mm-212 === system Adjunct Assistant Professor at the University of Montana.>Does the following system have a solution:1024 o x1 mod y2048 o x2 mod y8192 o x3 mod y16834 o x4 mod yx1+x2+x3+x4+33 =yI'm not sure what the o means. But if it means congruent to, thensurely just letting x1 = 1024, x2 = 2048, x3=8192, x4=16834, and y =x1+x2+x3+x4+33 will === puzzle.xzy +xyz---yzxI have to find the values of x y and === - it looks to me like my poster plotzed.]> Iam trying to solve a puzzle.xzy +> xyz> ---> yzx> I have to find the values of x y and z?> SSCis odd and x < 5. Enumeration of possibilities is then simple, and you can pick out the right combination. Actually, now I see that all we have to do is solve (2x+1) + (x+1) = 10 + x.If ts is homework, you should cite your sources in submitted work, RH.-- Stephen J. Herschkorn === lot. Actually ts wasn't my homework. I got ts questionfrom a looks to me like my poster plotzed.] > Iam trying to solve a puzzle.xzy +> xyz> ---> yzx> I have to find the values of x y and z?> SSC> is odd and x < 5. Enumeration of possibilities is then simple, and you > can pick out the right combination. Actually, now I see that all we > have to do is solve (2x+1) + (x+1) = 10 + x. If ts is homework, you should cite your sources === trying to solve a puzzle. xzy +> xyz> ---> yzx> I have to find the values of x y and z? > SSCcolumn you see, that 2x=y or 2x+1=y (in case that you have a carry). Henceyou know x=0,1,2,3,4. Then y can be 2x or 2x+1 (as mentioned prviously).Then z can be easily computed. Hence you have to look at 5*2=10 possiblecombinations and see, whether the summation is correct.In ts case the following way is more easy: look at the middle column:z+y=z. Now there are only two possibilities: y=0 or y=9.y=0: then from the trd column you know that x=z, and from the first columnx=0. Hence the (pretty boring) solution is 000+000=000.y=9: then from the first column you know x=4 and then from the last columnz=5, hence the === trying to solve a puzzle. xzy +> xyz> ---> yzx> I have to find the values of x y and z?Try x = 4, y =9 z = 5Then 459 + 495 = === Iam trying to solve a puzzle. xzy +> xyz> ---> yzx> I have to find the values of x y and z? > SSCLook at the second column. y could be 0, or y could be 9 with 1carried from the right column. Since the sum is yzx, let's assume y isnot 0. xz9 +x9z --- 9zx9 is odd, so the left column must be 4+4 with one carry. === ConjectureMaybe I _should_ have written:hcf(pm, p(m+M)) | hcf(pm, M) rather than hcf(pm, p(m+M))=hcf(pm, M)The rest of the argument still follows anyway... (in fact the rest of theargument becomes slightly more immediate)[though for pointing ts out??]J> Hmm.. You claim that hcf (pm, p(m+M)) = hcf (pm, M). Let's take n => 8, m = 2, M = 12, pm = 3, p(m+M) = 2. Now hcf(pm, p(m+M)) = hcf(3, 2)> = 1, and hcf(pm, M) = hcf(3, 12) = 3, and where I come from 1 != 3.> As ts statement is completely false, I don't see how the rest of the> argument is to hold together. Perhaps you meant to make some> stipulations on the choices of pm. If so, what are === of that, but there are still some problems. Here is the most important example: Let n = 22. Then p_n and p_(n+2) can be 2, as you say, but thenwe can let p_(n+3) = 5 and p_(n+5) = 3; however, you state thatp_(n+5) can't equal 3. In other words, I have switched the 3 andthe 5 in the divisor list. The example here is kind of silly, becausefor very small numbers there are very small intervals between primenumbers. However, as numbers get larger, ts is not the case at all,and I could possibly switch the list so that p_(n+11) = 5. In tscase, n + 1 = (n + 11) - 10, and so p_(n+1) = 5. Note that ts kindof switcng could happen basically anywhere on the list; for example,if p_(n+7) = 3, then so does p_(n+1), or if p_(n+57) = 5, then so doesp_(n+37). In ts way we can continue re-using the same primes overand over again. Your argument that p_m and p_(m+P) must either beequal or different and unequal to P does not take into account thefact that the relationsp between p_(m + p1) and p_(m + p2) is adifference of a composite number. You may claim, Okay, well obviously that won't happen. You maybe able to swap around a couple of the prime numbers, but you can'tmove them *that* far. Well, if so, prove it. Also, notice that ifyou try to redefine p_(n+k) as the smallest prime that divides n+k butdoesn't divide n+j for j <> Maybe I _should_ have written:> hcf(pm, p(m+M)) | hcf(pm, M) rather than hcf(pm, p(m+M))=hcf(pm, M)> The rest of the argument still follows anyway... (in fact the rest of the> argument becomes === of interest, snipped] -- To make invalid valid, reverse the === R^2 -> Zlet Z : R*{0} U {0}*R define quotient topology of Z such that function g is quotient map.>(nt : At ts time, any two points don't separated to open set.)>I read the problem statement carefully; unfortunately, I don't understandwhat the problem is about. You could try restating the problem withmore === youSigh. Yes, the following, if the reasoning as (sic) actually>correct, can be easily formalized in ZF.... note that for example the in operator below>is not going to correspond to the in in ZF, it's>going to be just some predicate, with axioms>involving it.> C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)>>Classification>> C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]>> (Equi-membered classes are identical iff these are sets.)Put your money where your mouth is. Let's see your formalization. It's clear to anyone with half a clue that your reasoning, if> correct, can easily be formalized in ZFC. (Um: I should have> said with some predicate in place of Ôin' and also some> predicate in place of Ô=', instead of just pointing out that> your Ôin' would not correspond to the Ôin' in ZFC.) Ts> is because there's notng non-standard about your> _reasoning_, all that's non-standard is your _axioms_> regarding Ôin' and Ô='. In case you don't have half a clue, the part of the> formalization corresponding to those two axioms> might be C3 EyAx[ni(x, y) <-> Et(ni(x,t)) & A] (with y not free in A)> Classification> > C4 AyAx[Az(ni(z,y) <-> ni(z,x)) -> {(S(y) & S(x)) <-> I(x,y)}]> (Equi-membered classes are identical iff these are sets.)-- Anybody with a fraction of a brain knows that ZFC has only twoPRIMITIVE predicates: in and =. (Some claim that ZFChas only one primitive predicate: in.) But you are a bot witha CPU instead of a brain, and no capacity for actions thatare not part of your program. So even though you may be dimlyaware that only in and = are primitive predicatesof ZFC, you are unaware that if you use (e.g) ni and I in yourformalization of (C3,C4) in ZFC, you must provide DEFINITIONS wchallow you to transform every wff of your formalization in wchni or I occur, into wffs into wch only in or = occur. C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)Classification C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}] (Equi-membered classes are identical iff these are sets.)(A) Anybody on sci.math/sci.logic with a fraction of a brain andan ounce of integrity will take you to task for the piece of crapthat you have posted and called a formalization of C3,C4 in ZFC.(B) Nobody on sci.math/sci.logic will take you to task for thepiece of crap that you have posted and called a formalizationof C3,C4 in ZFC. Ullrich's HomeworkExplain why (A) and (B) are consistent. You may do tsinformally--that is, without trying to formalize (A) and(B) in any of the systems that you do not know.--A performance system is designed to work in a defined task domain,accepting particular goals and seeking to reach them by some kind ofghly selective search. The system must be told what goal is to bereached and must be given a description of the structure andcharacteristics of the task domain in wch it is to operate: itsproblem space. [...] In contrast, a learning system is capable ofacquiring a problem space, in whole or part, by interacting with theexternal environment and without being instructed about it directly.--Herbert Simon === JSH: Humanity needed youhttp://www.giganews.com/info/dmca.html>[...](A) Anybody on sci.math/sci.logic with a fraction of a brain and>an ounce of integrity will take you to task for the piece of crap>that you have posted and called a formalization of C3,C4 in ZFC.(B) Nobody on sci.math/sci.logic will take you to task for the>piece of crap that you have posted and called a formalization>of C3,C4 in ZFC. Ullrich's HomeworkExplain why (A) and (B) are consistent.Well duh, of course they're consistent. You seem to tnkthat they're both true. Probably they are - glad you pointedts out, now we get to see whether anyone on sc.math/sci.logichas a fraction of a brain and an ounce of integrity.I've wondered about ts for years.> You may do ts>informally--that is, without trying to formalize (A) and>(B) in any of the systems that you do not know.--A performance system is designed to work in a defined task domain,>accepting particular goals and seeking to reach them by some kind of>ghly selective search. The system must be told what goal is to be>reached and must be given a description of the structure and>characteristics of the task domain in wch it is to operate: its>problem space. [...] In contrast, a learning system is capable of>acquiring a problem space, in whole or part, by interacting with the>external environment and without being instructed about it directly.>--Herbert Simon> === JSH: Humanity needed you>>>Sigh. Yes, the following, if the reasoning as (sic) actually>correct, can be easily formalized in ZF.>>... note that for example the in operator below>is not going to correspond to the in in ZF, it's>going to be just some predicate, with axioms>involving it.>>C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)>>Classification>>C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]>> (Equi-membered classes are identical iff these are sets.)>>Put your money where your mouth is. Let's see your formalization.>>It's clear to anyone with half a clue that your reasoning, if>>correct, can easily be formalized in ZFC. (Um: I should have>>said with some predicate in place of Ôin' and also some>>predicate in place of Ô=', instead of just pointing out that>>your Ôin' would not correspond to the Ôin' in ZFC.) Ts>>is because there's notng non-standard about your>>_reasoning_, all that's non-standard is your _axioms_>>regarding Ôin' and Ô='.>>In case you don't have half a clue, the part of the>>formalization corresponding to those two axioms>>might be>>C3 EyAx[ni(x, y) <-> Et(ni(x,t)) & A] (with y not free in A)>>Classification>>C4 AyAx[Az(ni(z,y) <-> ni(z,x)) -> {(S(y) & S(x)) <-> I(x,y)}]>> (Equi-membered classes are identical iff these are sets.)>-->>>> > Anybody with a fraction of a brain knows that ZFC has only two> PRIMITIVE predicates: in and =. I have a === needed you >>>>Sigh. Yes, the following, if the reasoning as (sic) actually>correct, can be easily formalized in ZF.>>... note that for example the in operator below>is not going to correspond to the in in ZF, it's>going to be just some predicate, with axioms>involving it.>>C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)>>Classification>>C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]>> (Equi-membered classes are identical iff these are sets.)>>Put your money where your mouth is. Let's see your formalization.>>It's clear to anyone with half a clue that your reasoning, if>>correct, can easily be formalized in ZFC. (Um: I should have>>said with some predicate in place of Ôin' and also some>>predicate in place of Ô=', instead of just pointing out that>>your Ôin' would not correspond to the Ôin' in ZFC.) Ts>>is because there's notng non-standard about your>>_reasoning_, all that's non-standard is your _axioms_>>regarding Ôin' and Ô='.>>In case you don't have half a clue, the part of the>>formalization corresponding to those two axioms>>might be>>C3 EyAx[ni(x, y) <-> Et(ni(x,t)) & A] (with y not free in A)>>Classification>>C4 AyAx[Az(ni(z,y) <-> ni(z,x)) -> {(S(y) & S(x)) <-> I(x,y)}]>> (Equi-membered classes are identical iff these are sets.)>-->>>> > Anybody with a fraction of a brain knows that ZFC has only two> PRIMITIVE predicates: in and =. I have a fraction, and I don't know. f you don't know, it's because there is no reason you should know. Ts isnot the case for === ordered pair (x,y) would be defined as>{{x}{x,y}}. I understand that a set has no order but therefore, wouldn't>that be the same as {{x,y} {x}}.It would.>If anyone has any good links to ts just for some background information,>that would be realy helpfuly.As long as one has a reasonable definition for anobject defined by two arguments, and identifying the twoarguments, one can use the term ordered pair, and itmakes no difference wch one is used. Ts is not thefirst, but it is the shortest I have seen and works.The Quine definition would even give the ordered pair oftwo classes, but would not work === square, and you choose two points on it,uniformly and independently. What is the probability that the distance(euclidean distance, or more generally any distance on === Re: Probability question> Suppose you have an unit square, and you choose two points on it,> uniformly and independently. What is the probability that the distance> (euclidean distance, or more generally any distance on R^2) betweenthem> is less than K fixed?The simple part is considering the circle around the first point withradius K. The probability of placing a second point in that circle is asimple ratio of the area of the circle to the area of the square. Themore difficult part is assessing the probability of placing the firstpoint at a place where a full circle cannot be drawn and then mergingthese === Re: polysigned numbers Robin. I just thought I'd offer some general help here.Ts polysigned math is very simple. The operations of summation andmultiplication are well defined. Here are the general concepts.The signs -, +, *, and # represent signs 1, 2, 3, and 4 respectively.There is a clear mnemonic here; minus has one line, plus two lines,etc.For a given sign math (say three-signed) sign multiples move the valuearound the number branch the number of branches of the sign. So, forexample in 3-signed: (-1)(+2) = *2. (+2)(*3) = +6. (-3)(-4) = +12. (*4)(*5) = *20.Unfortunately the identity operator moves around depending on how manysigns you are working with.All of the standard behaviors of real arithmetic work with polysignednumbers.That is, the commutative, associative, and distributive properties allwork just fine as if the math were just two-signed (the reals). So,for example in 4-signed: (-1+2)(*3) = (-1)(*3)#(+2)(*3) = #3-6just as in two-signed: (-1+2)(-3) = (-1)(+3)+(+2)(-3) = +3-6 = -3.That brings you to the question of why #3-6 did not resolve to asimple value.The answer is that the sum law for two signed values to go to zero is: -a+a = 0.Therefor all real valued sums can be reduced to a single value. Forthree-signed values: -a+a*a = 0.So a three-signed number like -1+2*3 can be reduced: -1+2*3 = -1+1*1 +1*2 = +1*2just as in the two-signed reals: -2+3 = -2+2 +1 = +1.Ts zero-sum rule causes three-signed numbers to be two-dimensional.Four-signed numbers are three-dimensional.It is shown on the three-signed thread that 3-signed math exactlyequivalent to complex math for product and sum.What is the meaning of the four-signed numbers with their simplearithmetical product? Does it have an equivalent like complex math?Ts is a puzzle I hope you will work on. Since the three-signednumbers match complex math exactly then there may be some value tothese four-signed numbers for three-dimensional space, the space weall seem to live in.>>-1(a,b,c,d) = (d,a,b,c)>>>+1(a,b,c,d) = (c,d,a,b)>>>*1(a,b,c,d) = (b,c,d,a)>>>#1(a,b,c,d) = (a,b,c,d)>>>>Bonkers: so #1 is the multiplicative identity not +1 :-(>>Correct.>>>>If my assumption is correct these 4-signed numbers form a ring>>isomorpc to R x C: *1 will correspond to (-1, i) etc.>>>>No.>>>>Yes: Still isomorpc to R x C.>>>>>You'll have to apply trig functions to get the exact values. You>>>are going from norm 1 to norm sqrt(2). The norm is preserved under the>>>isomorpsm.>>>>Sorry, that makes no sense?>>You're right, I had forgotten how multiplication in RxC works.>>So #1 = (1,1)>What would -1,+1,*1 be?>> (-1,-i), (-1,-1), (-1,i).(-1)(-1) = +1 should mean +1 is (1,-1). Yeah sorry. Other than that, it makes sense. Interesting. I'm not === refresher>Bingo! Got it!In several dimensions, analogues of the fundamental theorem of>calculus are of the form:Int[DifOp(f),{inside}] = Int[f,{boundary}],Where DifOp is differential operator.Such tngs had occured to me before. Assuming there is also an>analogue to the product rule in the form DifOp(fg) = fDifOp(g) +>gDifOp(f), you are all set to integrate by parts., adam. It's only 9 a.m. Saturday morning, and you've already>made my day! Now the rest of the day may be spent in sloth. ;-)You're welcome. I'm glad I could help.>BTW: I've adopted your notation. Is that Mathematica or some such, or>sometng you invented on the ßy for ts post?>Yeah, the notation is a bastardization of Mathematica's language. === - k_2/x)^.5>>Where the k's are positive constants.>>Is it proper to claim that Y is proportional to x^-.5? In the>presentation, the symbol alpha was used in place of the words is>proportional to and I was wondering if that was a rigorous use of>that symbol.>>You've already been told the answer is no. However, coming from a>>physics background I'll add that under some circumstances, one might>>say sometng like Y varies as x^0.5 for small x, if that was the>>regime of interest. The physicist would write sometng like>> Y ~ x^-0.5 for small x Coming from a physics background myself, I must disagree. A good>physicist would not write sometng like y ~ x^-.05 _for small x_>because it doesn't matter if x itself is small...Of course it does.If I have y = cx/(b + x) then y ~ x for x small, but y is about b/cfor x large. It matters very much if x is small.Approximations are for specific ranges of the independent variable.How can you say that an approximation does not depend on the variablebeing large or small? Tnk about simple pendulums (small angleapproximation), far-field electromagnetic fields (distance largecompared to wavelength), bulk approximations of matter (scale largecompared to molecular size), etc. Physics is FULL of approximationsthat depend very much on the size of the dependent variable. === Google for G.9adel number. He represented statements as natural> numbers. The theorems he proved about statements were also theorems> about the integers representing them.However, Goedel numbering schemes are somewhat arbitrary. They are merely artifacts for self reference, hence any theorems proven about a particular set of numbers used as Goedel encoding are unlikely to be of any fundemental importance in number === comes first?