather than circles. When I decided to try and use basic algebra to find a short proof of Fermat's Last Theorem, which I really, really, really hoped existed, I made a conscious decision not to bother with overdone approaches. I like simple. What happened is that I came upon a rather basic, straightforward approach which boils down to factoring x^p + y^p - z^p indirectly. However, that approach revealed that mathematicians hadn't discovered enough mathematical infrastructure to handle factorizations at that level. So I was forced to work out that infrastructure myself, which I call object mathematics. While thinking about such things, I found myself chatting about simple polynomials like x+1, and (x+1)(x+3), which got me to thinking about prime numbers, and a few weeks later I had a way to count them that mathematicians got close too, but never quite got the full thing. I know they didn't because I can look at their work where they got close, and see how close they came to what I have. And also I can see what my discovery does that what they have cannot do. I played with prime counting for a while, including working out a partial differential equation that follows from my functional way to count prime numbers, and then went back to thinking about my FLT work. For a while I was convinced by others that I needed algebraic integers, which are numbers defined to be the roots of monic polynomials with integer coefficients. You know, like x^2 + 2x + 2, as the polynomial is monic because the first coefficient is 1. So I put object mathematics to the side, hoping that maybe mathematicians had indeed built up the infrastructure needed for my FLT work, but then a little later I found out that no, they hadn't, and in fact there was this intriguing little problem with algebraic integers. That forced me to go back and blow the dust off of my work on object mathematics, and I finally worked it out thoroughly within the last few weeks, as part of the polishing process. Then I was surprised to find that mathematicians seemed to not know basic things about their own work, which thinking back to Ptolemy, doesn't surprise me now, as when you have a lot of excess, based on unnecessary rules, people can learn things by rote, and not understand. So mathematicians apparently don't understand that including fractions like 1/2 with numbers like integers gives you the field of reals. Their belief comes from arbitrary rules where they exclude infinite sums on an ad hoc basis. Seeing that is easy. Consider that 1/(k-1) = 1/k + 1/k^2 +... when k is a nonzero integer other than 1 or -1, which is easy to prove in the classic way using S. S = 1/k + 1/k^2 +... = 1/k(1 + 1/k + 1/k^2+...) = 1/k(1+ S), so kS = 1 + S, S = 1/(k-1). So if you add 1/2 in with integers, you have 1/4 = 1/2(1/2), so you get 1/3, and now you can have 1/12 = 1/4(1/3), which gives you 1/11, both from the formula above, and that process leads you on and on until you have the field of reals. I say that such infinite sums are decidable since you can get an answer, and there's no reason to exclude them. Mathematicians want them excluded so they yelp, and start tossing out arbitrary rules. And I think back to Ptolemy. So that's an overview of my work and for LOTS of mathematics you can check http://groups.msn.com/AmateurMath where I go into a lot of detail, giving a short proof of FLT, my prime counting function and its associated partial differential equation, and I have a paper outlining the problem with algebraic integers. Also I have the framework object mathematics with discussion on why I unearthed it, and I even connect back to Gauss's Fundamental Theorem of Algebra. Basically I do a lot in a few pages which is what you can do with concise and potent mathematics. If you can make it through and understand it all, you are at least a hundred years ahead of current mathematicians. But don't tell them that as they seem to get upset very easily. James Harris === Subject: Re: Connecting the Dots: Overview of my work > So mathematicians apparently don't understand that including fractions > like 1/2 with numbers like integers gives you the field of reals. > Their belief comes from arbitrary rules where they exclude infinite > sums on an ad hoc basis. > Seeing that is easy. Consider that > 1/(k-1) = 1/k + 1/k^2 +... > when k is a nonzero integer other than 1 or -1, which is easy to prove > in the classic way using S. > S = 1/k + 1/k^2 +... = 1/k(1 + 1/k + 1/k^2+...) = 1/k(1+ S), so > kS = 1 + S, S = 1/(k-1). > So if you add 1/2 in with integers, you have 1/4 = 1/2(1/2), so you > get 1/3, and now you can have 1/12 = 1/4(1/3), which gives you 1/11, > both from the formula above, and that process leads you on and on > until you have the field of reals. The trouble is that you can't evaluate any of this unless you know that S is convergent. And to do this, you need an axiom in your field structure that says something like a bounded monotonic sequence is convergent. Well, guess what. *The presence of such an axiom is what distinguishes the rationals from the reals.