mm-3939 === Subject: Re: Maxwellize the Uncertainty Principle; and is a form of the FusionBarrier Principle Re: Proof of the Fusion Barrier Principle The spherical reaction is compressed by being accelerated to near light speeds. The Cylinder confining the reactive spherical zones can be non-constricted and even harmonically amplified (probably like how stars work in their cores..) Of course Exxon and the CIA will blot out this post. Hugs Aliens - we were smarter than most of us looked. Please don't think Bush was many of our .. blackout * === Subject: Re: Maxwellize the Uncertainty Principle; and is a form of the FusionBarrier Principle Re: Proof of the Fusion Barrier Principle This is why rich Jews and the Royal Family institutionalize their defective genomic snake-eyed offspringers. There is a fusion barrier for all nuclei below 5 atomic weight - it is something to do with the birth of the universe. If you could sort deuterons on an age basis - you could do fusion with a AAA battery. Pick out protons from the first pico universe and you have infinite power because.. Nano Nano, RICK === Subject: Re: The Holy Shroud The Mystery of the Holy Shroud print,which couldn't be made by men >> at all, must be correlated to the Mystery of the Resurrection of >> Jesus,Who was wrapped in the Holy Shroud after being buried and rose >> from the death in the Holy Shroud. > You have no direct evidence to support this assertion. How to you know > the famouns shroud did not wrap up the body of Polani ben Polani the > camel salesman? > Bob Kolker The documented providence of the Shroud dates back only to AD 1357. denouncing the Shroud as a forgery (a painting), designed to attract tourists and their money. http://en.wikipedia.org/wiki/Turin_Shroud === Subject: Re: Probability of a subset of orthogonal matrices >Let X,Y be n x k (n - 1 > k) real matrices with LI columns such that >there exists >an n x n, real, orthogonal matrix M where MX = Y. >Let On = {M: n x n, orthogonal, real matrices} >Let A = {M: nxn, orthogonal, real matrix s.t. MX = Y} >With respect to the Haar prob. measure, what is the probability that a >matrix drawn from >On will land in A? >Seems to me that it should be zero, but, being a measure theory duffer, >I'm not sure how to >prove it. Any ideas? >>If you do an SVD of matrices X and Y, it seems to me that you reduce the >>problem to the case when X=Y=truncation of nxn identity matrix to only k >>columns. In that case you are looking at matrices whose first k columns >>match the identity matrix. Then it is obvious. Since this thread is getting old, it is likely I won't see it again. So you may be best off emailing me personally (my return address should work). > Even if what you suggest about the first k columns matching the > identity matrix is true, the result is not obvious to me (remember, I > am a measure theory duffer!) Could you point out some source material > that would have the proofs of the obvious result? I don't have a reference, but it seems obvious to me. It is kind of like saying that the equator is zero measure in the sphere. Since you are a measure theory duffer, how detailed a proof do you want? Why do you want it? Would Herman Rubin's answer be sufficient? Otherwise I guess I could dream up a rigorous proof, but I would have to spend a bit of time thinking about it. > Along the lines of whether what you suggest about the first k columns > is true, > I am able to prove that there exist n x k, matrices U, V with > orthogonal columns such that any nxn, orthogonal matrix, M with MX = Y > can be written as > Y(X'X)^-1X' + UPV > where is a n-k x n-k, orthogonal matrix. In effect A is related to > O(n-k) by a translation and multiplication by fixed matrices. === Subject: infimum Hi there, im following a proof which says that if i have a sequence of measurable functions then the inf f_n(x) where the inf is taken over n, is also a measurable function in the proof i have here, it states g(x) = inf_n f_n (x) then for all c in R {x in X s.t g(x)< or = c}= intersection _n {x in X s.t f_n(x)< or = c} (*) the problem im having is in the understanding the of the last line for example take f_n (x) = (x)^n x in [-1,1] now consider just when x=-1 so f_n(-1)=-1 odd, 1 even from this i think g(-1) =-1 but if i consider (*) let c=0, well when n is odd the set contains x as -1, but when n is even, it equals the empty set (as 1 is not less that 0) so the intersection of anything with the empty set is...the empty set, but this is nonsense, as it *should* somehow contain -1 if anyone can shed some light on this..i would very much appreciate it! === Subject: Re: Probability Question Back to your original question. Given X and Y n by k matrices. Let A = {M in SOn: MX=MY} wlog X and Y are n by 1 matrices (simply pick a non-zero column from the matrix X, then this only makes A bigger). By the rotational invariance of Haar measure on SOn, wlog X=Y=[1,0,...,0]. Thus A is the set of matrices whose first column is [1,0,...,0]. Now let P_t be an element of SOn whose leading 2x2 submatrix is the usual rotation by t, and the rest of the matrix is the diagonal with all entries equal to 1. Then P_t A = {P_t M:M in A} has the same measure as A (by rotational invariance of Haar measure on SOn), and they are all disjoint for 0>On Mon, 10 Apr 2006 02:22:43 -0400, Stephen J. Herschkorn >> >Let f be a positive, continuous function on the reals such that f(-x) > >>f(x) for all positive x. For nonnegative t, let a(t) be the >> >abscissa of the centroid of the region {(x,y) in R^2: |x| <= t, 0 <= >y <= f(x)}. It is intuitive that a is strictly decreasing. Is the >formal proof of this obvious? (A sincere question.) If so, what is >this proof? > >>I think the claim is false. >>A counterexample can be created as follows ... >>Let f have a central piece for which the centroid is strongly biased >>left. Then define f on the left and right of that central piece to be >>almost symmetrical (hence almost unbiased). Thus, as a function of t, >>the outer piece will force the x-value of the centroid to the right, >>back towards 0. Hence for the values of t corresponding to the outside >>pieces, the x-value of the centroid will be an increasing function of >>t. >>Here's an actual counterexample based on the above idea ... >>Define f as a piecewise function as follows: >>f(x) = >> 5 if x <= -1 >> 3 - 2*x if -1 < x <= 1 >> 9*x - 8 if 1 < x <= 4/3 >> 4 if x > 4/3 >>Then a(t) is increasing on the interval [1,2]. >I think you mean on the interval [4/3,2], since a(4/3) < a(1) (if I >was correct the two times I did the calculations). Yes, I meant [4/3,2], not [1,2]. quasi === Subject: Re: R2 = ? > OK. > Restating (note I accidentally left out the brackets before). > I am noting the actual measured values in a test unit. > The basic equation again (with brackets): > I = E / [(R1+R2) + jX] > where > I = current = 0.209 amp > E = voltage = 1.41 volts > R1 = resistance = 6.413 ohms > X = reactance = 1.759 ohms > What do you get for R2? TIA Unless the reactance is zero at the frequency of the power source (either E or I), there must be a complex component, or phase shift, between the voltage and current waveforms. You have the reactive component of the impedence as 1.759 Ohms. Are the current and voltage instantaneous values, or RMS values as measured by a meter? If the latter, I think that you could use the absolute value of the impedance: E/I = |(R1+R2) + jX| (E/I)^2 = (R1+R2)^2 + X^2 Solve for R2. You'll get a quadratic with one positive and one negative root. The positive root should be what you're looking for if R2 represents a real resistor. === Subject: triple integral Hello I need to evaluate this triple integral int(1,x= -sqr(R^2-z^2-y^2)..sqr(R^2-z^2-y^2),y=-sqr(R^2-z^2-a^2)..sqr(R^2-z^2-a^2),z= -R..h) to know the volume bounded by x^2 + y^2 + z^2 =R^2 , z=h and x=a .a,h,R parameters anybody know the comand in maple? === Subject: Re: Is this a proof? for f(n) given. Just calculate 7 f(n+1) - 10 f(n) using the formula > for f(n) When I did 7 f(n+1) I got -98 and 10 f(n) I got -10 = -88 <-- doesn't look right When I did 7 f(n) I got -7 and 10 f(n-1) I got 7 = 0 <--This looks right but I am SURE that you guys know more than I do. >and collect the powers of 2 together and ditto the powers of 5 > using a little algebra and see if it doesn't match your answer. I'm not quite sure how to collect the powers..(2,4,8,16-5,25,125) that kind of thing? >SIMPLER apply (S-5)(S-2) to f(n) = 3 2 - 4 5 >where S(n) = n+1, S f(n) = f(n+1) is (linear) shift n by 1 I don't know where the S came from, if you explain what the S represents I'll try to apply it to the function. I really appreciate the help guys, I just hope I understand it by tomorrow, I'm trying... Joe === Subject: Re: Is this a proof? where S(n) = n+1, S f(n) = f(n+1) is (linear) shift n by 1 > I don't know where the S came from, if you explain what the S > represents I'll try to apply > it to the function. > I really appreciate the help guys, I just hope I understand it by > tomorrow, I'm trying... > Joe Joe: Umm.... don't worry about the S. I'm almost positive that you haven't seen this *yet* (it's called an operator). Focus on just plugging in r_{n-1} and r_{n-2} with the formula you are given (plug k = n - 1 and k = n - 2 into r_{k} ). The algebraic manipulation is up to you, sorry to say. Good luck (start your assignments earlier :-) ) (aside: Clever thing with the operator :-) ) M> Was I right with the 7 and -7? I really have been working on this all weekend, my wife has to think I'm an idiot by now. Normally I catch on much quicker, this discreet === Subject: Re: Analysis question > if a map f(g)=Sum (g^(m))/m, where g^(m) is the m'th derivative of g, then why is this not a distribution on R? > Can this be a distribution on a subset of R? > (Note: Sum is infinite) Because f(g) acting on a test function w would be = sum (-1)^m /m. Although w(m) is defined for all m, the quantity sum w(m)/m is not necessarily going to converge to a test function - indeed it should be quite easy to find an example w where this completely fails. === Subject: Re: Analysis question >> if a map f(g)=Sum (g^(m))/m, where g^(m) is the m'th derivative of g, then why is this not a distribution on R? >> Can this be a distribution on a subset of R? >> (Note: Sum is infinite) >Because f(g) acting on a test function w would be > = sum (-1)^m /m. >Although w(m) is defined for all m, the quantity sum w(m)/m is not >necessarily going to converge to a test function - >indeed it should be >quite easy to find an example w where this completely >fails. Hmm... ok So it will work on (0, oo) right? === Subject: Re: Analysis question >if a map f(g)=Sum (g^(m))/m, where g^(m) is the m'th derivative of g, then why is this not a distribution on R? >Can this be a distribution on a subset of R? >(Note: Sum is infinite) >>Because f(g) acting on a test function w would be >> = sum (-1)^m /m. >>Although w(m) is defined for all m, the quantity sum w(m)/m is not >>necessarily going to converge to a test function - >indeed it should be >>quite easy to find an example w where this completely >fails. > Hmm... ok > So it will work on (0, oo) right? I don't see why this should be any easier. I do remember seeing a theorem somewhere (maybe in the book by Friedlander) to the effect that if you take any distribution, then if you take the anti-derivative a sufficiently large finite number of times, you end up with a regular function. Basically it is the choice of test functions that define what allowable distributions you can have. If you can find a set of test functions which allow these kinds of infinite sums, then you are going to be able to do it with the distributions. But I don't think that a set of test functions with the required properties is going to exist. Stephen === Subject: Re: Logarithm of transfinite numbers David R Tribble said: >> the set of hyperintegers and the set of reals have the same >> cardinality, c, which is a larger cardinality than the set of naturals. > Well, that's because they're actually infinite, rather than unboundedly large > but finite. I'm sorry. I guess countable just can't be finite for me. It's > like, mesobigulous or xenobigulous, or malbigulous, or something. In any > case, the equivalence between these internal and external infinities (see last > post) even more lends credence to the notion that infinite naturals should not > be excluded in the standard theory, don't you think? If you can demonstrate how the Peano axioms generate infinite naturals, then you can demonstrate why set theory should include them. === Subject: Re: Logarithm of transfinite numbers It doesn't seem to you that the infinite number line has an infinite length, or >> that we can put a name on this infinite length? Which of those seems >> unreasonable? David R Tribble said: >> What leads you to conclude that the real number line (i.e., the set >> of real numbers) is infinite? Could it be that for any real x we >> choose, there always exists an x+1? Or did you have something >> less obvious in mind? > Observe: > For x,y,z e S: > x x y -> x x x As long as the number of iterations of element generation is not limited to a > finite number, each will produce an infinite set, and together produce all the > reals. > But, that's obvious, right? :) Well, your observation says nothing about length, or a line for that matter. And what is the number of iterations of element generation? Is that hidden somewhere in the '<' order operator mentioned in your observation? === Subject: Re: Logarithm of transfinite numbers I think that, as an assumption, it is not inherently wrong, but sensible, >> thatan infinitely long oline has an infinite length. David R Tribble said: >> So a line with an infinite length is infinitely long? Sounds circular >> to me. Perhaps you could define the length of a line. > How about we say that the number of unit intervals on the line is assumed to be > greater than any finite number, and that length is defined as the number of > contiguous units in an interval? Then this line is greater than any finite > length, and has some infinite length. How is that any different from saying that: the numbers of finite members in a set is assumed to be greater than any finite number. Then the set size is greater than any finite size, and the set has an infinite size. === Subject: Re: Two real cachalots at the current modern symbolic market, by the end of the CAS Dark Age The long standing competition is coming to the its natural end. === Subject: Re: Harvey Friedman on Cantorian pseudomathematics > A bright high school student might also notice that ZF is a fabrication >> going way beyond reality, and the set theorists would have no way to >> tell him he's wrong except, perhaps, by appealing to authority, or >> baffling him with nonsense, >> No, the set theorist could refer the student to the vast history of the >> philosophy of mathematics in which set theory, >I fail to see how that's different from appealing to authority and >baffling him with nonsense. > How is giving the history of the development of a subject an appeal > to authority? If the teacher humbly said something like, I don't know how to respond to your assertion that ZF is a fabrication going far beyond reality, but I do know that philosophers have discussed that idea in the past and you may be interested in reading what they have written; that's the best answer I have to offer, then it wouldn't be an appeal to authority. === Subject: Re: Harvey Friedman on Cantorian pseudomathematics >> A bright high school student might also notice that ZF is a fabrication >> going way beyond reality, and the set theorists would have no way to >> tell him he's wrong except, perhaps, by appealing to authority, or >> baffling him with nonsense, > >> No, the set theorist could refer the student to the vast history of the >> philosophy of mathematics in which set theory, > >I fail to see how that's different from appealing to authority and >baffling him with nonsense. > How is giving the history of the development of a subject an appeal > to authority? > If the teacher humbly said something like, I don't know how to respond > to your assertion that ZF is a fabrication going far beyond reality, > but I do know that philosophers have discussed that idea in the past > and you may be interested in reading what they have written; that's the > best answer I have to offer, then it wouldn't be an appeal to > authority. The teacher may say that, or the teacher may say that he does have his own views about mathematical ontology but that the student start best by reading some of the history of the philosophy of mathematics. In any case, you objection has been met. An introductory course in set theory only requires that a student can understand and perform proofs. If a student has philosophical objections to the axioms, then the instructor can discusss the philosophy with the student or refer the student to a bibliography of the philosophy of mathematics or both. MoeBlee === Subject: Re: Harvey Friedman on Cantorian pseudomathematics On 9 Apr 2006 14:25:02 -0700, david petry said: >> They use abstruseness to hide from others and even from >> themselves the lack of concrete content in their mathematics. >> Abstruse? A bright high school student can understand basic ZF set >> theory. > A bright high school student might also notice that ZF is a > fabrication going way beyond reality, ZF is obviously not a fabrication; we have the axioms. What you are objecting to is a certain philosophical interpretation of the axioms, which one can take or leave. For genuine fabrications, I think your world of computation is a much better example, as, unlike set theory, you have provided no actual mathematics to spell the idea out, just lots of talk. > and the set theorists would have no way to tell him he's wrong except, > perhaps, by appealing to authority, or baffling him with nonsense, or > intimidating him with accusations of being a crackpot, or threatening > him with poor grades and reduced prospects for future success in life. Few if any mathematics professors would ever threaten a student with poor grades for disliking a realist view of the metaphysics of set theory. They award bad grades for poor mathematics. Nobody has to accept realism to do good mathematics. Indeed, most of the mathematicians I know don't really give much thought at all to the ontological status of mathematical objects and would simply find your objections (and most of the responses to you, for that matter) so much philosophical twaddle. === Subject: Re: Harvey Friedman on Cantorian pseudomathematics <44356e78$0$2019$ba620dc5@text.nova.planet.nl> fabrication going way beyond reality, > ZF is obviously not a fabrication; we have the axioms. What you are > objecting to is a certain philosophical interpretation of the axioms, > which one can take or leave. Mathematics is not merely a game. For the vast majority of mathematics, and all of the mathematics relevant to understanding the world in which we live, philosophy has nothing to offer. === Subject: Re: Harvey Friedman on Cantorian pseudomathematics <44356e78$0$2019$ba620dc5@text.nova.planet.nl> A bright high school student might also notice that ZF is a > fabrication going way beyond reality, > ZF is obviously not a fabrication; we have the axioms. What you are > objecting to is a certain philosophical interpretation of the axioms, > which one can take or leave. > Mathematics is not merely a game. For the vast majority of mathematics, > and all of the mathematics relevant to understanding the world in which > we live, philosophy has nothing to offer. That one can reject certain philosphical interpretations of the axioms does not make set theory a mere game. And what is the world in which we live? Among our experiences are contemplation of ideas. Philosophy MoeBlee === Subject: Re: Harvey Friedman on Cantorian pseudomathematics >>Professors of set theory and mathematics that uses set theory are >>priests only if their mathematics is a religion, which it is not. Your >>'priestly class' is question begging. >Mathematics isn't concerned about the reality of material things, >right? Therefore mathematics isn't concerned about things that >matter, right? >> Another example of argument by bad pun. > My bad pun is also called the etymology of words in some circles. > Han de Bruijn But you conveniently ignore etymology when it suits you. Afterall, 'infinite' literally means 'not finite'. Yet you have decided it means 'finite, but very large' despite its etymology. Stephen === Subject: Re: Harvey Friedman on Cantorian pseudomathematics <44356e78$0$2019$ba620dc5@text.nova.planet.nl> A bright high school student might also notice that ZF is a fabrication > going way beyond reality, and the set theorists would have no way to > tell him he's wrong except, perhaps, by appealing to authority, or > baffling him with nonsense, > No, the set theorist could refer the student to the vast history of the > philosophy of mathematics in which set theory, > I fail to see how that's different from appealing to authority and > baffling him with nonsense. That's a pretty big failure. Recommending that someone read about the intellectual history of a field - which includes sharp and systematic disagreements from all kinds of points of view - is very different from appeal to authority or intentional obfuscation. > Sometimes children notice correctly that their elders all full of bull. > It happens. And that supports exactly what point of yours? MoeBlee === Subject: Re: Harvey Friedman on Cantorian pseudomathematics <44356e78$0$2019$ba620dc5@text.nova.planet.nl> <87y7ygpn59.fsf@phiwumbda.org > a key question is, what is the meaning of a mathematical > statement? That's a question I address. > You're narrowing meaning to practical use. > Not true. Why do you post nonsense like that? > Recall that for most people, mathematics is something that can be > applied. It's not