> at 01:53 PM, mattias_wikst71@hotmail.com (Mattias Wikstr?m) said:You have a point, two points actually, but I did not say that proof>is a mathematical concept, talk about are Mathematical concepts.Mathematicians make frequent use of proofs, but the study of proofsbelongs to logic rather than mathematics, so in one sence they arepart of mathematics, and in another sence they are not.Any that follow from the definition of natural number I give below. You don't give a definition.Here is a definition wch I hope will satisfy you: The naturalnumbers are the tngs obtainable from the two operations (the firstof wch is actually a constant) 0:N and s:N->N.Ts is an example of an inductive definition.> And you certainly haven't answered> my question as to what properties you believe they have that do not> follow from the Peano postulates.I have (see above).In some cases it is easy to tell; Then tell.If it can be proven in the axiom system that the Peano Postulates areinconsistent, then the axiom system is unsound.The Peano Postulates are an example of a sound axiom system.In s paper G.9adel showed how, for any sound formal system for>natural numbers (that is, one in wch all provable statements are>true), The word true in that context is a technical term quite different> from what is generally understood by true.Technical definitions of truth had not been invented by the time G.9adelstatements about natural numbers is different from true applied toother statements.>G.9adel showed how to find true statements about>natural numbers not provable from the (first-order) Peano>Postulates. He showed to to prove additional theorems by adjoining to the Peano> Postulates an additional axiom, essentially asserting that the Peano> Postulates were consistent.Yes, and he showed how more and more axioms not provable from previousaxioms can be added ad infinitum.I hope we agree that the Peano Postulates are consistent. Here youhave an example of a statement about natural numbers wch we agree istrue, but wch cannot be proven from the Peano Postulates.>The natural>numbers are the numbers>0, 1, 2, 3, 4, ....>When spelled out ts becomes>0, s(0), s(s(0)), s(s(s(0))), s(s(s(s(0)))), ... . That's not a definition.Maybe not.> That doesn't even say that they obey the> Peano Postulates.I will show how the Peano Postulates can be proven from the followingdefinition:The natural numbers are the tngs obtainable from the two operations0:N and s:N->N.Ts definition is complete if we understand operation the rightway. Namely, there should be no sence of equality for the tngs weobtain by theese operations other than that two expressions built upfrom signs denoting theese operations denote the same tng iff theyare equal. Ts is vaguely formulated, but I hope it is clear what Imean. For example, since s(s(0)) is not the same expression ass(0), s(s(0))/=s(0), and since s(s(s(0))) is the same expressionas s(s(s(0))), s(s(s(0)))=s(s(s(0))).Alternatively, stick to ordinary operations and add appropriateaxioms for equality.Here is how the Peano Postulates can be proven:1) 0 is a natural number.Follows directly from definition.2) For each natural number x, s(x) is a natural number.Follows directly from definition.3) There is no natural number x such that s(x)=0.s(x) has s as its outermost operator, whereas 0 has 0 as itsoutermost operator. If two objects have different outermost operators,then they are different.4) x/=y => s(x)/=s(y)s(x) and s(y) have the same outermost operator (namely s), but sincethe tngs it is applied to are not the same (x/=y), s(x) and s(y)cannot be the same.5) {P(0) ^ Ax[P(x) => P(s(x))]} => AyP(y)Suppose we know that P(0) and that Ax[P(x) => P(s(x))]. We wish toprove P(y) for arbitrary values of y. Since y can be obtained from 0and s there is a list [a_0, a_1, a_2, ..., a_n=y], where each a_i canbe immediately obtained from previous elements of the list by means of0 and s. For example, a list for the number s(s(s(0))) is [0, s(0),s(s(0)), 0, 0, 0, s(0), s(s(0)), s(s(s(0))), s(s(s(s(0)))),s(s(s(0)))]. Either P holds for all elements of the list, in wchcase P(y) holds, or there is a first element of the list, say a_m,such that P(a_m) does not hold. Either a_m=0 or a_m=s(a_l), where l I'm trying to show the 1 - 1 correspondence between R and the power set of N> without resorting to the binary decimal argument. I know I can use the> Schroeder-Bernstein Theorem. There is no problem finding an injection from> the power set of N into R. I'm having difficulty finding an injection from> R into the power set of N. Help! - L For each real number r, let U(r) be the unique set of rationals greater than r (Dedekind cut upper sets), so U:R -> P(Q) is an injection (but not a bijection).Let F:Q -> N be any bijection from the rationals to the integers.Such bijections arer known to exist since Q and N have the same cardinality.Then F induces a set function on the power sets G: P(Q) -> P(N), defined, for any S in P(Q) by G(S) = {f(s) : s in S} in P(N) , and G is also a bijection.Then the composite function GU:R -> P(N), where (GU)(r) = G(U(r)), is an injection followed by a bijection, so is itself an === Natural No's>>>I'm trying to show the 1 - 1 correspondence between R and the power set of N>>without resorting to the binary decimal argument. I know I can use the>>Schroeder-Bernstein Theorem. There is no problem finding an injection from>>the power set of N into R. I'm having difficulty finding an injection from>>R into the power set of N.>>Help!>> ->>L>> For each real number r, let U(r) be the unique set of rationals >greater than r (Dedekind cut upper sets), so U:R -> P(Q) is an >injection (but not a bijection).Let F:Q -> N be any bijection from the rationals to the integers.>Such bijections arer known to exist since Q and N have the same >cardinality.>Or let F be a specific injection: F(p/q) = 2^(p+) 3^(p-) 5^q, where p and q have no common factor, q > 0, and p+ and p- are respectively the positive and negative parts of p.Then F induces a set function on the power sets G: P(Q) -> P(N), >defined, for any S in P(Q) by G(S) = {f(s) : s in S} in P(N) , and >G is also a bijection.>Then the composite function GU:R -> P(N), where (GU)(r) = G(U(r)), >is an injection followed by a bijection, so is itself an injection.-- Stephen J. Herschkorn === can post questions here if we need help? Am I not allowed topost to sci.math? Where is a more appropriate newsgroup that I can post nonworthy questions? in advance...With the function y = 8/x, gives 2^4 = 4^2, I'm wondering if there areother> functions wch would have more than ts. in advanceIf I had a dollar for every time those old chestnut has been posted to> === I thought we can post questions here if we need help? Am I not allowed to> post to sci.math? Where is a more appropriate newsgroup that I can post non> worthy questions? in advance...One tng you can do before posting is to see if ts === Re: x^y=y^x> I thought we can post questions here if we need help? Am I not allowed to> post to sci.math? Where is a more appropriate newsgroup that I can post non> worthy questions? in advance...I will cut you a break.let x = s^(1/1-s) and y = === <8Ojkb.1615$tz.1113@newssvr29.news.prodigy.com> post to sci.math? Where is a more appropriate newsgroup that I can post non> worthy questions? in advance...I will cut you a break.let x = s^(1/1-s) and y = s^(s/1-s). Try it outA more interesting parametrization is x = (1 + 1/p)^p and y = (1 + 1/p)^(p+1)where p>0 covers all pairs 0 < x < y(wch turns out to imply 1 < x < e < y).Find out what happens when p is an integer,and when p tends to infinity.(Ts classic was known to === Function Theory by Axler et al. and came across ts:Let K be a compact set. Fix x in K. Harnack's inequality shows there is aconstant C, 1 < C < oo, s.t.u_m(y)-u_k(y) <= C[u_m(x)-u_k(x)]for all y in K, whenver m > k. Ts implies {u_m} is uniformly Cauchy onK... (p. 50).In ts example, we also know {u_m} is harmonic and pointwise increasing.Maybe I'm missing sometng obvious, but how do you get Cauchyness from === sequencehttp://www.giganews.com/info/dmca.html>I'm reading Harmonic Function Theory by Axler et al. and came across ts:Let K be a compact set. Fix x in K. Harnack's inequality shows there is a>constant C, 1 < C < oo, s.t.u_m(y)-u_k(y) <= C[u_m(x)-u_k(x)]for all y in K, whenver m > k. Ts implies {u_m} is uniformly Cauchy on>K... (p. 50).In ts example, we also know {u_m} is harmonic and pointwise increasing.Maybe I'm missing sometng obvious, but how do you get Cauchyness from that>statement?Well, knowing that Axler et al are no dummies, I deduce you must haveleft sometng out. It must be that in the section you quote they've already made the assumption that u_n(x) tends to a finite limit,right?If so then (u_n(x)) is === sequenceWe're also given u_n is finite everwhereI'm reading Harmonic Function Theory by Axler et al. and came acrossts:Let K be a compact set. Fix x in K. Harnack's inequality shows there isa>constant C, 1 < C < oo, s.t.u_m(y)-u_k(y) <= C[u_m(x)-u_k(x)]for all y in K, whenver m > k. Ts implies {u_m} is uniformly Cauchy on>K... (p. 50).In ts example, we also know {u_m} is harmonic and pointwise increasing.Maybe I'm missing sometng obvious, but how do you get Cauchyness fromthat>statement?Well, knowing that Axler et al are no dummies, I deduce you must have> left sometng out. It must be that in the section you quote they've> already made the assumption that u_n(x) tends to a finite limit,> right?If so then (u_n(x)) is Cauchy, so for any epsilon > 0 ... === finite everwhereNo, you're given sup u_n is finite === sequencehttp://www.giganews.com/info/dmca.html> We're also given u_n is finite everwhereNo, you're given sup u_n is finite === Re: cauchy sequence> Ah, that was one of my guesses. , al.What is ts guess business? You mean you don't own the book? -- === sequencehttp://www.giganews.com/info/dmca.html> Ah, that was one of my guesses. , al.What is ts guess business? You mean you don't own the book? -- alOh dear. Um, my dog ate it, yeah, === does ts get you Cauchyness?We're also given u_n is finite everwhereNo, you're given sup u_n is finite === how does ts get you Cauchyness?An increasing sequence that is bounded above converges. So u_n(x) converges, hence is Cauchy === sequencehttp://www.giganews.com/info/dmca.html>We're also given u_n is finite everwhereThat doesn't answer the question I asked.If you want to clarify ts you could post the _entire_ textof the theorem and proof in question. Or you could waituntil Wade notices ts thread - I believe he's somewhatfamiliar with that book...>>I'm reading Harmonic Function Theory by Axler et al. and came across>ts:>>Let K be a compact set. Fix x in K. Harnack's inequality shows there is>a>>constant C, 1 < C < oo, s.t.>>u_m(y)-u_k(y) <= C[u_m(x)-u_k(x)]>>for all y in K, whenver m > k. Ts implies {u_m} is uniformly Cauchy on>>K... (p. 50).>>In ts example, we also know {u_m} is harmonic and pointwise increasing.>>Maybe I'm missing sometng obvious, but how do you get Cauchyness from>that>>statement?>> Well, knowing that Axler et al are no dummies, I deduce you must have>> left sometng out. It must be that in the section you quote they've>> already made the assumption that u_n(x) tends to a finite limit,>> right?>> If so then (u_n(x)) is Cauchy, so for any epsilon > 0 ... === Function Theory by Axler et al. and came acrossts:Let K be a compact set. Fix x in K. Harnack's inequality shows there is a> constant C, 1 < C < oo, s.t.u_m(y)-u_k(y) <= C[u_m(x)-u_k(x)]for all y in K, whenver m > k. Ts implies {u_m} is uniformly Cauchy on> K... (p. 50).In ts example, we also know {u_m} is harmonic and pointwise increasing.Maybe I'm missing sometng obvious, but how do you get Cauchyness fromthat> statement?It would be awkward in ascii, but it's spelled out in pretty good detail inAhlfors _Complex === Math: Image processing)Are there any guidelines on how to design a good filter to enhance image?I am facing the following problem that I need your help:We have in our experiments some special images need to be processed. My taskis to look at these images and see how to improve/enhance them. In fact we/Ihave no idea on how much enhancement we can get.So what I've done in the past month is to play with different kind offilters to try on the images. We use PSNR as judgement. It turns out I founda Gaussian 3x3 filter has particularly gh enhancement to the images,comparing with other filters provided by Matlab, such as laplacian,averaging...So my focus was sfted to Gaussian-like filters: after playing with manyGaussian-like filters, it turns out that there are other non-Gaussianfilters(but numerically similar to Gaussian filter) wch performed betterthan Gaussian filter. So I designed a stupid search program to search forit. After one week running, it gives a small 3x3 filter wch can be said asbest... with best PSNR...But ts procedure seems very ad-hoc. And the best filter on one image getschanged on another image. In reality there is no source image available toallow computing PSNR and search one week on a super-computer to get anotherfilter.Are there any guidelines on how to design good filters that applicable tomost(if not all) images? Discussion should be in two cases: a) with sourceimage and can get PSNR to compare results; b) with no source image and onlyhave the reconstructed images, so no PSNR can be computed, how to design agood filter fit into ts case?I believe there are some great treatment, the problem is just I am a laymanand not even know where to locate === Re: (Applied Math: Image processing)Are there any guidelines on how to design a good filter to enhance image?> But ts procedure seems very ad-hoc. And the best filter on one image > gets> changed on another image. In reality there is no source image available > to> allow computing PSNR and search one week on a super-computer to get > another> filter. Are there any guidelines on how to design good filters that applicable to> most(if not all) images? Discussion should be in two cases: a) with > source> image and can get PSNR to compare results; b) with no source image and > only> have the reconstructed images, so no PSNR can be computed, how to design > a> good filter fit into ts case?You might try asking your question on comp.dsp.I found in my work in signal detection that those who knew muchmore than I would often choose an algorithm based on a model ofthe system under consideration. The models I was interested intypically included models of signal, clutter, and noise.-- Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/ Bukharin.html To solve Linear Programs: .../LPSolver.htmlr c A game: .../Keynes.html v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question fit perhaps to be discussed by n e . slaves in the hearing of their masters, but ghly @ r c m unbecoming to reasonable and free men in === argument, core error proven... > > Then give a *single* function in ALL of mathematics wch behaves as > > you wish. > > You're making up some wacky mathematics. Oh, come on. Do you know the Moebius function? > mu(n) = 1 if n = 1 > 0 if n is divisible by a square > (-1)^r if n = the product of r distinct primes. > Wacky enough? But indeed, not the function we seek, see below. > ... > Remember I have > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) > Variables: m, f, x, u E Ring of Algebraic integers > and the factorization > P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf) > Variables: a_1, a_2, a_3, roots of cubic defined as follows. > > Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m) The poster wishes for a variable function wch has the property of > being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1 > has w_1(m) as a factor for all integer m.Pray note that w_1(m) is also a function of f. > Then a_1(m) x + uf must have w_1(m) as a factor, so dividing through > gives a_1(m)/w_1(m) + uf/w_1(m) where uf/w_1(m) can't be an algebraic integer for all integer m.Please show us *why not*! You are assuming properties of a yet undefinedfunction. > It might help for me to put in actual numbers for u and f, wch I can > do as the variables are independent of each other, so let u=2, f=7, > then its a_1(m)/w_1(m) + 14/w_1(m) and clearly, if w_1(m) varies with m, then 14/w_1(m) is not an > algebraic integer for all integer m.Why not? > For those who STILL need help, consider that if you had 14/w_1(m) = r(m) introducing r(m) for the result of the division, then w_1(m) r(m) = 14 so w_1(m) r(m) - 14 = 0 wch would force zeroes for m.Yes, so what, what is the problem with that? > That is, you can't have algebraic integer functions, that is functions > that give algebraic integer results, and not have only certain values > of m that would work. That is, you can have sometng like 2m+ 7 = 21, that works for a > particular value of m, but you can't have functions in algebraic > integers that will multiply to give 14 for all integer m.Such functions have been given sufficiently I tnk. > To get such functions, you have to go outside the ring into a field.Eh? > That refutes the position of Dik T. Winter, and note that as I've > pointed out that poster clearly either has limited mathematical > ability, or he's been lying now for some time.I tnk you do not have any === argument, core error proven> ...> > > Then give a *single* function in ALL of mathematics wch behaves as> > > you wish.> > > > You're making up some wacky mathematics.> > Oh, come on. Do you know the Moebius function?> > mu(n) = 1 if n = 1> > 0 if n is divisible by a square> > (-1)^r if n = the product of r distinct primes.> > Wacky enough? But indeed, not the function we seek, see below.> ...> > Remember I have> > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f)> > Variables: m, f, x, u E Ring of Algebraic integers> > and the factorization> > P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf)> > Variables: a_1, a_2, a_3, roots of cubic defined as follows.> > Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m)> > The poster wishes for a variable function wch has the property of> > being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1> > has w_1(m) as a factor for all integer m. Pray note that w_1(m) is also a function of f.It turns out that w_1(m) = f.> > Then a_1(m) x + uf must have w_1(m) as a factor, so dividing through> > gives> > a_1(m)/w_1(m) + uf/w_1(m)> > where uf/w_1(m) can't be an algebraic integer for all integer m. Please show us *why not*! You are assuming properties of a yet undefined> function.You're stupid Dik Winter. > > It might help for me to put in actual numbers for u and f, wch I can> > do as the variables are independent of each other, so let u=2, f=7,> > then its> > a_1(m)/w_1(m) + 14/w_1(m)> > and clearly, if w_1(m) varies with m, then 14/w_1(m) is not an> > algebraic integer for all integer m. Why not?You are an ignoramus Dik Winter. You're just a stupid, dumbass. > > For those who STILL need help, consider that if you had > > 14/w_1(m) = r(m)> > introducing r(m) for the result of the division, then> > w_1(m) r(m) = 14> > so> > w_1(m) r(m) - 14 = 0> > wch would force zeroes for m. Yes, so what, what is the problem with that?You dumbass, don't you realize that if m has zeroes then it can't varyover all over algebraic integers!!!Its values would be constrained you fool!!!You are just a ing dumbass Dik Winter. You are just === ...> > > Then give a *single* function in ALL of mathematics wch behaves as> > > you wish.