mm-210 === Subject: Re: Math factorization, other side> ...> Notice that you can go from> (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) => 300125 x^3 - 18375 x^2 - 360 x + 22 to (5 g_1 x+ 1)(5 g_2 x + 1)(5 g_3 x + 22) => 300125 x^3 - 18375 x^2 - 360 x + 22 where the g's *should* be algebraic integers, and, of course, Again, why *should*? Well, g_3 clearly is an algebraic integer, but wouldn't you argue that> g_1 and g_2 are not? Now then, why would you so argue?There's no need to argue about it. Nor is there any need to make claims about what the Ôg's *shouldbe*. The facts are amenable to proof.> For others, yes the intelligence test continues.This thread appears to reßect more on *your* intelligence than on others.> James Harris--There are two things you must never attempt to prove: the unprovable -- and the obvious.--Democracy: The triumph of popularity over === factorization, other side Now I've given enough time for quite a few of you to think about the> factorization I talked so much errors. When you fix> them you don't have anything remaining. Just how inordinately refractorily stooopid are you, Harris? Are you> some kind of Black History advocate mindlessly incessantly spewing> dingleberries about Hemet while any North African Arab accused of> being Black would rip your lungs out through your ears for the> insult? Are you so stooopid you cannot even do algebra consistently? http://www.crank.net/harris.html> It's not every braying jackass that gets a whole page at crank.net Well, not surprisingly, Uncle Al failed the intelligence test. Here's more to help the rest of you along. Notice that you can go from (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 to (5 g_1 x+ 1)(5 g_2 x + 1)(5 g_3 x + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 where the g's *should* be algebraic integers, and, of course, f_1(x) = g_1 x, f_2(x) = g_2 x, and f_3(x) = g_3 x, so they're just> linear functions!!! If you realized that simple step, then at least you know you may have> an intellect at a certain level, if you did not, then you are already> out of your league. For those completely lost, but looking for help, notice that 300125 = 2401(125), and of course, 125 is just 5^3. Can you work out what's going on now? Do any of you think you can> give the g's?> James Harris> My math discoveries, found for profit> http://mathforprofit.blogspot.com/Why are you more cocky than usual? There is no need to be this cocky. Mostof the people here don't dispute things as you claim; such as I don't thinkanyone well-versed in mathematics will dispute the distributive property. Itis clear your mathematical education === factorization, other side [JSH]Uncle Al Now I've given enough time for quite a few of you to think about the> factorization pointed out your errors. When you fix> them you don't have anything remaining.True, but JSH never runs out of material. A typical sequence:1. Harris makes some absurd blunder about factorizing a _quadratic_polynomial. Someone reminds JSH of the formula for factorizing quatratics.2. JSH changes the subject to his latest advanced technique, this timeabout factorizing a cubic over Q. Someone points out to Harris that if acubic can be factored, one of the factors is linear! This makes the problemtrivial (which is not to say that JSH got the right answer).3. No problem. JSH looks next at 4 x^3 + a x^2 + b x + c(some specific a, b, and c) and starts shufßing the ring of coefficients.When this latest claim is fully refuted, with more patience than I would beable to give it, we next get:4. JSH starts talking about non-polynomial and non-symmetricfactorizations. These are his own coinages, and refer to notions too cosmicfor anyone except himself to understand...Just look at the titles of some of JSH's threads in October 2001:http://mathquest.com/discuss/sci.math/a/200110-200111LH== === =Subject: Re: Central Limit Theorem (Lindenberg) questionX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>Does anyone here know why the Lindenberg version of the Central Limit>>Theorem implies the traditional Central Limit Theorem?>>I.e.>>Lindenberg version :>>Let X_{n,k} be independent, Surely all you need is that for each n the variables>X_{n,1}, ... X_{n,n} are independent. (Not that that>actually makes a difference, but it eliminates a certain>technicality below.)>E(X_{n,k}) = 0 and Sum (from k = 1 to n) of>>Variance(X_{n,k}) = (s_n)^2>>Then lim P[ (X_{n,1} + .... + X_{n,n}) / s_n <= x] = phi(x) where phi>>is the cumulative normal distribution and <= denotes less than or equal to>>provided that>>lim Sum (from k = 1 to n) of E[ (X_{n,k})^2 * 1(|X_{n,k}| > e*s_n) ] = 0>>for all e > 0>>and E is expectation>>Tradictional version :>>If X_k are i.i.d and E(X_k) = 0 and Var(X_k) = 1, then>>lim P [ (X_1 + ... + X_n) / sqrt(n) <= x ] = phi(x)Took me a second to realize that this was not a stupid>question: It's clear that the second would follow from the >first if Var(X) = 1 implied lim n * E[ X^2 * 1(|X| > e*sqrt(n)) ] = 0for all e > 0, but simple examples show that that's>not so.Ah. ArtfDodger says you stated the Lindenberg theorem wrong;with what he says is the correct version it's trivial toget the iid theorem.>This must be in Chung Looked it up, turns out it's an exercise in Chung.>- if I recall correctly you>define X_{n,k} (1 <= k <= n) to be X_k truncated at>a certain height. Like if you said X_{n,k} = 0>when |X_k| > n^(1/3), X_{n,k} = X_k elsewhere then>the X_{n,k} satisdfy the hypothesis of the Lindenberg>theorem - then you show that the conclusion of the>Lindenberg theorem implies the conclusion of the>traditional version (for example, with my >conjectured definition of X_{n,k} it seems to>me that the variance of >sum(X_k)/sqrt(n) - sum(X_{n,k}/sqrt(n) will tend>to zero...)That may not be exactly right but I'm pretty sure>something like that works (and I'm not sure that>exactly what I said doesn't === %-nbwyHlYn=yh4r^* v|!,o}OFN$97k ?cjI1!x?l>5*VZ)c/:of{IPQt> Does anyone here know why the Lindenberg version of the Central Limit> Theorem implies the traditional Central Limit Theorem?> I.e.> Lindenberg version :> Let X_{n,k} be independent, E(X_{n,k}) = 0 and Sum (from k = 1 to n) of> Variance(X_{n,k}) = (s_n)^2> Then lim P[ (X_{n,1} + .... + X_{n,n}) / s_n <= x] = phi(x) where phi> is the cumulative normal distribution and <= denotes less than or equal to> provided that> lim Sum (from k = 1 to n) of E[ (X_{n,k})^2 * 1(|X_{n,k}| > e*s_n) ] = 0> for all e > 0> and E is expectationThe condition above needs to be emended slightly:lim (s_n)^{-2} Sum (from k = 1 to n) of E[ (X_{n,k})^2 * 1(|X_{n,k}| > e*s_n) ] = 0> Tradictional version :> If X_k are i.i.d and E(X_k) = 0 and Var(X_k) = 1, then> lim P [ (X_1 + ... + X_n) / sqrt(n) <= x ] = phi(x)In the iid mean zero variance 1 case the Lindeberg condition reduces tolim_n E[X_1^2 * 1(|X_1|>e*sqrt{n})] = 0.This is true by virtue of the Dominated Convergence Theorem: the r.v. X_1^2 * 1(|X_1|>e*sqrt{n}) in the above expectation converges to 0 almost surely and is dominated by the integrable r.v. X_1^2.-- === questionDoes anyone here know why the Lindenberg version of the Central Limit>Theorem implies the traditional Central Limit Theorem?I.e.Lindenberg version :Let X_{n,k} be independent, Surely all you need is that for each n the variables> X_{n,1}, ... X_{n,n} are independent. (Not that that> actually makes a difference, but it eliminates a certain> technicality below.)E(X_{n,k}) = 0 and Sum (from k = 1 to n) of>Variance(X_{n,k}) = (s_n)^2>Then lim P[ (X_{n,1} + .... + X_{n,n}) / s_n <= x] = phi(x) wherephi>is the cumulative normal distribution and <= denotes less than or equaltoprovided thatlim Sum (from k = 1 to n) of E[ (X_{n,k})^2 * 1(|X_{n,k}| > e*s_n) ] = 0>for all e > 0>and E is expectationTradictional version :If X_k are i.i.d and E(X_k) = 0 and Var(X_k) = 1, then>lim P [ (X_1 + ... + X_n) / sqrt(n) <= x ] = phi(x) Took me a second to realize that this was not a stupid> question: It's clear that the second would follow from the> first if Var(X) = 1 implied lim n * E[ X^2 * 1(|X| > e*sqrt(n)) ] = 0 for all e > 0, but simple examples show that that's> not so. This must be in Chung -Which book you> define X_{n,k} (1 <= k <= n) to be X_k truncated at> a certain height. Like if you said X_{n,k} = 0> when |X_k| > n^(1/3), X_{n,k} = X_k elsewhere then> the X_{n,k} satisdfy the hypothesis of the Lindenberg> theorem - then you show that the conclusion of the> Lindenberg theorem implies the conclusion of the> traditional version (for example, with my> conjectured definition of X_{n,k} it seems to> me that the variance of> sum(X_k)/sqrt(n) - sum(X_{n,k}/sqrt(n) will tend> to zero...) That may not be exactly right but I'm pretty sure> something like that works (and I'm not sure that> exactly what === (Lindenberg) questionX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>Does anyone here know why the Lindenberg version of the Central Limit>>Theorem implies the traditional Central Limit Theorem?>>I.e.>>Lindenberg version :>>Let X_{n,k} be independent,>> Surely all you need is that for each n the variables>> X_{n,1}, ... X_{n,n} are independent. (Not that that>> actually makes a difference, but it eliminates a certain>> technicality below.)>>E(X_{n,k}) = 0 and Sum (from k = 1 to n) of>>Variance(X_{n,k}) = (s_n)^2>>Then lim P[ (X_{n,1} + .... + X_{n,n}) / s_n <= x] = phi(x) where>phi>>is the cumulative normal distribution and <= denotes less than or equal>to>>provided that>>lim Sum (from k = 1 to n) of E[ (X_{n,k})^2 * 1(|X_{n,k}| > e*s_n) ] = 0>>for all e > 0>>and E is expectation>>Tradictional version :>>If X_k are i.i.d and E(X_k) = 0 and Var(X_k) = 1, then>>lim P [ (X_1 + ... + X_n) / sqrt(n) <= x ] = phi(x)>> Took me a second to realize that this was not a stupid>> question: It's clear that the second would follow from the>> first if Var(X) = 1 implied>> lim n * E[ X^2 * 1(|X| > e*sqrt(n)) ] = 0>> for all e > 0, but simple examples show that that's>> not so.>> This must be in Chung ->Which book and what is chung's first name?Don't recall. Let's see, the book's at the office andI'm not... ah, there's www.amazon.com:A Course in Probability Theory Revised by Kai X_{n,k} (1 <= k <= n) to be X_k truncated at>> a certain height. Like if you said X_{n,k} = 0>> when |X_k| > n^(1/3), X_{n,k} = X_k elsewhere then>> the X_{n,k} satisdfy the hypothesis of the Lindenberg>> theorem - then you show that the conclusion of the>> Lindenberg theorem implies the conclusion of the>> traditional version (for example, with my>> conjectured definition of X_{n,k} it seems to>> me that the variance of>> sum(X_k)/sqrt(n) - sum(X_{n,k}/sqrt(n) will tend>> to zero...)>> That may not be exactly right but I'm pretty sure>> something like that works (and I'm not sure references or proofs,>>Mike>> David C. Ullrich>David C. === questionX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>Does anyone here know why the Lindenberg version of the Central Limit>Theorem implies the traditional Central Limit Theorem?I.e.Lindenberg version :Let X_{n,k} be independent, Surely all you need is that for each n the variablesX_{n,1}, ... X_{n,n} are independent. (Not that thatactually makes a difference, but it eliminates a certaintechnicality below.)>E(X_{n,k}) = 0 and Sum (from k = 1 to n) of>Variance(X_{n,k}) = (s_n)^2>Then lim P[ (X_{n,1} + .... + X_{n,n}) / s_n <= x] = phi(x) where phi>is the cumulative normal distribution and <= denotes less than or equal toprovided thatlim Sum (from k = 1 to n) of E[ (X_{n,k})^2 * 1(|X_{n,k}| > e*s_n) ] = 0>for all e > 0>and E is expectationTradictional version :If X_k are i.i.d and E(X_k) = 0 and Var(X_k) = 1, then>lim P [ (X_1 + ... + X_n) / sqrt(n) <= x ] = phi(x)Took me a second to realize that this was not a stupidquestion: It's clear that the second would follow from the first if Var(X) = 1 implied lim n * E[ X^2 * 1(|X| > e*sqrt(n)) ] = 0for all e > 0, but simple examples show that that'snot so.This must be in Chung - if I recall correctly youdefine X_{n,k} (1 <= k <= n) to be X_k truncated ata certain height. Like if you said X_{n,k} = 0when |X_k| > n^(1/3), X_{n,k} = X_k elsewhere thenthe X_{n,k} satisdfy the hypothesis of the Lindenbergtheorem - then you show that the conclusion of theLindenberg theorem implies the conclusion of thetraditional version (for example, with my conjectured definition of X_{n,k} it seems tome that the variance of sum(X_k)/sqrt(n) - sum(X_{n,k}/sqrt(n) will tendto zero...)That may not be exactly right but I'm pretty suresomething like that works (and I'm not sure thatexactly === self-identityobjects to be equal to themselves. Despite my admittedly casualefforts at understanding the arguments and assertions, I find thetopic to be hard to motivate, and harder to follow.Apparently a big deal is made of the existence of separateentities having the same elements, as though that should createa ripple of consternation among mathematicians. What I findodd about that position (and if I've misrepresented theposition, I would appreciate being set straight), is that we*typically* deal with entities that cannot be distinguishedby a mere listing of the elements. After all, all finite-dimensional manifolds can be viewed as constituting differententities existing on a set of cardinality c. Similarly, allLie groups are among the entities existing on that same set.Virtually all the mathematics I've seen involves entitiesthat are *more* than the underlying sets, Surely we do notidentify the real line with the complex projective plane,nor do we identify the cyclic group of order 4 with theKlein 4-group, despite the seeming similarities at the levelof set theory, which is blind to differences due to topologyand algebra.That given, one might expect that there is a fundamental problemat distinguishing objects in these somewhat richer theories, andI believe that such a fundamental problem exists. Take, for example,the N-dimensional Poincare Conjecture, or the problem of classificationof groups of order N (finite). In each case, the solution exists forsome range of N, and not for others.Each is of fundamental importance, but not of foundational importance:it is a Good Thing to know how to distinguish different things in atheory. It is not essential in the development of a theory, to be ableto discriminate between every possible pair of distinct objects.As an example that I have a passing familiarity with, DifferentialTopology has ßourished during the gradual elimination of cases of thePoincare Conjecture (which addresses detection of the N-dimensionalsphere from among all candidate spaces: it is known to be true for N>=4,and unknown for N=3, although there are claims of a potential proof).The inability to answer the question is A = B?is vexing, but not disabling. Note that there is no question aboutthe answer to is A = A?in Differential Topology. A more puzzling situation may be affordedby consideration of the existence of the 28 differential structureson the 7-dimensional sphere S^7. Of these, only the standard roundsphere embeds in R^8, a number are equivalent (i.e., diffeomorphic)to their negatives (same sphere, different orientation), whileothers aren't (the set forms an abelian group of order 28, in whichchange of orientation corresponds to taking the additive inverse; anelement diffeomorphic to itself with reversed orientation would needto be an element of order dividing 2, and a group of order 28 cannothave all elements being of order 2). In this system of spaces, everyspace is topologically equivalent (I think even PL-equivalent) to thestandard sphere, but with the additional data of the differentialstructure, there are surprising differences.In this latter case, the question is S_i = S_j? (i,j denoting differential structures on S^7)is answered in the differential category, according to the details ofwhich structures are chosen, but the same question in the topologicalcategory becomes is S^7 = S^7?,with the obvious answer.I'm sorry, I got a little long-winded.I guess all I really wanted to say was this:If all the tempest is about the claim that there exist thingsthat are not *just* sets, it seems a bit overblown.Am I totally misreading the === equal to themselves. Despite my admittedly casual> efforts at understanding the arguments and assertions, I find the> topic to be hard to motivate, and harder to follow.> I guess all I really wanted to say was this: If all the tempest is about the claim that there exist things> that are not *just* sets, it seems a bit overblown. Am I totally misreading the argument?