necessarily about proving theorems. For most people > then, a key question is, what is the meaning of a mathematical > statement? That's a question I address. Perhaps I misunderstood you point. Anyway, what I meant is that that context narrows meaning to practical use. So I correct my comment to: In the context of what you say about most people, you are focusing narrowly on practical use. Then I'd have to make sure I didn't misunderstand what you're driving at in that part of the disucssion to say whether the rest of my remarks in my post still pertain. MoeBlee === Subject: Re: Where is the Galois group? > On Mon, 10 Apr 2006 11:24:05 +0300, Jyrki Lahtonen > T^3 - c*T^2 - b*T -a = 0 (p); > > Moreover M+N+P = c, M*N+M*P+N*P = -b, M*N*P = a > > So we can conclude that the field generated by M, N and P generate the > splitting field of the polynomial at (p). > >>I'm having trouble with this step. What is the dimension of the algebra >>generated by M,N,P? Ok, your first equation tells us that one of them, >>say P, is a linear combination of I,M, and N. They all satisfy a third >>degree equation, so we can write M^3 and higher powers in terms of the >>lower ones. It seems to me that the dimension of the algebra is at >>least 6, so it can be a field only, if its Galois group is all of S_3. >>A basis would then be 1,M,M^2,N,M*N,M^2*N, but is N^2 in the span of >>these? In fact, N satisfies the polynomial equation T^2+T*(M-c)+(M-c)*M -b=0 (p'), that's just how N was determined, namely as the companion matrix of this polynomial (with coefficients in |Q[M]). Remark: (p') is derived by Euclidean division of T^3-c*T^2-b*T-a by T-M. Now two cases are possible: 1) (p') splits as a quadratic over |Q[M] as two linear factors over |Q[M], in this case the Galois group is A_3 and N belongs to |Q[M], just like it would in the case of a quadratic polynomial over |Q. 2) (p') is irreducible over |Q[M], in this case N lies outside of |Q[M], so the splitting field of f(x) has dimension 6, and the Galois group is S_3. > My problem is that I can't find out what the Galois group of this field > is (as extension of the scalar rational matrixes - to be identified with > the field of rationals - ). >>Your field is too generic, as you can get any cubic polynomial >>by choosing a, b and c suitably. The Galois groups of cubics vary, you >>know, so you cannot get a blanket answer in this manner :) May be >>you want to choose a,b,c to be transcendental? In that case we >>do know that the Galois group is S_3. > If a,b,c are specified rational numbers such the polynomial p(T) is > irreducible, then you can actually get an explicit representation of > the Galois group of p(T) as a subgroup GL(36,Q). The resulting group > will have 3 elements (if isomorphic to Z_3) or 6 elements (if > isomorphic to S_3). Either way, the group elements will be represented > explicitly as 36 x 36 matrices. > If you choose a,b,c as indeterminates, then the representation will be > in GL(36,Q(a,b,c)). In this case, as you observed above, the Galois > group will be isomorphic to S_3. > I'm not saying this is an efficient way to construct the Galois group, > but it does provide one way to do it. That's the bad news, but the good news is that the involved matrices have a huge amount of zeroes in them, especially if one goes into higher degrees. A companion matrix of a polynomial of degree n has n^2-2*n +1 zeroes in it, so adequate algorithms only get a complication of O(2*n). > If I have time this weekend, I may try to implement this method in > Maple, just for fun. > quasi === Subject: Re: Where is the Galois group? > On Mon, 10 Apr 2006 11:24:05 +0300, Jyrki Lahtonen > T^3 - c*T^2 - b*T -a = 0 (p); > > Moreover M+N+P = c, M*N+M*P+N*P = -b, M*N*P = a > > So we can conclude that the field generated by M, N and P generate the > splitting field of the polynomial at (p). > >>I'm having trouble with this step. What is the dimension of the algebra >>generated by M,N,P? Ok, your first equation tells us that one of them, >>say P, is a linear combination of I,M, and N. They all satisfy a third >>degree equation, so we can write M^3 and higher powers in terms of the >>lower ones. It seems to me that the dimension of the algebra is at >>least 6, so it can be a field only, if its Galois group is all of S_3. >>A basis would then be 1,M,M^2,N,M*N,M^2*N, but is N^2 in the span of >>these? > My problem is that I can't find out what the Galois group of this field > is (as extension of the scalar rational matrixes - to be identified with > the field of rationals - ). >>Your field is too generic, as you can get any cubic polynomial >>by choosing a, b and c suitably. The Galois groups of cubics vary, you >>know, so you cannot get a blanket answer in this manner :) May be >>you want to choose a,b,c to be transcendental? In that case we >>do know that the Galois group is S_3. > If a,b,c are specified rational numbers such the polynomial p(T) is > irreducible, then you can actually get an explicit representation of > the Galois group of p(T) as a subgroup GL(36,Q). The resulting group > will have 3 elements (if isomorphic to Z_3) or 6 elements (if > isomorphic to S_3). Either way, the group elements will be represented > explicitly as 36 x 36 matrices. See my reply further in this thread. > If you choose a,b,c as indeterminates, then the representation will be > in GL(36,Q(a,b,c)). In this case, as you observed above, the Galois > group will be isomorphic to S_3. > I'm not saying this is an efficient way to construct the Galois group, > but it does provide one way to do it. > If I have time this weekend, I may try to implement this method in > Maple, just for fun. > quasi === Subject: .81@.8cc.8eq.82Å.82· by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with SMTP id k3B09f315314 for ; Mon, 10 Apr 2006 20:09:42 -0400 .81.9d.82¨.90K.82[Eth].8fo.82·.82Ì.82 ª.89õ.8a´.82Å.82· .96.bc.91O.81F.8cc.8eq.81i.83T.83C.83Y.81F.82s165 50kg.81j .90E.8bÆ.81F.90l.8dÈ http://www.gokinjsan.net?angel .81w.82±.82Ì.81A.83T.83C.83g.82[Copyright ].82ç.82[NonBreakingSpace].82È.82.bd.82[Capi talIGrave].83A.83h.83.8c.83X.82[Eth].8f[CapitalEth].89î.82[ Micro].82Ä.82à.82ç.82¢.82[Ca pitalUDoubleDot].82[Micro].82.bd.81B.8bß.82[Hyphen].82[CapitalE Acute].82¨.8fZ.82Ü.82¢.82È20 2ñ.82Å.82·.82Á.82[Capita lADoubleDot].82Ë.81H .8eÀ.82Í.81A.8e.84.81A.8c.8b.8d¥ 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ñ6.96.9c.88È.8fã.82Æ.82[Cent ].82¤.82±.82Æ.82Å.82[Capi talCCedilla].82¤.82©.82[Micro].82ç.81c2 01B.82».82ê.82É.82Â.82 ¢.82Ä.82Í.81A.8cã.82[Ca pitalUGrave].82Ç.81A.82¨.89ï.82¢2 02[Micro].82.bd.82Æ.82«.82É23 0b.82[Micro].82Ü.82[Micro].82å.82¤ .81B .82¨.8c[CapitalYAcute].82¢.81A.8eh.8c.83.93I.82[CapitalE Grave].8ay.82[Micro].82¢.8e.9e.8aÔ.82[Eth].89[S Z].82.b2.82[Micro].82.bd.82¢.82Å.82· 202Ë.81I .82».82ê.82Å.82Í.81A .82¨.95Ô.8e.96.82[Eth].90S.91Ò.82[Do wnQuestion].82É.82[Micro].82Ä.82[Cent ].82Ü.82·(^-^) .8dd.82[Hyphen].82È.82ç.82¸.82[CapitalEA cute].81A.8bC.8ay.82É.97.88.82Ä.82 Ë.81I.81@.81x http://www.gokinjsan.net?angel .81.a618.8dÎ.88È.8fã.82[CapitalIG rave].95û.82Ì.82[CapitalYAcute].82Ì 203T.83C.83g.82Å.82·.81.a6 .94z.90M.92â.8e~.82Í.82±.82[DownQues tion].82ç.82Ü.82Å.82¨.8a[ EGrave].82¢.82[Micro].82Ü.82·.81B iranai@gokinjsan.net === Subject: Re: I need some help with a proof Ulrich, I was with you untill: >For n in N^+ you can apply (I): >(2n)^2 - n^2 = sum_{i=1}{2n}left(2i-1right) - sum_{i=1}{n}left(2i-1right) Could you help a bit more? Bill, the functions f(n+1) and f(n). Are you talking about f(n+1) = 3(n+1)^2 and f(n) = 3n^2? Joe === Subject: Re: JSH: Prime counting should be easier > I don't need the non-polynomial factorization result. > I think that the reason I can't get anywhere with my prime counting > research is that mathematicians are worried about letting all my > research in, so if I didn't have that find, then maybe I'd be making > progress. > Interesting! Another possible reason you can't anywhere is that > you're dumb. We know the folks at Alltel thought so! That's > why they fired you! What have you been doing for a living since > then? Outing personal information on Usenet is NEVER permissible except in the most extreme cases where a serious crime is involved. Go away. You are not welcome here. === Subject: Re: JSH: Prime counting should be easier >> I don't need the non-polynomial factorization result. >> I think that the reason I can't get anywhere with my prime counting >> research is that mathematicians are worried about letting all my >> research in, so if I didn't have that find, then maybe I'd be making >> progress. >> Interesting! Another possible reason you can't anywhere is that >> you're dumb. We know the folks at Alltel thought so! That's >> why they fired you! What have you been doing for a living since >> then? > Outing personal information on Usenet is NEVER permissible except in the > most extreme cases where a serious crime is involved. > Go away. You are not welcome here. But you send out your personal information with every one you post: === Subject: Re: JSH: Prime counting should be easier sv3-mYNiloOFQHMlB/vg8L1CyAn5P6RVXKpd+cjRiILKwIP/NooUwMeAF0fNYN7MHM+BcBcIhtgv ze8f+q1!8sW41TCAcQGOJ84qZXuiWwVP2r7Q/WkkNN57qjNap2fjv+P5L8FlYFuxoBvdiABajnIXf /yD0iCD!ULZxw/+ZnFHg78Zzt0s0EmMRzHy5ZkWzVar7v4S7 properly === Subject: Re: JSH: Prime counting should be easier >> I don't need the non-polynomial factorization result. >> I think that the reason I can't get anywhere with my prime counting >> research is that mathematicians are worried about letting all my >> research in, so if I didn't have that find, then maybe I'd be making >> progress. >> Interesting! Another possible reason you can't anywhere is that >> you're dumb. We know the folks at Alltel thought so! That's >> why they fired you! What have you been doing for a living since >> then? > Outing personal information on Usenet is NEVER permissible except in the > most extreme cases where a serious crime is involved. > Go away. You are not welcome here. > But you send out your personal information with every one you post: === > Subject: Re: JSH: Prime counting should be easier > sv3-mYNiloOFQHMlB/vg8L1CyAn5P6RVXKpd+cjRiILKwIP/NooUwMeAF0fNYN7MHM+BcBcIhtgvz e > 8f+q1!8sW41TCAcQGOJ84qZXuiWwVP2r7Q/WkkNN57qjNap2fjv+P5L8FlYFuxoBvdiABajnIXf/y D > 0iCD!ULZxw/+ZnFHg78Zzt0s0EmMRzHy5ZkWzVar7v4S7 > properly Hmmm ... I'm a Comcast user. Well there's shame enough in that ... === Subject: Re: JSH: Prime counting should be easier >> In what way does his employment status add to his credibility? > > The same way that being a veteran adds to his credibility. > You care to explain that? > Did anyone here claim his veteran status is a reason to accept his > mathematical arguments? Or to reject them? Actually he has introduced it in the discussions. Het had (being a veteran) easy access to the army and that would help him getting his point across. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Prime counting should be easier <87slol9zv0.fsf@phiwumbda.org> <443a9c94$0$79787$892e7fe2@authen.yellow.readfreenews.net> <87ek059rit.fsf@phiwumbda.org> In what way does his employment status add to his credibility? > The same way that being a veteran adds to his credibility. > You care to explain that? > Did anyone here claim his veteran status is a reason to accept his > mathematical arguments? Or to reject them? I belive it was JSH himself who first brought up the subject of his being a veteran to enhance his credibility. > -- > Jesse F. Hughes > The Cantorians are conducting a campaign of psychological warfare > against humanity. > -- David Petry, on why set theory is evil. === Subject: Re: JSH: Prime counting should be easier >> if he is unemployed and just making stuff up at the computer, that is >> one >> thing. >> If he is fully employed that adds to his credibility. > Bullshit. Let his mathematics (and rants) speak for themselves. > Mathematical arguments don't require pedigrees. > If you don't believe he's credible and choose not to go through the > arguments, fine. You have no duty to do so. But this silliness of > asking for a public account of his work history is pathetic. >> you are quite wrong, sir. >> What then is the point of a Resume? > Sorry, are you deciding whether to accept his arguments or offer him a > job? Would it make any difference if his Resume had Senior Math Professor at MIT or Sandgrain Quality Inspector at Galveston Beach ? >> If he worked for Bell Labs, or MIT Math department, or had a PHD in Math, >> people would tend to believe him more. > This fact has nothing to do with validity of his arguments. there is no such thing as a JSH validity of argument That is his point, not to have any. >> I think he should post his resume, his vita, to provide additional >> weight >> to his credibility. > Provide additional weight to his credibility? your right, guess he could eat 70 hotdogs and add to his credibility as well. > Anyway, I think that you should post your own, to provide additional > weight to your requests. Maybe also you could post a transcript > showing you've taken a course in critical thinking. You should post, no we all should post, we have taken sufficient Engrish courses to understand each other. > Mathematical arguments are either valid or not. They don't depend on > employment histories. With Mathematical Proofs that is so, but arguments are open ended, and experience histories count, That is why he needs to post his resume in this news group to end all arguments on weather he is just a troll or just fishing around, or he has the intellegence to really discover something. === Subject: Re: JSH: Prime counting should be easier <87slol9zv0.fsf@phiwumbda.org> <443a9c94$0$79787$892e7fe2@authen.yellow.readfreenews.net> <87hd519rl2.fsf@phiwumbda.org> <443ab0d3$0$51839$892e7fe2@authen.yellow.readfreenews.net> <87k69x882e.fsf@phiwumbda.org> <443adbb9$0$22410$892e7fe2@authen.yellow.readfreenews.net [...] > Mathematical arguments are either valid or not. They don't depend on > employment histories. > With Mathematical Proofs that is so, but arguments are open ended, and > experience histories count, > That is why he needs to post his resume in this news group to end all > arguments on weather he is just a troll or just fishing around, or he has > the intellegence to really discover something. His posts themselves are enough to determine that he doesn't have anything new. Actually, I looked at his prime-counting function, and it is correct. However, it is not new, and it is not an improvement on the existing algorithm. (Actually, he had made it more inefficient.) If JSH had the mathematical background, he would have been able to determine this on his own. If JSH had the mathematical maturity, he wouldn't be harping about this, many years on. --- Christopher Heckman === Subject: The Impossible Hexahedron It is known that from four given side-lengths a,b,c,d > 0 that satisfy 2*max(a,b,c,d) < a + b + c + d, one can form (up to congruence) exactly three different concircular quadrilaterals, and these quadrilaterals have equal areas and radii of circumcircles. The following problem resulted from an attempt to 'build a three-dimensional object' out of such quadrilaterals: Apart from the trivial example - cubes and other rhombic hexahedra - do there exist a hexahedra ABCDEFGH, with planar faces ABCD, EFGH, ABFE, BCGF, CDHG, DAEH, whose side lengths satisfy: AB = FG = DH = a BF = CD = EH = b AD = EF = CG = c BC = GH = AE = d ? I figure that such a construction is probably impossible if one demands in addition, that all six faces of the hexadedron be concircular - unless we are in the case a = b = c = d. Any hints and suggestions are welcome! === Subject: Re: Polynomials, general factorization, distributive property <090420060113268337%plsperry@sc.rr.com [...] > > f(x) + 7 = 4 - x + sqrt(9 - x - 6x^2) > > g(x) + 1 = 4 - x - sqrt(9 - x - 6x^2) > > (f(x) + 7)(g(x) + 1) = (4 - x)^2 - (9 - x - 6x^2) > = (16 - 8x + x^2) - (9 - x - 6x^2) > = 7 - 7x + 7x^2 > = 7(1 - x + x^2) > = 7*Q(x) > > You shouldn't object to this; after all, it's an instance > of YOUR nonpolynomial factorization. > Yes, but you don't have the crucial rule that f(0) = g(0) = 0, as the > value of the functions is ambiguous at x=0. > [...] > Find another example where the functions equal 0 at x=0 without > ambiguity. > What about f(x) + 7 = 4 - x + |sqrt(9 - x - 6x^2)| and > g(x) + 1 = 4 - x - |sqrt(9 - x - 6x^2)|? > -- > Paul Sperry > Columbia, SC (USA) Do you think that's really different from the original? The issue here is that the square root gives two answers, so for instance for sqrt(4) both 2 and -2 are correct. For most mathematics people can throw away the -2 and happily believe that the sqrt(4) is just 2, but with my research it changes the answer! So just saying the square root has only one answer doesn't work, no matter what tricks you try. The reality is, what the argument is about, is that the square root returns TWO VALUES so sqrt(4) has 2 and -2 as answers as (-2)(-2) = 4. Now you can say take the positive, but it doesn't change the square root. That is, no matter what you say, or how people define it, it is still true that (-2)(-2) = 4. The -2 solution cannot be destroyed. Get it yet? Or are you people going to argue over trivialities until the day you die? James Harris === Subject: Re: Polynomials, general factorization, distributive property Regarding: : > What about f(x) + 7 = 4 - x + |sqrt(9 - x - 6x^2)| and : > g(x) + 1 = 4 - x - |sqrt(9 - x - 6x^2)|? : Do you think that's really different from the original? Yes it is different. The original didn't have the absolute values wrapped around it. At present it satisfies all of your hypotheses yet does not fit your conclusion. In mathematics this is called a counterexample. : The issue here is that the square root gives two answers, so for : instance for sqrt(4) both 2 and -2 are correct. Yes, and this is taken care of by the example. Stick to the mathematics, James, test the above examples in light of your claim. This is not a social game and if you can't argue against the mathematics, don't bring up a straw man in order to divert attention from your failings. It's all in the math. Justin === Subject: Re: Polynomials, general factorization, distributive property > The issue here is that the square root gives two answers, so for > instance for sqrt(4) both 2 and -2 are correct. > For most mathematics people can throw away the -2 and happily believe > that the sqrt(4) is just 2, but with my research it changes the answer! > So just saying the square root has only one answer doesn't work, no > matter what tricks you try. Do you understand the notion of a convention? That everybody in the world recognizes that there are two real numbers satisfying x^2 = 2; and that by convention, we say that the symbol sqrt selects the positive one. Nobody has forgotten or thrown away anything. The solution set of x^2 = 2 is the set {-sqrt(2), sqrt(2)}. You have no research that changes this. === Subject: Re: Naive tangent bundle space question ... >The union of all these tangent spaces is a tangent bundle >which is also a space. >> You have to be careful talking about the union of all these >> tangent spaces. >I will go back and reread the section in light of your notes >but that specific claim was (part of what is) driving me >crazy. >Both the word union in the text description and the >Union symbol for the equation version were used (separately). >The source seemed adamant about this point -- if that is >the key element you are correcting then I must resolve my >problem with this book. The source of your problem is not the union, it's what you mean by all these tangent spaces. There are various really different things that are called the tangent space to a manifold at a point. Much of the time it's acceptable, even helpful, to confound the various different things, since they can be naturally identified (james dolan can tell you what I mean by that better than I can); but this is one case when (at least at first) it's unhelpful, if not unacceptable. === Subject: Re: Naive tangent bundle space question > On a differential manifold there is a tangent space at > every point. > The union of all these tangent spaces is a tangent bundle > which is also a space. > I am having a very hard time getting my head around > what such tangent bundles look like or what their > meaning AND VALUE (usefulness) would be. > (Applications are probably further along in my books > but I tend to get stuck a bit at this point.) > It would (naively) seem that the tangent bundle space > for a 2-Sphere would be all of R^3 except the interior > of the Sphere. Is this correct? > But then it would seem the tangent bundle space of a > 2-Torus (genus 1) would be all of R^3 since there is > some tangent vector at SOME point that can 'reach' > all of R3 > What's the point (of the tangent bundle concept)? > I can memorize the definitions but they just seem > arbitrary at this point. A tangent space to a point on a manifold is a vector space attached to that point in some abstract manner - the notion that the manifold is embedded in a higher dimensional vector space can be misleading. So when the author says union he means disjoint union because the vector spaces at two different points are completely different. Thus the tangent bundle to a manifold of dim n is 2n dimensional. For this reason you are going to have a really hard time fully visualizing the tangent bundle to any 2d manifold. The tangent bundle to the circle can be represented as an infinite cylinder in 3-dim space, it being a 2-dim manifold. === Subject: Re: Naive tangent bundle space question >> On a differential manifold there is a tangent space at >> every point. >> The union of all these tangent spaces is a tangent bundle >> which is also a space. that point in some abstract manner - the notion that the manifold is > embedded in a higher dimensional vector space can be misleading. So when > the author says union he means disjoint