> > > > You're making up some wacky mathematics.> > Oh, come on. Do you know the Moebius function?> > mu(n) = 1 if n = 1> > 0 if n is divisible by a square> > (-1)^r if n = the product of r distinct primes.> > Wacky enough? But indeed, not the function we seek, see below.> ...> > Remember I have> > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f)> > Variables: m, f, x, u E Ring of Algebraic integers> > and the factorization> > P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf)> > Variables: a_1, a_2, a_3, roots of cubic defined as follows.> > Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m)> > The poster wishes for a variable function wch has the property of> > being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1> > has w_1(m) as a factor for all integer m. Pray note that w_1(m) is also a function of f. It turns out that w_1(m) = f. > Then a_1(m) x + uf must have w_1(m) as a factor, so dividing through> > gives> > a_1(m)/w_1(m) + uf/w_1(m)> > where uf/w_1(m) can't be an algebraic integer for all integer m. Please show us *why not*! You are assuming properties of a yet undefined> function. You're stupid Dik Winter.Ok, yeah, I got frustrated. These people are *lying* so badly, andthe lie is needed to de ts esoteric problem, but somehow, theykeep getting away with their lying!!!Maybe using complex numbers will help.So the poster needsw_1(m) w_2(m) w_3(m) = f^2where all the w's give values in the ring of algebraic integers foralgebraic integers m and f, wle varying with m.But do sometng like e^{x(m)} = w_1(m), with all the function, andyou have a sum of exponents, likex(m) + y(m) + z(m) = ln f^2 but, if any of the terms of *any* of the functions have m, then theremust exist other terms that will subtract them off.You know like if you have m^2 + 1 with x(m), then you need -m^2,somewhere to handle it. Hmmm...you'd need -1 as well to get rid ofthat 1, as the final result is ln f^2.Don't any of you realize that part of the game for people like Wintermay be in getting so many to nod along with their b.s. assertions?It's === error proven> ...> > > Then give a *single* function in ALL of mathematics wch behaves as> > > you wish.> > > > You're making up some wacky mathematics.> > Oh, come on. Do you know the Moebius function?> > mu(n) = 1 if n = 1> > 0 if n is divisible by a square> > (-1)^r if n = the product of r distinct primes.> > Wacky enough? But indeed, not the function we seek, see below.> ...> > Remember I have> > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f)> > Variables: m, f, x, u E Ring of Algebraic integers> > and the factorization> > P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf)> > Variables: a_1, a_2, a_3, roots of cubic defined as follows.> > Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m)> > The poster wishes for a variable function wch has the property of> > being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1> > has w_1(m) as a factor for all integer m. Pray note that w_1(m) is also a function of f. It turns out that w_1(m) = f. > Then a_1(m) x + uf must have w_1(m) as a factor, so dividing through> > gives> > a_1(m)/w_1(m) + uf/w_1(m)> > where uf/w_1(m) can't be an algebraic integer for all integer m. Please show us *why not*! You are assuming properties of a yet undefined> function. You're stupid Dik Winter. Ok, yeah, I got frustrated. These people are *lying* so badly, and> the lie is needed to de ts esoteric problem, but somehow, they> keep getting away with their lying!!! Maybe using complex numbers will help. So the poster needs w_1(m) w_2(m) w_3(m) = f^2 where all the w's give values in the ring of algebraic integers for> algebraic integers m and f, wle varying with m. But do sometng like e^{x(m)} = w_1(m), with all the function, and> you have a sum of exponents, like x(m) + y(m) + z(m) = ln f^2 but, if any of the terms of *any* of the functions have m, then there> must exist other terms that will subtract them off.And yeah, it's possible, wch I guess I should have realized but someposter on sci.math happily pointed it out.The simple way is like using some polynomial like x^2 + mx - f^2, youhave two funtions of m that multiply to give f^2 over all === argument, core error proven> ...> > > Then give a *single* function in ALL of mathematics wch behaves as> > > you wish.> > > > You're making up some wacky mathematics.> > Oh, come on. Do you know the Moebius function?> > mu(n) = 1 if n = 1> > 0 if n is divisible by a square> > (-1)^r if n = the product of r distinct primes.> > Wacky enough? But indeed, not the function we seek, see below.> ...> > Remember I have> > P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f)> > Variables: m, f, x, u E Ring of Algebraic integers> > and the factorization> > P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf)> > Variables: a_1, a_2, a_3, roots of cubic defined as follows.> > Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m)> > The poster wishes for a variable function wch has the property of> > being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1> > has w_1(m) as a factor for all integer m. Pray note that w_1(m) is also a function of f. It turns out that w_1(m) = f. > Then a_1(m) x + uf must have w_1(m) as a factor, so dividing through> > gives> > a_1(m)/w_1(m) + uf/w_1(m)> > where uf/w_1(m) can't be an algebraic integer for all integer m. Please show us *why not*! You are assuming properties of a yet undefined> function. You're stupid Dik Winter. Ok, yeah, I got frustrated. These people are *lying* so badly, and> the lie is needed to de ts esoteric problem, but somehow, they> keep getting away with their lying!!! Maybe using complex numbers will help. So the poster needs w_1(m) w_2(m) w_3(m) = f^2 where all the w's give values in the ring of algebraic integers for> algebraic integers m and f, wle varying with m. But do sometng like e^{x(m)} = w_1(m), with all the function, and> you have a sum of exponents, like x(m) + y(m) + z(m) = ln f^2 but, if any of the terms of *any* of the functions have m, then there> must exist other terms that will subtract them off. And yeah, it's possible, wch I guess I should have realized but some> poster on sci.math happily pointed it out.And yet you PROVED it's not possible! A very simple proofthat only a liar or an idiot would disbelieve. The simple way is like using some polynomial like x^2 + mx - f^2, you> have two funtions of m that multiply to give f^2 over all algebraic> integers from its roots.> Doesn't ts make you want to question your belief thatyour proofs are infallible? Doesn't it make you want tolook at your own proof on ts very point to see where,in a chain of steps you were absolutely sure of, youmade a mistake? Wouldn't you like to === argument, core error proven> You're stupid Dik Winter. Ok, yeah, I got frustrated. These people are *lying* so badly, and> the lie is needed to de ts esoteric problem, but somehow, they> keep getting away with their lying!!! And yeah, it's possible, wch I guess I should have realized but some> poster on sci.math happily pointed it out.I'm sure it was just an oversight that you forgot to mentionthe some poster was Dik Winter, who you have called stupid,an idiot, and who knows what else in the same thread. Infact you called m arrogant, cldish, and a dummy 10minutes before telling everybody else === core error proven... > You're stupid Dik Winter. Ok, yeah, I got frustrated. These people are *lying* so badly, and > the lie is needed to de ts esoteric problem, but somehow, they > keep getting away with their lying!!! Maybe using complex numbers will help. So the poster needs w_1(m) w_2(m) w_3(m) = f^2 where all the w's give values in the ring of algebraic integers for > algebraic integers m and f, wle varying with m. But do sometng like e^{x(m)} = w_1(m), with all the function, and > you have a sum of exponents, like x(m) + y(m) + z(m) = ln f^2 but, if any of the terms of *any* of the functions have m, then there > must exist other terms that will subtract them off. You know like if you have m^2 + 1 with x(m), then you need -m^2, > somewhere to handle it. Hmmm...you'd need -1 as well to get rid of > that 1, as the final result is ln f^2.-1-m^2+2.pi.i will also work. You assume that from: exp(a) === argument, core error proven>As I have written already multiple times, it is trivially monic in other>rings, the cubic (x - b1)(x - b2)(x - b3) serves just fine. But you>want to go to a ring where your divisibility claims work. It includes>the ring of algebraic integers. But wch of the two numbers> (-sqrt(7) + sqrt(15))/2 and (-sqrt(7) - sqrt(15))>is a unit in that ring? And why precisely that number?> ...> (B) In the Object ring, ts marvellous construction that fixes the> ring of algebraic integers by allowing numbers that should be in,> there are polynomias with integer coefficients, irreducible over Q,> wch have ->SOME<- roots in the ring but not all of them. And in case (B), how do we know wch one is there, as you ask?I tnk (B) is indeed precisely the case. James wants all those numbers>in s ring such that the cubics in the b's are monic with numbers in>s rings for all possible polynomials P(m). So when we look at the>polynomial in Ôa' for m=1 and f=2 we get (in the algebraic integers)>three roots that are obviously not coprime to f. To get at s claim>that exactly *one* of the a's is coprime to f he has to make one of>those common factors (that I have written above) units. They are both>roots of an irreducible polynomial with integer coefficients of degree 4.>Actually sometime ago he made a remark that suggested ts.The main problem is that he has to make ts choice for each and every>polynomial that comes up. Now I am not sure, but it may be possible>to have a ring where only a single root of a quadratic is in s>ring, Depends on the quadratic and the factors involved (in order for the> ring A[w], with A the algebraic integers and w an algebraic number) to> intersect Q exactly on the integers, certain conditions on w must be> met; not any old w will do. So he would have to prove that there is> always one of the factors in s situation where ts condition is> met.I've brought up the obvious wch is that the poster Arturo Magidinhas all along been arguing for an algebraic integer function definedby the ***multiplicative inverse*** of another algebraic integerfunction.It's such a ludicrous position that I can now continue to loudlyproclaim just how corrupt mathematicians must be to continue with sucha position.Here's the math to back me up.Remember I haveP(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) Variables: m, f, x, u E Ring of Algebraic integersand the factorization P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf) Variables: a_1, a_2, a_3, roots of cubic defined as follows. Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m)The poster wishes for a variable function wch has the property ofbeing a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1has w_1(m) as a factor for all integer m.So letting r(m) = w_2(m) w_3(m), you have w_1(m) = f^2/r(m)wch is the requirement by wch an algebraic integer function isdefined by the multiplicative inverse of another *algebraic integer*function.It's such disgustingly bad mathematics that Arturo Magidin *should*hang s head in shame, but here he is still posting as if he were inthe clear.He couldn't do that without the support of math society.Mathematicians ARE corrupt.Then a_1(m) x + uf must have w_1(m) as a factor, so dividing throughgivesa_1(m)/w_1(m) + uf/w_1(m)where uf/w_1(m) can't be an algebraic integer for all integer m.It might help for me to put in actual numbers for u and f, wch I cando as the variables are independent of each other, so let u=2, f=7,then itsa_1(m)/w_1(m) + 14/w_1(m)and clearly, if w_1(m) varies with m, then 14/w_1(m) is not analgebraic integer for all integer m.For those who STILL need help, consider that if you had 14/w_1(m) = r(m)introducing r(m) for the result of the division, thenw_1(m) r(m) = 14sow_1(m) r(m) - 14 = 0wch would force zeroes for m.You see, basic algebra refutes the claims of the poster ArturoMagidin, but why should he bother with *correct* mathematics?After all, you all believe m no matter how stupid the mathematics hepresents because you're corrupt. You're lost souls with no belief inanytng--not even === proven Adjunct Assistant Professor at the University of Montana.>>As I have written already multiple times, it is trivially monic in other>>rings, the cubic (x - b1)(x - b2)(x - b3) serves just fine. But you>>want to go to a ring where your divisibility claims work. It includes>>the ring of algebraic integers. But wch of the two numbers>> (-sqrt(7) + sqrt(15))/2 and (-sqrt(7) - sqrt(15))>>is a unit in that ring? And why precisely that number?>> ...>> (B) In the Object ring, ts marvellous construction that fixes the>> ring of algebraic integers by allowing numbers that should be in,>> there are polynomias with integer coefficients, irreducible over Q,>> wch have ->SOME<- roots in the ring but not all of them.>> And in case (B), how do we know wch one is there, as you ask?>>I tnk (B) is indeed precisely the case. James wants all those numbers>>in s ring such that the cubics in the b's are monic with numbers in>>s rings for all possible polynomials P(m). So when we look at the>>polynomial in Ôa' for m=1 and f=2 we get (in the algebraic integers)>>three roots that are obviously not coprime to f. To get at s claim>>that exactly *one* of the a's is coprime to f he has to make one of>>those common factors (that I have written above) units. They are both>>roots of an irreducible polynomial with integer coefficients of degree 4.>>Actually sometime ago he made a remark that suggested ts.>>The main problem is that he has to make ts choice for each and every>>polynomial that comes up. Now I am not sure, but it may be possible>>to have a ring where only a single root of a quadratic is in s>>ring,>> Depends on the quadratic and the factors involved (in order for the>> ring A[w], with A the algebraic integers and w an algebraic number) to>> intersect Q exactly on the integers, certain conditions on w must be>> met; not any old w will do. So he would have to prove that there is>> always one of the factors in s situation where ts condition is>> met.I've brought up the obvious wch is that the poster Arturo Magidin>has all along been arguing for an algebraic integer function defined>by the ***multiplicative inverse*** of another algebraic integer>function.Huh?Oh, you mean, if h_1(m) is the gcd of g_1(m) and f, then the functionI am arguing about is g_1(m)/h_1(m)?I thought I was talking about gcd(a_1(m),f), but whatever.>It's such a ludicrous position that I can now continue to loudly>proclaim just how corrupt mathematicians must be to continue with such>a position.What is ludicrous about it?Is the functionf(x) = gcd(x,7) with x integers, a ludicruous function?>Here's the math to back me up.Remember I haveP(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f)Variables: m, f, x, u E Ring of Algebraic integersand the factorizationP(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf)Variables: a_1, a_2, a_3, roots of cubic defined as follows.Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m)The poster wishes for a variable function wch has the property of>being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1>has w_1(m) as a factor for all integer m.And you claim notng like that could possibly exist.>So letting r(m) = w_2(m) w_3(m), you have w_1(m) = f^2/r(m)wch is the requirement by wch an algebraic integer function is>defined by the multiplicative inverse of another *algebraic integer*>function.So what?Take the function g(x) = gcd(x,14), where x ranges over allintegers. Then one can define the functionh(x) = 14/g(x)and ts is a perfectly well defined function of integer values withinteger variable: it is equal to 1 when x is divisible by 14; it isequal to 7 when x is divisible by 2 but not by 7; it is equal to 2when x is divisible by 7 but not by 2; and it is equal to 14 when x isodd and not divisible by 7.Is there sometng horribly wrong with such a function that I do notknow about? [.presonal attacks removed.]>Then a_1(m) x + uf must have w_1(m) as a factor, so dividing through>givesa_1(m)/w_1(m) + uf/w_1(m)where uf/w_1(m) can't be an algebraic integer for all integer m.Why not? Can 14/g(x) be an integer for all integer values of x above?>It might help for me to put in actual numbers for u and f, wch I can>do as the variables are independent of each other, so let u=2, f=7,>then itsa_1(m)/w_1(m) + 14/w_1(m)and clearly, if w_1(m) varies with m, then 14/w_1(m) is not an>algebraic integer for all integer m.Why not? I already posted a function of COMPLEX variable that has theproperty that for every complex number m, 14/w_1(m) is an algebraicinteger, and wch is not constant. [.snip more rambling.]=Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of s power to do miscef. Mr. Smith has untiring energy, wch does sometng; self-evident honesty of conviction, wch does more; and a long purse, wch does most of all. He has made at least ten publications, full of figures few readers can criticize. A great many people are staggered to ts extent, that they imagine there must be the indefinite sometng in the mysterious all ts. They are brought to the point of suspicion that the mathematicians ought not to treat all ts with such undisguised contempt, at least. -- A Budget of === Finisng argument, core error proven> ...> > > Yes. And so what?> > > > Readers should note that the poster admits that s position requires> > > a function that is either a unit or has f as its only non-unit factor> > > depending on certain conditions as m varies from 0 to positive> > > infinity. A condition follows.> > Yes, I still see no problem with such a function.> > Then give a *single* function in ALL of mathematics wch behaves as> > you wish.> > You're making up some wacky mathematics. Oh, come on. Do you know the Moebius function?> mu(n) = 1 if n = 1> 0 if n is divisible by a square> (-1)^r if n = the product of r distinct primes.> Wacky enough? But indeed, not the function we seek, see below. > I've found a simpler refutation of ts poster's position. Remember I have P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) Variables: m, f, x, u E Ring of Algebraic integers and the factorization P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf) Variables: a_1, a_2, a_3, roots of cubic defined as follows. Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m) The poster wishes for a variable function wch has the property of> being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1> has w_1(m) as a factor for all integer m. Ts may come as a shock not only to , but alsomaybe to Dik and Arturo. In general for most values of m, one expects that w_1(m) = f^{2/3}. {My apologies to Dik and Arturoif they already knew ts.