>It is important to understand the history of the arguments, and, that is ahard thing to do when the communities involved ascribe to thephilosophy-last-if-at-all approach discussed by Stewart Shapiro in theintroduction of Philosophy of Mathematics.First things first. I cannot speak for John Correy's sense of thesematters, although I respect what he has done and believe that I have someidea how manipulations with formal calculi will always lead to suchresults. As for myself, I am one of the outsiders whose own views areconsidered heretical on sci.logic.For my part, I agree with your statement, If all the tempest is about theclaim that there exist thingsthat are not *just* sets, it seems a bit overblown. The problem, however,is that we are teaching students of mathematics to look to set theory as afoundation for their subject. When they do, they find that their subject isthen grounded in philosophic principles that can seem at odds with how theyunderstand mathematics.To see how the identity predicate plays into this, visit http://plato.stanford.edu/entries/identity-relative/Then, I would also recommend the paper, http://www.cfh.ufsc.br/~nel/nel_prints/KraCoeFEV02.pdfIf you do visit these sites, keep in mind that topology was developing inthe nineteenth century in parallel with logic. There are specific claimsmade by the logic community which are not necessarily held bymathematicians. With regard to the various debates of the modern period,these questions resolve to the question of intuitionism. While recentlyreading posts on the Foundations of Mathematic archives, I ran across onespecifically attributing Brouwer's beliefs to his understanding of ImmanuelKant's Critical Philosophy. In turn, Kant was responding to David Hume'sAn Inquiry Concerning Human Understanding.I can assure you that your seemingly obvious conclusion is not trivial. Thevery nature of Hume's question precludes one from being able to justify yourstatement as easily as you might believe.As another matter of historical reference, classical logic has historicalground in the philosophical community. To give you some sense of thesituation, here is a quote from Husserl concerning his perception of howphilosophical logicians viewed mathematics at the time foundationalism was apopular area of inquiry:The scorn with which philosophical logicians liketo speak of mathematical theories of inference, doesnot alter the fact that the mathematical form oftreatment is in their case (as in the case of all strictlydeveloped theories in the proper sense of this word)the only scientific one, the only one that offers ussystematic closure and completeness and a surveyof all possible questions together with the possibleforms of their answers.There is certainly more than you wanted to know in this post. So, I shouldprobably stop right now. If you do have additional questions, I will beglad to respond to the best of my ability.With regard to my personal interest in these matters, I studied Critique ofPure Reason prior to studying mathematics at the University of Chicago. Ifell into the trap of these identity puzzles (see http://plato.stanford.edu/entries/frege/#puzzles )because of my familiarity with Kant and utter ignorance of Frege.I hope some of this has helped explain some of the complexity behind === people have enough food to survive for 25 days. Given that only 43people are left, for how many days will they now be able to survive(with the same amount of food)? The correct answer is supposed to be27. It seems to me that the question sounds a little easier than itreally is.. I'm puzzled.--Espen === people have enough food to survive for 25 days. Given that only 43> people are left, for how many days will they now be able to survive> (with the same amount of food)? The correct answer is supposed to be> 27. It seems to me that the question sounds a little easier than it> really is.. I'm puzzled.48*25/43 = 27.907-- Alec === polynomial~> hello.......> matrix A => 2 1 0 0> 0 2 0 0> 0 0 1 1> 0 0 -2 4> find minimal polynomial of A> ---------------------------------> my solving process is........> characteristic polynomial of A is> f(x)={(x-2)^2}(x-1)(x-4)The characteristic and minimal polynomial for the block [[ 2 1 ] [ 0 2 ]]is (2-x)*(2-x) - (0*1) = (x-2)^2The characteristic and minimal polynomial for the block [[ 1 1 ] [ -2 4 ]]is (1-x)*(4-x) - (-2) = x^2 - 5*x + 6 = (x-2)*(x-3)so I find characteristic polynomial of (x-2)^3*(x-3),and LCM and minimal polynomial of (x-2)^2*(x-3).> thus> minimal polynomial of A is> m(x)=(x-2)(x-1)(x-4) or m(x)={(x-2)^2}(x-1)(x-4)> thus> i want find m(x) such that m(A)=0 or m(A)=0> but i don't find that.> please.....point out an my error. sir.....thank you> === 0 0> 0 0 1 1> 0 0 -2 4 find minimal polynomial of A --------------------------------- my solving process is........ characteristic polynomial of A is f(x)={(x-2)^2}(x-1)(x-4)It's wrong:f(x)=(x-2)^2*((x-1)(x-4)+2)=(x-2)^2*(x^2-5x+6)=(x-2)^3(x -3)thus m(x)=(x-2)^i(x-3), where i=1, 2 or 3You can now compute (A-2)(A-3), (A-2)^2(A-3) and (A-3)^(A-3) to find === and M(2,R) [was: Re: Complex numbers and 2x2 matrices]> I've written at least twice about this subject in the past, without> receiving any feedback. I'd be glad to read any kind of comment!>Exponentials, and logarithms of invertible elements, exist in any Banach>>algebra. Look up holomorphic functional calculus.I should qualify that. The holomorphic functional calculus exists in>complex Banach algebras, while the usual quaternions are an algebra>over the reals. Of course is certainly OT wrt the OP, I would like to expand to some> extent on the relationships between C (the complex *field*) and the> the *algebra* M(2,R) of 2x2 real matrices.I didn't see your previous post, but here's my 2 cents on your comments inthis post.If you pick up some book on finite dimensional (associative) algebras andstudy it, you will find that this is not so amazing after all. In generalany n-dimensional algebra over a field F which has an identity can beembedded in the algebra M(n,F) of nxn matrices over F. In this case C is a2-dimensional algebra over R and as a consequence can be embedded in M(2,R).Just as the real quaternions H being 4 dimensional can be embedded inM(4,R). As it happens H can also be embedded in M(2,C).See, e.g.Associative Algebrasby Richard S. Pierceand for more on the mysteries of quaternions and generalizations to Cliffordalgebras and other beautiful stuff seeTopological Geometryby I. R. Porteousan online discussion of these matters and more can be found in theinteresting paperThe Octonionshttp://math.ucr.edu/home/baez/Octonions/ octonions.htmlby John Baez(The preliminaries has a brief history of the subject R ->C->H->O that isfun to read and may be informative if you are new to this stuff.)Also recommended online source:Go to Dave Rusin's websitehttp://www.math-atlas.org/welcome.htmland look for 16: === ??hello....find the area of the region between the graphs of the given equations.xy = 4xy = 8x(y^3) = 5x(y^3) = 15hint : u=xy , v=x(y^3) then surface Integral (so I heard . I don't knowfor certain. )answer : 2(ln3)-----------------------i can't solve this problem.........help.....me === v, the Jacobian is 1/(2xy^3) = 1/2v and theintegralArea = Int(x,y) dx dy(means double integral in x and y for the function f(x,y)=1)is nowArea = Int(4 hello.... find the area of the region between the graphs of the given equations. xy = 4> xy = 8> x(y^3) = 5> x(y^3) = 15 hint : u=xy , v=x(y^3) then surface Integral (so I heard . I don't know> for certain. ) answer : 2(ln3)> ----------------------- i can't solve this === Problem with Euler's function> Please can anyone help me? How can i prove that phi(n) is even if and only if n>=3?> I am right in saying:> 1) suppose n>=3, and want to prove phi(n) is even:This can be proved more easily without using the prime factorization of n:For n>1, phi(n) is the number of integers x in the interval [1,n-1] suchthat x is relatively prime to n. Note that x is r.p. to n if and only ifn-x is r.p. to n. So these numbers come in pairs, except for x=1 when n=2.Dean === Halton Arp>{snip} date November 1966, in what I see as a not subtle attempt to skew> reader opinion. As I've said, Dr. Arp has DATA. Check for yourself.> Don't be so stooopis Harris--Arp's anomalies are just that.LOL! Yes, you are correct. And an anomaly is defined as an observationthat doesn't fit standard theory.--greywolf42ubi dubium ibi === Reconsidering Halton Arp> LOL! Yes, you are correct. And an anomaly is defined as an observation> that doesn't fit standard theory.> Glad you are laughing... Arp's statistical analysis didn't discreditthe fact that the object are not connected. === Reconsidering Halton Arp Arp's anomalies are just that> LOL! Yes, you are correct. And an anomaly is defined as an observation> that doesn't fit standard theory. Glad you are laughing... Arp's statistical analysis didn't discredit> the fact that the object are not connected. I'm still glad you got a> laugh out of it.How was the Ôfact' of the object are not connected determined? Do youeven know the meaning of the word Ôanomaly?'--greywolf42ubi dubium ibi libertas{remove planet === You see, Dr. Arp is a scientist, a world renowned scientist and hehas> *data*, real, hard astronomical data, which is more substantive in> disproving the commonly taught Big Bang Theory, than the data usedto> support that theory. Arp's data is presented in the _Atlas of Peculiar Galaxies_,> Astrophysical Journal Supplement Number 123, Volume 14, November 1966. In the Atlas Arp presents galaxies that appeared abnormal. Follow-up> observations showed that some, not all, of the galaxies were in fact> two galaxies that are apparently interacting. What caused the doubt> about the Big Bang was that some of these pairs have very different> red-shifts. If the galaxies are close to each other the different> red-shifts would sound the death knell for expansion and the BB. However, as observing technique has improved we've determined that> most of these pairs are simply close in the line of sight and are> at very different distances. There are a few cases that have not> been elucidated, the necessary obsrvations are, at best, difficult. These remaining cases do not constitute an overthrow of the BB,> to do that would require high quality observations of difficult> objects; big results require big data, obscure, difficult cases> do not provide that. I went to Google, and found a relevant link. with a redshift value of 0.029. Object 1 is a quasar with z = 0.057.> Objects 2 and 3 are quasar-like objects with z values of 0.243 and> 0.391 respectively. As L.97pez-Corredoira and Guti.8errez noted:> Everything points to the four objects being connected among> themselves, but how to explain the different redshifts? (p. L17). How> to explain indeed? Gribbin lamented: That strikes at the foundation> stone of received cosmological wisdom (p. 65). It certainly does! As> case where we once again are experiencing a situation where data get> thrown out if they don't fit the theory. Big Bang cosmology simply> cannot explain Arp's anomalies.> reader opinion. As I've said, Dr. Arp has DATA. Check for yourself. examples have already been disproved.An example, please. AFAIK, the argument against Arp's data is purelystatistical. (The old Ôcoincidence' claim.)> What is left are some vague cases.Meaning there are still many without Ôstandard' explanation.> As Tom Kirke said:> big results require big data, obscure, difficult cases do not provide> that.>I don't consider Kirke an authority. And his statement is both unscientificand fallacious.--greywolf42ubi === Re: Reconsidering Halton Arp> James,> Most professional astronomers would be truly excited to find that the> standard interpretation of redshift is incorrect --- it would be a> huge discovery. They aren't dogmatically holding to the Hubble> interpretation. Rather, they know that this interpretation has a lot> of supporting data and describes many observed phenomena. They also> know that it requires very clear and compelling evidence to make such> a big change to our physical description of the universe.> Most astronomers are not willing to overthrow the standard> interpretation of redshift (and all of the other data that support it)> because of a handful of peculiar galaxies out of the billions of> observable galaxies in the sky.How much data does it take to refute a theory? > When we take deep images, there are many mishapen galaxies --- some> with extended spiral arms, some with long tails, etc. Sometimes there> is no readily apparent neighbor to cause the asymmetry. Sometimes> there is a background galaxy or quasar which appears to be> interacting. But at the moment, there is no compelling data to prove> that these associations happen more frequently than we expect by> chance. Of course, we don't have complete statistics on the> distributions of quasars and galaxies in redshift or space. As we> gather more data, we should be able to determine how significant (or> insignificant) these Arp associations are.Here's some data for you to comment on at the following link:http://perso.wanadoo.fr/lempel/red_shift_NGC_7603_ uk.htmNow then, what about the legacy of today's physicists and astronomers?Fight the truth now and researchers could find their entire workdiscredited, like someone like Hawking could be seen as a dark stainon physics and astronomy.What do we today think of cultures known for fighting science?Remember, science ultimately is what is correct, which is what fitsthe real world, not the fantasy world, or the wished-for world.So now you're confident in your beliefs and in your group, but so wereso many others who now are known to us as backwards people fightingagainst science.We don't have to wait for history though, as we can go with the *data*today.James === comprises two galaxies that look like they areinteracting even though they are at much different redshifts. So whereis your proof that they are indeed interacting? Because of theprojection of looking through 3-space, we expect that objects atdifferent distances can appear close to each other or even on top ofeach other. (During a solar eclipse, it looks like the moon iscolliding with the sun, but of course, they aren't even touching). Thequestion is whether occurences like NGC7603 occur more often than weexpect. (The current answer is we don't think so but we need moreand better datasets to improve this answer.) We could also ask whetherthere is some other technique or observation we could take in order todetermine if some of these objects are truly interacting. Forinstance, if NGC7603a is at redshift z1 and NGC7603b is at redshiftz2, we might expect that the material in the apparent bridge betweenthem would span the whole range of redshifts between z1 and z2. Nowthat would be a pretty compelling dataset!Anyway, I think to overthrow the Hubble interpretation of redshiftwhich has a lot of other observational support, you need to show clearstatistical anomalies in the occurence of these apparent overlaps orsome other test indicating they are === Yes, NGC 7603 comprises two galaxies that look like they are> interacting even though they are at much different redshifts. So where> is your proof that they are indeed interacting? Because of the> projection of looking through 3-space, we expect that objects at> different distances can appear close to each other or even on top of> each other. (During a solar eclipse, it looks like the moon is> colliding with the sun, but of course, they aren't even touching). The> question is whether occurences like NGC7603 occur more often than we> expect. (The current answer is we don't think so but we need more> and better datasets to improve this answer.) We could also ask whether> there is some other technique or observation we could take in order to> determine if some of these objects are truly interacting. For> instance, if NGC7603a is at redshift z1 and NGC7603b is at redshift> z2, we might expect that the material in the apparent bridge between> them would span the whole range of redshifts between z1 and z2. Now> that would be a pretty compelling dataset!Yes it would. I wonder if any work has been done in that area.Anybody know if redshifts have been checked over every piece of thegroup? > Anyway, I think to overthrow the Hubble interpretation of redshift> which has a lot of other observational support, you need to show clear> statistical anomalies in the occurence of these apparent overlaps or> some other test indicating they are truly interacting.Well then, let's expand out the list as readers may think with all myemphasis on one nice to show example that there's not more.Here the problem is with the observations of jets of plasma.Varshni (1974) has shown that quasar redshift is merely an emptynumber without physical significance, quasars are stars within thegalaxy. However, despite the overwhelming amount of contradictorydata, the astronomical community still persists in assuming that theredshift is a valid distance indicator from which they incorrectlydeduce that quasars are extra-galactic. The gross overestimation ofquasar distance has led to spurious paradoxical properties such assuperluminal motion, one of four paradoxes of Kellermann (1972), whichwe now discuss:http://home.achilles.net/~jtalbot/news/3C345. htmlHere is yet more evidence. After all, per Einstein, matter doesn'tmove faster than the speed of light.James === comprises two galaxies that look like they are> interacting even though they are at much different redshifts. So where> is your proof that they are indeed interacting? Because of the> projection of looking through 3-space, we expect that objects at> different distances can appear close to each other or even on top of> each other. (During a solar eclipse, it looks like the moon is> colliding with the sun, but of course, they aren't even touching). The> question is whether occurences like NGC7603 occur more often than we> expect. (The current answer is we don't think so but we need more> and better datasets to improve this answer.) We could also ask whether> there is some other technique or observation we could take in order to> determine if some of these objects are truly interacting. For> instance, if NGC7603a is at redshift z1 and NGC7603b is at redshift> z2, we might expect that the material in the apparent bridge between> them would span the whole range of redshifts between z1 and z2. Now> that would be a pretty compelling dataset!