* Jack Rudd === Subject: Re: Connecting the Dots: Overview of my work > Since I can go from talking about FLT, to discussing an esoteric error > in the ring of algebraic integers, to my prime counting function, to > partial differential equations, to what happens when you add 1/2 to > the ring of integers, [snip 1) http://w0rli.home.att.net/youare.swf 2) http://www.apa.org/journals/psp/psp7761121.html 3) http://www.mazepath.com/uncleal/sunshine.jpg -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Connecting the Dottiness. > .... If you've picked up a textbook on abstract > algebra, you might notice there are all these little, dry rules. Well you don't want to both yourself with those little, dry rules. (Oh... wait a minute... you don't do you?) -- G.C. === Subject: Re: Connecting the Dottiness. > .... If you've picked up a textbook on abstract > algebra, you might notice there are all these little, dry rules. > Well you don't want to both yourself with those little, dry rules. ^^^^ Meant bother sorry. > (Oh... wait a minute... you don't do you?) > -- > G.C. -- G.C. === Subject: Re: Connecting the Dottiness. > .... If you've picked up a textbook on abstract > algebra, you might notice there are all these little, dry rules. > Well you don't want to both yourself with those little, dry rules. > (Oh... wait a minute... you don't do you?) > -- > G.C. I think he prefers large, wet rules. === Subject: Re: Connecting the Dots: Overview of my work >Since I can go from talking about FLT, to discussing an esoteric error >in the ring of algebraic integers, to my prime counting function, to >partial differential equations, to what happens when you add 1/2 to >the ring of integers, some might get a little lost, or not realize >that all of it is connected by a rigid, logical framework based >primarily on modern ideas. >And in fact the modern ideas I use have a lot to do with your reading >this post as object-oriented thinking is quite important in computer >programming. >What I want to do in this post is give a roadmap, connect all the >dots, so that the big picture people understand how it all relates. >First off, while object-oriented thinking permeates modern computer >programming, and has always been in the sciences, mathematicians seem >to have missed the train. Suggesting that you've remedied this deficiency is utterly silly. Ignoring for a second the fact that you've never given a _coherent_ definition of what the Object Ring actually is: Your Objects have nothing in common with the notions involved in object-oriented programming - the only connection is the _word_ object. > If you've picked up a textbook on abstract >algebra, you might notice there are all these little, dry rules. >Being someone who focuses on the concrete, when I picked up a text on >abstract algebra, back when I was fourteen at Duke University, I put >it back down after a few minutes, and instead went back to playing >with calculus. And, as we see below, you still have no understanding of the most basic concepts of abstract algebra, or even of the most basic concepts of mathematics itself, like what a _definition_ is. But the fact that you know nothing about a field doesn't stop you from lecturing on what's wrong with it. Good for you. >That forced me to go back and blow the dust off of my work on object >mathematics, and I finally worked it out thoroughly within the last >few weeks, as part of the polishing process. It's only been worked out thourougly in the last few weeks? So the people who you called incompetent liars for not understanding it at various times during the last year or so actually had good reason for not understanding it? Imagine that. >Then I was surprised to find that mathematicians seemed to not know >basic things about their own work, which thinking back to Ptolemy, >doesn't surprise me now, as when you have a lot of excess, based on >unnecessary rules, people can learn things by rote, and not >understand. >So mathematicians apparently don't understand that including fractions >like 1/2 with numbers like integers gives you the field of reals. >Their belief comes from arbitrary rules where they exclude infinite >sums on an ad hoc basis. This is the part where you show that you don't even know what a _definition_ is. Also the part where you show that you believe that the square root of 2 is rational. (No, you didn't _say_ that. But the rationals are a ring containing the integers and 1/2, so it follows from what you just said that the rationals are the same as the reals, hence sqrt(2) is rational.) Fascinating how things develop over time. Some years ago you were insulting people because they were talking about mathematics even though they were so ignorant they didn't realize that integers were irrational. Now it's just the opposite: the fact that we just learn things by rote without understanding them is why we don't realize that sqrt(2) is rational! >Seeing that is easy. Consider that > 1/(k-1) = 1/k + 1/k^2 +... >when k is a nonzero integer other than 1 or -1, which is easy