} Here is the reasoning. Consider the polynomial Q(m), Q(m) = A * x^3 + B * x + u^3*f,where A = m^3 f^4 - 3m^2 f^2 + 3m,and B = -3(-1+mf^2)*u^2. Assume a factorization of Q(m) of the form Q(m) = (b1*x + c1)*(b2*x + c2)*(b3*x + c3),where b1, b2, b3, and c1, c2, c3 are algebraicintegers. Here is the key. We may assume that[2] c1 = c2 = c3 = u * f^{1/3}. [I tnk Arturo can verify ts.] Therefore [3] Q(m) = (b1*x + u*f^{1/3})*(b2*x + u*f^{1/3})*(b3*x + u*f^{2/3}). Note that b1*b2*b3 = A, and that, if m isrelatively prime to f and f <> 3, A is relativelyprime to f. Therefore b1, b2, and b3 areall *relatively prime* to f [in general: see below]. Now one factors P(m) as follows: P(m) = f^2*Q(m) = (f^{2/3}*b1*x + f^{2/3}*u*f^{1/3}) *(f^{2/3}*b2*x + f^{2/3}*u*f^{1/3}) *(f^{2/3}*b3*x + f^{2/3}*u*f^{1/3}) = (f^{2/3}*b1*x + u*f) *(f^{2/3}*b2*x + u*f) *(f^{2/3}*b1*x + u*f). Note that, given [2] and [3], ts is the ONLY way toproduce a factorization of the form that Harris assumes, i.e., of the form P(m) = (a1*x + u*f)*(a2*x + u*f)*(a3*x + u*f). Note also that a1 = f^{2/3}*b1, a2 = f^{2/3}*b2, a3 = f^{2/3}*b3. Thus if w1(m) = f^{2/3}, one has a1/w1(m) = b1,an algebraic integer. Moreover of course u*f/f^{2/3} = u*f^{1/3}, also an algebraic integer.Therefore *each of a1, a2, and a3 has a factor in commonwith f, namely w_1 = f^{2/3}, and NONE of them are divisible by f*. Ts is all contingent on the following polynomialR(a) being irreducible: R(a) = u^3*f*a^3 - B*a^2 - A,where A and B are as defined above. In general R(a) is irreducible. We have given examples.One is when u = 1, m = 1, f = 5: R(a) = 5*a^3 + 72*a^2 - 553. All of Harris's discussion below is academic, sincewhen R(a) is irreducible, w_1(m) = f^{2/3}. > Then a_1(m) x + uf must have w_1(m) as a factor, so dividing through> gives a_1(m)/w_1(m) + uf/w_1(m) where uf/w_1(m) can't be an algebraic integer for all integer m. It might help for me to put in actual numbers for u and f, wch I can> do as the variables are independent of each other, so let u=2, f=7,> then its a_1(m)/w_1(m) + 14/w_1(m) and clearly, if w_1(m) varies with m, then 14/w_1(m) is not an> algebraic integer for all integer m. For those who STILL need help, consider that if you had 14/w_1(m) = r(m) introducing r(m) for the result of the division, then w_1(m) r(m) = 14 so w_1(m) r(m) - 14 = 0 wch would force zeroes for m. That is, you can't have algebraic integer functions, that is functions> that give algebraic integer results, and not have only certain values> of m that would work.> No - Arturo has given a perfectly good example where w_1(m) * r(m) = 14for all complex numbers m, and, when m is an integer,w_1(m) and r(m) are both algebraic integers. Ts is somewhat academic anyway in view of the result I gave above.> That is, you can have sometng like 2m+ 7 = 21, that works for a> particular value of m, but you can't have functions in algebraic> integers that will multiply to give 14 for all integer m. To get such functions, you have to go outside the ring into a field. That refutes the position of Dik T. Winter, and note that as I've> pointed out that poster clearly either has limited mathematical> ability, or he's been lying now for some time.> I must comment on ts. Dik in my view is a very able mathematician.As for lying: every controversy in Usenet eventually reaches the very predictable common denominator of one person calling another a liar. Harris goes right to it immediately in every argument. In ts case, as in === Finisng argument, core error proven Adjunct Assistant Professor at the University of Montana. [.snip.]>> That is, you can't have algebraic integer functions, that is functions>> that give algebraic integer results, and not have only certain values>> of m that would work.>> No - Arturo has given a perfectly good example where w_1(m) * r(m) = 14for all complex numbers m, and, when m is an integer,>w_1(m) and r(m) are both algebraic integers.In fact, in the example I gave, w_1(m) and r(m) are both algebraicintegers for ANY complex value of m, not just the integer values. Justthought I would pick that nit. It is trivial to come up with suchfunctions, I am surprised James has argued for over 24 hours that theycannot exist.=Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of s power to do miscef. Mr. Smith has untiring energy, wch does sometng; self-evident honesty of conviction, wch does more; and a long purse, wch does most of all. He has made at least ten publications, full of figures few readers can criticize. A great many people are staggered to ts extent, that they imagine there must be the indefinite sometng in the mysterious all ts. They are brought to the point of suspicion that the mathematicians ought not to treat all ts with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. === argument, core error proven Adjunct Assistant Professor at the University of Montana.>> ...>> > > Yes. And so what?>> > > > > Readers should note that the poster admits that s position requires>> > > a function that is either a unit or has f as its only non-unit factor>> > > depending on certain conditions as m varies from 0 to positive>> > > infinity. A condition follows.>> > > Yes, I still see no problem with such a function.>> > > Then give a *single* function in ALL of mathematics wch behaves as>> > you wish.>> > > You're making up some wacky mathematics.>> Oh, come on. Do you know the Moebius function?>> mu(n) = 1 if n = 1>> 0 if n is divisible by a square>> (-1)^r if n = the product of r distinct primes.>> Wacky enough? But indeed, not the function we seek, see below.>> I've found a simpler refutation of ts poster's position.>> Remember I have>> P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f)>> Variables: m, f, x, u E Ring of Algebraic integers>> and the factorization>> P(m) = (a_1(m) x + uf)(a_2(m) x + uf)(a_3(m) x + uf)>> Variables: a_1, a_2, a_3, roots of cubic defined as follows.>> Cubic: a^3 + 3(-1+mf^2)a^2-f^2(m^3 f^4 - 3m^2 f^2 + 3m)>> The poster wishes for a variable function wch has the property of>> being a factor of f, so I'll use w_1(m) w_2(m) w_3(m) = f^2, where a_1>> has w_1(m) as a factor for all integer m.>> Ts may come as a shock not only to , but also>maybe to Dik and Arturo. In general for most values of m, one >expects that w_1(m) = f^{2/3}. {My apologies to Dik and Arturo>if they already knew ts.}I sent an e-mail in response to your query, noting that ts is nottrue in general. Feel free to post that message. But ifw_1(m)=f^{2/3}, then by Galois Theory you know that all of the a_i aremultiples of f^{2/3}, wch would imply that the quadratic coefficientof the cubic would be a multiple of f^{2/3}, but if f is coprime to 3,then it is coprime to f. So ts cannot be true at all, regardless ofwhether the polynomial is irreducible or not.When the cubic is irreducible, expect f to factor as a product of 3coprime algebraic integers, f = r1*r2*r3, and for a1, a2, a3 to bedivisible by r2*r3, r1*r3, and r1*r2 respectively.=Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of s power to do miscef. Mr. Smith has untiring energy, wch does sometng; self-evident honesty of conviction, wch does more; and a long purse, wch does most of all. He has made at least ten publications, full of figures few readers can criticize. A great many people are staggered to ts extent, that they imagine there must be the indefinite sometng in the mysterious all ts. They are brought to the point of suspicion that the mathematicians ought not to treat all ts with such undisguised contempt, at least. -- A Budget of === How much longer must physics put up with F=ma?Expires: 28 days>> The total force [F] used to move an object against friction or any otherIf I push against a brick wall, notng will move. Am I really>exerting a force, or is ts only tension, or stress?But when we consider an object in a frictionless state, there is>another question that arises. Newton said in s trd law of motion:To every action there is always opposed an equal reaction; or the>mutual actions of two bodies upon each other are always equal, and>directed to contrary parts.Now when a force is applied to a mass in a frictionless state, the>inertial force opposes the motive force, wch limits the acceleration>according to a=F/m. But if the inertial force is an equal and>opposite force to the motive force, as Newton's 3rd law seems to say,>then why do we have acceleration at all?Consider an object suspended on a long cord. If you push it with your hand, it initially accelerates more or less withoutexternal restriction. However the object's inertia resists acceleration and aforce of reaction is exerted on your hand. Ts reaction is transmitted throughyour body to the ground. So there are two pairs of Ôequal and opposite' forces involved. 1) Your body is compressed by equal and opposite forces on your hand and feet.2) The Earth is accelerated very slightly backwards by the inertial reaction ofthe object, acting through your (rigid) body. Henri Wilson.Read all about my H-aether theory and see the funny side of === Subject: Re: How much longer must physics put up with F=ma?> The total force [F] used to move an object against friction or any other>If I push against a brick wall, notng will move. Am I really>exerting a force, or is ts only tension, or stress?But when we consider an object in a frictionless state, there is>another question that arises. Newton said in s trd law of motion:To every action there is always opposed an equal reaction; or the>mutual actions of two bodies upon each other are always equal, and>directed to contrary parts.Now when a force is applied to a mass in a frictionless state, the>inertial force opposes the motive force, wch limits the acceleration>according to a=F/m. But if the inertial force is an equal and>opposite force to the motive force, as Newton's 3rd law seems to say,>then why do we have acceleration at all? Consider an object suspended on a long cord. If you push it with your hand, it initially accelerates more or less without> external restriction. However the object's inertia resists acceleration and a> force of reaction is exerted on your hand. Ts reaction is transmitted through> your body to the ground. So there are two pairs of Ôequal and opposite' forces involved. 1) Your body is compressed by equal and opposite forces on your hand and feet.> 2) The Earth is accelerated very slightly backwards by the inertial reaction of> the object, acting through your (rigid) body. > Henri Wilson. Read all about my H-aether theory and see the funny side of relativity:> http://www.users.bigpond.com/HeWn/index.htmYes, it's easy to === Seeing through the veils: vectors, matrices, polynomials, quaternions... all the same tngConsider the polynomial 3x^3 + 2x^2 + x + 1;Consider the vector (3,2,1,1);Consider the quaternion 3+2i+j+k;Consider the matrix 3 2 1 1;Consider the matrix 3 2 1 1 0 0 0 0 0;Consider the vector (3,2,1,1,0,0,0,0);Consider the ordered 4-tuplet (3,2,1,1);Consider the real number 3+2sqrt(2)+sqrt(3)+sqrt(5);...There is *NO DIFFERENCE* between any of the above tngs. They areall different names of the same exact tng.All that changes is the context. And based on that context we givedifferent definitions to such tngs as addition, multiplication,etc. In some contexts we have extra notions such as the degree ofthe polynomial or the value of the polynomial when x=5 or theroots of the polynomial or the angle of the vector and some othervector or the location of the tuplet in 4 dimensional space or thedeterminant of the 3x3 matrix.Really it is just the same tng with many different kinds ofmultiplication and addition associated with it, and === the veils: vectors, matrices, polynomials, quaternions... all the same tng> Consider the polynomial 3x^3 + 2x^2 + x + 1;> Consider the vector (3,2,1,1);> Consider the quaternion 3+2i+j+k;> Consider the matrix> 3 2> 1 1; Consider the matrix> 3 2 1> 1 0 0> 0 0 0; Consider the vector (3,2,1,1,0,0,0,0); Consider the ordered 4-tuplet (3,2,1,1);> Consider the real number 3+2sqrt(2)+sqrt(3)+sqrt(5);> ... There is *NO DIFFERENCE* between any of the above tngs. They are> all different names of the same exact tng. All that changes is the context. And based on that context we give> different definitions to such tngs as addition, multiplication,> etc. In some contexts we have extra notions such as the degree of> the polynomial or the value of the polynomial when x=5 or the> roots of the polynomial or the angle of the vector and some other> vector or the location of the tuplet in 4 dimensional space or the> determinant of the 3x3 matrix. Really it is just the same tng with many different kinds of> multiplication and addition associated with it, and many different> other properties.-- They're isomorpc (even canonically isomorpc) as real vector spaces but that doesn't make them the same. You may as well say that the real and complex numbers are the same because they're isomorpc as sets. Or that a squre and a circle are the same because they're isomorpc as topological spaces. Or that 4 gallons is the same as 4 miles.Have a tolerable === vectors, matrices, polynomials, quaternions... all the same tng> Consider the polynomial 3x^3 + 2x^2 + x + 1;> Consider the vector (3,2,1,1);> (SNIP other isomorpc tngs) > Really it is just the same tng with many different kinds of> multiplication and addition associated with it, and many different> other properties. They're isomorpc (even canonically isomorpc) as real vector spaces but > that doesn't make them the same. You may as well say that the real and > complex numbers are the same because they're isomorpc as sets. Or that a > squre and a circle are the same because they're isomorpc as topological > spaces. Or that 4 gallons is the same as 4 miles.If they are not the same, then what is different about them?Your skepticism is understandable- it is rooted deep in a lifetime ofdefining tngs like numbers, vectors (the calc 3 vectors, not theabstract algebra ones), matrices, etc., with basically notng morethan hand waving gestures.We are taught what a matrix is in very elementary college algebra,where it is given a very handwavy definition because 99% of the peoplein the class are just there so they can get their liberal artsdegrees. Then in later math classes when we reencounter it we justassume a familiarity with it. I defy you to offer a definition of the2x2 matrices of R and a definition of the vectors suchthat the two are different whatsoever. (Have you ever even TRIED toformulate a mathematically sturdy definition of a matrix or of thevectors taught in calc 3?)Everytng is really just a bunch of sets- and beyond that it's justcontext and notation, endless context and === matrices, polynomials, quaternions... all the same tng> Consider the polynomial 3x^3 + 2x^2 + x + 1;> Consider the vector (3,2,1,1);> (SNIP other isomorpc tngs) > Really it is just the same tng with many different kinds of> multiplication and addition associated with it, and many different> other properties. They're isomorpc (even canonically isomorpc) as real vector spaces but > that doesn't make them the same. You may as well say that the real and > complex numbers are the same because they're isomorpc as sets. Or that a > squre and a circle are the same because they're isomorpc as topological > spaces. Or that 4 gallons is the same as 4 miles. If they are not the same, then what is different about them?> Your skepticism is understandable- it is rooted deep in a lifetime of> defining tngs like numbers, vectors (the calc 3 vectors, not the> abstract algebra ones), matrices, etc., with basically notng more> than hand waving gestures. We are taught what a matrix is in very elementary college algebra,> where it is given a very handwavy definition because 99% of the people> in the class are just there so they can get their liberal arts> degrees. Then in later math classes when we reencounter it we just> assume a familiarity with it. I defy you to offer a definition of the> 2x2 matrices of R and a definition of the vectors such> that the two are different whatsoever. (Have you ever even TRIED to> formulate a mathematically sturdy definition of a matrix or of the> vectors taught in calc 3?) Everytng is really just a bunch of sets- and beyond that it's just> context and notation, endless context and notation.Have you ever seen the definition of a vector space?There is more to it than context and notation - there are sets with_operations_ defined on them that satisfy certain _axioms_. What youhave pointed out is, indeed, a canonical isomorpsm of _vectorspaces_. But it doesn't necessarily extend to canonical isomorpsmsof, say, algebras.Anyway, here's a rigorous definition of an mxn matrix over a field F:An mxn matrix over a filed F is a function from {1,...,m}x{1,...,n} === vectors, matrices, polynomials, quaternions... all the same tngEverytng is really just a bunch of sets- and beyond that it's just> context and notation, endless context and notation. Have you ever seen the definition of a vector space?> Yes.There is more to it than context and notation - there are sets with> _operations_ defined on them that satisfy certain _axioms_. What you> have pointed out is, indeed, a canonical isomorpsm of _vector> spaces_. But it doesn't necessarily extend to canonical isomorpsms> of, say, algebras.> When we define a vector space by associating operations with sets,are we saying at the same time that no other operations exist uponthose sets? No, we are not. A vector space is still a vector spaceif we assign a thousand different operations onto it, so long as atleast one of them satisfies the axioms.Anyway, here's a rigorous definition of an mxn matrix over a field F:> An mxn matrix over a filed F is a function from {1,...,m}x{1,...,n} to> F.>Alright. In ts definition, the matrix becomes a set of the form{(a,b,x)|a = 1,2,..,m, b=1,2,..,n, x an arbitrary element of F andwhenever two elements of the set are (a,b,x) and (a,b,y) for the samea,b, then x=y}So let's look a little deeper at what we've just said. Basically, itis just a set of mn elements of F-- what purpose do the integersserve? Answer: they give those elements of F an order, so that weknow where to put each element when constructing our matrix.Now if we ask ourselves whether such an ordering scheme is unique,it most certainly is not. And that is the only possible differencebetween our mxn matrix and an mn-tuplet. For an mn-tuplet is againjust a set of mn elements of F, associated with an order.The difference at first glance lies in the fact that we can accomplishts ordering scheme in different ways. For example our mn-tupletmight actually be a set of the form{(a,x)|a=1,2,..,mn, x in F and for two (a,x), (a,y) with the same a,x=y}.But need it be? We could just as well write it as{(a,b,x)|a=1,2,..,m, b=1,2,..,n, x an arbitrary element of F andwhenever two elements of the set are (a,b,x) and (a,b,y) for the samea,b, then x=y}We would still have an order to our mn elements, and that is what isimportant, *and you'll note ts definition makes an mn-vector equalto an mxn matrix*. We would have to adjust the operations but that isa non-issue. They would still yield the various algebraic,topological, etc. properties and be just as easy to compute.So we have found that the only difference between mXn matrices andmn-tuplets or mn-vectors in the calc3 sense, is the scheme we use toorder them, and ts scheme is really completely arbitrary.In other words, we could choose the ordering schemes in such a waythat all these different structures become the same tng, and what,then, do we lose? *Absolutely notng*. What do we gain? A huge newbunch of very well-behaved operations (for example the determinant ofa 4-vector <1,2,3,4> or the cross product of a quaternion and apolynomial:)We actually do ts kind of tng in some instances. Usually in theform of characteristic equations where we take some non-polynomialtng (like a differential equation) and look at it as a polynomial. And obviously we treat 1xn matrices and nx1 matrices as n-vectors withlittle hesitation.