> Yes it would. I wonder if any work has been done in that area.> Anybody know if redshifts have been checked over every piece of the> group?> Anyway, I think to overthrow the Hubble interpretation of redshift> which has a lot of other observational support, you need to show clear> statistical anomalies in the occurence of these apparent overlaps or> some other test indicating they are truly interacting.> Well then, let's expand out the list as readers may think with all my> emphasis on one nice to show example that there's not more.> Here the problem is with the observations of jets of plasma.> number without physical significance, quasars are stars within the> galaxy. However, despite the overwhelming amount of contradictory> data, the astronomical community still persists in assuming that the> redshift is a valid distance indicator from which they incorrectly> deduce that quasars are extra-galactic. The gross overestimation of> quasar distance has led to spurious paradoxical properties such as> superluminal motion, one of four paradoxes of Kellermann (1972), which> we now discuss:> http://home.achilles.net/~jtalbot/news/3C345.html> Here is yet more evidence. After all, per Einstein, matter doesn't> move faster than the speed of light.> James HarrisIt's unlikely that quasars are stars in our galaxy. For one thing,they don't exhibit proper motion. There is a simple geometrical andspecial relativity explanation for the superluminal jets in quasars.The material is not moving faster than c; rather, it is moving at asignificant fraction of c and at an angle close to the line of sight.When these conditions are met, the jet appears to recede from thecentral object at faster than c. You can find the simple derivation inan intro astro book or on many astro web pages. Of course, we don'talways expect the quasar jet to be pointed right at us, along the lineof sight. It is more probable that the jet is pointing some otherdirection, and in fact, most quasars and radio galaxies do not exhibitsuperluminal jets.Do you agree that Cepheid variables establish a redshift-distancerelationship (i.e. the Hubble law) at least for low z? or do youpropose some other explanation for === data does it take to refute a theory? Enough to pursuade sufficiently many scientists that thegrass is greener on the other side.There is no agreed-upon threshold.-- === > We don't have to wait for history though, as we can go with the *data*> today.> Today's data supports the BB === it's settled. The data says that it's not.> I want to emphasize that point! The issue here is that certain people> are working to dismiss alternative theories, other explanations in> favor of what is now the current Big Bang Theory.> That is a natural defensive mechanism for established theories.> If the fundamental assumptions had to be evaluated all of the> time, science would be reduced to discussions about science,> without any time to do real science.Real science is data first.The problem occurs when people start pushing dogma, so they challengehard data because they don't like it, not because it's questionable.The way science works is that DATA comes first.That's it. No need to philosophize. No need for protestations orclaims about discussion versus doing as experimentalist will *happily*test the data endlessly, and that's science.Or in astronomy, people will look MORE not less, so it's like yourargument is pathologically twisted, as when DATA invokes morediscussion, it also pushes more observation!!! > You may find some answers to your questions about how science> works in a good book on the theory of science. I recommend> What is this thing called Science? by Chalmers. You may in> particular find the chapters on Kuhn and Lakatos interesting.Oh, so you think you know how science works from *one* book??!!!What's your background?James === Reconsidering Halton Arp>Message-id: HarrisJames, why have you given up your elite sci.math troll position to go slummingin sci.physics where trolls are a dime === Halton Arp> Real science is data first.Galileo used thought-experiments, and only very little empiricaldata, to device his mechanics. Galileo's mechanics was used topursuade the scientific community that the Earth was not at thecenter of the Universe.I agree with your point that data should not be ignored. However,as data may be inaccurate or incorrect it can be a good idea todefer decisions based of the data until more data has beenobtained or better sensors have been built. To quote Popper: I have always stressed the need for some dogmatism: the dogmatic scientist has an important role to play. If we give into criticism too easily, we shall never find where the real power of our theories lies.How much dogmatism is necessary is probably in a subjectivedecision.>> You may find some answers to your questions about how science>> works in a good book on the theory of science. I recommend>> What is this thing called Science? by Chalmers. You may in>> particular find the chapters on Kuhn and Lakatos interesting.> Oh, so you think you know how science works from *one* book??!!!all answers. Furthermore, reading one well-considered bookon the subject is probably better than zero.I was trying to be helpful to offer a reference which canexplain many of the questions that seem to puzzle you. I have,however, become unsure if you are genuinely interested inanswers, or if you just use the questions for the sake ofargument. If it is the latter, please count me out of thisdiscussion.-- === > Real science is data first.> Galileo used thought-experiments, and only very little empirical> data, to device his mechanics. Galileo's mechanics was used to> pursuade the scientific community that the Earth was not at the> center of the Universe.Galileo became *famous* in his day, and did so by dramaticalyimproving the telescope and introducing it to the military.As for going against the earth-centered ideas of his time, Galileo hadhelp from outside sources, like the book by Copernicus, banned, by,wasn't it the Catholic Church?And his *observations*, notably watching the changing phases of Venusthrough his telescopes, were what ultimately convinced him thatCopernicus had to be right.His *data* was challenged by others who were firmly set on the notionthat the sun went around the earth, and couldn't care less if theDATA, like phases of Venus proved otherwise.Galileo was frustrated enough about the craziness of his society thatcharacter questioning the data represented the Pope.> I agree with your point that data should not be ignored. However,> as data may be inaccurate or incorrect it can be a good idea to> defer decisions based of the data until more data has been> obtained or better sensors have been built. To quote Popper:> I have always stressed the need for some dogmatism: the> dogmatic scientist has an important role to play. If we> give into criticism too easily, we shall never find where> the real power of our theories lies.> How much dogmatism is necessary is probably in a subjective> decision.When data that refutes a theory is ignored based on the notion thatjust because the theory is liked by a lot of people the data must bewrong.If data appears to refute a theory, then you check the data.Ultimately, if the data stands with the state of the art at the time,then the theory has to be junked, or revised.That's because science ultimately has to give the right answers in thereal world.You go with the state of the art at the time as if scientists stoptrusting their instruments, then what do they have?>> You may find some answers to your questions about how science>> works in a good book on the theory of science. I recommend>> What is this thing called Science? by Chalmers. You may in>> particular find the chapters on Kuhn and Lakatos interesting.> Oh, so you think you know how science works from *one* book??!!!> all answers. Furthermore, reading one well-considered book> on the subject is probably better than zero.Your proposal is condescending, especially given that I have a degreein physics, so I have *four* years of training on what is science froma major university.That's part of the problem with dogmatic thinking. You want tobelieve you have the answers, right? So you read something, love it,and then proselytize.Physics students are taught to rely on the data, not on dogma.> I was trying to be helpful to offer a reference which can> explain many of the questions that seem to puzzle you. I have,> however, become unsure if you are genuinely interested in> answers, or if you just use the questions for the sake of> argument. If it is the latter, please count me out of this> discussion.Well, I hope you're being honest and you'll quit chattering becauseyou sound like a Creationist or some other fundamentalist anyway.I get sick of people who believe they have all the answers willing towaste a lot of other people's time pushing their truth.And I notice you deleted out my question asking about your background.Just part of my background is a Bachelor's of Science in physics. Ispent my four years and got my degree.What have you done besides read a === *data* was challenged by others who were firmly set on the notion> that the sun went around the earth, and couldn't care less if the> DATA, like phases of Venus proved otherwise.I was not talking about Galileo's observations (of which youforgot the moons of Jupiter), but about his mechanics, whichwas an essential precursor for the Newtonian mechanics.Galileo made fundamental errors in his mechanics, which couldeasily have been detected by experiments (see Petroski'sDesign Paradigms for further information), but the wentunnoticed by him. Regardless of this, his mechanics did pavethe way for Newton.As for Copernicus, there were many reasons why he did notsucceed in convincing the scientific community of his theory.One of those were the fact that he was unable to explain whyan object dropped from a tower fell parallel to the tower.At the time, inertia was an unknown concept, and therefore itwas believed that the object should move away from the towerif the Earth was rotating. Galileo's mechanics was able toexplain why this did not happen.> Ultimately, if the data stands with the state of the art at the time,> then the theory has to be junked, or revised.So Newton's theory should have been discarded or revised whenthe irregularities in the path of Uranus were discovered?The point I am trying to convey to you is that data may beßawed as well; be it due to problems with the experimentalsetup, the accuracy or precision of sensors, or ourinterpretation of the observations. Furthermore, much data istheory-dependent. For example, without several theories (suchas the Doppler effect) Arp's data would just be pretty dots inthe sky.> That's part of the problem with dogmatic thinking. You want to> believe you have the answers, right? So you read something, love it,> and then proselytize.No, I am trying to widen your horizon on a topic that is notusually taught in Physics by providing relevant references (itis taught in Philosophy.) Your apparent attempts to dodge thosereferences makes me wonder who is the dogmatic person here.> I get sick of people who believe they have all the answers willing to> waste a lot of other people's time pushing their truth.What truth have I been trying to push?> And I notice you deleted out my question asking about your background.My background is irrelevant to this discussion.-- === Arpon the other hand,Arp's stuff doesn't say any thing about Fermat's Last,or your algebra; does it?... I mean,I didn't read that much of his book!mea culpa, about challenging interpretations, dood. > http://perso.wanadoo.fr/lempel/red_shift_NGC_7603_uk.htm > Not that I'd have a problem with that, but I'd like to make sure that> you're *stating* clearly that you're challenging his === I NEED HELP BADLY (sorry, maths not psych)Expires: 28 days>Here is what I have told you many times,>>and what you once more have forgotten:>>time it passes through the accelerating field, regardless of its speed.>> Hmm!>>The gained energy does NOT approach zero (or any other value)>>asymptotically when the speed approached zero, because it is constant!>> Hmm!>>That is how much energy!>>The proof of that is that when the accelerator is in steady state,>>the lost energy in the bends is equal to the energy gained in>>the RF-cavities. The lost energy is radiated as synchrotron>>radiation, which is easy to measure.>> There is nothing sensational about that.>>This lost energy does NOT decrease when the speed of>>the electrons increases, quite the contrary.>> Does that mean Ôit increases'.Yes.>per cycle in the RF-cavities regardless of the speed.>But it will loose more and more energy per cycle in>the bends as the speed increases.>When the two are equal, the accelerator is in steady state.>when it is going at peak efficiency.gaps. During the rest of their travels they lose a little speed.However the question is not about the steady state condition - in which theinput energy goes into radiation. It is why increasingly more energy isbalance radiation. >Thus the gained energy does NOT decrease when the speed>>of the electrons approaches c.>>So whatever you think happens to the field in the RF-cavities>>when the speed approaches c, we KNOW for certain that>> But where does that energy go?Into kinetic energy, of course.>Einstein says: KE = m*c^2*(((1/sqrt(1 - v^2/c^2)) - 1)>Newton says: KE = 0.5*m*v^2Why do you find the second of these equations more natural>than the first?>Two different theories, the first is experimentally confirmed,>the second is experimentally falsified.Once again it is not the KE itself we have to worry about but changes in KE.Once again you get a term with Ôv.dm/dt'.>That's not what NATURE tells us. That's what SR says.Why do you have a problem with that?>Why not simply accept it?Accepting it means virtually nothing. What's the point?>Which proves you WRONG.>>The radiation from an accelerated charge!>>or the fields associated with a moving charge!>>or The ÔBack EMF' concept.>>any less when the speed approaches c.>> that does not conßict with what I said.Uh? :-)I repeat, THAT DOES NOT CONFLICT WITH WHAT I SAID.My argument is about what you claim is an apparent, Ômass increase'.