union because the vector > spaces at two different points are completely different. due to James, Lee, and a private email message from another very helpful responder. I *WAS* trying to put them into a (new) continuous union space. > Thus the tangent bundle to a manifold of dim n is 2n dimensional. For And even though I heard this, and knew this intellectually it was getting warped by trying to merge all of those tangent spaces into a continuous union (rather than a space made of disjoint subspaces.) > this reason you are going to have a really hard time fully visualizing the > tangent bundle to any 2d manifold. > The tangent bundle to the circle can be represented as an infinite > cylinder in 3-dim space, it being a 2-dim manifold. Ok, let me see if I have this (correct me if I blow it): It's a cylinder because it is a (disjoint) union of the tangents at every point on the S-2, BUT we just (mentally) twist them all so they line up like the fibers of bamboo cylinder pointing along the same axis. They are disjoint, i.e., like separate fibers, but we can stack or align, or BUNDLE them into a cylinder -- of infinite length since each fiber of the bundle is an infinite line (a single tangent). IF the above restatement is (informally) correct then perhaps this is the reason these things are called BUNDLES and also the reason that some types of bundles are called Fiber bundles? -- Herb Martin === Subject: Re: Naive tangent bundle space question >On a differential manifold there is a tangent space at >every point. >The union of all these tangent spaces is a tangent bundle >which is also a space. > A tangent space to a point on a manifold is a vector space attached to >>that point in some abstract manner - the notion that the manifold is >>embedded in a higher dimensional vector space can be misleading. So when >>the author says union he means disjoint union because the vector >>spaces at two different points are completely different. > due to James, Lee, and a private email message from > another very helpful responder. > I *WAS* trying to put them into a (new) continuous union > space. >>Thus the tangent bundle to a manifold of dim n is 2n dimensional. For > And even though I heard this, and knew this intellectually it was > getting warped by trying to merge all of those tangent spaces into > a continuous union (rather than a space made of disjoint subspaces.) >>this reason you are going to have a really hard time fully visualizing the >>tangent bundle to any 2d manifold. >>The tangent bundle to the circle can be represented as an infinite >>cylinder in 3-dim space, it being a 2-dim manifold. > Ok, let me see if I have this (correct me if I blow it): > It's a cylinder because it is a (disjoint) union of the tangents > at every point on the S-2, BUT we just (mentally) twist them > all so they line up like the fibers of bamboo cylinder pointing > along the same axis. > They are disjoint, i.e., like separate fibers, but we can stack > or align, or BUNDLE them into a cylinder -- of infinite length > since each fiber of the bundle is an infinite line (a single tangent). > IF the above restatement is (informally) correct then perhaps > this is the reason these things are called BUNDLES and also > the reason that some types of bundles are called Fiber bundles? Yes, I think you have it. Also, there is a manifold structure upon the tangent space bundle, and the manifold structures in the two cases are identical. (For example, it is possible to put a rather strange manifold structure on this disjoint union which makes it look a bit like a Mobius strip, if you get my meaning - you won't be able to embed this into three space, but you may well get what I am talking about.) Stephen === Subject: Re: Naive tangent bundle space question >> They are disjoint, i.e., like separate fibers, but we can stack >> or align, or BUNDLE them into a cylinder -- of infinite length >> since each fiber of the bundle is an infinite line (a single tangent). >> IF the above restatement is (informally) correct then perhaps >> this is the reason these things are called BUNDLES and also >> the reason that some types of bundles are called Fiber bundles? > Yes, I think you have it. > Also, there is a manifold structure upon the tangent space bundle, and the > manifold structures in the two cases are identical. I don't see the above distinction (or maybe meaning) manifold structure upon the tangent space bundle versus the 'OTHER' of the two manifold structures. Where are there two manifold structures? > (For example, it is possible to put a rather strange manifold structure on > this disjoint union which makes it look a bit like a Mobius strip, YES, you have anticipated my NEXT question as soon as it seems that I understand the basic tangent bundle concept.... In Penrose's Road to Reality he spends quite a bit of time trying to explain Fiber bundles that use a Moebius structure instead of the cylinder (I understand Moebius strips just never understood what he was getting at in terms of fiber bundles OR why OR even WHERE he pulled that from....) Is this merely a CHOICE of manifold structure that is an alternative to a cylinder (because you can do that by choosing to align the fibers with a flip) or is there some more fundamental explanation of the source of the Moebius strip structure on the bundle? Probably either way, but especially if it just a choice then is there some benefit to this choice in either Quantum theory, Relativity, or other physics? > if you get my meaning - you won't be able to embed this into three space, > but you may well get what I am talking about.) I think so, but why can't I embed this in 3-space? Is it due to the infinite length causing problems when we try to insert the twist or to some other reason that I am missing? The Moebius replacement for the cylinder would still be a 2-surface in 3-space, would it not? -- Herb Martin === Subject: Re: Naive tangent bundle space question > YES, you have anticipated my NEXT question as soon as > it seems that I understand the basic tangent bundle concept.... > In Penrose's Road to Reality he spends quite a bit of time > trying to explain Fiber bundles that use a Moebius structure > instead of the cylinder (I understand Moebius strips just never > understood what he was getting at in terms of fiber bundles > OR why OR even WHERE he pulled that from....) > Is this merely a CHOICE of manifold structure that is an > alternative to a cylinder (because you can do that by choosing > to align the fibers with a flip) or is there some more fundamental > explanation of the source of the Moebius strip structure on the > bundle? Yes, it is just a choice. > Probably either way, but especially if it just a choice then > is there some benefit to this choice in either Quantum theory, > Relativity, or other physics? I don't know. >>if you get my meaning - you won't be able to embed this into three space, >>but you may well get what I am talking about.) > I think so, but why can't I embed this in 3-space? > Is it due to the infinite length causing problems when > we try to insert the twist or to some other reason that I am > missing? Yes, it is the infinite length that causes the problems, nothing else. I think you got exactly what I am saying. > The Moebius replacement for the cylinder would still be > a 2-surface in 3-space, would it not? I don't see how to get the embedding because all the fibers of the bundle will get tangled up because of the infinite lengths. But I might be wrong. === Subject: Re: Naive tangent bundle space question > Yes, it is just a choice. >> Probably either way, but especially if it just a choice then >> is there some benefit to this choice in either Quantum theory, >> Relativity, or other physics? > I don't know. Maybe it is related to Spinors where one must circle twice (4pi) to get back to the same value or something similar....maybe it was just a (poorly explained) example. Or maybe I was so lost that the explaination whooshed right by me... >> The Moebius replacement for the cylinder would still be >> a 2-surface in 3-space, would it not? > I don't see how to get the embedding because all the fibers of the bundle > will get tangled up because of the infinite lengths. But I might be > wrong. I don't know enough to qualify for an opinion (yet) but your statement seems perfectly reasonable; my thinking was that if we put the flip in (perhaps by going outside the 3-space) then it is just there and can live quite well in 3-space but I certainly see how it might not be possible to perform the twist. I don't know topology rules well enough to make any type of claim though -- for now I can accept it either way. others who stuck in here with me on this. -- Herb Martin === Subject: Re: Naive tangent bundle space question >A tangent space to a point on a manifold is a vector space attached to >that point in some abstract manner - the notion that the manifold is >embedded in a higher dimensional vector space can be misleading. So >when the author says union he means disjoint union because the >vector spaces at two different points are completely different. Well, now. Yes, it's a disjoint union, in the sense that its underlying set is the disjoint union (coproduct?) of the underlying sets of the various vectorspaces. But it's misleading to Herb (I think) to say the vector spaces at two different points are completely different. They're completely *disjoint*, that's all. The vectorspaces are topological spaces, not just sets, and the tangent bundle is NOT the coproduct in the category of topological spaces (much less of vectorspaces); it's (no doubt) some kind of fiber product. (Dolan, help me out here, I'm sinking fast.) I mean, I wouldn't call R^2 the disjoint union of the vertical lines {x}times R (unless I wanted to give it the lexicographic order topology, maybe); would you? Lee Rudolph === Subject: Re: Naive tangent bundle space question >>A tangent space to a point on a manifold is a vector space attached to >>that point in some abstract manner - the notion that the manifold is >>embedded in a higher dimensional vector space can be misleading. So >>when the author says union he means disjoint union because the >>vector spaces at two different points are completely different. > Well, now. Yes, it's a disjoint union, in the sense that its > underlying set is the disjoint union (coproduct?) of the underlying > sets of the various vectorspaces. But it's misleading to Herb (I > think) to say the vector spaces at two different points are completely > different. They're completely *disjoint*, that's all. but the correction is also helpful. The key for me was realizing that this is a disjoint set of spaces (the tangent spaces) that can be bundled into a single larger space (not separate space now, but disjoint elements of the larger space) that was the key. I was naively trying to make this all a continuous/smooth (whatever the technical word is) bundle space. I now understand (please confirm) that since