to prove >in the classic way using S. > S = 1/k + 1/k^2 +... = 1/k(1 + 1/k + 1/k^2+...) = 1/k(1+ S), so > kS = 1 + S, S = 1/(k-1). >So if you add 1/2 in with integers, you have 1/4 = 1/2(1/2), so you >get 1/3, and now you can have 1/12 = 1/4(1/3), which gives you 1/11, >both from the formula above, and that process leads you on and on >until you have the field of reals. >I say that such infinite sums are decidable since you can get an >answer, No, you say such sums are decidable because you don't know what the word means. >and there's no reason to exclude them. Mathematicians want >them excluded so they yelp, and start tossing out arbitrary rules. Arbitrary rules. Like the arbitrary rule that says a tomato is not an automobile. Or the arbitrary rule that says a kumquat is not an airplane. Hint: You should _really_ confine your errors to things that involve long complicated strings of equations, so that the reader has to do a little work to see why it's nonsense. The stuff you're going on about these days is going to embarass you in _exactly_ the same way as the integers are irrational episode does, once you sober up... >And I think back to Ptolemy. >So that's an overview of my work and for LOTS of mathematics you can >check > http://groups.msn.com/AmateurMath >where I go into a lot of detail, giving a short proof of FLT, my prime >counting function and its associated partial differential equation, >and I have a paper outlining the problem with algebraic integers. >Also I have the framework object mathematics with discussion on why I >unearthed it, and I even connect back to Gauss's Fundamental Theorem >of Algebra. >Basically I do a lot in a few pages which is what you can do with >concise and potent mathematics. If you can make it through and >understand it all, you are at least a hundred years ahead of current >mathematicians. But don't tell them that as they seem to get upset >very easily. >James Harris ************************ David C. Ullrich === Subject: Re: Connecting the Dots: Overview of my work > Hint: You should _really_ confine your errors to things that > involve long complicated strings of equations, so that the > reader has to do a little work to see why it's nonsense. Hee. Yes, he should limit himself to stuff that is not even wrong, rather than plain wrong. V. -- mail me at lastname at cs utk edu === Subject: Re: Gravitons are electrons, Electrons are gravitons > In sci.physics, Satan the Devil > > Devil Satan, > > Whatever you do Lucifer, DO NOT, and I repeat, DO NOT TELL THEM that > in the central levels, the radius (r), even the Radius (R), IS OF OF > EACH PARTICLE PARTICLE, not the square of the distance between their > two centers. > Actually, you have a good point. I'm not sure how much difference > it makes to back-of-napkin orbital calculations but it's clear > gravitational attraction. Therefore, a good approximation would > be an integral over the volume of the planet. (An even better > one would be the sum of several integrals, each one modeling > a specific part: one for the crust, one for the mantle, one for > the liquid core, one for the solid core. And even then one > has issues regarding the continents.) > F = integral(sphere)( GmM/d^2 dV) > where d^2 and dV are linked (e.g., d^2 = x^2+y^2+z^2 and dV = dx dy dz; > however, most would probably want to use polar coordinates Thou shalt not say; (+: :-) > (As for the original subject: I can't say. Besides, electrons > are charged, making for some problems.) Thou shalt not say; (-: :+) Thou shalt not say; Gravitons are electrons in charged and curved space Thou shalt not say; Electrons are gravitons in charged and curved space -- And; Thou shalt not say; Thou shalt not say Offspring of David... F=Gm^2/w^2 for a wave Thou shalt not say; -- And; Thou shalt not say; 2) F=Gm^2/w^2 (for a wave) -- And; Thou shalt not say; No thou shalt not. -- I am Satan the Devil I am === Subject: Re: How to prove (conclude) this? > a -> b [alpha] > is equivalent to > ~a V b [beta] > ...but how can one conclude the one from the other? a -> b |= -a v b 1 (1) a -> b Assumption 2 (2) -(-a v b) Assumption for RAA 3 (3) a Assumption 1,3 (4) b 3,1, MPP 1,3 (5) -a v b 4, vI 1,2,3 (6) (-a v b) & -(-a v b) 5, 2, &I 1,2 (7) -a 3, 6, RAA 1,2 (8) -a v b 7, vI 1,2 (9) (-a v b) & -(-a v b) 9, 2, &I 1 (10) --(-a v b) 2, 9, RAA 1 (11) -a v b 10, DN MPP = modus ponendo ponens vI = or introduction &I = and introduction RAA = reductio ad absurdum DN = double negation. === Subject: Re: Looking for Spare Cycles Mensanator Try this link and tell me if it works any better: ---> http://www.gridontap.com/grid.zip I think the other server has had it... === Subject: Re: Origin of Ben Franklin's Force Equations This one or the last one? Satan the Devil -- > In sci.physics, Satan the Devil > > Devil Satan, > > Whatever you do Lucifer, DO NOT, and I repeat, DO NOT TELL THEM that > in the central levels, the radius (r), even the Radius (R), IS OF OF > EACH PARTICLE PARTICLE, not the square of the distance between their > two centers. > Actually, you