> mean. Is it converted to 1/2mv^2? Or does some of it go into the Ôrelativistic>> mass increase'? If so, how and why?Answered above.Not answered. What is dE/dt?>You know this, and have admitted to know this.>> You have misinterpreted my statements again.Another step in Henry Wilson's eternal cycle of ßeeing>statements ßed before?rubbishPaul>Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)>Here is what I have told you many times,>>and what you once more have forgotten:>>time it passes through the accelerating field, regardless of its speed.>> Hmm!>>The gained energy does NOT approach zero (or any other value)>>asymptotically when the speed approached zero, because it is constant!>> Hmm!>>That is how much energy!>>The proof of that is that when the accelerator is in steady state,>>the lost energy in the bends is equal to the energy gained in>>the RF-cavities. The lost energy is radiated as synchrotron>>radiation, which is easy to measure.>> There is nothing sensational about that.>>This lost energy does NOT decrease when the speed of>>the electrons increases, quite the contrary.>> Does that mean Ôit increases'.Yes.>per cycle in the RF-cavities regardless of the speed.>But it will loose more and more energy per cycle in>the bends as the speed increases.>When the two are equal, the accelerator is in steady state.>when it is going at peak efficiency. gaps. During the rest of their travels they lose a little speed. However the question is not about the steady state condition - in which the> input energy goes into radiation. It is why increasingly more energy is> balance radiation.The answer is that Nature tells us that the KE is: m*(gamma-1)*c^2>>Thus the gained energy does NOT decrease when the speed>>of the electrons approaches c.>>So whatever you think happens to the field in the RF-cavities>>when the speed approaches c, we KNOW for certain that>> But where does that energy go?Into kinetic energy, of course.>Einstein says: KE = m*c^2*(((1/sqrt(1 - v^2/c^2)) - 1)>Newton says: KE = 0.5*m*v^2Why do you find the second of these equations more natural>than the first?>Two different theories, the first is experimentally confirmed,>the second is experimentally falsified.You didn't answer this crucial question, Henry.Why are you willing to accept that KE = 0.5*m*v^2as nature tells us is wrong,and NOT willing to accept KE = m*c^2*(((1/sqrt(1 - v^2/c^2)) - 1)as Nature tells us is correct?> Once again it is not the KE itself we have to worry about but changes in KE.> Once again you get a term with Ôv.dm/dt'.No you don't.You obviously didn't read what I said!The momentum is m*gamma*v where m is invariant.so dp/dt = m*d/dt(gamma*v)The KE is m*(gamma-1)*c^2 where m and c are invariant.so dE/dt = m*c^2*d/dt(gamma)=m*c^2*gamma^3*v*dv/dt That's not what NATURE tells us. That's what SR says.That's what Nature through experiments tells us.That SR says the same is why SR isn't falsified.That NM says otherwise is why NM is falsified.>Why do you have a problem with that?>Why not simply accept it? Accepting it means virtually nothing. What's the point?This illustrates your problem.You refuse to accept that Nature works as it actually does.And again I must ask:Why don't you say that accepting KE = 0.5*m*v^2means vertually nothing. What's the point?>>Which proves you WRONG.>>The radiation from an accelerated charge!>>or the fields associated with a moving charge!>>or The ÔBack EMF' concept.>>any less when the speed approaches c.>> that does not conßict with what I said.Uh? :-) I repeat, THAT DOES NOT CONFLICT WITH WHAT I SAID.> My argument is about what you claim is an apparent, Ômass increase'.You were talking about:radiation from an acceleraed charge!,which ONLY is relevant when the charge is accelerated,and: The ÔBack EMF' concept. which only is relevantwhen there is an EMF (electic field).And now you are insisting that you were NOT talkingabout what happens in the acceleration part, but howaccelerated in an electric field?That doesn't make sense.You are of course free to speculate where the KEConsider the consequence of that!>> mean. Is it converted to 1/2mv^2? Or does some of it go into the Ôrelativistic>> mass increase'? If so, how and why?Answered above. Not answered. What is dE/dt?It === NEED HELP BADLY (sorry, maths not psych)Expires: 28 days>> It is interesting because the term Ôv.dm/dt' is explains the energy increase> that is normally associated with Ô'relativistic' mass increase.>>Of course.>>The question is what is the momentum?>>If the momentum is mv, then the mass _must_ increase with the speed.>>In 1905 this was the accepted definition, and was why Einstein>>said that the mass increased with speed.>>However, the modern approach is to say that>>momentum = m*f(v) where m is the invariant mass>>and f(v) =v/sqrt(1 - v^2/c^2)> It actually supports my argument that this energy really goes into the Ôreverse> field bubble' that forms around a moving charge.>>Why not call your enigmatic bubble a fairy?>>more invisible but massive fairies clings to it.>>When the fairies loose their grip in the bends of the accelerator,>>their mass is transformed to synchrotron radiation.>>Make perfect sense, doesn't it?>> It makes just as much sense as saying Ômass increases' for no apparent reason.>> Where is your supporting PHYSICAL evidence? How can mass simply appear to>> increase? What does Ômass increase' mean Paul?>> You people are really funny.You are not very good at reading, are you? :-)>I am not saying mass is increasing.>I am saying the mass is invariant.>I am saying the momentum is m*v/sqrt(1-v^2/c^2)Which is meaningless unless you provide a reference by which v can be measured.And as you pointed out before, what really matters is the Ôchange in momentum',dp/dt.What does the Ôv.dm/dt' term imply, Paul?How can mass appear to change if it DOES NOT change?A mass increases with speed in one frame can just as easily be a decrease inanother. Please explain. And you don't really have to ask for the physical evidence>for that, do you? Of course not.>You know that this is verified all the time.I don't think the physical implications of what is commonly termed the'relativistic mass increase' are understood at all. I have suggested a perfectly sound theory to explain it. As the chargeappraoches c, a Ôreverse field bubble' forms around it, thus making it harderand harder to accelerate further.The equation for this is no doubt very similar (maybe even identical) to the SRone. And know what, Henry?>Physical evidence doesn't go away by stating:>I see no reason why Nature should behave as it does,> so it doesn't.Don't misquote me please Paul.You are the one who believes rod lengths and clock rates physically change whenthey are accelerated - even though you strongly deny the fact.Paul>Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)>> It is interesting because the term Ôv.dm/dt' is explains the energy increase> that is normally associated with Ô'relativistic' mass increase.>>Of course.>>The question is what is the momentum?>>If the momentum is mv, then the mass _must_ increase with the speed.>>In 1905 this was the accepted definition, and was why Einstein>>said that the mass increased with speed.>>However, the modern approach is to say that>>momentum = m*f(v) where m is the invariant mass>>and f(v) =v/sqrt(1 - v^2/c^2)> It actually supports my argument that this energy really goes into the Ôreverse> field bubble' that forms around a moving charge.>>Why not call your enigmatic bubble a fairy?>>more invisible but massive fairies clings to it.>>When the fairies loose their grip in the bends of the accelerator,>>their mass is transformed to synchrotron radiation.>>Make perfect sense, doesn't it?>> It makes just as much sense as saying Ômass increases' for no apparent reason.>> Where is your supporting PHYSICAL evidence? How can mass simply appear to>> increase? What does Ômass increase' mean Paul?>> You people are really funny.You are not very good at reading, are you? :-)>I am not saying mass is increasing.Read this again:>I am saying the mass is invariant.THE MASS IS INVARIANT.>I am saying the momentum is m*v/sqrt(1-v^2/c^2) Which is meaningless unless you provide a reference by which v can be measured. And as you pointed out before, what really matters is the Ôchange in momentum',> dp/dt. What does the Ôv.dm/dt' term imply, Paul? How can mass appear to change if it DOES NOT change? A mass increases with speed in one frame can just as easily be a decrease in> another. Please explain.There is no point in answering when you cannot read, is there?And you don't really have to ask for the physical evidence>for that, do you? Of course not.>You know that this is verified all the time. I don't think the physical implications of what is commonly termed the> Ôrelativistic mass increase' are understood at all.The physical implication of kinetic energy is quite obvious in the real world.> I have suggested a perfectly sound theory to explain it. As the charge> appraoches c, a Ôreverse field bubble' forms around it, thus making it harder> and harder to accelerate further.> The equation for this is no doubt very similar (maybe even identical) to the SR> one.And the Ôreversed field bubble' around a baseball reallyhurts when you get it in your face, doesn't it?>And know what, Henry?>Physical evidence doesn't go away by stating:>I see no reason why Nature should behave as it does,> so it doesn't. Don't misquote me please Paul.I didn't quote you. I am telling you:Physical evidence doesn't go away by stating:I see no reason why Nature should behave as it does, so it doesn't.> You are the one who believes rod lengths and clock rates physically change when> they are accelerated - even though you strongly deny the fact.Displaying your confusion again, Henry?Unnecessary. We all know === psych)Expires: 28 days>You are not very good at reading, are you? :-)>>I am not saying mass is increasing.Read this again:>I am saying the mass is invariant.THE MASS IS INVARIANT.>I am saying the momentum is m*v/sqrt(1-v^2/c^2)>> Which is meaningless unless you provide a reference by which v can be measured.>> And as you pointed out before, what really matters is the Ôchange in momentum',>> dp/dt.>> What does the Ôv.dm/dt' term imply, Paul?>> How can mass appear to change if it DOES NOT change?>> A mass increases with speed in one frame can just as easily be a decrease in>> another. Please explain.There is no point in answering when you cannot read, is there?>And you don't really have to ask for the physical evidence>>for that, do you? Of course not.>>You know that this is verified all the time.>> I don't think the physical implications of what is commonly termed the>> Ôrelativistic mass increase' are understood at all.The physical implication of kinetic energy is quite obvious in the real world.> I have suggested a perfectly sound theory to explain it. As the charge>> appraoches c, a Ôreverse field bubble' forms around it, thus making it harder>> and harder to accelerate further.>> The equation for this is no doubt very similar (maybe even identical) to the SR>> one.And the Ôreversed field bubble' around a baseball really>hurts when you get it in your face, doesn't it?>And know what, Henry?>>Physical evidence doesn't go away by stating:>>I see no reason why Nature should behave as it does,>> so it doesn't.>> Don't misquote me please Paul.I didn't quote you. I am telling you:>Physical evidence doesn't go away by stating:>I see no reason why Nature should behave as it does,> so it doesn't.> You are the one who believes rod lengths and clock rates physically change when>> they are accelerated - even though you strongly deny the fact.Displaying your confusion again, Henry?>Unnecessary. We all know it.Paul>The SRian self-delusion is really so pathetically amusing.In the beginning there was the Ôrelativistic mass increase'. Now they call itthe Ôrelativistic momentum increase' - which just happens to have a termv.dm/dt hidden hust beneath the surface.Paul, have you ever considered what Ômass increase' might mean? How can mass'increase'? Binding energy? More Ôfield' asociated with it? Objects don't get'heavier' just becasue an observer moves past them you know.You people think you have all the answers but all you are doing is preventingphysics from progressing. The term Ôm.gamma' was around long before relativity.It is probably correct because it matches observation. But it is just anequation. It tells us nothing about the physics behind the Ôapparent massincrease'. If mass REALLY DOES increase with velocity, how might that happen and what isthe significance?Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)Expires: 28 days>How long after the electron enters the field will>>it be affected?>>Randy, haven't you noticed that whenever Paul is in trouble, he always attempts>to hijack the converstaion by diverting the subject down a side track.>>I have noticed this in conversations between you and Paul. Between you>>and me, too. However, it isn't Paul who does not. Nor is it me.>>When does the velocity start changing from 100 m/sec? How>>far does the electron get before this happens?>>That is not the question I have raised.>>I know. It's the question he was asking you, that you didn't answer.>>At least you admit that rather than answer the question, you changed>>the subject. You raised this question in lieu of giving an answer.>> - Randy>> BULL!But you know Randy is right, don't you?You know bloody well the subject of the conversation was>specifically:>How long time does it take before a force act on a charged>is already there?cannot feel the force at the same instant as it enters the field,>but that there must be some action time.Paul, the question I want you to answer is not the one you stated above. Youappear to be obsessed with the charge being at rest.It is, when the charge is moving, is there a time lag due to the charge'smovement?You know bloody well that in order to defend this claim,>YOU tried to hijack the converstaion by diverting the subject> down a side track, namely the following:>| Let us consider a charged sphere somewhere in the universe. It exerts a force>| on every other charge. If we can arrange for it to lose that charge somehow,>| you are claiming that all those forces disappear INSTANTLY.You know bloody well that my only claim was:> as it enters a static electric field.Which was originally made by you to divert the conversation away from the mainissue. The question you are trying to evade involves the possible time lag ofthe field on the charge when it is moving at high speeds.You know bloody well that I NEVER claimed what you>in your side track said I had claimed, namely:>If a charge somewhere in the universe somehow suddenly>disappears, all charges in the universe will instantly be affectedWell, how would you answer that question?In light of this, your statement:> Randy, haven't you noticed that whenever Paul is in trouble, he always attempts> to hijack the conversation by diverting the subject down a side track.>appears rather pathetic.It is so obvious who is in trouble and is desperate to divert>the subject down a side track, that you can be sure that>Randy is not the only person who has noticed it.As the late Louis Savain would say, Randy just wants to kiss your arse.Paul, on the main track>Paul Andsernon - off the rails.Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)>>How long after the electron enters the field will>>it be affected?>>Randy, haven't you noticed that whenever Paul is in trouble, he always attempts>to hijack the converstaion by diverting the subject down a side track.>>I have noticed this in conversations between you and Paul. Between you>>and me, too. However, it isn't Paul who does not. Nor is it me.>>When does the velocity start changing from 100 m/sec? How>>far does the electron get before this happens?>>That is not the question I have raised.>>I know. It's the question he was asking you, that you didn't answer.>>At least you admit that rather than answer the question, you changed>>the subject. You raised this question in lieu of giving an answer.>> - Randy>> BULL!But you know Randy is right, don't you?You know bloody well the subject of the conversation was>specifically:>How long time does it take before a force act on a charged>is already there?cannot feel the force at the same instant as it enters the field,>but that there must be some action time. Paul, the question I want you to answer is not the one you stated above. You> appear to be obsessed with the charge being at rest.