> 2x2 matrices have a natural product structure that makes them into> an algebra. 4-vectors do not.4-vectors are not an algebra by the specific operation wch weassociate with them to make them a vector space. But there isabsolutely notng to stop us from associating other operations tothem, for example, the products of the 2x2 matrices. Your slip up isunderstandable- but what we must realize is that there is not someethereal one true multiplication wch is defined differently fordifferent tngs. There are many divers kinds of multiplication andaddition and there is notng stopping us from defining more than oneof these over any given bunch of objects. (Ts one truemultiplication cult is very common and is very frustrating to anyonewho truly loves abstract tngs)> you could say that R^4 is the set of all functions from {0,1,2,3} to R> and GL(2,R) is the set of all functions from {0,1}x{0,1} to R.> I don't know whether you see {0,1,2,3} and {0,1}x{0,1} as being the same,> but if they are different then R^4 and GL(2,R) are differentQuite right, quite right. You have a solid grasp on these tngs. And no, by no means are {0,1,2,3} and {0,1}x{0,1} the same. However,there is in ts case notng intrinsically important about {0,1,2,3}or {0,1}x{0,1}. We can use either to define R^4 and lose notng, oreither to define GL(2,R) and again lose notng, or we could use{90,103,70001,Graham's Number} to define R^4 and again lose notng. All we are doing is giving our 4 elements an order- and we can do tsin many different ways, and lose notng. And gain much. Why, then,deprive ourselves and make our mathematics destitute just because ofpickiness over arbitrary ordering schemes that we never even use whendoing the calculations === matrices, polynomials, quaternions... all the same tngFirst, let me say how much I am enjoying ts debate.> When we define a vector space by associating operations with sets,> are we saying at the same time that no other operations exist upon> those sets? No, we are not. A vector space is still a vector space> if we assign a thousand different operations onto it, so long as at> least one of them satisfies the axioms.If a set has the proper cardinality, there ALWAYS EXISTS operations that make it into a vector space. In fact, there will be lots of such operations. So by your reasoning, all sets of the same cardinality are really the same. But the pure existence of a vector space stucture doesn't make the underlying set a vector space. In order to make a set into a vector space, you have to actually CHOOSE the operations. If you choose different operations, you put a different vector space structure on the set and hence create a different vector space. For certain familiar spaces like R^n, the choice of operations is implicit but nevertheless definite. For isomorpc vector spaces with different underlying sets, there are lots of different isomorpsms between them and often there is no obvious choice of wch isomorphsim is the right one. In order identiy the two vector spaces, you have to choose the isomorpsm. For example, to identify R^n with GL(2,R) in a nice way, you have to make the fair arbitrary choice of wch of the ordered pairs <0,1> and <1,0> is the greater. To say that two vector spaces are the same, there should be no arbitrariness in choosing they isomorpsm.> So let's look a little deeper at what we've just said. Basically, it> is just a set of mn elements of F-- what purpose do the integers> serve? Answer: they give those elements of F an order, so that we> know where to put each element when constructing our matrix.Now if we ask ourselves whether such an ordering scheme is unique,> it most certainly is not. And that is the only possible difference> between our mxn matrix and an mn-tuplet. For an mn-tuplet is again> just a set of mn elements of F, associated with an order. The difference at first glance lies in the fact that we can accomplish> ts ordering scheme in different ways. For example our mn-tuplet> might actually be a set of the form> {(a,x)|a=1,2,..,mn, x in F and for two (a,x), (a,y) with the same a,> x=y}.> But need it be? We could just as well write it as> {(a,b,x)|a=1,2,..,m, b=1,2,..,n, x an arbitrary element of F and> whenever two elements of the set are (a,b,x) and (a,b,y) for the same> a,b, then x=y}> We would still have an order to our mn elements, and that is what is> important, *and you'll note ts definition makes an mn-vector equal> to an mxn matrix*. We would have to adjust the operations but that is> a non-issue. They would still yield the various algebraic,> topological, etc. properties and be just as easy to compute. So we have found that the only difference between mXn matrices and> mn-tuplets or mn-vectors in the calc3 sense, is the scheme we use to> order them, and ts scheme is really completely arbitrary.But the ordering of {0,...,mn} is NOT arbitrary. It has a canonical ordering wch is kind of the whole point of have natural numbers in first place. The set {0,...,m}x{0,...,n} has a pair of orderings that both seem more natural than any other orderings but both seem equally natural so deciding on one of them is a fairly arbitrary choice. You have to say that left is greater than right or right is greater than left. There is no way to make ts choice rationally without reference to your specific cultural standards or sometng like that. If you made ts decision once and for all, that still wouldn't solve your problem because then you would still to decide whether rows were greater than columns or vice versa.> We actually do ts kind of tng in some instances. Usually in the> form of characteristic equations where we take some non-polynomial> tng (like a differential equation) and look at it as a polynomial.> And obviously we treat 1xn matrices and nx1 matrices as n-vectors with> little hesitation.When we do ts, we are adding more structure to these objects so we don't end up with quite the same tng. > 4-vectors are not an algebra by the specific operation wch we> associate with them to make them a vector space. But there is> absolutely notng to stop us from associating other operations to> them, for example, the products of the 2x2 matrices. Your slip up is> understandable- but what we must realize is that there is not some> ethereal one true multiplication wch is defined differently for> different tngs. There are many divers kinds of multiplication and> addition and there is notng stopping us from defining more than one> of these over any given bunch of objects. (Ts one true> multiplication cult is very common and is very frustrating to anyone> who truly loves abstract tngs)There is sometng that stops us and that is arbitrariness. Two objects that can be identified only in an arbitrary way, can also be distinguished in some other way. > Quite right, quite right. You have a solid grasp on these tngs.> And no, by no means are {0,1,2,3} and {0,1}x{0,1} the same. However,> there is in ts case notng intrinsically important about {0,1,2,3}> or {0,1}x{0,1}. We can use either to define R^4 and lose notng, or> either to define GL(2,R) and again lose notng, or we could use> {90,103,70001,Graham's Number} to define R^4 and again lose notng.> All we are doing is giving our 4 elements an order- and we can do ts> in many different ways, and lose notng. And gain much. Why, then,> deprive ourselves and make our mathematics destitute just because of> pickiness over arbitrary ordering schemes that we never even use when> doing the calculations themselves?You make them look the same if you ignore any larger context but only then. For example, by your reasoning R^4=R^{0,1,2,3}x{<0,0,0,0>}={<0,0,0,0>}xR^{4,5,6,7} since both definitions give you a naturally ordered basis. But look at ts in the larger context of R^8={0,1,2,3,4,5,6,7}. These two R^4's are 4D subspaces of R^8 and together they span R^8 so they must be different.Have a tolerable existence. === matrices, polynomials, quaternions... all the same tng>> Consider the polynomial 3x^3 + 2x^2 + x + 1;>> Consider the vector (3,2,1,1);>> (SNIP other isomorpc tngs)>> Really it is just the same tng with many different kinds of>> multiplication and addition associated with it, and many different>> other properties.>> They're isomorpc (even canonically isomorpc) as real vector spaces>> but>> that doesn't make them the same. You may as well say that the real and>> complex numbers are the same because they're isomorpc as sets. Or that>> a squre and a circle are the same because they're isomorpc as>> topological>> spaces. Or that 4 gallons is the same as 4 miles. If they are not the same, then what is different about them?> Your skepticism is understandable- it is rooted deep in a lifetime of> defining tngs like numbers, vectors (the calc 3 vectors, not the> abstract algebra ones), matrices, etc., with basically notng more> than hand waving gestures. We are taught what a matrix is in very elementary college algebra,> where it is given a very handwavy definition because 99% of the people> in the class are just there so they can get their liberal arts> degrees. Then in later math classes when we reencounter it we just> assume a familiarity with it. I defy you to offer a definition of the> 2x2 matrices of R and a definition of the vectors such> that the two are different whatsoever. (Have you ever even TRIED to> formulate a mathematically sturdy definition of a matrix or of the> vectors taught in calc 3?) Everytng is really just a bunch of sets- and beyond that it's just> context and notation, endless context and notation.-- .2x2 matrices have a natural product structure that makes them into an algebra. 4-vectors do not. Polynomials whose order is at most 3, have a natural product also that completely different from the product on GL(2,R). Quaternions have yet another product. You could define or redefine the product on these objects but then you wouldn't really have the same object. You would've changed the object's algebraic structure.However, if you're only interested in these objects only as sets, (wch is not usual) you could say that R^4 is the set of all functions from {0,1,2,3} to R and GL(2,R) is the set of all functions from {0,1}x{0,1} to R. I don't know whether you see {0,1,2,3} and {0,1}x{0,1} as being the same, but if they are different then R^4 and GL(2,R) are different. Polynomials are functions from R to R.Sometimes people include the operations as part of the set such as R^4 = {V,+,*} where V=R^{0,1,2,3}, +:VxV-->V, and *:RxV-->V. That would REALLY distinguish it from GL(2,R}={V,+,*_1,*_2} where *_1:RxV-->V is scaler multiplication and *_2:VxV-->V is matrix multiplication. I admit that ts method strikes me as kind of icky. I'm somewhat uncomfortable with foundations as they stand currently. Have a tolerable existence. Eli electron-dot-cloud are === galaxiesSubject: Re: should Gauss's Law of Magnetism be: without monopoles int B dot dA must vanish,> since it's directly related to del dot B by Gauss's divergence> theorem.Igor, how have you managed to stay clear of the gang of 5? Ts is a gang ofscience jerks on the Internet who spend 99% of their time attacking other peoplewhether direct toGraham Lee as he has ceased posting to ts thread. The Gang-of-5 have even forged myemail address to intimidate and threaten people who respond to my Internet threads.Some people fall into the trap of believing that I intimidated and threatened peoplevia email. Such is the primitive and prank condition of the Internet like the lawlessWest of the 1800s where bad people can committ their badness. Alot of people inscience can easily realize or figure out that forgery email impersonating A.Plutonium is a forgery. And it seems as though about the only people willing toengage in a discussion of the science in my threads are people who mask their realidentity odd to me that so many people exist with a mentality of suppression hatredrather than a mentality of engage and discuss science. Over 50% of the activeparticipants in the science newsgroups have a mentality not of doing-science but personality. Sort of like apsychology therapy in logging on to the newsgroups not for science content but torelease and relieve themselves of their inbuilt hatred.One of the great puzzles to me and to psychology is that one would tnk that peoplewho can do science and do logical procedures would be people most apt to refrain fromthe degenerating into a ofthe Internet, that the people in science are more apt at degenerating into hatredthan are people outside of science. And ts is a very sad revelation, in that thepeople who get into science are the people most prone into devolving or degeneratinginto the illogic of hatred. Sorry for the long spiel Igor, but it seems as though thehatemongers have all but cut off and severed everyone that engages in a discussionwith Arcmedes Plutonium. And I can go on posting to the Internet with never anyresponse. But that is a gross shame, not on any of my account. But a gross shame thathatred dominates these newsgroups and not that of the subject concerned.Question Igor: when did the Meissner Effect come out storically and when did Diracbegin with s road of finding a monopole? The reason I ask ts question is not thatI want to belittle the giant of physics that Dirac was. Dirac realized that theMaxwell Equations were asymmetrical and Dirac thus realized that if a monopoleexisted that the Maxwell Equations would restore their symmetry. I do not supposeDirac ever battled ts issue of the asymmetry of the Maxwell Equations by simplychanging the Gauss Law of Magnetism, but then again Dirac may have done that and Isimply am not aware of it.So I wonder if the Meissner Effect had existed a long time when Dirac was playingwith the asymmetry of the Maxwell Equations. And whether Dirac examined the two sideby side. Examined the Maxwell Eq. and the Meissner Effect and Dirac saying sometngAh, ts is curious, in that if I alter the Gauss Law of Magnetism to accomodate theMeissner Effect then symmetry is restored to the Maxwell Equations.I would be awfully surprized that the Meissner Effect co-existed a long time and thatDirac's adventure into the monopole-defect was missed by Dirac. Surprized that Diracdid not put Meissner Effect and monopole-defect together. Dirac was a giant inphysics and gaps and holes of logic do escape the purview of even giants.What I am saying, Igor, is that the monopole that Dirac was looking for is the entirephenomenon of the Meissner Effect. And the resolution of the problem is that theGauss Law of Magnetism needs altering to fit the Meissner Effect. Once it is alteredthen the Maxwell Equations end up as completely symmetrical.And it was symmetry that lead Dirac down the road to monopole theory. Dirac detestedasymmetry, and rightfully he should. Asymmetry made Dirac look for a monopole, butcuriously Dirac did not try to alter the Maxwell Equations first. Someone who knowsthe detailed story of Dirac can perhaps tell us why Dirac never put the MeissnerEffect alongside the asymmetry of the Maxwell Equations.Arcmedes Plutonium, a_plutonium@hotmail.comwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are === Integral B dot dA = q/p + q/p_t> The problem is that, without monopoles int B dot dA must vanish,>> since it's directly related to del dot B by Gauss's divergence>> theorem.Igor, how have you managed to stay clear of the gang of 5? Ts is a gang of>science jerks on the Internet who spend 99% of their time attacking other people>whether direct followups or whether has ceased posting to ts thread. The Gang-of-5 have even forged my>email address to intimidate and threaten people who respond to my Internet threads.>Some people fall into the trap of believing that I intimidated and threatened people>via email. Such is the primitive and prank condition of the Internet like the lawless>West of the 1800s where bad people can committ their badness. Alot of people in>science can easily realize or figure out that forgery email impersonating A.