have a good point. I'm not sure how much difference > it makes to back-of-napkin orbital calculations but it's clear > gravitational attraction. Therefore, a good approximation would > be an integral over the volume of the planet. (An even better > one would be the sum of several integrals, each one modeling > a specific part: one for the crust, one for the mantle, one for > the liquid core, one for the solid core. And even then one > has issues regarding the continents.) > F = integral(sphere)( GmM/d^2 dV) > where d^2 and dV are linked (e.g., d^2 = x^2+y^2+z^2 and dV = dx dy dz; > however, most would probably want to use polar coordinates Thou shalt not say; (+: :-) > (As for the original subject: I can't say. Besides, electrons > are charged, making for some problems.) Thou shalt not say; (-: :+) Thou shalt not say; Gravitons are electrons in charged and curved space Thou shalt not say; Electrons are gravitons in charged and curved space -- And; Thou shalt not say; Thou shalt not say; Offspring of David... F=Gm^2/w^2 for a wave Thou shalt not say; -- And; Thou shalt not say; 2) G=Fw^2/m^2 (for a wave) -- And; Thou shalt not say; No thou shalt not. -- I Am; Satan the Devil I Am; === Subject: Re: The sum = product sequence in the complex plane > [reply address is baloglouAToswego.edu] > Consider the sequence An defined by A1*A2*...*An = A1+A2+...+An and, > consequently, A(n+1) = A1*A2*...*An/(A1*A2*...*An-1). When A1 is real, > it is possible to show that the infinite sum/product converges to 0 for > A1 < 1 (with An --> 0) and diverges to +oo for A1 > 1 (with An --> 1). A(n+1) = (A1+..+An)/(A1*..*An - 1) > For complex An the convergence region seems to be less tractable, and > I have already noticed some oscillation at A1 = 1/2 + i/2; there could > be an 'oscillation curve' enclosing the convergence region, which may > not be easy to determine analytically: could someone do so graphically? === Subject: Re: The sum = product sequence in the complex plane > [reply address is baloglouAToswego.edu] > Consider the sequence An defined by A1*A2*...*An = A1+A2+...+An and, > consequently, A(n+1) = A1*A2*...*An/(A1*A2*...*An-1). When A1 is real, > it is possible to show that the infinite sum/product converges to 0 for > A1 < 1 (with An --> 0) and diverges to +oo for A1 > 1 (with An --> 1). > A(n+1) = (A1+..+An)/(A1*..*An - 1) So, what's the difference? The two sequences defined recursively by B(n+1) = B(1)*...*B(n)/(B(1)*...*B(n)-1) A(n+1) = (A(1) + ... + A(n))/(A(1)*A(2)*...*A(n)-1) are identical, if A(1) = B(1) (the point being that this makes A1+...+An = A1*A2*...*An). --Ron Bruck === Subject: Re: The sum = product sequence in the complex plane > Consider the sequence An defined by A1*A2*...*An = A1+A2+...+An and, > consequently, A(n+1) = A1*A2*...*An/(A1*A2*...*An-1). When A1 is real, > it is possible to show that the infinite sum/product converges to 0 for > A1 < 1 (with An --> 0) and diverges to +oo for A1 > 1 (with An --> 1). > For complex An the convergence region seems to be less tractable, and > I have already noticed some oscillation at A1 = 1/2 + i/2; there could > be an 'oscillation curve' enclosing the convergence region, which may > not be easy to determine analytically: could someone do so graphically? Yes, indeed, there does appear graphically to be such a curve, and not suprisingly it is a fractal. Let's just consider the sequence of products p[n] = product(A[i], i= 1..n). Then this sequence satisfies the recurrence p[n+1] = p[n]^2/(p[n]-1) = p[n]+1+1/(p[n]-1). This sequence has orbits of all finite lengths. For each n, there are 2^n-1 points (including 0) that are in orbits of length n (including lengths that divide n). 1/2+1/2*i and 1/2-1/2*i are an orbit of length 2. I plotted p in the rectangular region with corners 3-2*i and 1.5+2*i. I used 500x500 = 250,000 initial points p[0] evenly spaced in this rectangle. I computed up to 200 points of the sequence for each p[0]. I decided (somewhat arbitarily) that the sequence diverged if any of the 200 had magnitude greater than 100. I used this time-to-divergence to assign a color to the initial point. There is a somewhat cigar-shaped fractal boundary opening to the left from 1, passing through i and -i, and lying between the lines -2*i and 2*i. I will guess that inside this boundary, p -> 0, and outside the boundary, p -> infinity. On top of this plot, I plotted all orbits of lengths 2 through 7. They all lie on the boundary. The point 1 is a special case: it can be considered an orbit of length 0. It appears that the points in finite orbits may be dense on the boundary. So what happens to the points on the boundary that are not in these finite orbits (there are only countably many that are in finite orbits)? Some of them get sucked into a finite orbit: For example, i and -i get sucked into the orbit of length 2. Letting f(z) = z^2/(z-1), and given any point p in a finite orbit and any positive integer n, we can solve the polynomial equation (f@@n)(z) = p to find up to 2^n point that will get sucked to p. Doing this for