> It is, when the charge is moving, is there a time lag due to the charge's> movement?They are going close to c!I think you just derailed, Henry.>You know bloody well that in order to defend this claim,>YOU tried to hijack the converstaion by diverting the subject> down a side track, namely the following:>| Let us consider a charged sphere somewhere in the universe. It exerts a force>| on every other charge. If we can arrange for it to lose that charge somehow,>| you are claiming that all those forces disappear INSTANTLY.You know bloody well that my only claim was:> as it enters a static electric field.> Which was originally made by you to divert the conversation away from the main> issue. The question you are trying to evade involves the possible time lag of> the field on the charge when it is moving at high speeds.I claim that there is no time lag in order to evadethe question if there is a time lag?You are SO funny when your desperate, Henry! :-)You are not on a side track any more.You went off the track.>You know bloody well that I NEVER claimed what you>in your side track said I had claimed, namely:>If a charge somewhere in the universe somehow suddenly>disappears, all charges in the universe will instantly be affected Well, how would you answer that question?See?You want me to follow your irrelevant side track, won't you? :-)>In light of this, your statement:> Randy, haven't you noticed that whenever Paul is in trouble, he always attempts> to hijack the conversation by diverting the subject down a side track.>appears rather pathetic.It is so obvious who is in trouble and is desperate to divert>the subject down a side track, that you can be sure that>Randy is not the only person who has noticed it.Paul, on the track, looking for Henry in === psych)Expires: 28 days>cannot feel the force at the same instant as it enters the field,>>but that there must be some action time.>> Paul, the question I want you to answer is not the one you stated above. You>> appear to be obsessed with the charge being at rest.>> It is, when the charge is moving, is there a time lag due to the charge's>> movement?They are going close to c!I think you just derailed, Henry.Paul you become quite irrational when you are desperate for answers.>You know bloody well that in order to defend this claim,>>YOU tried to hijack the converstaion by diverting the subject>> down a side track, namely the following:>>| Let us consider a charged sphere somewhere in the universe. It exerts a force>>| on every other charge. If we can arrange for it to lose that charge somehow,>>| you are claiming that all those forces disappear INSTANTLY.>>You know bloody well that my only claim was:>> as it enters a static electric field.> Which was originally made by you to divert the conversation away from the main>> issue. The question you are trying to evade involves the possible time lag of>> the field on the charge when it is moving at high speeds.I claim that there is no time lag in order to evade>the question if there is a time lag?You are SO funny when your desperate, Henry! :-)You are not on a side track any more.>You went off the track.>You know bloody well that I NEVER claimed what you>>in your side track said I had claimed, namely:>>If a charge somewhere in the universe somehow suddenly>>disappears, all charges in the universe will instantly be affected>> Well, how would you answer that question?See?>You want me to follow your irrelevant side track, won't you? :-)>In light of this, your statement:>> Randy, haven't you noticed that whenever Paul is in trouble, he always attempts>> to hijack the conversation by diverting the subject down a side track.>>appears rather pathetic.>>It is so obvious who is in trouble and is desperate to divert>>the subject down a side track, that you can be sure that>>Randy is not the only person who has noticed it.Paul, on the track, looking for Henry in the abyss>Is the force on a moving charge equal to qE?How do you know?Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)Expires: 28 days>>Let it be a hot wire in the hole in the electrode.>Thermionic emission of electrons.>When one of these electrons gets out of the hole>and into the static field between the electrodes,>how long time will it take before a force start acting on it?>> I could probably write a whole book answring that.>> However I would conficently say that, before it starts to move, the force is> instant.>>It's settled then.>> Paul, it is by no means settled.>>We agree:>> as it enters a static electric field.>>One can but wonder why it took you so long to realize>>the obvious.>> Read what I said! Can you see the words before it starts to move?>> Do you know what they mean?>> Do you also know what the electron is doing for the rest of the time? IT IS>> MOVING!OK.>So let's change the scenario a little.>We still have two electrodes 1 km apart, with a million>volts potential difference between them.>Behind the cathode, there is an electron gun shooting>electrons with an initial speed v_o through a small hole>in the cathode.>The question is still:>When does a force start to act on the electron?>My answer is:>as it enters a static electric field.What is your answer?The forces from both electrodes are acting on the electron even before itenters the field. They are insufficient to dislodge it from wherever it is.If you consider a hot cathode, then the force probably does act immediately itis emitted.However that is not the main question.[..]>> If a highly charged sphere is moved, say, backwards and forwards between two> electrodes, what happens to the force on those electrodes. Does it change> instantly or is there a time lag?>>Why do you have to ask?>>There is obviously a time lag.>> Why didn't you say so earlier. One can but wonder why it took you so long to>> realize the obvious.>> But are you quite sure of your answer?>> After all, you DID say that electrostatic forces acted instantaneously. Have>> you changed your mind?I find this line of arguing rather irritating.Are you really so desperate that you have to try to make>it appear that I have said something I never did in order>to refute statements I never made?Isn't this technique below you?>Have you no personal integrity to take care of?I probably learnt that tactic from you.> Can you refer to an experiment that shows the time lag?Henry, the physical consequence of moving a charged sphere>backwards and forwards is that an EM-wave is emitted.>Do you have to ask if there are experiments showing that it>propagates at a finite speed?>OK, instead of moving the sphere, try moving the plates.<-|->--------O----------<-|->Now there is NO change in the EM radiation emitted from the sphere. In thiscase, is an effect felt immediately at the plates or is there a time lag?>Paul>Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)>Let it be a hot wire in the hole in the electrode.>Thermionic emission of electrons.>When one of these electrons gets out of the hole>and into the static field between the electrodes,>how long time will it take before a force start acting on it?>> I could probably write a whole book answring that.>> However I would conficently say that, before it starts to move, the force is> instant.>>It's settled then.>> Paul, it is by no means settled.>>We agree:>> as it enters a static electric field.>>One can but wonder why it took you so long to realize>>the obvious.>> Read what I said! Can you see the words before it starts to move?>> Do you know what they mean?>> Do you also know what the electron is doing for the rest of the time? IT IS>> MOVING!OK.>So let's change the scenario a little.>We still have two electrodes 1 km apart, with a million>volts potential difference between them.>Behind the cathode, there is an electron gun shooting>electrons with an initial speed v_o through a small hole>in the cathode.>The question is still:>When does a force start to act on the electron?>My answer is:>as it enters a static electric field.What is your answer? The forces from both electrodes are acting on the electron even before it> enters the field. They are insufficient to dislodge it from wherever it is.Don't be ridiculous!> If you consider a hot cathode, then the force probably does act immediately it> is emitted.And wht the heck has the temperature of the cathode to with ait it?Stop the side tracking, and answer the question, please.The cathode isn't hot. But it has a small hole.It is utterely irrelevant from where the electon comes,but it comes out of the hole with a high speed.The question is still:When does a force start to act on the electron?No more silly evasions, please.> However that is not the main question.It is the only question.Have you forgotten the issue?It is that YOU claimed that there is a delay beforea force start acting on electron when it enters the field.[..]>Henry, the physical consequence of moving a charged sphere>backwards and forwards is that an EM-wave is emitted.>Do you have to ask if there are experiments showing that it>propagates at a finite speed?> OK, instead of moving the sphere, try moving the plates. <-|->--------O----------<-|-> Now there is NO change in the EM radiation emitted from the sphere. In this> case, is an effect felt immediately at the plates or is there a time lag?How do you accelerate the plates? Kick them?Do you feel the kick === BADLY (sorry, maths not psych)Expires: 28 days>What is your answer?>> The forces from both electrodes are acting on the electron even before it>> enters the field. They are insufficient to dislodge it from wherever it is.Don't be ridiculous!> If you consider a hot cathode, then the force probably does act immediately it>> is emitted.And wht the heck has the temperature of the cathode to with ait it?Try running a CRO tube with a cold cathode.Stop the side tracking, and answer the question, please.The cathode isn't hot. But it has a small hole.>It is utterely irrelevant from where the electon comes,>but it comes out of the hole with a high speed.>The question is still:>When does a force start to act on the electron?No more silly evasions, please.I answered that question twice Paul. Can't you read?> However that is not the main question.It is the only question.>Have you forgotten the issue?>It is that YOU claimed that there is a delay before>a force start acting on electron when it enters the field.[..]Readers note: SNIPPER ANDSERNON DOES IT AGAIN.>Henry, the physical consequence of moving a charged sphere>>backwards and forwards is that an EM-wave is emitted.>>Do you have to ask if there are experiments showing that it>>propagates at a finite speed?>> OK, instead of moving the sphere, try moving the plates.>> <-|->--------O----------<-|-> Now there is NO change in the EM radiation emitted from the sphere. In this>> case, is an effect felt immediately at the plates or is there a time lag?How do you accelerate the plates? Kick them?>Do you feel the kick immediately, or is there a lag?Stuck for an answer again eh, Paul?Paul>Henri Wilson. See the Stupidity of === I NEED HELP BADLY (sorry, maths not psych)>Why not call your enigmatic bubble a fairy?>>more invisible but massive fairies clings to it.>>When the fairies loose their grip in the bends of the accelerator,>>their mass is transformed to synchrotron radiation.>>Make perfect sense, doesn't it?>> Neat! Hot fairies.Indeed.>And since synchrotron radiation may appear bluish,>it also supports the notion that invisible fairies are blue.I can only tell my blues by contrast. So where do the unicornscome in?/BAHSubtract a hundred and four for === How many possible combinationsIf this posting appears twice my apologies its the second time I sent it the first appears to have got lost>>can you point to a site that perhaps has a good description> No, but here's a page with a couple examples:> http://members.aol.com/mensanator666/fun/playing.htm> Hi are well written and illuminating.using your example....C(m.n) = m!/((n!)*(m-n)!)How many 8-bit binary numbers have exactly 4 ones?C(8,4) = 8!/(4!)*(4!) = 8*7*6*5*4*3*2 / 4*3*2 * 4*3*2 = 8*7*6*5 / 4*3*2 = 7*6*5 / 3 = 7*2*5 = 70 I used the following to calculate the possible combinations of 16 bits out of 256. in other words How many 256-bit binary numbers have exactly 16 ones?.from your example I got256!/(16!*(256-16)!)I plugged this into maxima (only downloaded yesterday :)and the answer was 10078751602022313874633200my final step was to calculate how many bits was required to store this numberlog2(10078751602022313874633200) = 83.05951932 bits( had to use excel as I couldn't find how to change the base in Maxima )So I can store the original 256 bit pattern into 84 bitsDoes === possible combinations> log2(10078751602022313874633200) = 83.05951932 bits> ( had to use excel as I couldn't find how to change the base in Maxima )> So I can store the original 256 bit pattern into 84 bits> Does that sound reasonable ?> ChrisYou're exactly right. You can encode bit patterns of length 256 thathave exactly 16 ones using only 84 bits (and with some leftover, too). There are many such encodings. One of the more obvious is to takethe set of all bit patterns of length 256 with exactly 16 ones, andtake the set of all bit combinations of length 84, order each of themas if they represented binary integers. Encode the first one oßength 256 to the first one of length 84, the second to the second,and so on. Your above calculations show that there are enough of thestrings of length 84 to accomodate all of the strings of === C^3Technological SurpriseIt has never happened that seemingly abstract advances in mathematical physics have not had practical applications to military and commercial technology. Kaiser's use of physical wavelets, a kind of Penrose twistor technique in complex space-time is of immediate use in the radars and undersea sonars that show detailed shapes of the moving accelerating target not merely blips on a screen as in the old WWII era. The AST in n dimensions is actually a special case of the Radon Transform used in tomagraphy MRI imaging.Kaisers approach to quantum field theory avoids the algebra of canonical commuation rules of second quantized local field operators whose products create renormalization issues. He starts with the abstract Lie algebra of observables and finds unitary representations of the Lie group to go directly to quantum theory of fields. The mother wavelet scale-dependent description of solutions of field equations in complex space-time as a kind of 8-dim phase space allows a basis for coherent states that are eigenstates of the non-Hermitian positive and negative frequency parts of the quantized operator formalism. This clean separation to + & - frequencies fails for quantum gravity due to obstructions from curvature (Hawking).Some of Kaisers assumptions like positive energy and keeping to the mass shell may leave out some important physics like the vacuum structure and near induction Tesla field communication. I am not sure yet if that is a real problem.Kaiser also points out an interesting subtlety in taking the limit c infinity of the 10 parameter Poincare Group. One does not get the Galilean Group of Newtons Mechanics without an additional step.Kaiser also starts in Euclidean signature ++++ and then continues to Light Cone signature +--- in some of his examples saying this route is better than the opposite at least for some problems. Also he seems to have a deeper physical picture for the physics of imaginary time than does Hawking et-al. Thus the physics of the gravitational instanton also important in the self-dual Penrose twistor version of Einsteins gravity may become more intuitive leading to new insights and applications to the practical problem of metric engineering as applications of V.I. Arnolds A-D-E mathematics (Saul-Paul Sirag).arXiv:math-ph/0108006 v1 15 Aug 2001Invited paper presented at the NATO Advanced Research Workshop on CliffordAnalysis and its Applications, Prague, October 30 - November 3, 2000.Communication via Holomorphic Green FunctionsGerald Kaiser1The Virginia Center for Signals and Waveskaiser@wavelets.com _ www.wavelets.comAbstractLet G(xr - xe) be the causal Green function for the wave equation in fourspacetime dimensions, representing the signal received at the spacetime pointxr due to an impulse emitted at the spacetime point xe. Such emission andreception processes are highly idealized, since no signal can be emitted orreceived at a single (mathematical) point in space and time. We