these are disjoint subspaces that comprise the bundle space it is permissible to just rotate/align/bundle them into a larger ABSTRACT space conceptually. > The > vectorspaces are topological spaces, not just sets, and the > tangent bundle is NOT the coproduct in the category of topological > spaces (much less of vectorspaces); it's (no doubt) some kind of > fiber product. Cool. I think this and what I worked through elsewhere confirms my suspicious that this leads naturally to the fiber bundles and WHY they are called FIBER bundles. > (Dolan, help me out here, I'm sinking fast.) > I mean, I wouldn't call R^2 the disjoint union > of the vertical lines {x}times R (unless I wanted to give it > the lexicographic order topology, maybe); would you? Either I am REALLY lost or you guys are helping me immensely. I have spent a couple of weeks on and off trying to get this straight -- and not knowing enough to know what to ask. -- Herb Martin === Subject: Re: Naive tangent bundle space question > I mean, I wouldn't call R^2 the disjoint union > of the vertical lines {x}times R (unless I wanted to give it > the lexicographic order topology, maybe); would you? Yes, that is what I intended. It is an abstract, but correct, way to think about it. === Subject: Re: Naive tangent bundle space question >> I mean, I wouldn't call R^2 the disjoint union >> of the vertical lines {x}times R (unless I wanted to give it >> the lexicographic order topology, maybe); would you? >Yes, that is what I intended. It is an abstract, but correct, way to >think about it. It is in no sense a correct way to think about it! For instance, we want to say that a (continuous) vectorfield is a *continuous* section of the tangent bundle. To do that, we must put the topology on it that it gets when considered as I originally suggested that Herb consider it, sitting inside the tangent bundle of the ambient R^n of the manifold. If we put on the lexicographic order topology, that fails terribly. Honestly, now. I *know* you're an analyst, not a topologist, by trade. But can you *really* say with a straight face that R^2 is the disjoint union of its vertical lines? Or, down one dimension, that R is the disjoint union of its singleton subsets? Forgetful functor, heck--that's the Alzheimer functor. Lee Rudolph === Subject: Re: Naive tangent bundle space question >I mean, I wouldn't call R^2 the disjoint union >of the vertical lines {x}times R (unless I wanted to give it >the lexicographic order topology, maybe); would you? >>Yes, that is what I intended. It is an abstract, but correct, way to >>think about it. > It is in no sense a correct way to think about it! For instance, > we want to say that a (continuous) vectorfield is a *continuous* > section of the tangent bundle. To do that, we must put the > topology on it that it gets when considered as I originally > suggested that Herb consider it, sitting inside the tangent > bundle of the ambient R^n of the manifold. If we put on the > lexicographic order topology, that fails terribly. > Honestly, now. I *know* you're an analyst, not a topologist, > by trade. But can you *really* say with a straight face that > R^2 is the disjoint union of its vertical lines? Or, down > one dimension, that R is the disjoint union of its singleton > subsets? Forgetful functor, heck--that's the Alzheimer functor. Well my explanation did seem to help Herb. You might find my description distasteful, but it is nevertheless correct. === Subject: proof regarding 3-SAT I've been told that for any instance of 3-SAT there is an interpretation that satisfies at least 7/8 of the clauses. So if there are m clauses then 7m/8 can be satisfied. I am trying to prove this and need some help getting started. Any help or hints would be appreciated. Tom === Subject: Re: proof regarding 3-SAT > I've been told that for any instance of 3-SAT there is an > interpretation that satisfies at least 7/8 of the clauses. So if there > are m clauses then 7m/8 can be satisfied. > I am trying to prove this and need some help getting started. Any help > or hints would be appreciated. Flip a coin for each variable. --- Christopher Heckman P.S. It has been proven that if 7/8 can be improved, then P = NP. === Subject: Weapons of Math Distraction! Numbers! Weapons of Math Distraction! Numbers! -- Casey === Subject: complex function, congugate and partial derivative Hi All, If f(x,y) is a complex function and f*(x,y) is its congugate. Then will f(x,y) multiplied by the partial derivative of f*(x,y) wrt x be always equal to f*(x,y) multiplied by the partial derivative of f(x,y) wrt x If not, when will it be and when will it not be so. TIA, Ash === Subject: Re: sum of the normal distributions >In one paper I fond some strange definition of the normal distribution >F(x) = sum_{k=-infty}^{infty} e^{(x-r)^2} (I removed some >constatnts) >May anybody explain what sum from minus infinity to plus infinity >stands for? How this distribution relates to the standard normal >distribution? I do not see k in the contents of the summation. If the r is replaced by k, and the exponential is multiplied by a complex number with negative real part, this is an elliptic modular function. The argument x can be any complex number. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: how we think >When you read advanced math and come back to more elementary subjects in the same domain, you have impression that everything is more clear. > It's like after you reach high peaks, the lower ones will be more affordable. Understanding the concepts makes the applications clear. The new math tried to teach concepts before arithmetic, and it worked when those who understood the concepts did the teaching. Teachers who are good at arithmetic have great difficulty with the concepts. I believe it would be easier if the ordinal concept was presented before the cardinal one, as it uses less. Learning variables in full generality, which should be done with beginning reading, is easier that starting out with variables for numbers, and generalizing. The concept is linguistic, but of a type not used by linguists. With this, formulation of equations, etc., of any complexity, becomes trivial. Logic can then follow, and the main rule for solving problems is that the same operation performed on equals gets equal results. ONE rule, not the dozens usually given. Measure and integration, although not some of the proofs, are high school level. Start with discrete; everything else is limits. Limits belong before infinite decimals. Abstract algebra is easier than the ideas of linear algebra, and belong first. Development of real analysis likewise makes calculus trivial; calculus first makes analysis hard. Especially the one who is NOT a mathematician needs to know what derivatives are, not how to calculate them. Learn theory, THEN apply it. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Average path length in random walk >The Wikipedia entry for Random Walk >(http://en.wikipedia.org/wiki/Random_walk) >states the following under Properties: >following rules: > * There is a starting point. > * The distance from one point in the path to the next is a >constant. > * The direction from one point in the path to the next is chosen at >random, and no direction is more probable than another. >The average straight-line distance between start and finish points of a >random walk of n steps is on the order of sqrt{n}, or more precisely, >its asymptote converges to sqrt{2 n over pi} approx 0.8 sqrt{n}. >Determine the coefficients A, B, C so that the curve y= Ax^2 + Bx + C >will pass through the point (1,3) and be tangent to the line 4x + y = 8 >at the point (2,0). Plugging in 1 for x and 3 for y will give you one equation on A, B, C. Knowing that it has to pass through the point (2,0) gives you another: plug in 2 for x and 0 for y. Finally, you want the line 4x+y=8 to be the tangent. The slope of this line is -4. That means that you need the derivative of y to have value -4 at x=2. The derivative of y will be y' = 2Ax + B. Plug in 2 for x and -4 for y' to get a third equation involving A, B, and C (C with coefficient zero). That will give you a system of three linear equation in three unknowns. Solve it to get the values. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Gifted math student >> Have you considered homeschooling him? >If you are thinking of withdrawing him from school completely and having all >lessons at home, my thoughts are that creates more problems than it solves. >There was a case in England about 10 years ago when a young girl, who I think >was 8 at the time, wanted to do maths at Cambridge. But they turned her down, >despite the academic qualifications were as good as they would expect of anyone >entering as ungrad. I think another uni took her. I can't find a reference to >this on the web, so perhaps someone else recalls the details better. >I can't help feeling sorry for someone like that. >Whilst I realise the original poster was not intending this course of action, >children need to grow up and experience life for themselves and perhaps too much > encouragement (pushing?) is a bad thing. It is NOT pushing; it is allowing them to go at their own pace. As for experiencing life, they are NOT like others mentally, they are likely not to see things the way most others see them, and they need to get used to this when they have achieved basic understanding. Anyone who keeps children with their age group must be considered an enemy of education. Marvin Mirsky, in one of his columns, claimed that it is bad socially to keep children with their age group, regardless of the educational aspects. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Gifted math student >(x - 1)(x + 1) = x * x - 1 >The fact that that is the case and that your son was able to find that >is good, bt does he know why? >(x - g)(x + g) = x * x - g >(a). +x * +x >(b). (+x * +g) + (-g * x) = 0 >(c). -g * +b = -(g*g) >a + b + c >For your son to go to a college or university he has to know not just >high school math but all of his subjects that are covered in high >school. He can learn on his own. The high schools teach little. Someone with mathematical ability, NOT what is now mistaught in the high schools, can get admitted, but not by the normal means of application. Get a mathematical scholar on the case. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Gifted math student >The book Mathematics for the Nonmathematician and Mathematics: It's >content, methods, and meaning should greatly help him. I would expect >that he should read a section many times. >It is important to note that logic is important and I think can be >formally studied at age 10 or after. It can be studied at age 6 or earlier by a gifted student. My son used some preliminary materials plus Suppes, _Introduction to Logic_, a college text. My late wife's college text can be used by literate children of any age, but some of the applications may use material which would not be known, and can be omitted without loss. >What I would do is get a book like Arithmetic Refresher, run him >through it, and make sure he has it all. >Then use a book on the SAT test. >Thrid use a book such as How to Prove It. Forget about arithmetic; it is irrelevant to understanding mathematics at any level. It also detracts from any attempt to understand the integers, as it deals only with a particular representation, with no intrinsic properties. A number is a number whether it is in base 10, in base 2, in base 60, in scientific notation in any of these bases, if it is represented as a number of tick marks, or in any other manner. >It also important to note that arithmetic and mathematics is not the >same. Mathematics is two parts: slove equations, and prove theorems. There are other more important parts. Understanding the concepts, which is almost completely eliminated now, and was mostly vague in the past. Peano's Postulates and their development are an excellent introduction to the ordinal concept of the integers, but the cardinal one has to be put into it. I consider trying to do it all with the cardinal concept in the new math to have been a mistake, as there is no way to define finite without at some point using ordinals. The second part is to understand the structure of proofs. Proving theorems is what a mathematician does, but first one must know what it means, and this applies to those who are not mathematicians as well. And for applying mathematics to anything, one needs to understand the concepts to formulate the problem. Non-mathematicians do not have to know how to solve problems, but how to formulate them. Even mathematicians often have to use computer packages to solve problems. >Have you considered homeschooling him? -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Calculus XOR Probability <729c8$44325afd$82a1e228$15581@news1.tudelft.nl> <72900$443a6855$82a1e228$31779@news2.tudelft.nl> > If you have n possibilities all mutually exclusive and equally likely, = > and one > of them must occur, then the chances that any given one will occur is 1= > /n, so > that the probabilities of all will sum to 1, as expected. As n increases > without bound, this relationship is preserved and functions without iss= > ue, given > expected results. So, the question here is, why would you expect this > relationship between n 1/n's summing to 1 to break down at n=3Doo? > > Well put! Why would you expect a result for arbitrary finite n to break > down at n=3Doo? Natura non facit saltus. (Nature does not jump): Leibniz. > Precisely! And from this we can also deduce that the diagonal of a unit > square has length 2. Why would you expect a result for arbitrary finite > n to break down at n=3Doo? Mit der Dummheit k=E4mpfen G=F6tter selbst > vergebens. - Friedrich Schiller > What are you talking about it breaking down at oo for, if it never worked for > any finite n? What do you mean never worked? As far as I can see, it /always/ works: the stair case has length n*2/n = 2 for every finite n, just like the sum n*1/n = 1 for every finite uniform distribution on n. Han argues to go from the finite to the infinite; this implies that when we take the limit, the length should stay being 2, instead of somehow magically jumping around to become sqrt(2). > Something's breaking down, but it's not the uniform probability > distribution of the infinite set, but your handle on what it means to be a > troll. What's really cheeky is those trolls who go so far as to suggest that even though n*1/n = 1 for all finite n, when we get to infinity, the answer somehow magically jumps around and becomes 0 instead of 1. Amazing what some people will believe, eh? === Subject: Re: Calculus XOR Probability <729c8$44325afd$82a1e228$15581@news1.tudelft.nl> cbrown@cbrownsystems.com said: > > What does surprise me is that you don't see the obvious parallel > > between my reasoning that since lim (n*(2/n))=2, therefore the diagonal > > has length 2; and your reasoning that since the lim (n*(1/n))=1, that > > therefore there must exist a uniform distribution on the naturals. > > If you use 2 steps, you still get 2. > If you use a 2 element set, you still get a total probability of 1. > Uh, yeah, except that 1 is the value you want for the probability of the entire > sample. 2 is not the value you want for the diagonal. Han seems unbothered by the fact that 1 is not the value you get from the sum of a countable number of 0's; he simply wants it to be 1. And lo and behold, that is what his argument affirms, just as mine affirms that the length of the diagonal is exactly what I want it to be: 2. > As you use more and more > steps, it looks more and more like a diagonal line, but the length stays the > same as when it did not. > As you use more and more elements, it looks more and more like a > uniform distribution on the naturals, but the total probability stays > the same as when it did not (sic; i.e., whatever that means). > The sum of all outcomes should be 1 if they are mutually exclusive, equally > probable, and one must be chosen. So, it's GOOD that n*1/n equals 1 for all > (finite) n. Similarly, if the sum of all steps should be 2, that's what I get as I travel along the tiny stair steps. So, it's GOOD that n*2/n equals 2 for all (finite) n; in fact, I find it extremely good evidence that in the limit, it will be 2 as well. > You are not approximating the length of the diagonal. That sum is never > equal nor getting closer to the answer. > You are not approximating a uniform distribution on the naturals. That > sum (1) is never equal to or getting closer to the answer (which is > either 0, since the sum of a countable number of 0's is 0; or infinite > by the archimedean property of the reals). > Uh, no. If one of the outcomes is to be selected, then the probability of all > outcomes should sum to 1. The fact that you get a sum of 0 in the infinite case > indicates an error. The sum is correctly 1 for all finite n. The sum is > correctly 1 for infinite n as well, but the the probability of each outcome is > 1/n, which in the infinite case is infinitesimal, not absolute zero. By exactly the same logic: If I travel along the stair steps, no matter how small, then sum of all the stair steps should sum to 2. The fact you seem to think that the distance in the infinite case is actually sqrt(2) indicates an error. The sum is correctly 2 for all finite n. The sum is correctly 2 for infinite n as well, but the length of each stair step is 2/n, which in the infinite case is an infinitesimal hodon [*], not absolute 0. Is the parallel making sense to you yet? Do you see /any/ difference in these /arguments/, besides the fact that one gives an answer you want or should get, whereas the other doesn't? [*] e.g., http://plato.stanford.edu/entries/geometry-finitism/ > So, the question here > is, why would you think it gives a correct answer at oo, if it gives an equally > incorrect answer for all finite nnumber of steps? The limit of the error is not > 0. > So, the question here is, why would you think it gives a correct answer > at oo, if it gives an equally incorrect answer for all finite number of > steps? The limit of the error is not 0. > Uh, excuse me. Maybe I'm getting confused, but maybe not. If there is a set of > n equally likely outcomes, and one is picked, is not the chance of one being > picked equal to 1, and is this not the sum of n individual probabilities of > 1/n? Is the chance of picking any given natural from 1 to a million 1 > millionth? What error is there in n*1/n=1 for any finite n? None, so what is > the parrot act all about? Is this what you call analysis? Please try to answer > the question, sincerely. Alright, I'll stop - but I encourage you to do it for yourself: Just substitute in your argument trillions of tiny steps and What error is there in n*2/n for any finite n? Note that: * No one is claiming anything other than that, in a finite set of n equal outcomes, each outcome has probability 1/n; just as no one would claim anything other than that if there are (finite) n steps, each tread and riser has length 2/n. * No one is claiming that for finite n, n*1/n equals something other than 1, anymore than than any one would claim that n*2/n equals something other than 2. So there's no reason to repeat these assertions; they are accepted. Each of the claims you have made so far for the correctness of Han's argument can be mirrored as an equally valid claim for correctness of my argument; just substitute 2 for 1 and sqrt(2) for 0, and finite number of stair steps for finite set of outcomes. But my /conclusion/ is so obviously false (by appeal to Pythagoras), there must be something wrong with my /argument/. I claim that if you can figure out why the /argument/ is wrong (not just the conclusion), then you will also see why Han's argument is wrong (above and beyond the fact that his conclusion is also independently false, although perhaps not as obviously, by appeal to Archimedes). > > If you have n possibilities all mutually exclusive and equally likely, and one > of them must occur, then the chances that any given one will occur is 1/n, so > that the probabilities of all will sum to 1, as expected. > If you have n steps in the diagonal, the length of each tread/riser is > 2/n, so that the total length will sum to 2, as expected. > So, you EXPECT the diagonal to be equal to 2? Whatever. You're not discussing > things constructively at all. This is much closer to the problem: Why would you think that the answer is anything BUT 2? In fact, the proof shows it /must/ be 2! Equally, why would the sum of a countable number of equal reals not possibly be 1? In fact, the proof shows it /must/ be 1! > As n increases > without bund, this relationship is preserved and functions without issue, given > expected results. > As n increases without bund (sic), this relationship is preserved and > functions without issue, given expected results. > So, you DO think the diagonal of a unit square is 2. I see. And you think Han > and I are cranks..... No, I don't think that the diagonal is really 2. But the logic is /exactly/ the same. Han uses this logic to conclude that there is a uniform distribution on the naturals. Someone might claim that Han's conclusion is obviously wrong, because the Archimedean Principle /clearly/ implies that the sum of a countable number of equal real numbers must either be 0 or infinite. Han replies that the Archimedean Principle therefore indicates a flaw in the real numbers. I uses the same logic to conclude that the diagonal of a unit square has length 2. Someone might respond that my conclusion is obviously wrong, because the Pythagorean Theorem /clearly/ implies that the diagonal is sqrt(2). I reply that the Pythagorean Theorem therefore indicates a flaw in Euclidean geometry. But there' no need to go off on a tangent into some kind of bizzare alternative mathematics here. Our grandiose counterclaims are both just /hot air/. Neither reply is valid; because