p=1 gives the points at the vertices of the divots in the boundary. For example, f(1/2+sqrt(3)/2*i) = 1 and f(f((1+sqrt(2*sqrt(13)-5) + i*(sqrt(3)-sqrt(2*sqrt(13)+5)))/4)) = 1. I plotted the points that get sucked to 1 in 7 or fewer steps in green. Still, this only accounts for countably points on the boundary. It could be that the other points on the boundary have infinite orbits that lie entirely on the boundary and perhaps are dense on the boundary. Can anyone find analytically a single point on the boundary that is not on a finite orbit and does not get sucked into a finite orbit? The GIF plot and the Maple worksheet to generate it are at http://group.yahoo.com/group/meg-sugarbush in the files area. Under brand-new Yahoo rules, you will have to join the group (which means you need to create a free Yahoo ID if you don't already have one) before you can access the files area. This group is only for exchanging Maple files through the web site -- there is no email associated with it. === Subject: Re: quaternions-- what's the point? >>it seems to me that a quaternion is nothing more than a simple >>ordered quadruplet. And that multiplication/addition is simply >>extended in such a way that it is defined over such ordered >>quadruplets >So far, so good. But then you go way off into left field: >>It's as if I were to write a geometry book where I >>called points binarinions and write them as a#{(b}) instead of >>(a,b).. >No, because there you are inventing nonstandard nomenclature for an >existing concept. WRH invented a new concept, and needed new >nomenclature to describe it. The fact that in the 21st Century we have >more useful tools than Quaternions does not take away any of the >importance the concept had when Hamilton invented them. Quaternions are still important, as a Clifford algebra, and therefore as a natural setting for SO(3) spinors, and as a division algebra over R (the only division algebras over R being R, C and H). The result that the only finite-dimensional simple algebras over R are the matrix algebras over R, C and H (i.e. M_n(R), M_n(C) and M_n(H)) is a case in point, and also the identification of real forms of certain real simple Lie algebras and real simple Lie groups, where their simplest description is in terms of modules over the quaternions, including SU^*(2n), SO^*(2n), and Sp(2p,2q), the last being the group of symplectic matrices which are pseudounitary. The simple compact classical groups are: SO(n) for n > 2 (the simplest description of SO(n) is in terms of real matrices), SU(n) for n > 1 (the simplest description of SU(n) is in terms of complex matrices), Sp(2n) for n >= 1 (the simplest description of Sp(2n) is in terms of quaternion matrices), where a >= b means a is greater than or equal to b. All the simple compact classical groups (i.e. groups which are simple as Lie groups) are locally isomorphic to the above examples. All such groups are isomorphic to quotient groups of Spin(n), SU(n) or Sp(2n) (SO(n) is isomorphic to Spin(n)/Z_2 - Spin(n) is simply connected for all n > 2). While on the topic of Lie groups and Lie algebras, G_2 is the automorphism group for the octonions. Similarly, in the field of random matrices, the three important distributions - the GOE (Gaussian Orthogonal Ensemble), the GUE (Gaussian Unitary Ensemble) and the GSE (Gaussian Symplectic Ensemble) - have their simplest descriptions in terms of real matrices, complex matrices and quaternion matrices respectively. In a quantum mechanical system, the underlying Hilbert space is a complex Hilbert space, but for a system with a time-reversal invariant Hamiltonian, the Hamiltonian becomes a Hamiltonian over a real Hilbert space or a quaternion Hilbert space. The three principal models of quantum logic are quantum mechanics on real Hilbert spaces, complex Hilbert spaces and quaternion Hilbert spaces. In representation theory, Schur's Lemma tells us that an operator commuting with an irreducible complex representation is a complex multiple of the identity operator. In the case of an irreducible real representation, the space of such operators falls into one of three classes: real multiples of the identity operator; real linear combinations of the identity operator I and an operator E such that E^2 = -I; real linear combinations of the identity operator I and three operators E, F, G such that E^2 = F^2 = G^2 = EFG = -I, so that EF = -FE = G, FG = -GF = E, GE = -EG = F. The first such space is isomorphic to R, the second space is isomorphic to C, and the third space is isomorphic to H. The importance of quaternions does not come from them being a tool, but rather from the fact that they are of interest in their own right, and in that way, they are just as important now as they ever were. >>the point being that it would be arbitrary and accomplish nothing >>and, in fact, greatly obfuscate something for no reason. >Your nomenclature, yes; Quaternions, no. >>This, it seems, is the entire nature of quaternions. >No, just the nature of your understanding of them. At the time they >were a useful tool, better than what was available, and even today >they are useful in special cases. Instead of prejudging their utility >and rationale from ignorance, you would do better to do some reading. It is amazing what ignorance of a topic will make some people believe, isn't it. David McAnally -------------- === Subject: Re: quaternions-- what's the point? > it seems to me that a quaternion is nothing more than a simple > ordered quadruplet. (SNIP) > So Hamilton spent years pondering a trivial problem, and subsequently > came up with a entirely trivial result? The point is rotations. (SNIP) I apologize, my letter was misinterpreted. Of course I acknowledge that the theorems of Hamilton are very good, etc. What I was saying was that these results could all be proven just as easily with only the most trivial modifications, if you skipped the whole quaternion detour and just defined multiplication and addition of ordered quadruplets. And it seems to me (though I am certainly no authority) that all the applications of the quaternions, to rotations and anything else, would be just as applicable if the entire theory simply used ordered quadruplets. So what I meant to say is that it seems that the quaternion apparatus merely makes the results unnecessarily opaque. I apologize for all the offense this seems to have caused and assure you I meant no dishonour to Hamilton. === Subject: Re: quaternions-- what's the point? >> it seems to me that a quaternion is nothing more than a simple >> ordered quadruplet. >(SNIP) >> So Hamilton spent years pondering a trivial problem, and subsequently >> came up with a entirely trivial result? The point is rotations. >(SNIP) >I apologize, my letter was misinterpreted. Of course I acknowledge >that the theorems of Hamilton are very good, etc. What I was saying >was that these results could all be proven just as easily with only >the most trivial modifications, if you skipped the whole quaternion >detour and just defined multiplication and addition of ordered >quadruplets. So what is the essential difference between quaternions and your ordered quadruplets? You can just define the space of quaternions as the set of ordered quadruplets with the appropriate addition and multiplication. All that I can see is that you are objecting to the name that is given. What is the difference between complex numbers and ordered pairs of real numbers (a,b) subject to (a,b)+(c,d) = (a+c,b+d), and (a,b)(c,d) = (ac-bd,ad+bc)? Answer: no difference whatsoever. The structures are isomorphic. In the same way, there is no difference between the structure of the space of quaternions and your space of quadruplets subject to the addition and multiplication laws. The structures are isomorphic. On the other hand, the term complex number is much more specific than ordered pair, and much more concise than ordered pair subject to the appropriate addition and multiplication. Similarly, quaternion is much more specific than ordered quadruple and much more concise than ordered quadruple subject to the appropriate addition and multiplication. As far as I can see, your objection rests solely on the fact that these important objects have been given a special name. David McAnally -------------- === Subject: Polynomial problem Hello I'm trying to prove the following statement, but got really stuck. Does anyone have a suggestion? Let P be a polynomial of the n_th degree such that P(x) is in [-1,1] for every x in [-1, 1]. Then, P'(x) is in [-n^2, n^2] for every x in [-1, 1]. Apparently, this is not that hard, but I couldn't get trough. Amanda PS: I think this is true only if the coefficients of P are real. === Subject: Re: Bob's Positive Integer Pages === >>Subject: Re: Bob's Positive Integer Pages >>Message-id: Glad you enjoyed it! >>FYI, ANS was also published in the American Mathematical Monthly by >>another author after i published it in the Southwest Journal of Pure and >>Applied Mathematics. Both are peer reviewed journals. > Were they aware of your claim that zero is not a number? > Who were the peers, guys like Nico Benschop? > Are you sure you didn't mean beer reviewed? > Don't forget that even Benschop had a bogus paper published in a > peer reviewed journal: Strange this, isn't it? Being peer reviewed is supposed to be a Good Thing, but if a nutter is peer reviewed that tells us nothing except that there are plenty of nutters around. -- G.C. === Subject: Re: Bob's Positive Integer Pages > Zero is Not a number. It is the absence of number. IMHO. How shall we signify the absence of stuff? If I have a bag of sweets and I eat them all there are no sweets left in the bag--0 sweets. That there are no numbers in the bag is irrelevant, there never were any numbers in the bag. > Just as infinity is not a number but is rather the idea of unlimited. There are lots of numbers infinity, and infinity is also (in other contexts) what Russell called an incomplete symbol. > I do not wish to discuss this any further. The refusal to discuss it any further is just a sign of your irrationality. > There are two schools of thought that will argue endlessly just as they > argue that 0.99999=1 There may be two schools of thought, but only one is right: 0.9999 does not = 1 (but I suspect that you didn't mean 0.99999=1, that's ok, sloppiness and imprecision always appeal to nutters). -- G.C. === Subject: Re: Bob's Positive Integer Pages >Zero is Not a number. It is the absence of number. IMHO. >Just as infinity is not a number but is rather the idea of unlimited. >I do not wish to discuss this any further. >There are two schools of thought that will argue endlessly just as they >argue that 0.99999=1 Tee-hee. Nobody has ever argued that 0.99999=1, they're clearly different numbers. People _do_ argue about whether 0.999...