present asimple model for extended emitters and receivers by continuing G analyticallyto a function G*(zr - ze), where ze = xe - iye is a complex spacetime pointrepresenting a circular pulsed-beam emitting antenna dish centered at xeand emitting in the direction of ye, and zr = xr - iyr represents a circularpulsed-beam receiving antenna dish centered at xr and receiving from thedirection of yr. The holomorphic Green function G*(zr - ze) represents thecoupling between the emission from ze and the reception at zr. To preservecausality and give nonsingular coupling, the orientation vectors ye and yrmust belong to the future cone V+ in spacetime. Equivalently, ze and zrbelong to the future and past tubes in complex spacetime, respectively. Thespace coordinates of ye and yr give the spatial orientations and radii of thedishes, while their time coordinates determine the duration and focus of theemission and reception processes. The directivity D(y) of the communicationprocess is a convex function on V+, i.e., D(yr + ye) < or = D(yr) + D(yr). Thisshows that the efficiency of the communication can be no better than thesum of its emission and reception components.1Supported by AFOSR Grant #F49620-01-1-0271. December 20, 20011 IntroductionI begin by summarizing earlier work. Physical wavelets were defined in [K94]as wavelet-like bases for spaces of solutions of the homogeneous wave equation(acoustic wavelets) or Maxwell's equations (electromagnetic wavelets). Thiswas motivated by the observation that information is often communicatedby acoustic or electromagnetic waves, and this fact should be taken intoaccount when processing the resulting signals. All such wavelets can beobtained from a single mother wavelet by translations, scaling, rotationsand Lorentz transformations.This part is important in relation to the Desitter Group extension of the Conformal Group.One would also need field equations for gravity waves and possibly for torsion waves from generalized local Einstein field equations.The construction of physical wavelets was based on a holomorphic extensionF*(x + iy) of solutions F(x) to complex spacetime, with the imaginaryspacetime variables y interpreted as singling out approximate directions andfrequencies of propagation. Thus F*(x+iy) is a description of the wave intermediatebetween the spacetime domain (where exact positions and times areknown but no directional or frequency information is given) and the Fourierdomain (where exact directional and frequency information is known but nolocal spacetime information is given). This is an extension to spacetime ofcontinuous wavelet analysis of one-dimensional time signals, whose wavelettransform is intermediate between the time domain and the frequency domainrepresentations.The physical wavelets of the homogeneous wave- and Maxwell equations werethen shown to split into a sum of causal and anticausal wavelets. Essentially,the causal wavelets are holomorphic extensions, in the sense of positive frequencyanalytic signals, of the causal (retarded) Green function, and the anticausal ones are similar extensions of the anticausal (advanced) Green function for the appropriate equation. The causal wavelets are pulsed-beam solutions emitted by disk-like sources. That is, that they represent well directed acoustic or electromagnetic beams that are pulsed in time rather than going on forever. The direction, pulse width, and duration of these beams are determined by the imaginary spacetime variables y. Such objects have appeared previously in the engineering literature under the name complex-source pulsed beams (see Heyman and Felsen, 1989, and the references therein).In this paper we further develop the above analysis by showing that the holomorphic extension of the causal Green function describes not only the emission but also the reception of a pulsed beam, and so represents a communication between the emitting and receiving antenna dishes.Kaiser shows how dynamical fields F are extended to complex space-time using an Analytic Signal Transform whose AST Fw is a wavelet, i.e. eq 14 p. 6 of 48 whereFw = F(x [CapitalEth] sy/u) i.e. multi-scale resolution WAVELET!Z = x [CapitalEth] iyThe Kernel in the AST integral is (s [CapitalEth] iu)^-1s is integrated from [CapitalEth] to + infinityu is imaginary part of complex y.Some provocative excerpts fromPhysical wavelets and their sources:Real physics in complex spacetime_Gerald KaiserCenter for Signals and Waveswww.wavelets.comWaves, wavelets, and complex spacetimeI begin with a brief review of my past efforts to extend classical and quantum theories tocomplex spacetime and interpret the results physically. By that I mean that the imaginaryspacetime coordinates, and any other extras associated with analyticity, are to be understood directly in terms of common observable attributes and not merely as a technical device for proving theorems or exotic higher dimensions inaccessible to mortals stuck in the real world like the poor souls in Platos cave. I have tried not to impose an a priori grand vision but, rather, interpret the imaginary coordinates in each theory by understanding their effects within that theory. Consequently, the interpretations vary somewhat from one theory to another. But they all have in common the following theme. In the extended theory, certain singular points (evaluation maps on fields or wave functions, source points, etc.) become inßated to extended objects. This transformation is determined by analyticity and the particular theory. In every case, the structure of the objects is shaped by the equations of the theory and their degrees of freedom are specified precisely by the complex spacetime coordinates. The real coordinates give thecenter, and the imaginary coordinates the extent and orientation of the object in space andtime. These ideas are similar in spirit to wavelet analysis, where a function of one variable (time, say) is expressed in terms of an additional variable describing the scale or resolution in the first. This analogy goes farther in the treatment of massless than massive fields, since the latter have an intrinsic scale and thus cannot be scaled arbitrarily. For relativistic fields with mass, spacetime orientation includes velocity, and this makes the complex spacetime an extended phase space. The relativistic coherent-state representations for massive Klein- Gordon and Dirac fields constructed in [K77, K78] interpolate between time-frequency and wavelet descriptions, behaving like the former in the nonrelativistic regime and like the latter in the ultrarelativistic one. In fact, there is a very close correspondence between the nonrelativistic limit in physics and the narrow-band approximation in signal theory; see [K90, K94, K96]. Although the results cited in this section are not new, I believe they have acquired some currency because of substantial progress recently in the understanding of the sources associated with retarded holomorphic fields. The new results focus on massless fields, but it is likely that similar computations exist for massive fields where the integrals are more difficult. Sources describe the breakdown of analyticity due to natural singularities and physically necessary branch cuts. What I find especially fascinating is that such branch cuts behave much like real matter. Depending on the theory, they carry charge, mass and spin, and they emit and absorb radiation. In spite of their simple origins, they turn out to have surprising and complex (pardon the expression) properties, the pursuit of which has the feeling of exploring hitherto unknown forms of matter and not merely the mathematical properties of branch cuts. The results of this search have intrigued and inspired me, and I hope to share this excitement with the reader. At any rate, the dismissal of ict in [MTW73] may have been premature. Even in generalrelativity, analytic continuations in time and space have borne some rich fruit, even if thephysical basis of the procedure is often ill-understood. For example, an exquisitely simplegeometric derivation of the Hawking temperature for Schwarzschild black holes is obtained [HI79] by analytically continuing the metric in time, interpreting the Euclidean time coordinate as an angle, and choosing its period to make the horizon a coordinate singularity like the origin in polar coordinates. The reciprocal of the imaginary time period is interpreted in the usual (KMS) way as a temperature, and this turns out to be nothing but the Hawking temperature! I confess that I do not understand this derivation in more than a formal way, but analytic continuation has, in any case, become the main strategy of black-hole thermodynamics, as explained in[Kr03]. The analytic continuation of spatial coordinates also has an honorable history in relativity, having played a major role (and conceptually an equally obscure one) in the discovery of charged spinning black holes by Newman et al. [N65]; see also [N73, NW74, K01a, N02]. And then there are the theories of twistors and H-spaces 11 ConclusionsAnalytic continuations to complex time and complex spacetime abound in physics, although the terminology of Wick rotations is, in my opinion, sometimes used too casually, without any mathematical justification or even any basis for justification (not even wrong). There have been times while reading papers (or even books) on string theory, for example, when was unable to tell whether the author was working in a Euclidean or Lorentzian signature. But even when justified, the extensions are usually regarded as mathematical methods without any particular physical significance. Here is a non-exhaustive list of examples known to me. In the correspondence between quantum field theory and statistical mechanics, the imaginary time (more precisely, its period) is related to the reciprocal temperature. But this is regarded as an analogy between the two theories, albeit a precise and very useful one. To make it more than analogy one might, for example, interpret the complex time as a combination of evolution and thermal parameters for a system in a local equilibrium state, something like the complex combination of the (also incompatible) position and momentum observables occurring in coherent-state representations. (These need not be eigenstates of a corresponding combinations of operators, as they are in the Bargmann-Segal representation. For example, the relativistic coherent states ez (29) do not depend on the existence of covariant spacetime operators, which do not in fact exist within the usual framework. In Wightman field theory, n-point functions are extended to tube domains in their difference variables and powerful methods of complex analysis are used to prove theorems like PCT and the connection between spin and statistics about the original fields in real spacetime [SW64]. There is no attempt to interpret the complex coordinates z = x - iy, although the interpretation of y as (proportional to) an expected energy-momentum in relativistic coherent states, proved for free fields in [K77, K78, K87], extends to general axiomatic fields [K90, Section 5.3]. In constructive quantum field theory [GJ87], the Euclidean region is used to correlate the n-point functions of a given theory by rigorous (Feynman-Kac) path integral methods. Then they are continued back to real spacetime and used to construct interacting fields. Again there is no attempt to interpret complex spacetime because the quantized field exists only in the Minkowskian region while the random field exists only in the Euclidean region. In between, there are only n-point (Wightman) functions. There have been various efforts to represent spacetime as a Shilov boundary of a complex domain (see [G01] and references therein), but I am not aware of any claiming to do physics directly inside these domains. Complex spacetime plays a prominent role in twistor theory [PR86, P87] and the theories of Heaven or H-spaces [HNPT78, BFP80], but again no direct interpretation is generally given to the complex coordinates. To the best of my knowledge, the only examples (aside from the relativistic coherent states and physical wavelets covered here) where complex spacetime coordinates are given a direct physical significance have appeared in the works of Newman et al. [N73, NW74], who have proved the following very intriguing result. Consider an isolated classical relativistic system in ßat spacetime with positive total mass The total angular momentum splits into orbital and spin parts L + s, and L is made to vanish by translating to the center of mass. Similarly, if the system has total charge e =/= 0 and a magnetic moment [Micro], then its dipole tensor is reduced to [Micro] by translating to the center of charge. However: With a further imaginary translation by is/mc, the spin can be made to vanish. Thusspin may be identified with an imaginary center of mass. With an imaginary translation by i[Micro]/e, the magnetic moment can be made to vanish.Thus magnetic moment may be identified with an imaginary center of charge. If the centers of mass and charge coincide, then the spin and magnetic moment can betransformed away simultaneously by an imaginary translation. The necessary and sufficient condition for that is that the gyromagnetic ratio of the system have the Dirac value: [Micro] = (e/mc)s.In the massless case, the world lines with complex center of mass are replaced by a totally null complex plane if the spin (in real Minkowski space) is nonzero. This idea, although proved in ßat spacetime, was inspired by the Kerr-Newman solution to the Einstein equation [N65], which is the universal model for spinning, charged black holes.It was discovered by performing a somewhat mysterious complex coordinate transformation on the spherically symmetric solution with mass and charge (Reissner-Nordstrom) which is, roughly, a general-relativistic version of extending the Newtonian potential from R3 to C3 The Kerr-Newman solution was soon realized to have the Dirac gyromagnetic ratio. Recently, an old debate was re-ignited with A. Trautman whether the Dirac value necessarily depended on the nonlinear character of the equations. Newman settled the question by showing that the Dirac ratio was obtained as well for the linearized solution [N02]. In the related work [K01a], the charge-current distribution for a (real, static) electromagnetic field defined as in [N73] by a holomorphic Coulomb potential was computed and shown to represent a rigidly spinning disk so that the rim moves at the speed of light. This is consistent with the fact that (191)represents the electromagnetic part of the linearized Kerr-Newman black === Change, line 1:> Mathematics is the study of proportion and change.> No, that's not what mathematics is. This line does not belong> there.This is my definition and its beautiful. What's yours? That whichthe math department teaches.> Line 2: Anything we can measure is a dimension. Again, this is an> incorrect use of dimension, and the meaning of measure is> unclear. This line is either content-free or wrong.Its a general definition of dimension and is correct. Maths do notown the word. If biologists grow bugs and counts them over time, thenBugs and Time are the two dimensions they work with. Measure is likemeasure distance with a tape measure, count beans, read a dial, andgenerally find a number.> Line 2: The measurements we take are simply relative differences> in proportion. No, they are not.Oh, yes they are!> Line 2: Proportion is the quality that forms a continuum from zero> to infinity with one being the base to compare the proportion.> Unclear at best, incorrect at worst. I will make it clear; Q:What does two mean if you don't Know what one is? A:Nothing> > Is that what you want? Sort of but I would like you to stick to the good stuff. The FTC.>...It will take several hours of my time, which I> am not at liberty to spend for free. If you are willing to pay, I'll> be happy to provide you with a complete list of errors and suggested> corrections. But I doubt that you are. er,...sorry but I am not at liberty to pay for it.> A more serious problem is that you are claiming that your method is> a more intuitive of understanding the Fundamental Theorem of> Calculus. I disagree. There is no intuition behind M(x) or R(x). They> just simply happen to work. No man, your missing the point, they happen to work (you actuallyagree with me) because of two similar triangles formed by nothing morethan the function and its derivative. And! by division of the twoalone, I get M(x). Wee!