>Plutonium is a forgery. And it seems as though about the only people willing to>engage in a discussion of the science in my threads are people who mask their real>identity in order that so many people exist with a mentality of suppression hatred>rather than a mentality of engage and discuss science. Over 50% of the active>participants in the science newsgroups have a mentality not of doing-science but of>that of Sort of like a>psychology therapy in logging on to the newsgroups not for science content but to>release and relieve themselves of their inbuilt hatred.One of the great puzzles to me and to psychology is that one would tnk that people>who can do science and do logical procedures would be people most apt from indications of the sci newsgroups of>the Internet, that the people in science are more apt at degenerating into hatred>than are people outside of science. And ts is a very sad revelation, in that the>people who get into science are the people most prone into devolving or degenerating>into the illogic of hatred. Sorry for the long spiel Igor, but it seems as though the>hatemongers have all but cut off and severed everyone that engages in a discussion>with Arcmedes Plutonium. And I can go on posting to the Internet with never any>response. But that is a gross shame, not on any of my account. But a gross shame that>hatred dominates these newsgroups and not that of the subject concerned.Question Igor: when did the Meissner Effect come out storically and when did Dirac>begin with s road of finding a monopole? The reason I ask ts question is not that>I want to belittle the giant of physics that Dirac was. Dirac realized that the>Maxwell Equations were asymmetrical and Dirac thus realized that if a monopole>existed that the Maxwell Equations would restore their symmetry. I do not suppose>Dirac ever battled ts issue of the asymmetry of the Maxwell Equations by simply>changing the Gauss Law of Magnetism, but then again Dirac may have done that and I>simply am not aware of it.So I wonder if the Meissner Effect had existed a long time when Dirac was playing>with the asymmetry of the Maxwell Equations. And whether Dirac examined the two side>by side. Examined the Maxwell Eq. and the Meissner Effect and Dirac saying sometng>Ah, ts is curious, in that if I alter the Gauss Law of Magnetism to accomodate the>Meissner Effect then symmetry is restored to the Maxwell Equations.I would be awfully surprized that the Meissner Effect co-existed a long time and that>Dirac's adventure into the monopole-defect was missed by Dirac. Surprized that Dirac>did not put Meissner Effect and monopole-defect together. Dirac was a giant in>physics and gaps and holes of logic do escape the purview of even giants.What I am saying, Igor, is that the monopole that Dirac was looking for is the entire>phenomenon of the Meissner Effect. And the resolution of the problem is that the>Gauss Law of Magnetism needs altering to fit the Meissner Effect. Once it is altered>then the Maxwell Equations end up as completely symmetrical.And it was symmetry that lead Dirac down the road to monopole theory. Dirac detested>asymmetry, and rightfully he should. Asymmetry made Dirac look for a monopole, but>curiously Dirac did not try to alter the Maxwell Equations first. Someone who knows>the detailed story of Dirac can perhaps tell us why Dirac never put the Meissner>Effect alongside the asymmetry of the Maxwell Equations.Arcmedes Plutonium, a_plutonium@hotmail.com>whole entire Universe is just one big atom where dots>of the electron-dot-cloud are galaxiesI'm not as well versed on the Meissner effect as I probably should be,but I do know a bit about it. I know it relates to superconductingcurrents. But that's the problem with your idea, as far as I cantell. We're still talking about current based fields, where del dot Bstill vanishes and that gives us Gauss's law for magnetism. Ampere'slaw will still give the field based on the current. electron-dot-cloud are === galaxiesSubject: Re: should Gauss's Law of Magnetism be: on the Meissner effect as I probably should be,> but I do know a bit about it. I know it relates to superconducting> currents. But that's the problem with your idea, as far as I can> tell. We're still talking about current based fields, where del dot B> still vanishes and that gives us Gauss's law for magnetism. Ampere's> law will still give the field based on the current.The Meissner Effect geometrically is that it exludes an external magnetic field from itsvolume. So, in a sense, geometrically the Meissner Effect is a monopole. For when weconsider Gauss's Law of Magnetism del dot B, geometrically, the vanisng is because the+ pole or - pole vanishes the lines of force of one to the other. You cannot have amonopole with del dot B simply because you cannot make the lines of force vanish into thesecond pole.But the Meissner Effect is an obstruction of ts vanisng. And although it is not amonopole in terms of north and south magnetic poles or + or - poles, the MeissnerEffect is not a example of del dot B. Gauss's Law of Magnetism does not describe theMeissner Effect, nor do the other 3 Maxwell Equations.So to fix and remedy the problem is that the 4 Maxwell Equations are missing a term. Aterm that belongs in the Gauss Law of Magnetism that will incorporate the MeissnerEffect.In their most general form the Maxwell Equations look like ts: curl E = -c B curl B = c E + c J div E = p div B = 0To correct the Maxwell Equations by incorporating the Meissner Effect we need them tolook like ts in most general form: curl E = -c B curl B = c E + c J div E = p div B = -pu + qunote: not sure exactly where the negative sign goesThe Meissner addition makes the Maxwell Equations fully symmetrical.Note: The difference between Classical Physics and Quantum Physics is that if one canwrite out mathematics for the inner workings such as the Maxwell Eq then one is dealingwith Classical Physics but if one is unable to write out any math equations for theworkings of a piece of physics (example: Heisenberg Uncertainty relationsp, or theStrongNuclear Force) then one is dealing with Quantum Physics. If a piece of physics isable to be penetrated by math equations then it is Classical Physics. If all we can do isattach probabilities then it is Quantum Physics. If BCS and Cooper pairing are correctthen we will never have math equations describing superconductivity. But if I am correctthat superconductivity is merely maximum electronegativity plus conduction bands, well,then we shall have math equations for a full description of superconductivity.And also, if the BCS and Cooper pairing are not fake theories, then the Meissner Effectwould also be Quantum Physics and not Classical Physics. And the Meissner Effect wouldnever be attached to the Maxwell Equations in any manner or degree to wch I amattacng. You simply cannot have the Meissner Effect as part of Maxwell theory and therest of superconductivity with a BCS and Cooper pairing explanation.Igor, the Meissner Effect is not described by the div B = 0, and neither by the other 3Maxwell Equations. That leaves us with the problem of incorporating.The remedy is simple in that the exclusion of an external magnetic field fromvolume is the add on of a new term to the Gauss Law of Magnetism.I am not sure of the negative sign where it goes. But if the other Gauss law ists: div E = c pthen the new Gauss Law of Magnetism would look like ts: div B = - c p u + c q uwhere the second term is the exclusion of magnetism of the first term. And where thesecond term is like a monopole although not a true monopole. If I had to give it a name Iwould call it a psuedomonopole. It imitates a monopole.Arcmedes Plutonium, a_plutonium@hotmail.comwhole entire Universe is just one big atom where dotsof the === knowing that I can now catch you in your pathetic lies by showingthey resolve to the stupid position that with xy=2, x and y can bothbe algebraic integers wle x varies over all of algebraic integers.YES, now that I know that none of you can be so stupid as to actuallybelieve you are correct but MUST know that the error I found is real.YES, knowing that I WILL make certain that some of you are ended,finished, cast down in shame as the world finds out that truth.Yes...the sadness at looking at you broken for no good reason at all,as fighting mathematics is the worst kind of stupidity, the mostfoolish of actions, the most surprising tng--from mathematicians. Ullrich, Arturo Magidin, Nora Baron and the others betrayedyou, and now I have you caught betraying mathematics and logic.Continue to === Re: JSH: Fun timewhere was teh transition from the fake@MSN,to theRealJSH@MSN -- if that correspondentof the 10-year programme to prove the last theorem o'Fermat,can be proven to be for real?anyway, isn't ts first example, just the sameas the definition of ideals, that I citedfrom a story o'math book? > they resolve to the stupid position that with xy=2, x and y can both> be algebraic integers wle x varies over all of algebraic integers.--UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?...La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto:(FOSSILISATION [McCainanites?] (TM/sic))/BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm.Http://www.tarpley.net/ bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP === timehttp://www.giganews.com/info/dmca.html>Yes, knowing that I can now catch you in your pathetic lies by showing>they resolve to the stupid position that with xy=2, x and y can both>be algebraic integers wle x varies over all of algebraic integers.Uh, nobody has claimed ts.>YES, now that I know that none of you can be so stupid as to actually>believe you are correct but MUST know that the error I found is real.YES, knowing that I WILL make certain that some of you are ended,>finished, cast down in shame as the world finds out that truth.Yes...the sadness at looking at you broken for no good reason at all,>as fighting mathematics is the worst kind of stupidity, the most>foolish of actions, the most surprising tng--from mathematicians. Ullrich, Arturo Magidin, Nora Baron and the others betrayed>you, and now I have you caught betraying mathematics and logic.Continue to fight and you lose everytng, including your souls.So when we start raving like a === Fun time> Yes, knowing that I can now catch you in your pathetic lies by showing> they resolve to the stupid position that with xy=2, x and y can both> be algebraic integers wle x varies over all of algebraic integers.Step 1: Misrepresent your opposition,> YES, now that I know that none of you can be so stupid as to actually> believe you are correct but MUST know that the error I found is real.Step 2: Engage in character assassination (argumentum ad hominem),> YES, knowing that I WILL make certain that some of you are ended,> finished, cast down in shame as the world finds out that truth.Step 3: Appeal to mob,> Yes...the sadness at looking at you broken for no good reason at all,> as fighting mathematics is the worst kind of stupidity, the most> foolish of actions, the most surprising tng--from mathematicians.Step 4: Argument from intimidation,> Ullrich, Arturo Magidin, Nora Baron and the others betrayed> you, and now I have you caught betraying mathematics and logic.Step 5: Identify opponents (finally!),> Continue to fight and you lose everytng, including your souls.Step 6: Issue threats.All in all, ts is a typical example of Ô work. Rage,vitriol, contempt, hatred -- no valid math.QED--There are two tngs you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of === Re: JSH: Fun time Yes, knowing that I can now catch you in your pathetic lies by showing> they resolve to the stupid position that with xy=2, x and y can both> be algebraic integers wle x varies over all of algebraic integers. Step 1: Misrepresent your opposition, YES, now that I know that none of you can be so stupid as to actually> believe you are correct but MUST know that the error I found is real.I will DESTROY THE OPPOSITION!!! > Step 2: Engage in character assassination (argumentum ad hominem), YES, knowing that I WILL make certain that some of you are ended,> finished, cast down in shame as the world finds out that truth.HELL YEAH!!!> Step 3: Appeal to mob, Yes...the sadness at looking at you broken for no good reason at all,> as fighting mathematics is the worst kind of stupidity, the most> foolish of actions, the most surprising tng--from mathematicians.Um, HELL YEAH!!! > Step 4: Argument from intimidation, Ullrich, Arturo Magidin, Nora Baron and the others betrayed> you, and now I have you caught betraying mathematics and logic. Step 5: Identify opponents (finally!), Continue to fight and you lose everytng, including your souls. Step 6: Issue threats. All in all, ts is a typical example of Ô work. Rage,> vitriol, contempt, hatred -- no valid math. YOU you stupid LOSER!!!HELL YEAH!!!I am === === time>Subject: Re: JSH: Fun time>Message-id: that I can now catch you in your pathetic lies by showing>> they resolve to the stupid position that with xy=2, x and y can both>> be algebraic integers wle x varies over all of algebraic integers.>> Step 1: Misrepresent your opposition,>> YES, now that I know that none of you can be so stupid as to actually>> believe you are correct but MUST know that the error I found is real.>I will DESTROY THE OPPOSITION!!!> Step 2: Engage in character assassination (argumentum ad hominem),>> YES, knowing that I WILL make certain that some of you are ended,>> finished, cast down in shame as the world finds out that truth.>HELL YEAH!!!> Step 3: Appeal to mob,>> Yes...the sadness at looking at you broken for no good reason at all,>> as fighting mathematics is the worst kind of stupidity, the most>> foolish of actions, the most surprising tng--from mathematicians.Um, HELL YEAH!!!> Step 4: Argument from intimidation,>> Ullrich, Arturo Magidin, Nora Baron and the others betrayed>> you, and now I have you caught betraying mathematics and logic.>> Step 5: Identify opponents (finally!),>> Continue to fight and you lose everytng, including your souls.>> >> Step 6: Issue threats.>> All in all, ts is a typical example of Ô work. Rage,>> vitriol, contempt, hatred -- no valid math. YOU you stupid LOSER!!!HELL YEAH!!!I am the MAX. So bow down before me worm.Did you buy TWO six-packs === lose everytng, including your souls.There is notng with wch I === knowing that I can now catch you in your pathetic lies by showing>they resolve to the stupid position that with xy=2, x and y can both>be algebraic integers wle x varies over all of algebraic integers.Does anyone know what James actually means here? Is the stupid positionthere are algebraic integers x and y such that xy = 2? Or is itfor all algebraic integers x and all complex numbers y, if xy=2, theny is an algebraic integer? Or is it there are functions x, y from thealgebraic integers to the algebraic integers such that x(m) y(m) = 2 for all algebraic integers m?>[drivel deleted]-- -- Peter van Rossum, | Universal law of linearity: for allDept. of Mathematics, New Mexico | f : R -> R and for all x, y in R:State University, Las Cruces, NM, USA. | f(x + y) = f(x) + === timehttp://www.giganews.com/info/dmca.html>>Yes, knowing that I can now catch you in your pathetic lies by showing>>they resolve to the stupid position that with xy=2, x and y can both>>be algebraic integers wle x varies over all of algebraic integers.Does anyone know what James actually means here? Heh-heh. Figuring out what he means is a perennial problem.Actually ts one is not that hard to translate: The stupid positionhe's referring to is for every algebraic integer x there existsan algebraic integer y such that xy = 2.Wch is indeed an abysmally stupid position. Now if he hadany evidence that any of us evil ones had actually been _espousing_such a stupid position that would be interesting.>Is the stupid position>there are algebraic integers x and y such that xy = 2? Or is it>for all algebraic integers x and all complex numbers y, if xy=2, then>y is an algebraic integer? Or is it there are functions x, y from the>algebraic integers to the algebraic integers such that x(m) y(m) = 2 for all >algebraic integers m?>[drivel === makes a donkey happier than a rut in the barnyard, as tshee-hawing hoofer knows!javascript:amz_js_PopWin(Ô/exec/obidos/tg/stores/recs/ radio/krex/-/track/B00004OCGQ001003/ref=pd_krex_dp_t/102- 0871328-0308963',winName,winArgs);http://www.rubylane.com/ni/ === Notng makes a donkey happier than a rut in the barnyard, as ts> hee-hawing hoofer knows!javascript:amz_js_PopWin(Ô/exec/obidos/tg/stores/recs/ radio/krex/-/track/B00004OCGQ001003/ref=pd_krex_dp_t/102- 0871328-0308963',winName,winArgs);Does anyone really run a newsreader that executes javascript? If so,why on earth would you want to?Not enough popup ads when browsing the web?> http://www.rubylane.com/ni/shops/viperswife/iteml/GW-684-- Jesse HughesSo far as ts negative attitude toward life is concerned, Buddsmis merely Taoism a little touched in its === VPxu1EZYZedQ8NzJQbBHtZcOhAk0Mb93FucfYQNcVMGQ0+tAN8-ooV>Yes, knowing that I can now catch you in your pathetic lies by showing>>they resolve to the stupid position that with xy=2, x and y can both>>be algebraic integers wle x varies over all of algebraic integers. > Does anyone know what James actually means here? Is the stupid position[...]> for all algebraic integers x and all complex numbers y, if xy=2, then> y is an algebraic integer? [...] I tnk he means the === Xu3hRsZCYevATy1ezVCe+YriLo6yL2hAEKtnkKt8FIV0GlE9l6F3Ea James,> Yes, knowing that I can now catch you in your pathetic lies by showing> they resolve to the stupid position that with xy=2, x and y can both> be algebraic integers wle x varies over all of algebraic integers. Please do that![...]> Continue to fight and you lose everytng, including your souls. Could you further === Yes, knowing that I can now catch you in your pathetic lies by showing > they resolve to the stupid position that with xy=2, x and y can both > be algebraic integers wle x varies over all of algebraic integers.Pray provide a reference to that. > Continue to fight and you lose everytng, including your souls.You are === definitionAgapito ez...[Arturo handled ts part:)]> There must be some general utility in defining open sets as the> elements of a properly defined topology.Indeed. Some structures, wch don't look very geometric, can be given auseful topology, enabling us to prove ts or that by using an argument bycontinuity, or a passage to the limit, or an argument about connectedness.You might Google around for Zariski topology or p-adic distance to see acouple of such topologies in === ...[Arturo handled ts part:)]> There must be some general utility in defining open sets as the> elements of a properly defined topology.> Indeed. Some structures, wch don't look very geometric, can be given a> useful topology, enabling us to prove ts or that by using an argument by> continuity, or a passage to the limit, or an argument about connectedness.> You might Google around for Zariski topology or p-adic distance to see a> couple of such topologies in action.> LHMany to you and === Proportionality>>Coming from a physics background myself, I must disagree. A good>>physicist would not write sometng like y ~ x^-.05 _for small x_>>because it doesn't matter if x itself is small...Of course it does.If I have y = cx/(b + x) then y ~ x for x small, but y is about b/c>for x large. It matters very much if x is small.>what _really_ matters (as I said in my first post) is if>One more time. what matters is whether or notk_1 << k_2/xThat is; x must be much less than k_2/k_1 for the approximationto be good. So let me give an example.k_2 = 10^6k_1 = 1The original poster's equation was: Y(x)=(k_1 - k_2/x)^.5y = Sqrt[1 - 10^6/x]so even if x is 200 (wch is large according to you)y is Sqrt[1 - 5000] wch is just about equal to Sqrt[-5000]ts works out even though x is large === Proportionality>>Coming from a physics background myself, I must disagree. A good>physicist would not write sometng like y ~ x^-.05 _for small x_>because it doesn't matter if x itself is small...>>Of course it does.>>If I have y = cx/(b + x) then y ~ x for x small, but y is about b/c>>for x large. It matters very much if x is small.>>what _really_ matters (as I said in my first post) is ifOne more time. what matters is whether or notk_1 << k_2/xRight. Ts defines small. I can't say that x < 1 mm is small forall approximations. It's large for many, for instance as to whether Ican consider a one micron diffraction grating to contain infinitelymany slits.So I have no disagreement that small means small in comparison tosome other tng, sure. Usually in physics contexts the smallnesstest is independent of the system of units in use. It better be, sincethe physics doesn't change depending on your system of units. So thetest for x is small will be some dimensionless === experiment where I randomly draw a number from the interval[0,1], is the probability that I pick a number in the interval [0,1) equalto one?Likewise, if I remove a countable number of points, will the probability ofpicking in the === I have an experiment where I randomly draw a number from the interval> [0,1], is the probability that I pick a number in the interval [0,1) equal> to one? Likewise, if I remove a countable number of points, will the probability> of picking in the interval be equal to one?-- The answer is yes. In fact, certain UNcountable sets have probability 0 too. And for some sets, the probability isn't defined at all. If you're interested in more details, look up Lebesgue measure. It seems really weird to me to have an event be possible but with probability 0. It's one of those examples where infinite sets just don't work the same way finite sets do.Have a tolerable === really weird to me to have an event be possible but with> probability 0. It's one of those examples where infinite sets just> don't work the same way finite sets do.