=1, but the fact that people argue about this doesn't show that there is any problem, it just shows that there are a lot of ignorant people posting messages on the internet. The argument over whether 0 is a number has the same status. >> http://my.tbaytel.net/forslund/index.html >> Enjoy - Bob >> An Alternate Number System (ANS) - A number system that has no need for >> the digit zero. Fair enough, the ANS has no need for the digit zero; >> but _we_ do to symbolize the number nought. >> -- >> G.C. ************************ David C. Ullrich === Subject: Re: Bob's Positive Integer Pages >> Zero is Not a number. It is the absence of number. IMHO. >> Just as infinity is not a number but is rather the idea of unlimited. >> I do not wish to discuss this any further. >> There are two schools of thought that will argue endlessly just as they >> argue that 0.99999=1 > Also, I don't know of any school that argues that 0.99999 = 1. > Actually, 0.99999 = 99999/100000 -- at least in the rational or real > number systems. If 0.99 is a repeating decimal then n = 0.overline{99} 100n = 99.overline{99} 99n = 99.00 n = 1 >> http://my.tbaytel.net/forslund/index.html >> Enjoy - Bob >> An Alternate Number System (ANS) - A number system that has no need for >> the digit zero. Fair enough, the ANS has no need for the digit zero; >> but _we_ do to symbolize the number nought. >> -- >> G.C. === Subject: Re: [Primes|Assymptotics] What is the order of Sum_{p <= n, p prime} Sum_{i=1..n} [n/p^i] p > > > [...] so that Sum_{p <= n, p prime} Sum_{i=1..n} [n/p^i] ln(p) > should simply be ln(n!) > > Why do you put ln(p) instead of p? Is this a typo error? > ln(n!)= Sum_{p <= n, p prime} Sum_{i=1..n} [n/p^i] ln(p) > This is different of S(n). > Let's suppose that n!= p1^e1*p2^e2*...pk^ek (the pi are prime numbers > between 2 and n with exponent ei) then : > ln(n!)= sum_i ei*ln(pi) > and > S(n)= sum_i ei*pi (pi <= n) > (S(n) is the nolog(n!) function of my previous link) > I gave this indication to help you to understand the nolog function of > the link and allow you to write S(n)= sum_i ei*pi > > Can you explain me why S(n) is ln(n!). In the number sequence link > that you gave me, I find that the assymptotics is > a(n)~(Pi^2/12)*n^2/ln(n). Is this a contradiction with your reasoning? > > No this should be right (I have no simpler expression for S(n)) > Hoping this will be clearer, > Raymond > Xan. Now I understand all. And you answered me to the second question: S(n) is c n^2/log n assymptotically following the link you provide me. I want not to abuse of you ;-), but you know anymore about nolog Xan. === Subject: Re: Questions for James Harris > > Well, you may *wish* to exclude 1/3 from the ring Z[1/2] but the > > definition of the ring does NOT exclude it, and in fact, in the ring > > > > 1/3 = 1/4 + 1/4^2 + 1/4^3 +.... > > > > I'm curious to know how you think you might exclude it. > There is no part of the definition of rings that says you have to > include infinite sums. There is even no part of the definition of > rings that says that the infinite sum above converges. Using some > other kinds of convergence we can even get a field (the 2-adics) > that contains Z[1/2] where 1/3 = -1 - 4 - 4^2 - 4^3 - ... It's interesting watching math people try to squirm out of including 1/3 = 1/4 + 1/4^2 + 1/4^3 +.... into Z[1/2], but here I'm going to talk about the even more interesting for now 1/(x+1). Some of you might not know why ring operations are addition and multiplication, where -1 in a ring de facto gives you subtraction, while division is left to fields. Well, dividing x+1 into 1 might give some clues. 1/(x+1) it seems equals 1 - x + x^2 - x^3 + ... which is not decidable for any integer x, other than x=0. So there's something wacky that happens when you have division, and in fact, breaking the rules of integers, gives you the field of reals or complex numbers--though some posters seem to get lost there--and in that arena you have 1/(x+1) = 1 - x + x^2 - x^3 + ... when x fits certain criteria, like not being an integer unless it equals 0. What is actually the case is that like polynomials are the analogs to integers, expressions like 1/(x+1) are the analogs to fractions like 1/2. Of course, if you have x+1 and an integer x, you get an integer. If you have 1/(x+1), and say, x=1, you get 1/2. Mathematicians worked backwards because numbers like 1/2 were already numbers and part of the system in a special way before mathematicians got around to fully formalizing. However, they forgot that numbers like 1/2 are actually used as operators in the real world. No one ever just gets 1/2, as they get 1/2 *of* something. But mathematicians formalized a system where 1/2 was just another number. But then they had to tweak the system. Working to keep certain things rational to them, they put in ad hoc rules, so that they could have things like the field of rationals, and exclude numbers like pi, since as Euler noticed pi^2/6 = 1 + 1/2^2 + 1/3^2+...