in both cases, the conclusions /don't follow/ from the arguments to start with. > So, the question here is, why would you expect this > relationship between n 1/n's summing to 1 to break down at n=oo? > So the question here is, why would you expect this relationship between > n 2/n's summing to 2 to break down at n=oo? > I wouldn't, but if you're using some technique to approaximate the diagonal, I > would expect it to get closer with successive iterations, not maintain the same > wrong value. What wrong value? My argument claims that the right value is 2, and that sqrt(2) is wrong (and therefore Euclidean geometry is flawed). Han claims the right value is 1, and that 0 is wrong (and therefore, the standard reals are flawed). We both maintain our respective right values, all the way out to the limit; keeping continuity between the finite and the potential infinite. But you're getting warmer. Consider your use of technique to approximate. Where in my argument do I actually /justify/ my assertion that the stairstep really is a valid technique of approximating the length of the diagonal? Now apply this same question to Han's argument regarding a distribution: Where is it actually /justified/ that approximating a uniform distribution on the naturals can be accomplished by looking at uniform distributions on finite sets of naturals? What does it mean, mathematically, to get closer with successive iterations to a uniform distribution on the naturals? Exactly how far away is a uniform distribution on {1..10} from a uniform distribution on the naturals? This requires /at least/ as much solid justification as claiming that the stair-step approach approximates the diagonal in the limit. And such justifications must be done carefully. For example, I claim that the stairstep approach approximates the diagonal because the error here can be defined as the total area of difference between the stairstep curve and the diagonal. This area certainly approaches 0 as n approaches infinity. So the stairstep curve, in the limit, will have a 0 area difference with the diagonal, and therefore is the same curve; thus they must have the same length, which by Han's logic must be 2; since Nature abhors skipping about like a little girl: Glibnits. QED. 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Good luck, don't forget to follow the instructions correctly and spend > the money wisely!! === Subject: arc cos is cos (arc cox) = arc cos (cos x) for all X and why? === Subject: Re: math development curiosity question <87odzf16dq.fsf@phiwumbda.org> <49lkuiFpf2h6U1@individual.net > Again, I really don't quite get what you're asking. > Just the same old same old - why should we do anything differently than > is already being done, since what we have demonstrably works extremely > well in the areas you seem interested in? > If we're going to transfer our understanding of mathematics to the > computer, then we need to be able to explain to the computer just what > a mathematical statement *means*. That's the issue I have addressed. What possible *meaning* will it is foundational that all mathematics capture uncertainty have to a being for whom uncertainty doesn't exist, since according to you, it lives in a perfectly error free world of computation? Why do you think an AI would find this foundation any more or less believable than Cantor's infinites? And as usual, your addressing of this issue continues to beg the question of how we are to ascertain that a computer actually understands our explanations. If you could actually provide some concrete responses to criticisms of your ideas, you could move your program forward; instead of just repeating the same old, same old. An operational definition of understands our explanations would be highly useful in this regard. === Subject: Re: math development curiosity question >> Hmm. Actually, I'm suggesting that the basic ideas could be taught to >> elementary school children. What you're doing is a little beyond that. >> Will you start by teaching them that the circumference of a circle is >> about 6 times its radius? Or about 44/7 times its radius? >> I would teach them that uncertainty is always a part of measurement, >> which is what we use real numbers for. I would teach them that when >> they compute using finite precision real numbers, they could and should >> keep track of the uncertainty, and I would teach them how to do that. Mathematics is the exact part. The real world is approximated by its mathematical representation. As to how to keep track of the uncertainty, neither you nor anyone else can avoid greatly overestimating it in complex situations. The basic ideas of mathematics are the language of variables, the structure of the integers and its various associated concepts, and logical development. Once they have this, they can really go on. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: math development curiosity question > The breakthrough came only about 40 years ago when J.H. Wilkinson started > with backward error-analysis. What David Petry proposes is forward > error-analysis, but many problems do not have the properties to allow > such [ ... snip ... ] > I think this is a rather crucial issue and would encourage David Petry > to take good notice of it. My own strategy has changed more or less in > concordance with this principle. Make an exact calculation. Look at > the end-result. What happened to the uncertainities in the end-result? > Postprocessing instead of preprocessing. No claim to mimick Wilkinson, > but you get the idea ... You have no understanding of what backward error-analysis entails. It does *not* mean that you do your error-analysis after the fact. You can do both forward and backward error-analysis before, during or after the fact. Forward error-analysis means that you start with values (with or without uncertainties) and you keep track of the propagation of errors (and possibly introduction of new errors) during the computations, and as a result you find what the uncertainties in the final result are. It does not matter whether you use interval methods, statistical methods, or whatever. This is not doable in most linear algebra questions. Doing this gives you blown-up uncertainties in the result, in such a way that (as I more unknowns was impossible (and he thought only about the rounding errors due to the inexact arithmetic). For that only backward error-analysis is possible. You start with the given values and assume they are infinitely precise (i.e. rationals on the computer). Next you do the calculations. Finally you show that the result you have is the *exact* solution to a slightly perturbed initial problem, and derive bounds on the perturbations. And this is the best that can be done. Consider the quite innocious looking: (1.0 0.0) (x) = (1.0) ( 0 1.0) (y) (1.0) where the integer 0 is exact and the others have errors bounded by 0.05 (or in the statistical have a standard deviation of 0.05). When I do my calculations correctly, I find that with interval analysis: x = (1.0 +- 0.7), y = (1.0 +- 0.1) and with statistical methods: x = (1.0 +- 0.13), y = (1.0 +- 0.07) And in both cases in the calculation of x only four operations are involved, but even with the statistical method the error has increased by a factor of nearly 3. The big problem with backward error-analysis is how to interprete the result. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: math development curiosity question > The breakthrough came only about 40 years ago when J.H. Wilkinson started > with backward error-analysis. What David Petry proposes is forward > error-analysis, > I really don't think I've said anything that would imply I'm proposing > only forward error-analysis. You're arguing with a straw man. You argue keeping track of the uncertainties. That is precisely what forward error-analysis does do. You want to do it using statistical methods. Fine. It remains forward error-analysis. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: math development curiosity question The breakthrough came only about 40 years ago when J.H. Wilkinson started > > with backward error-analysis. What David Petry proposes is forward > > error-analysis, > > I really don't think I've said anything that would imply I'm proposing > > only forward error-analysis. You're arguing with a straw man. > You argue keeping track of the uncertainties. That is precisely what > forward error-analysis does do. You want to do it using statistical > methods. Fine. It remains forward error-analysis. What I have been arguing is that (infinite precision) real numbers can be developed by first developing a theory of finite precision real numbers which include a notion of uncertainty, and then considering the limit as the uncertainty goes to zero, and thus avoiding the fantasies of classical set theory. I have been trying hard to avoid any discussion of how to do error-analysis, as that is not germane to the point I am raising. You are arguing with a straw man. === Subject: Re: math development curiosity question > > > > The breakthrough came only about 40 years ago when J.H. Wilkinson started > > with backward error-analysis. What David Petry proposes is forward > > error-analysis, > > > > I really don't think I've said anything that would imply I'm proposing > > only forward error-analysis. You're arguing with a straw man. > > You argue keeping track of the uncertainties. That is precisely what > forward error-analysis does do. You want to do it using statistical > methods. Fine. It remains forward error-analysis. > What I have been arguing is that (infinite precision) real numbers can > be developed by first developing a theory of finite precision real > numbers which include a notion of uncertainty, and then considering the > limit as the uncertainty goes to zero, and thus avoiding the fantasies > of classical set theory. I have been trying hard to avoid any > discussion of how to do error-analysis, as that is not germane to the > point I am raising. You are arguing with a straw man. You argued: > So what I propose is the we (mathematicians) first develop a theory of > finite precision real numbers, which can be represented as rationals > plus a measure of uncertainty. Then a theory of differential equations > can be developed using those finite precision reals, and the classical > theory can be recovered by considering the limit as that uncertainty > goes to zero. The first part is nothing more, nor less than traditional mathematics with forward error-analysis. Consider a simple system of linear result after computations with the uncertainties in the result. But as consider the limit of things that make no sense? Is there a limit in this case? Can you prove there is a limit and that it conforms to the result in traditional mathematics? Do you have any idea of the relation between the initial uncertainty and the final uncertainty? Is that relation continuous? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/