>Indeed, consider the universe. What's the probabilityit came from notng? Zero! Yet here it is.> Have a tolerable existence.>You too, even tho === New Online Movie of Star Gate UniverseOK use http://qedcorp.com/APS/StarGate1.mov 7 megsIt is a self-running movie and it loads effectively instantly on DSL .I mean it starts animating right away unlikehttp://qedcorp.com/APS/AustinTx102403.movwch is slide show you control with click each step of the way.New version has more stuff & more art and drama (for Two Culture's audience) on1. Wheeler IT FROM BIT3. Dark Matter Detectors as Much Ado About Notng.4. Oh say can you see by WMAP's early light?5. Slam Dunk easy to follow animated explanations of Why Inßation? Dynamical instability of Dirac Vacuum.6. How superconductors work - animated.7. How Alcubierre's warp drive works - animated.etc.Great stuff below Saul-Paul I need to study it in airports and on planes Thursday. :-)You saw that Max Tegmark et-al do not buy the closed finite multiply-connected Universe.One cosmologist seems to tnk the low power in the WMAP quadrupole mode is acontingent artifact of the Earth's location? In any case ts is a worthy issue - whateverthe outcome.A gh resolution foreground cleaned CMB map from WMAPhttp://arxiv.org/abs/astro-ph/0302496 http://www.hep.upenn.edu/~max/wmap.html If you got here from the New York Times soccer ball story or the story about the Nature paper suggesting that the Universe is a dodecahedron, you'll find our paper ruling it out here. The point is that the dodecahedron model predics matched circles exactly opposite each other in the sky just like a bagel (but six pairs instead of one), wch our paper ruled out. See also the more thorough upcoming circle paper bySpergel, Starkman and Cornish. Soccer enthusiasts will note that a soccer ball is made up of pentagons and hexagons and isn't a dodecahedron. http://www.hep.upenn.edu/~max/wmap.html http://www.hep.upenn.edu/~angelica/topology.html ... all toroidal universes (cubes and rectangles) are also ruled out... Can the lack of symmetry in the COBE/DMR maps constrain the topology of the universe? http://xxx.lanl.gov/abs/astro-ph/9510109Who is Indrid? He is getting more and more useful and relevant lately.He was probably one of Robert Anton Wilson's groupies in Berserkleyis my guess? Obviously quite bright and a bit of a polymath. Lou, Bob, et alia,The book *Homograpes, Quaternions, and Rotations* by Patrick Du Val(Oxford, 1964) is indeed a wonderful book. It provides the most detailedpictures of the polyhedral fundamental domains --indicating the gluing ofopposite faces with appropriate twists. Du Val's book is based on the quaternion nature of the action of thebinary polyhedral groups in creating the spherical orbit spaces SU(2)/g,where g is a finite subgroup of SU(2), i.e., a cyclic group Zn, or a binarypolyhedral group of type DDn, TD, OD, or ID. Thus I wish to present anaspect of ts quaternion nature not covered by Du Val --i.e., the A-D-Eclassification of these finite subgroups of the unit-length quaternions,SU(2). First a note on my labelling: I use DDn (for the Dihedral double group) rather than the more common Dn(for binary dihedral group). It is a (possibly useful, and possiblyconfusing) coincidence that in the A-D-E classification of these groups, DDngoes with Dn as a Coxeter graph (and Lie algebra) label.Moreover, I use the Crystallographers names for the double polyhedral groups(rather than the more common mathematician's name of binary polyhedralgroups). Just as SU(2) double covers SO(3) via the orbit spaceSU(2)/{+1,-1}, so the double polyhedral groups are double covers for thepolyhedral groups, wch are finite subgroups of SO(3). It is most interesting that the presentation formulas for each of thefinite subgroups of SU(2) can be read directly from the Coxeter graphassociated with ts group. In every case except the very simple cyclicgroups Zn, the famous (1843) formula of William Rowan Hamilton isgeneralized. In fact the D4 graph provides the structure of Hamilton's (8element) quaternion group, wch is the same as the dihedral double group oftype DD4, actually the smallest of these groups. i^2 = j^2 = k^2 = ijk = -1.Here I will use R,S, and T instead to accord with Du Val and H.S.M.Coxeter's nomenclature (as in *Regular Polytopes* 3rd edition, 1973). Coxeter graph: Group label: Presentation: No. of elements: An: *--*--*-...-* (n nodes) Zn+1 R^n+1 = 1 (special case) n + 1Dn: *--*--*-...-* (n nodes, DDn R^2 = S^2 = T^n-2 = -1 4(n-2) | n > 3) <2,2,n-2> *E6: *--*--*--*--* TD R^2 = S^3 = T^3 = -1 24 | <2,3,3> *E7: *--*--*--*--*--* OD R^2 = S^3 = T^4 = -1 48 | <2,3,4> *E8: *--*--*--*--*--*--* ID R^2 = S^3 = T^5 = -1 120 | <2,3,5> *Looking at these graphs and the group presentations especially as condensedin the form of Coxeter's label for these groups , it is easy to seehow to read off the presentations from the graphs: simply count nodes alongthe three legs of the graphs, always starting with the node common to allthree legs.Ts exercise in the classification of the subgroups of SU(2), suggests thatthere might be many other mathematical objects to classify. After all thesegraphs were first devised as part of the classification of crystallograpcreßection groups (Coxeter groups). They were next used to classify Liegroups and Lie algebras (Dynkin diagrams). Then V.I. Arnold used them toclassify a vast generalization of the Thom type catastrophe structures.Now more than 20 mathematical objects are classified by these truly magical(and very simple) graphs. The advantage is that these ojects must all besomehow related to each other -- though the relationsps my lie very deepin the mathematical object underlying all the possible classificationsprovided by these A-D-E graphs, the study of wch I call ADEX theory.Although, the above classification of finite subgroups of SU(2) was known toCoxeter (and others), it was not until 1979 that Mckay found a trulydeep connection between these finite groups and the A-D-E classified Liealgebras. Ts was presented at a Coxeter festscrift in 1979 and publishedin the proceedings under the title, *The Geometric Vein*, edited by ChandlerDavis, Branko Grunbaum, and F.A. Sherk (Springer, 1981). Ts should be inany math library. And besides the great paper Representations and CoxeterGraphs by Ford and McKay, there is a wonderful paper by PatricDu Val, Crystallography and Cremona Transformations -- and many othergoodies.The vision of the unification of disparate mathematical objects via theA-D-E Coxeter graphs is due largely to V.I. Arnold. s little book*Catastrophe Theory* (Springer, 1986). As he puts it: At first glance, functions, quivers, caustics, wave fronts and regularpolyhedra have no connection with each other. But in fact, correspondingobjects bear the same label not just by chance: for example, from theicosohedron one can construct the function x^2 + y^3 + z^5, and from it thediagram E8, and also the caustic and wave front of the same name. To easily checked properties of one of a set of associated objectscorrespond properties of the others wch need not be evident at all. Thusthe relations between all the A, D, E-classififications can be use for thesimultaneous study af all simple objects, in spit of the fact that theorigin of many of these relations (for example, of the connections betweenfunctions and quivers) remains and unexplained manifestation of themysterious unity of all tngs. There is now material on the web on ts subject. In particular you candownload McKay's A Rapid Introduction to ADE Theory at:http://math.ucr.edu/home/baez/ADE.htmlclassifications.Notes onHyperspace wch covers some of ts same material with application tounified field theory and string theory.All for === now.Saul-Paul----------Subject: RE: Dodecahedron & Sagan's reference but you found the rightbook. The reason it is to be receommended is that it discusses the tilingof the three dimensional sphere by 120 dodecahedra from wch one obtainsthe Dodecahedral space as an orbit space of the three sphere under thesubgroup of the quaternions corresponding to three space symmetries of thedodecahedron.Indidentally, if we let a.b and axb be the dot and cross products ofvectors in three space, then the quaternion product of these isab = -a.b + axbTs is the best way to line up the quaternions in the form that peoplegeneralize when they go to Clifford algebras. Note that ab is in fourspace wth scalar part -a.b and vector part axb. The nice miracle is thatab is associative wle axb is not.Very best,LouI couldn't find the reference you gave below, but I found a reference: DuVal, P. Homograpes, Quaternions, and Rotations. Oxford, England: OxfordUniversity Press, 1964 wch is, no doubt, the same tng. But I can'tfind it on amazon.com. I found it in the local (U. of Rochester) library.However, I now prefer Geometric Algebra over just the quaternions. Thequaternions are a sub-space of the 3-dimensional Geometric Algebramultivector space. See, for example, the paper Imaginary Numbers are notReal - the Geometric Algebra of Spacetime athttp://www.mrao.cam.ac.uk/~clifford/introduction/intro/ intro.htmland the new book Geometric Algebra for Physicists by Chris Doran, AnthonyLasenby.The geometric algebra has an invertible product, wch the dot, cross andexterior products, by themselves are not. (Its a shame that the vectorcross product became popular....wch is only usable in 3-d vector be just a footnotein the mathematical physics === Matlock, and LynnclaireDennis may not have been on the mailing list of ts discussion.It is worth noting the resonance of an image of the universe asDodecahedral Manifold and Lynnclaire's geometry. We have previously raisedthe question of the relationsp of Dodec space and The Pattern (as Iprefer to call what is now called Mereon). Since Dodec space is built (thefive fold cylic branched covering) from a trefoil knot and the Pattern isan interrelationsp of the trefoil knot and the regular solids, it istantalizing to tnk that there must be a deep relationsp here. The keybook for a lot of ts geometry is Quaternions Homograpes andRotations by Patrick du Val. In any case, ts idea/evidence for Universeas a Dodec manifold raises the question once again!Best,Lou K.Jack & Stan, Plato in the *Timaeus*, allows Timaeus in dialogue with Socrates todescribe God's construction of the world by way of triangular elements.Midway through s long dissertation, Timaeus describes the constructionthetetrahedron, the cube the octahedron, and the icosahedron. He does notdescribe the construction of the dodecahedron, but merely says: There wasyet a fifth combination wch God used in the delineation of the universewith figures of animals. [Transltion by Benjamin Jowett.] There is only a nt here that he is describing a 12 sided solid,sincethere were traditionally 12 signs of the zodiac. Probably the exactconstruction of the dodecahedron was, in Plato's day, a Pythagoreansecret.Each pentagonal face of the dodecahedron has the star pentagon (called thepentagram) consisting of the 5 diagonals, and ts structure was sacred tothe Pythagoreans as a symbol of their brotherhood. Construction of ts pentagram could have employed the fact that eachofthe 5 diagonal lines cuts two other diagonals as a golden section (wchinthe terminology of the Greeks was in mean and extreme ratio. Book XIII of Euclid's *Elements* deals in great detail with theconstruction of the 5 Platonic solids. It is believed that ts book waswritten by Theaetetus (c. 380 B.C.), and was included by Euclid (c. 300B.C.) as the last chapter of s *Elements*. (Cf., B.L. van derWaerden,*Science Awakening*, pp, 172-173). In any case the scolium No.1 toEuclid's Book XIII, says that ts book concerns the 5 so-called Platonicfigures, wch however do not belong to Plato, 3 of the 5 being due to thePythagoreans, namely the cube, the pyramid, and the dodecahedron, wletheoctahedron and the icosahedron are due to Theaetetus. I would guess that Theatetus constructed these figures as duals to thecube and the dodecahedron, the pyramid (i.e., the tetrahedron) beingself-dual. Proposition 1 of Euclid's Book XIII is a description of the goldensection: If a straight line be cut in extreme and mean ratio, the square onthegreater segment added to the half of the whole is five times the square ofthe half. Ts construction of the golden section is used over and over againthroughout the entire Book XIII. Proposition 6: If a rational straight line be cut in extreme and meanratio, each of the segments is the irrational straight line calledapotome. Proposition 8: If in an equilateral and equiangular pentagon straightlines subtend two angles taken in order, they cut one another in extremeandmean ratio, and their greater segments are equal to the side of thepentagon. Proposition 17: To construct a dodecahedron and comprehend it in asphere, like the aforesaid figures, and to prove that the side of thedodecahedron is the irrational straight line called apotome. [Translated from the Greek by Sir Thomas L. Heath] Of course, the great geometrical master of these objects is H.S.M. t = 2 cos (pi/5) = [(5^1/2) + 1]/2 = 1.6180339887... [using t as a replacement for Coxeter's tau symbol]wch is the positive root of the quadratic equation x^2 - x -1 = 0Writing ts equation as x = 1 + (1/x), we see that t = 1 + 1/1+1/1+1/1+/+ ...It is well known that, of all regualr continued fractions, ts convergesslowest. Its nth convergent is fn+1/fn, where f1, f2,...are the Fibonaccinumbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...Since t^n-1 + t^n = t^n = 1, the integral powers of t are given by theformulae 2t^n = fn(5^1/2) + (fn-1 + fn+1) [n odd] 2t^-n = fn(5^1/2) + (fn-1 + fn+1) [n odd] 2t^n = (fn-1 + fn+1) + fn(5^1/2) [n even] 2t^-n = (fn-1 + fn+1 - fn(5^1/5) [n even]e.g., t^3 = (5^1/2) + 2, t^-6 = 9 - 4(5^1/2)[On page 52, Coxeter points out that the twelve vertices of theicosahedroncan be obtained by dividing the twelve edges of an octahedron according tothe golden section. Then he goes on to provide the Cartesian coordinates for the verticesofthe 5 Platonic solids. Cube: (edge = 2): (+/-1, +/-1, +/-1) Tetrahedron: (edge = 2(2^1/2)): (1,1,1), (1,-1,-1), (-1,1,-1),(-1,-1,1) Octahedron: (edge = 2^1/2): (+/-1,0,0), (0,+/-1,0), (0,0,+/-1) Icosahedron: (edge = 2): (0,+/-t,+/-1), (+/-1,0,+/-t), (+/-t,+/-1,0) Dodecahedron: (edge = 2/t): (0,+/-t^-1,+/-t), (+/-t,0,+/-t^-1) (+/-t^-1,+/-t,0), (+/-1,+/-1,+/-1) (where the last set of vertices are the vertices of one of the 5cubes inscribed in the dodecahedron). There is much more use of the golden section throughout ts book,wchdeals with gher dimensional regular figures. The 4th dimension isespecialy rich, and the golden section plays a big role in Coxeter'sdescription of these 4D polytopes. as is evident from s detailed tablesonpp. 298-304.(1985),p. 222: The diagrams were published in an eight-volume Ôcoffe table' book setthat was soon reprinted worldwide. All over the planet people tried tofigure out the pictures. The dodecahedron and the quasi-biological formswere especially evocative. Many clever suggestions were made by the publicand carefully sifted by the Argus team. Many harebrained interpretationswere also widely available, especially in weekly newspapers. Whole newindustries developed--doubtless unforeseen by those who devised theMessage--dedicated to using the diagrams to bilk the public. The AncientandMystical Order of the Dodecahedron was announced. The Macne was a UFO.TheMacne was Ezekiel's Wheel. An angel revealed the meaning of the Messgeandthe diagrams to a Brazilian businessman, who distributed--at first at sown expense--s interpretation worldwide. With so many enigmatic diagramsto interpret, it was inevitable that many religions would recognize someoftheir iconography in the Message from the stars. And on page 310 the secrecy of the project was mentioned: The ancient Pythagoreans, who first discovered the dodecahedron, haddeclared its very existence a secret, and the penalties for disclosureweresevere. So perhaps it was only fitting that ts house-sized dodecahedron,halfway around the world and 2,600 years later, was known only to a few. Vagay and Eda were deep in the arcana of gravitationalphysics--twistors, renormalization of ghost propagators, time-like Killingvectors, non-Abelian gauge invariance, goedesic refocussing,eleven-dimensional Kaluza-klein treatments of supergravity, and of course,Eda' own quite different superunification. I say: write Eda backwards and you get adE, wch suggests A-D-ECoxeter graphs:-) Coxeter graphs: Lie group: SU(2) Finitesubgroup: An: *--*--*- ...-* (n nodes) SU(n+1) Cyclic (n+1) Dn: *--*--*- ...