+1/k^2+... to literally keep rationals, rational. A lot of effort from modern mathematicians came about in generating the ad hoc rules, teaching them, and then finding ways to actually discover mathematics around them. It's a full-time job, and requires YEARS of effort for the typical mathematician. Pity the poor math grad student, eh? But luckily some matematicians came up with rings, and separated out the ring operations of addition and multiplication, and mathematics is only then about a hundred years behind, thankfully. James Harris === Subject: Re: Maple, MatLab, MathCad, and Mathematica -- Decisions, Decisions > Maple is for symbolic manipulation. I understand it is also part > of MatLab, but I normally use MatLab for numerical algorithms. {snip} Erm...wrong. They're individual software systems. Maple has a worksheet-type inteerface to Matlab. They're good at different things; 'Matlab' is a contraction of 'Matrix lab', after all. Drieux === Subject: Re: Maple, MatLab, MathCad, and Mathematica -- Decisions, Decisions you can just download maple from kazaa. its only 40MB. the program is great. i use it for large computations and numerical analysis problems. === Subject: Re: Maple, MatLab, MathCad, and Mathematica -- Decisions, Decisions > you can just download maple from kazaa. its only 40MB. the program is > great. i use it for large computations and numerical analysis problems. Do you also steal your food, you little piece of filth? Who raised you, Saddam Hussein? === Subject: Re: Problem with derivate (RS+T) > When I enter the following > > 2. SIN (X) > 1. X > [RS+T] > > the result in Degree MODE -> (PI/180)*COS(X) > in Gradians -> (PI/200)*COS(X) > in RAD -> COS(X) > > This is totally correct! > Why there are 3 different solutions? > Mathematically speaking, if you differeniate sin(X) while working in > degrees, you must use the 'chain rule'. I don't have time to explain > why at the moment, but I hope someone else can :-) > cheers, > Al Manel. Hi Al Borowski, I try to diferenciate y=sin(x) in degrees MODE with DERIVE 5 and shows the same result as HP49G. y'=(pi/180)*cos(x). The diference betweet HP49 and Derive is that in DERIVE you can integrate in DEG MODE so you can do the following: When you integrate y=(pi/180)*cos(x) with DERIVE, it shows: SIN(x). It is correct. This result is acording with INTEGRATE(DIFFERENCIATE(f(X)))=f(X). With HP49g [with SILENT MODE ON and DEG MODE]: 2. SIN(X) 1. X [PUSH RS+T] 1.(PI/180)*COS(X) [PUSH LS-4 F6 (INTVX) <----- HP49 does a silent change to RAD. 1.(PI/180)*SIN(X) Something extrange happens. Could be the mistake in RS+T?. It would have to change to RAD mode. or HP49 could have a INTEGRATE command that allows me to integrate in DEG mode as DERIVE. Goood-bye. === Subject: Re: need help in algebra !!! > 1.solve equiation in set of natural numbers 1/x + 1/y + 1/z = 1 Just a matter of enumerating cases. Suppose we start by assuming x <= y <= z. Then clearly 1 < x <= 3 since the sum of the reciprocals of three numbers all > 3 is clearly < 1. So only x = 2 and x = 3 need consideration. For x = 2, similar reasoning shows that 2 <= y <= 4 and consideration of cases gives (2,3,6) and (2,4,4). Similarly x =3 leads to one more solution (3,3,3). > 2. Find natural numbers divisable by 30, so that they have > exactly 30 dividers. Let's see. Since 30 = 2*3*5 has three prime factors, such a number x must also have at least three prime factors, say x = p^a*q^b*r^c*y and then d(x) (the number of divisors) is (a+1)*(b+1)*(c+1)*d(y) which implies that y = 1. Say a+1 = 2, b+1 = 3, and c+1 = 5. So a = 1, b = 2, and c = 4. Also p, q, and r have to be 2, 3, and 5 in some order. Since there are 6 ways of ordering them, there are 6 distinct solutions, such as 2*3^2*5^4 or 2^4*3*5^2. > 3.Do m and n exist, writen with same digits > (as 1234 and 4132) such that m - n = 1995 No since 1995 is not divisible by 9. > 4. f(n-1) | f(n^n - 1) > f je Euler's function It is immediate from the fact that f is multiplicative and f(p^a) = p^{a-1}(p-1) that x|y implies f(x)|f(y). Since n-1|n^a-1, then f(n-1)|f(n^a)-1. The chief difficulty with this question is that it is insufficiently general. === Subject: Re: Dice and probability >((1/6)^n)^m D. === Subject: Re: Dice and probability > Do you really mean to roll the 10 dice 10 times?? If you roll the 10 dice > once then the answer is 1/(2^10). Those aren't dice. Those are coins.