> But, if I do not know the fundamental theorem of calculus, why on> Earth would I ever draw the line you suggest? Why on earth would I> ever consider the function M(x), or the function R(x)? What is their> relation to the problem I am ->supposed<- to be working on, namely,> that of finding the area under the curve of f(x)?Good point. Start them with the problem of the area under a curve,show them the Rieman sum route, tell them it gets better because ofthe FTC, Then hit'em with my proof! Boom, a funky area f(x)dx becomesa simple rectangle f(x)M(x). The answer is simple: you would not. There is absolutely no intuitive> reason for me to draw the tangent to F(x) at the point a if I am> interested in finding the area under the curve of f(x). In fact, there> is no reason for me to go looking at F(x) in the first place, UNLESS I> approach can hardly count as an intuitive way of getting to the> fundamental theorem of calculus, if the only justification you can> find for taking your first step (looking at an antiderivative of f(x))> is that you ->already<- know that the Fundamental Theorem of calculus> will connect integration with derivatives, and in what way it will> connect them. Ya got me. For those reading this, I (TR) did not discover the FTC;Leibniz and Newton fought over that one. I did, however, invent aproof for it that they would have loved.What's worse, you convert the problem of finding the> area under a curve into a problem of finding a volume of a> 3-dimensional solid. That's a step back, not a step forward. I prefer the three dimensional solid over the rectangle becausevisually it makes integraion perfectly clear.> Worse: you do not PROVE the fundamental theorem of calculus. You> ASSUME it in order to state how your M(x) and R(x) relate to the> integral of f(x).No, I do prove it. Ask someone else in your math department. You'rejust pissed off and don't want to see it, but its there.By the way. Can you name one theorem that someone attempted a prooffor without the theorem (or at least the hypothesis)?> So, why is this a better way to teach students the fundamental> theorem of calculus? You don't ->prove<- it, you ->use<- it. To> explain it, you introduce two completely ad-hoc functions, and try> to convert the simple problem of figuring out an area into a> discussion of figuring out a volume of a 3-dimensional solid. I do prove it. You're wrong and you're mad. Relax. There are noad-hoc functions here. The area of an integral is not simple, and itwasn't for a couple thousand years. Calculating the area ofrectangles and solids with strait sides has always(~) been simple.> Your argument seems to be that you were confused, and this explanation> helped you. Therefore, if someone is confused, this explanation will> help them. Alas, that is not a valid logical inference.> Finally, your final paragraph:> In essence, calculus is the method of using rules for simplifying> and differentiating a function and then applying them to your> advantage to solve problems involving changing quantities. > If that is what you think calculus is in essence, then I have news> for you: it is not that you ->were<- confused, you ->still are<-. You> have no clue what the subject matter of calculus is, and you are> confusing the method with the subject. I called it PreCalculus and guess who its written for? Students aboutto take Calculus. For them (not you of course) it is that indeed. What's your one liner on calculus?> So, why does your stuff belong in textbooks? It does not prove> anything, it is not intuitive, it is not illuminating, it is ad hoc,> it works only in a very limited situation, and it assumes as true the> thing you claim it explains. All those statements are wrong. Your not even trying to understandit. What can I say? When you read it in textbooks you'll know. :)> I accept as reality.> --- Calvin (Calvin and Hobbes)> === challenge Adjunct Assistant Professor at the University of Montana.>> (1) Proportions and Change, line 1:>> Mathematics is the study of proportion and change.>>> No, that's not what mathematics is. This line does not belong>> there.This is my definition and its beautiful. You find it beautiful; good for you. Unfortunately, it means that muchwhich is definitely mathematics is not included in your definition,and much which is not mathematics is. For instance, History studies change; therefore, History ismathematics.And mathematics studies things like abstract algebra, which does notinvolve proportion and does not involve change. Therefore, abstractalgebra is not mathematics.Therefore, while you may think this definition is beautiful, it is avery poor definition: it fails to include everything it shouldinclude, and it includes many things it should not.That, by definition, is a bad definition.By the way, its is a possessive. The contraction is spelled it's.> What's yours? That which>the math department teaches.You forgot the question mark at the end, unless you are pretending to->tell me<- what my definition is? Surely you aren't being thatpresumptuous?Mathematics encompasses a lot; but for the most part, it can be saidto be the study of necessary logical consequences. >> Line 2: Anything we can measure is a dimension. Again, this is an>> incorrect use of dimension, and the meaning of measure is>> unclear. This line is either content-free or wrong.Its a general definition of dimension and is correct.What the hell is your definition of correct?Why is it a general definition of dimension? It fails to explain whythe dimension of the plane is 2. It fails to explain what it meansfor something to be 3 dimensional. And it completely begs thequestion: if dimension is anything we can measure, then how can wetell if we can measure something? What does Ômeasuring' mean in thiscontext? Is the Ôcan' a physical limitation, so that if I don't have aruler on me then nothing is a dimension? > Maths do not own the word. You CLAIM to be writing about mathematics. Therefore, your usageshould ->conform<- to mathematics.> If biologists grow bugs and counts them over time, then>Bugs and Time are the two dimensions they work with. So, if there are too many to count, then they are not a dimension,right? Because your definition depends on OUR ABILITY TO MEASUREsomething. > Measure is like>measure distance with a tape measure, count beans, read a dial, and>generally find a number.So, the telephone directory is a measure, then? It's how I generallyfind a number.Your view is too simplistic to be of use. It is clear that it is basedon both limited experience and lack of vocabulary.>> Line 2: The measurements we take are simply relative differences>> in proportion. No, they are not.Oh, yes they are!Oh, well, that proves me wrong, I guess.>> Line 2: Proportion is the quality that forms a continuum from zero>> to infinity with one being the base to compare the proportion.>> Unclear at best, incorrect at worst. I will make it clear; >Q:What does two mean if you don't Know what one is? >A:NothingI think I begin to see the problem: you think that the above makes itclear. I think it is about as clear as Q: Why is a duck different?A: One of its legs is not the same.>> Is that what you want? Sort of but I would like you to stick to the good stuff. The FTC.You ASSUME the FTC. You provide no content that relates to the FTC,except an ad hoc construction that assumes it.But if you want a detailed critique, then I'll ask you to pay for mytime. How much are you offering?>>...It will take several hours of my time, which I>> am not at liberty to spend for free. If you are willing to pay, I'll>> be happy to provide you with a complete list of errors and suggested>> corrections. But I doubt that you are. er,...sorry but I am not at liberty to pay for it.Then I will not provide you with a detailed critique that will takeseveral hours of my valuable time.>> A more serious problem is that you are claiming that your method is>> a more intuitive of understanding the Fundamental Theorem of>> Calculus. I disagree. There is no intuition behind M(x) or R(x). They>> just simply happen to work. No man, your missing the point, they happen to work (you actually>agree with me) because of two similar triangles formed by nothing more>than the function and its derivative. No, they are formed by the function and the ANTI-DERIVATIVE.Keep it straight, man. You are talking about the Fundamental Theoremof Calculus. Thus, you START with f(x), NOT with F(x). The graph youconstruct is the graph of F(x), NOT the graph of f(x).Thus, your method ASSUMES that there is a point to the anti-derivativeand that it relates to integrals. Therefore, your method ASSUMES thefundamental theorem of calculus.> And! by division of the two alone, I get M(x). Wee!Which is unmotivated. Why would I look at M(x), when I have a problemabout f(x)?For no good reason whatsoever. I can also get it to work by using atriangle which has one side of length sin(x) and the other side oßength F(x)/sin(x). But it would be just as ad hoc as your method.So, why is this intuitive? Why did I look at the graph of F(x), whenI was looking for the area under the curve of f(x)?>> But, if I do not know the fundamental theorem of calculus, why on>> Earth would I ever draw the line you suggest? Why on earth would I>> ever consider the function M(x), or the function R(x)? What is their>> relation to the problem I am ->supposed<- to be working on, namely,>> that of finding the area under the curve of f(x)?Good point. Start them with the problem of the area under a curve,>show them the Rieman sum route, tell them it gets better because of>the FTC, Then hit'em with my proof!You are confused. YOU HAVE NO PROOF! Your claim that the area underthe curve is given by M(x) and R(x) ASSUMES the FTC. What you have isNOT a proof of the FTC. It is not even an application of the FTC. Itis a numerical artifact that happens to be true BECAUSE of the FTC.What proof?> Boom, a funky area f(x)dx becomes>a simple rectangle f(x)M(x).Only if you ->assume<- the FTC is true. You have no proof. You onlyhave an assertion whose only justification ->is<- the FTC.>> The answer is simple: you would not. There is absolutely no intuitive>> reason for me to draw the tangent to F(x) at the point a if I am>> interested in finding the area under the curve of f(x). In fact, there>> is no reason for me to go looking at F(x) in the first place, UNLESS I>> approach can hardly count as an intuitive way of getting to the>> fundamental theorem of calculus, if the only justification you can>> find for taking your first step (looking at an antiderivative of f(x))>> is that you ->already<- know that the Fundamental Theorem of calculus>> will connect integration with derivatives, and in what way it will>> connect them. Ya got me. For those reading this, I (TR) did not discover the FTC;>Leibniz and Newton fought over that one. I did, however, invent a>proof for it that they would have loved.No, you missed it: YOU DO NOT HAVE A PROOF OF THE FTC. Your argumentdoes NOT prove that the integral of f(x) from a to be is equal toF(b)-F(a). Your argument ASSUMES that fact.Do you even ->know<- what a proof is?>>What's worse, you convert the problem of finding the>> area under a curve into a problem of finding a volume of a>> 3-dimensional solid. That's a step back, not a step forward. I prefer the three dimensional solid over the rectangle because>visually it makes integraion perfectly clear.You must have some rather funky vision problems. About 99.5% of thestudents I have met find three-dimensional solids much morecomplicated than two dimensional figures.>> Worse: you do not PROVE the fundamental theorem of calculus. You>> ASSUME it in order to state how your M(x) and R(x) relate to the>> integral of f(x).No, I do prove it.No, you assume it. Your assertion that the integral is equal to theproduct assumes that the integral is equal to F(b)-F(a). > Ask someone else in your math department. Why should I waste anybody else's time? I am perfectly capable ofpassing judgement on your presentation: you have NOT proven the FTC,you have merely used it.> You're>just pissed off There you go again: here's a newsßash, sonny: I am not pissed off.Just because I am telling you that you are wrong and ignorant does notmean I am pissed off. It means that you are wrong and you areignorant, that's it.>and don't want to see it, but its there.The Fundamental Theorem of Calculus, Part 1, states that if F is ananti-derivative of f, then the integral from a to b of f(x) is equalto F(b)-F(a).Your proof assumes that the indefinite integral of f(x) is equal toF(x) + C; then expresses F(x) in terms of f(x), M(x), and R(x); andthen uses this expression to calculate F(b)-F(a). However, it does NOTprove that the integral from a to b of f(x) is equal to F(b)-F(a). It->assumes<- that.>By the way. Can you name one theorem that someone attempted a proof>for without the theorem (or at least the hypothesis)?for without the theorem (or at least the hypothesis)? You did notstart from the hypothesis and derive the conclusion: you started fromthe conclusion and derived a different expression for it.The problem is not that you know the Fundamental Theorem of Calculus(first part), or that someone else proved it before you. The problemis that your argument does not simply assume the hypothesis of thefundamental theorem of calculus, it assumes the ->conclusions<- of thefundamental theorem of calculus. Therefore, it cannot constitute aproof.>> So, why is this a better way to teach students the fundamental>> theorem of calculus? You don't ->prove<- it, you ->use<- it. To>> explain it, you introduce two completely ad-hoc functions, and try>> to convert the simple problem of figuring out an area into a>> discussion of figuring out a volume of a 3-dimensional solid. I do prove it. You have no clue what constitutes a proof. > You're wrong and you're mad. No, I'm neither wrong, nor mad. Do you keep telling yourself that,because you cannot fathom that someone would disagree with you unlessthey are angry?> Relax. There are no ad-hoc functions here. The area of an integral is not simple, and it>wasn't for a couple thousand years. Calculating the area of>rectangles and solids with strait sides has always(~) been simple.I have the following problem before me: Calculate the area under thegraph of y=f(x), from the point a to the point b.You tell me: Ah! Let F(x) be the anti-derivative of f(x); letM(r) = the distance between r and the intersection of the line y - F(r) = f(r)(x-r) and the X-axis. Let R(x) = M(x)/x. Then the area under the graph of y=f(x) from x=a to x=b isf(b)*b*R(b) - f(a)*a*R(a).First: the reason you can say that the area under the graph is equal tof(b)*b*R(b) - f(a)*a*R(a) is because you know (through an ad hocgeometrical argument that has nothing to do with the original problem)that R(x)*x*f(x) = F(x). And then you use to fundamental theorem ofcalculus when claiming that F(b)-F(a) is equal to the area under the curve.Second: You never PROVE that that area under the curve is equal toF(b)-F(a). ->THAT'S<- what the Fundamental Theorem of Calculusproves. Third: There is no motivation for considering F(x).Fourth: M(x) is completely ad hoc.Fifth: R(x) is completely ad hoc.So I count: two ad hoc functions; one function which we had no reasonto consider in the first place (absent the FTC); and a conclusion thatDEPENDS on the FTC. You have not proven the FTC (actually, the ->first part<- of theFTC; the FTC has ->two<- parts, and you have not addressed the secondpart at all). You have ->used<- it to convert the problem from onething to another. But then, that's what the first part of the FTC is->for<-: to convert the problem of finding an area into the curve intoa different problem. So you have done little or nothing, and you have->certainly<- not proven the first part of the FTC.>> Your argument seems to be that you were confused, and this explanation>> helped you. Therefore, if someone is confused, this explanation will>> help them. Alas, that is not a valid logical inference.