-* (n nodes) SO(2n) Dihedral double | * E6: *--*--*--*--* E(27) Tetrahedral double | * E7: *--*--*--*--*--* E(56) Octahedral double | * E8: *--*--*--*--*--*--* E(248) Dodecahedral double | * (There's that Dodecahedron again :-) Since these graphs classify (in the mathematical & not the securitysense) at least 20 different mathematical objects of great utility inunified field theory and especially string theory, the study of thesegraphs(wch I call ADEX theory) is indeed a different approach tosuperunification. For example the gauge groups are witn the A-D-Eclassification of Lie groups (and Lie algebras).Sagan also mentions the Penrose twistors. It happens that gravitationalinstantons are closely related to twistors (Cf. Ward & Wells, *TwistorGeometry and Field Theory, 1990). These gravitational instantons areclassified by the A-D-E graphs via the finite subgroups g of SU(2). Such agravitationa instanton is the orbit space C^2/g. Ts is a 4-d compactspacewith a 3-d membrane between two hemispheres like the equator. Tsmembrane provides tunneling between the two parts of the instanton. Thestructure of ts membrane is the orbit space SU(2)/g. Note that in thecaseof g being the dodecahedral double group, there is the Oct. 8 *Nature*coverpaper proposing ts as the structure of the cosmological space!Also the 11-D supergravity theory invokes the E7 group as summetry group,with the 7-d torus subgroup as the dden dimensions. Moreover ts 11-dstructure is included as a 6th substructure in the M-theorysuperunificationof the 5 competing superstring theories. See the cover story of theNovember issue of Scientific American, wch is an interview with Brianstring theory to be aired Oct. 28 and Nov. 4.All for now.Saul-Paul----------Tenen, Michael E. === Brandt, Ph.D. Subject: Re: CONTACT - Carl Sagan - Welcome to The Macne.Hey Saul-Paul is there any connection of GoldenMean to Dodecahedron asIndrid Cold below suggests or is that New Age CargoCult Ka Ka?Jack,All of the Platonics have golden mean proportionsbuilt in. There are threeintersecting golden rectangles that form anicosahedron, wch is of coursedual to the dodecahedron. There's notng wrong withthe golden proportionin ts situation. I have to brush up on my Euclid and Plato. :-) Everyone,Picking up on Bob Gray's comment about Geometric Algebra....One of the implications of the Geometric Algebra approach (it seems to me) would be that the same structures would appear at different scales. So I tnk we could expect to find geometric symmetries at the planetary scale as well as at the cosmological scale.Wle earlier ts year I had presented (statistical) evidence of great circle alignment and dodecahedral symmetry in the locations of sacred sites around the world, a more detailed look is leading me away from models that are symmetric in 3D to models that are projections of gher dimensional structures.The Planetary HologramMy current hypothesis is that information from a regular 4D geometry (such as the 24 cell or the Poincare Homology Sphere) is projected onto the surface of a 3D sphere, via a holograpc projection. The nodes would correspond to various places on the surface of the sphere and the lines connecting the nodes would show up as great circle arcs. I have collected a database of about 1500 sites and devised several algorithms to search for such circles.It turns out that there is a lot of evidence for such arcs.1. There are several arcs wch contain 20 sites witn 5km of a perfect great circle. (The same algorithm running with randomly positioned sites can finds few such alignments of even 10 sites.)2. Admitting circles with as few as 4 sites per circle generates models with around 220 circles that can fit over 99% of the data. A system of 220 great circles of 10km width will cover, in total, about 15% of the earth's surface, so much of the sacred space on earth is concentrated on less than about 1/7 of the available surface.3. The models found give a dramatic reduction in the degrees of freedom required to fit the data. We would expect to need 750 circles to perfectly fit 1500 randomly arranged points on a sphere. (Each pair of points determine a circle.)4. Some sites lie on several of the great circles. Stonehenge, for example, lies on 15 circles, each of wch contains 9 or more sites.I see all of ts as evidence that the data has some sort of projective symmetry.One of the interesting tngs about the latest models is that many of the constituent great circle arcs (derived from positions of sacred sites) also align with the mesoscale geology of the planet, and the most ancient and stable rock formations.The beauty of the hyperdimensional models is that they resolve two problems with traditional sacred geometry as an explanation for planetary structure... first, that the earth does not look like a regular crystal, and second, that, over geologic time, tngs have moved around, wch would disrupt any 3D symmetries.Starting with the observation that the oldest and hardest rock (and best preserved fossils) lie along great circles connecting certain crossing points, the holograpc paradigm suggests a new theory for what powers plate tectonics. If what we are experiencing in the 3D world is a slow (i.e. geologic speed) rotation of a gher dimensional object, then the control points (i.e. the images of the nodes of the hyperdimensional object) will move around relative to the spin axis of the planet. (To get an idea of what gher dimensional rotations look like when projected to 3d, check out Michael bs' 4D Polytope Viewer http://members.aol.com/jmtsbs/draw4d.htm ) Movement of the relative locations of the control points will cause the intermediate crust to stretch or compress, giving us the seaßoor spreading and mountain building that we know from geology.Peace,HughP.S. Here is a link to Jeff Weeks' site, from wch you can download a viewer for various sorts of spaces. I did not see a file for the dodecahedral space he is proposing, however.Computer Grapcs in Curved Spaceshttp://geometrygames.org/CurvedSpaces/index.htmlP.P.S. And now, a few links I came across related to the Poincare Homology Sphere.... BaezTs Week's Finds in Mathematical Physics (Week 164)You may recall from week163 that the Poincare homology 3-sphere is a compact 3-manifold that has the same homology groups as the ordinary 3-sphere, but is not homeomorpc to the 3-sphere. I explained how ts marvelous space can be obtained as the quotient of SU(2) = S3 by a 120-element subgroup - the double cover of the symmetry group of the dodecahedron. Even better, the points in S3 wch lie in ts subgroup are the centers of the faces a 4d regular polytope with 120 dodecahedral faces. That's pretty cool. But here's another cool way to get the Poincare homology sphere..E8 is the biggest of the exceptional Lie groups. As I explained in week64, the Dynkin diagram of ts group looks like ts: o----o---o----o----o----o----o | | oNow, make a model of ts diagram by linking together 8 rings: / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /Imagine ts model as living in S3. Next, hollow out all these rings: actually delete the portion of space that lies inside them! We now have a 3-manifold M whose boundary dM consists of 8 connected components, each a torus. Of course, a solid torus also has a torus as its boundary. So attach solid tori to each of these 8 components of dM, but do it via ts attacng map:(x,y) -> (y,-x+2y)where x and y are the obvious coordinates on the torus, numbers between 0 and 2pi, and we do the arithmetic mod 2pi. We now have a new 3-manifold without boundary... and ts is the Poincare homology sphere.We see here a strange and indirect connection between E8 and the dodecahedron. Ts is not the only such connection! There's also the McKay correspondence (see week65) and a way of getting the E8 root lattice from the icosians (see week20).Are these three superficially different connections secretly just different views of the same grand picture? I'm not sure. I tnk I'd know the answer to part of ts puzzle if I better understood the relation between ADE theory and singularities.But Diarmuid Crowley told me much more....http://math.ucr.edu/home/baez/week164.htmlRuth LawrenceThe PSU(3) invariant of the Poincare homology sphereAbstract: Using the R-matrix formulation of the sl_3 invariant of links, we compute the coloured sl_3 generalised Jones polynomial for the is obtained. Ts takes complex number values at roots of unity. The result obtained is formally an infinite sum, independent of the order of the root of unity, wch at roots of unity reduces to a finite sum. Ts form enables the derivation of the PSU(3) analogue of the Ohtsuki series for the Poincare homology sphere, wch it was shown by Thang Le could be extracted from the PSU(N) invariants of any rational homology sphere.http://www.ma.huji.ac.il/~ruthel/papers/psu3phs.html GillmanUCLAThe best picture of Poincare's homology sphere (video lecture)http://www.pims.math.ca/science/2002/cascade/gillman/= === ==Subject: Solving odd-even recurrence relationI am trying to find a method for solving linear recurrence relations of the form: a_2n = F(a_n) # Even terms a_2n+1 = G(a_n) # Odd termswhere F() and G() are linear functions.For example, a_2n = c * a_n + d * a_n-1 + b * a_n-2 + 1 a_2n+1 = k * a_n - 1Could someone point me to literature that will help me? === trying to find a method for solving linear recurrence relations of> the form: a_2n = F(a_n) # Even terms> a_2n+1 = G(a_n) # Odd terms where F() and G() are linear functions. For example, a_2n = c * a_n + d * a_n-1 + b * a_n-2 + 1> a_2n+1 = k * a_n - 1 Could someone point me to literature that will help me? --Look into generating functions. However, I tnk your kind of === === Relation between theorem and Rationals/Irrationals>Subject: Re: Relation between theorem and Rationals/Irrationals> I need to prove a theorem that states:>> If R is a number set and for any 2 numbers there exists a point of R> between them, then every number is a limit point of R.>... then every number of R? is a limit ...>>R = {0} union (1,2) satisfies the condition,>>yet 0 isn't a limit point of R.>> Do you mean {0} union ]1, 2]?>No, I said ]1,2[ , however ]1,2] , that is (1,2] , will suffice>when 0 is added.>I tnk the OP left out the condition>> that the number set be open/closed.>I'm supposed to be psycc?>Heck no, are students in such a hurry or so lazy that they>won't bother to ask coherent questions?Alas too many students need to learn how to state questions.The rationals, wch are neither open nor closed,>satisfies both the condition and the conclusion.>So premises other than Ôopen' could be divined.As for closed, { 1/n | n in N } / {0} is closed set>with only one limit point, namely 0. So closed doesn't>suffice as 1/2, 1/3, ... aren't limit points of the set.It's pointless to guess what poorly written posts intend.>If it's of worth, the OP will soon make amends.>I'm sorry if I ruined your day but the actual theorem is sometng that mychange it to try to trick anyone. I was just tnking === Relation between theorem and Rationals/Irrationals[...]|>It's pointless to guess what poorly written posts intend.I don't agree, not with all of them at least.At work we occasionally deal with clients who are a bit, shall we say,mediocre at explaining what they need. Deciding that they are not beingarticulate enough is not always a reasonable option.|>If it's of worth, the OP will soon make amends.||I'm sorry if I ruined your day but the actual theorem is sometng that my|didn't change it to try to trick anyone. It's true that we get a lot of poorly written posts on sci.math, sometimesto the point that all we can do is to try to get some kind of clarification.Once in awle people jump the gun a bit at deciding that a posting in toounclear. But don't let it bother you. The original posting was fine. Theintent was clear enough, and it was easy for someone practiced in doingproofs of ts === === Re: Relation between theorem and Rationals/Irrationals Adjunct Assistant Professor at the University of Montana.>I need to prove a theorem that states:If R is a number set and for any 2 numbers there exists a point of R between>them, then every number is a limit point of R.Is ts basically the same proof as proving that there is a rational number>between any 2 irrationals and there is an irrational between any 2 rationals? No, I don't tnk so...>Or am I way off base here? I guess I'm not too sure how I could easily make>the transition from that to ts theorem or if it's possible at all.You are assuming that:(*) Given any two numbers, x and y, there is a number r in R such that x0, there is a number r in R such that|x-r|0 be any positivenumber. Then both x and x+e are numbers, so what can you conclude from(*)? And given that conclusion, can you show that there must be some rin R with |x-r| < e?Or, if you defined limit point as the limit of a sequence, then wewant to show that for every number x, there exists a sequencer_1, r_2, r_3, ...., r_n, ....of elements of R such that lim(r_n) = x.So: for each n, consider the numbers x and x+(1/n). What can youconclude from (*) for those two numbers? How do you use thatconclusion === Re: simple question about subspaces|I was wondering about the minimal required definition. I was told the|inclusion of the zero vector must be explicit to avoid a problem where|the empty set could be a subspace of V, but 0 * v = 0 implies directly|that 0 must be in S so why should it be required explicitly?It would be enough to require that the set has an element in it, becausegiven that some v is an element of the set, the fact that the set is closedunder scalar multiplication would imply that 0*v=0 is an element too. So wecould in fact replace the condition that 0 is an element by the conditiononly that the set has an element in it. But we need a condition like ts;the empty set is closed under all operations, because there are no elementswch need to be operated on.It might be worth pointing out that ts is a case of the convention,always followed in mathematics, that a statement saying each element ofa set has a property is considered vacuously true when the set is empty.I wouldn't say that ts is consistently followed in natural language, butI tnk it would cause needless complications and a lot of confusion if wewere to depart from ts convention. So for every element v of the emptyset, 0*v is an element (because there aren't any elements of the emptyset).I find it interesting to ask myself what advantages certain conventionsgive us. We could, after all, leave ts condition out of the definitionof vector subspace, and also perhaps modify the definition of vector spaceto permit the empty set as a vector. Instead of having the usual axioms for0 and -, we could have (v+(-v))+w=w and so on. If we defined the span of aset of vectors as the minimal subspace containing them, as usual, then thespan of {} would be {}, and the span of {0} would be {0}. A set of vectorsis considered independent if removing any element shrinks the span, so {0}would count as independent. That would create a special case for thecriterion that a set of vectors be independent. As usual, a set of vectorswould be independent if none of them could be written as a (nonempty) linearcombination of other elements of the set. But that would no longer beequivalent to the more symmetrical condition that there not exist a way towrite 0 as a nontrivial linear combination of elements of the set.Tnking of the sum (or linear combination) of an empty set of vectors asbeing 0 seems simply to be the === question about subspaces> |I was wondering about the minimal required definition. I was told the> |inclusion of the zero vector must be explicit to avoid a problem where> |the empty set could be a subspace of V, but 0 * v = 0 implies directly> |that 0 must be in S so why should it be required explicitly? It would be enough to require that the set has an element in it, because> given that some v is an element of the set, the fact that the set is closed> under scalar multiplication would imply that 0*v=0 is an element too. So we> could in fact replace the condition that 0 is an element by the condition> only that the set has an element in it. But we need a condition like ts;> the empty set is closed under all operations, because there are no elements> wch need to be operated on.We are accustomed to tnking about addition in a vector space asdefined by a binary operation, +:VxV->V. But what we really need aretwo operations:0:V+:VxV->VHere 0 is not an operation proper, since it takes no arguments (maybeit is called a 0-ary operation). The empty set is /not/ closed underboth theese operations.Also, instead of using theese two operations we may use a operationwch to a finite set of vectors assigns the sum of the vectors. If wecall the operation Sum, then we have Sum{}=0 and Sum{v1, v2}= v1+v2.Again, the empty set is not closed under ts operation.Ts can be generalised. The operation Sum can be defined for amonoid, in wch case we must replace finite sets with finite lists,and in wch case we should call it Prod instead of Sum. If e is theneutral element, we have Prod[]=e and Prod[a1, a2]=a1a2. The functionProd can also be defined for a category. In ts case, Prod[] isambigous since there are (in general) many identity arrows, but tsambiguity gets resolved when it is used in expressions such as(Prod[])(Prod[a1, a2, a3])=a1a2a3.If we want to define Prod for a ring without unity, then its argumentsmust be /non-empty/ lists. Alternatively, we could define Prod[] to bean element that is not part of the ring, wch can be multiplied withelements of the ring but not added to them (Can a ring without unityalways be extended to a ring with unity?).The reason why the empty set is not a subspace of a vector space isthe same as why an arbitrary subset is not a subspace: it is notclosed under addition and scalar multiplication. But the closure ofany subset is a subspace, === question about subspaces Adjunct Assistant Professor at the University of Montana.>> |I was wondering about the minimal required definition. I was told the>> |inclusion of the zero vector must be explicit to avoid a problem where>> |the empty set could be a subspace of V, but 0 * v = 0 implies directly>> |that 0 must be in S so why should it be required explicitly?>> It would be enough to require that the set has an element in it, because>> given that some v is an element of the set, the fact that the set is closed>> under scalar multiplication would imply that 0*v=0 is an element too. So we>> could in fact replace the condition that 0 is an element by the condition>> only that the set has an element in it. But we need a condition like ts;>> the empty set is closed under all operations, because there are no elements>> wch need to be operated on.We are accustomed to tnking about addition in a vector space as>defined by a binary operation, +:VxV->V. But what we really need are>two operations:>0:V>+:VxV->V>Here 0 is not an operation proper, since it takes no arguments (maybe>it is called a 0-ary operation).It is a perfectly fine operation! It is a nullary or zeroaryoperation, indeed.In general, if I is a set, then an I-ary operation on A is afunction A^I -> A. Since 0 = {}, a 0-ary operation is a functionA^{}->A; the set A^{} consists of all function from {} to A; the onlyfunction from {} to A is the empty function, so a zero-ary operationtranslates to a map from {emptyset} to A, wch means a distinguishedelement of A.Ts is all part of General or Universal Algebra.But if you want to be technical, a vector space V over the field Kreally has more than those 2 operations. It has 3 operationsassociated to the abelian group structure of V, and then one operationfor every element of K:+:V x V -> V binary operation(-): V -> V unary operation sending v to its additive inverse0: V^{} -> V a nullary operation identifying the zero vector;and for each k in K,k: V->V a unary operation corresponding to scalar multiplication byk.If you are in a variety of algebras, then the empty algebra is analgebra in the variety if and only if the type has no nullaryoperations. So the empty set is not a vector space, since the type ofvector spaces includes a nullary === subspaces|If you are in a variety of algebras, then the empty algebra is an|algebra in the variety if and only if the type has no nullary|operations. So the empty set is not a vector space, since the type of|vector spaces includes a nullary operation. I considered mentioning that, plus the fact that one could eliminatethe references to 0 in the definition by having the axiom (x+(-x))+y=yand so on. Ts would make the usual universal algebra definitionscompatible with the definition of vector space and vector subspacebeing extended to permit the empty set.I tnk maybe the suggestion that addition really is an operationon finite multisets of elements serves to explain why such an approachworks out less elegantly than the === subspaces Adjunct Assistant Professor at the University of Montana.|If you are in a variety of algebras, then the empty algebra is an>|algebra in the variety if and only if the type has no nullary>|operations. So the empty set is not a vector space, since the type of>|vector spaces includes a nullary operation. I considered mentioning that, plus the fact that one could eliminate>the references to 0 in the definition by having the axiom (x+(-x))+y=y>and so on. Would that axiom not allow an underlying structure wch is aninverse monoid but not a group? Hmmm... Haven't thought it through;perhaps the scalar multiplication takes care of that...==