>>> Finally, your final paragraph:>>> In essence, calculus is the method of using rules for simplifying>> and differentiating a function and then applying them to your>> advantage to solve problems involving changing quantities. >>> If that is what you think calculus is in essence, then I have news>> for you: it is not that you ->were<- confused, you ->still are<-. You>> have no clue what the subject matter of calculus is, and you are>> confusing the method with the subject. I called it PreCalculus and guess who its written for? Students about>to take Calculus.Let me get this straight: You invoke the ->antiderivative<- insomething addressed at students about to take calculus?Do you also invoke Shakespeare to students about to learn how to read?And above you ->claimed<- I should present your nonexistent proof tomy calculus students AFTER talking about Riemann sums. So make up yourmind. Who is it addressed to?> For them (not you of course) it is that indeed. You are claiming that a student who has ->not<- yet taken calculus,already thinks that calculus is the method of using rules forsimplifying and differentiating [a concept they do not know yet] afunction, and then applying them to your advantage to solve problemsinvolving changing quantities? How did they acquire this opinion, ifthey don't know what calculus is?And, is it your opinion that it is a ->good idea<- to give thestudents a ->false<- impression of what calculus is, as a firstintroduction of what calculus is?You believe that teaching ->false<-, ->incorrect<-, and ->inaccurate<-information is the way to get a student to understand something?Man, are you lost.You think it is a good idea to tell students that Calculus is a bunchof rules for them to memorize. That's just a bad first step.>What's your one liner on calculus?You mean, what is my one-line description of what calculus is? I havetwo: one for differential and one for integral calculus (both onevariable). Differential Calculus is the study of change and of instantaneouschange. Integral Calculus is the study of how to calculate areas. Or, more humorously: Differential calculus is what physicists did astheir day job: is what they used to help them understand the motion ofthe heavenly bodies. Integral calculus is what they did for theirweekend: is what they used to help them figure out how much beer fitsin a barrel.>> So, why does your stuff belong in textbooks? It does not prove>> anything, it is not intuitive, it is not illuminating, it is ad hoc,>> it works only in a very limited situation, and it assumes as true the>> thing you claim it explains. All those statements are wrong. Your page does not contain a proof of the first part of the FTC. Yourbelief that it does indicates a complete and utter misunderstanding ofwhat a proof is, not to mention of what the FTC says.> Your not even trying to understand it.No. I did try. You just did not accomplish what you think youaccomplished. Consider that possibility.> What can I say? When you read it in textbooks you'll know. :)Fine. I'll wager dollars to donuts that it won't, because each andevery one of the statements I made above is true. As are the new onesI just added.This will be my last reply to you. 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 his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which 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 this extent, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de === think my Big point is that this stuff belongs in textbooks? Noway, I just said that to get your attention. Its true though.My Big Point is that when you say the area under the curvex^3=x^4/4+C, I say that's because there is a function, in this casex/4, that is multiplied by x^3 to get x^4/4. My proof is that a righttriangle with a height x^3 and a base of 1 is always similar to aright triangle with a height x^4/4 and a base x/4. That means thatthe height to base ratios are equal. Its all outlined in mychanging curve becomes the area of a rectangle, which we have noproblem calculating. No dx, delta-x, or infinitesimals required.In your two post you take exception to my use of general definitionsand say they are wrong. Since you give no counter I assume itsbecause you've used more resrtictive definitions so I must be wrong. It doesn't work that way.Also you contradict me and say I'm wrong, and use your own generalterms, ad hoc and unmotivated, but don't want to waste your timeexplaining. I guess.I know that you are pissed-off. But with the above and you concernfor your students I'll just add the following anyways.... I thinkwith shapes and curves and I'll bet you think with numbers andequations. That's not a put down, people are different. Everyone hasa different learning style too. If you ask your psych dept about abook on the subject you'll find you can do things in class that willmake your students more interested and seem smarter. You can showthem my proof too.Finally, no I don't beieve in infinitesimals, they are pure sciencefiction. You think I'm part of the ignorant masses and I standard approach. Why should it be in a textbook, absent>> ->actual evidence<- that it is useful for a significant proportion of>> students?> - Students don't already know where the calculus is heading, but>you do. > And I don't know where YOUR approach was heading either. Your approach> seems completely ad hoc and unmotivated. Why would an ad hoc> unmotivated approach make more sense to me? Or to anyone else? It> would not.> YOU find your approach better because YOU know where it is> heading. But the student will not know where you are heading> ->either<-. > That's why You find it annoying. > > I did not say I found your approach annoying. I said your approach> would be ANNOYING TO APPLY TO A GENERAL FUNCTION. Please do ->try<- to> read what I write, not what you think I am writing. >I know my approach is>limited but for a student two small steps are better than one big one.> I do not see two small steps. I see unmotivated jumps. Why would I> ever consider the quantities M(x) and R(x)? There is absolutely NO> motivation for the construction. They are geometric artifacts that> happen to give (apparently), the right answers. They are no more> intuitive than the complicated diagrams from Euclid, which students> ->certainly<- do not find intuitive.> And in your reply, you did not address the most important point: YOU> HAVE NO EVIDENCE WHATSOEVER THAT YOUR METHOD IS USEFUL FOR A> SIGNIFICANT NUMBER OF STUDENTS. That's just ->your<- guess. As such,> it is not sufficient to warrant the sort of claims you make.>> (No, I did not check it very carefully, though in general it seems>> about right; - c'mon, that's not a good lead to the following...> Take what you get.>>... your function M(x) is closely related to Newton's Method, - close doesn't count> Does not count for what?> The intersection of the line with the x-axis is the next approximation> for Newton's Method. One can derive M(x) directly from the Newton's> method formula approximation.>>... and a formula for it could be developed directly by using the>> derivative. - and that's what I did, I'm after the integral> Without the derivative, you end up having to do>> approximations: your claim that you can find it graphically really>> amounts to claiming that you can draw accurate tangents by>> hand. That's really hardly true in practice, so your claims towards>> the end really end up being that you can find M(x) and R(x) by using>> the derivative, which means that you are really just running around in>> circles.... - as above, my point is that the integral is a function [f(x)]>multiplied by a function [M(x)], no more, no less. Try that on your>students.> Sorry, but I am not about to experiment on ->MY<- students, and> certainly not with an ad-hoc method that has no motivation> whatsoever. Bring back solid data from experiments, and I may consider> it. But the say so from someone who admits having had trouble with the> concept, and who admits to not being well-versed on the subject, and> who has NOTHING but his own subjective impression to back up that say> so, is simply not sufficient for me to risk confusing the students. >Also, I may be wrong but I think you believe that limits and>infinitesimals are the same thing. > You are indeed wrong.>They are not. A limit is a fixed>or variable quantity that can be used to produce the derivative>amongst other things. > No. A limit is NOT a fixed or variable quantity that can be used to> produce the derivative amongst other things.> An infinitesimal is a creation like my little>strait(?) part of the curve.> No, an infinitesimal is not a creation like [your] little straiGht> line.> Since you are obviously ignorant about these basic concepts, why is> your say so of what is useful and what belongs in a textbook worth> ->anything<- at all? > I accept as reality.> --- Calvin (Calvin and Hobbes)> === Assistant Professor at the University of Montana.>You think my Big point is that this stuff belongs in textbooks? No>way, I just said that to get your attention.No, that is not what I think was your Big point. You are clearlyincapable of even the simplest reading comprehension.What I thought your big point was is your claim that yourpresentation gives a more intuitive understanding of the FundamentalTheorem of Calculus. It does not. Your presentation ASSUMES theFundamental Theorem of Calculus, and therefore cannot possibly giveyou any intuition about why the FTC would be true in the first place. > Its true though.That's your personal, subjective opinion. You are welcome to writeyour own textbook and shop for a publisher, but, frankly, youroverblown claims are simply that: overblown.>My Big Point is that when you say the area under the curve>x^3=x^4/4+C, NOBODY who knows what they are talking about says that the area underthe curve x^3 is x^4/4 + C. And certainly nobody who understands thatmathematical symbols are meant to be read out loud would ever writethe area undre the curve x^3=x^4/4+C, because that is literally,the area under the curve defined by the equation x^3=x^4/4 + C,which is something else entirely from what you are refering to.>I say that's because there is a function, in this case>x/4, that is multiplied by x^3 to get x^4/4. And I say that your motivation for how to find this function x/4 iscompletely ad hoc and unintuitive, and therefore useless for thepurpose of clarifying the Fundamental Theorem of Calculus. If I present you with the problem of finding the area under the curveof x^3 from the point (0,0) to the point (1,1), say, you tell me thatI should multiply the function by x/4, then look at the graph ofy=x^4*(x/4). Then draw a random tangent for no good reason, choose twopoints on that tangent (of which only one is a natural point tochoose). The compare to artifacts (the similar triangles), and fromthat obtain that x^4/4 = x^3*(x/4).But why did I look at x^4/4 IN THE FIRST PLACE? For no good reasonwhatsoever. And what did I gain from this little excursion? Absolutelynothing. And how do I know that the function x^4/4 is related to theoriginal problem of finding the area under the curve of x^4 from (0,0)to (1,1)?Because you INVOKE the fundamental theorem of calculus.> My proof is that a right>triangle with a height x^3 and a base of 1 is always similar to a>right triangle with a height x^4/4 and a base x/4. Which is a complete waste of time, and completely unmotivated, andcompletely obscure, and completely unintuitive. Unless you happen toknow why you would be looking at a triangle with height x^4/4 and basex/4 in the first place. Why would you do that? Remember that thetriangle you got by looking, NOT at the graph of y=x^3 (the function youSTART with), but at the graph of y=x^4/4 (the function you are tryingto FIND).That is: before your argument can make sense, you must know what theANSWER is!How can that possibly be useful?> That means that>the height to base ratios are equal. Its all outlined in my>PreCalculus.Which is ->not<- PreCalculus.>changing curve becomes the area of a rectangle, which we have no>problem calculating. No dx, delta-x, or infinitesimals required.Nonsense. To justify your claim you must INVOKE the FundamentalTheorem of Calculus. You cannot substitute your argument for it. Thearea of a right triangle of height x^3 and base 1 is not related tothe area under the graph of y=x^3 from (0,0) to (1,1); it is strictlylarger than that area. And while you may hide the differentials (of which you obviously havelittle or no understanding), rather than ->explain<- to a student whatis going on, you force them to memorize a new ad-hoc rule that hasaboslutely no intuition and no reason for being. You do not evenmanage to convince that the calculation you give will produce theright answer, unless you invoke the Fundamental Theorem of Calculus. I can see why you find it appealing: you have the wrong impressionthat mathematics (and calculus in particular) is simply a matter ofmemorizing rules that make no sense anyway, so what is one more?Well, that's because you do NOT understand calculus and you do notunderstand the stuff you are talking about.>In your two post you take exception to my use of general definitions>and say they are wrong. I take exception to you making claims that certain words mean thingsthat they simply do not mean. It has nothing to do with your use ofgeneral definitions, but it has to do with your attempting to tellstudents that dimension is something that it is not.> Since you give no counter I assume its>because you've used more resrtictive definitions so I must be wrong. You have an uncanny way of simply assuming that my problem with yourpresentation is whatever it is that pops into that little head ofyours. You might want to ask when you do not understand.No. Your usage of dimension is empty nonsense, and I explainedwhy. You talk about dimension in terms of being able to measure, butyou do not say what it means to be able to measure something. Thatmeans that, as a definition, what you have given is worthless.As for the usual meaning of dimension, it is a term of art that haslittle to do with measurements. In analytic geometry, it refers to thenumber of parameters that are needed to specify a point uniquely, forexample.>Also you contradict me and say I'm wrong, and use your own general>terms, ad hoc and unmotivated,Do you even know what ad hoc and unmotivated means, or did yousimply crib it from my response?> but don't want to waste your time>explaining. I guess.What is it you wish me to explain?>I know that you are pissed-off.You know nothing. I am not pissed-off. I find your attitudecondescending. And since you are demonstrating an appalling ignoranceon the subject to begin with, it seems all the more unjustified. > But with the above and you concern>for your students I'll just add the following anyways.... I think>with shapes and curves and I'll bet you think with numbers and>equations. Please explain, with shapes and curves, why a triangle with base 1and height x^3 is related with the area under the curve of y=x^3 fromthe point (0,0), to, oh, say, (10,1000). You claim your approach is better; then use it to PROVE theFundamental Theorem of Calculus. Your page does not do so; your pagemerely invokes it and presents an unmotivated way of obtaining twofunctions, one of which happens to be the anti-derivative. But inorder to ->obtain<- those functions in the first place, you must FIRSTknow what the anti-derivative is. Your approach is not only restrictedand unmotivated, it is ->circular<-. >That's not a put down, people are different. Everyone has>a different learning style too. Then what makes you ->so sure<- that your approach will benefit mystudents? I certainly have no doubt that your approach benefited you:but you are claiming that your approach is OBJECTIVELY SUPERIOR to thestandard approach. Yet your approach must ASSUME the standard resultsin order to work. > If you ask your psych dept about a>book on the subject you'll find you can do things in class that will>make your students more interested and seem smarter. I don't want my students to seem smarter. I want my students to->learn the material<-, to ->learn it correctly<- and to know how toapply it. That is the purpose of a service course, which is whatcalculus is. You may be under the impression that your approach makesyou seem smarter, but I should tell you that your impression is incorrect.> You can show>them my proof too.You can show them yourself.>Finally, no I don't beieve in infinitesimals, they are pure science>fiction. Let's see: first you claimed that ->I<- believed that limits andinfinitesimals were the same thing, and went on to provide anincorrect explanation of what limits were to justify this claim. So,when I said I did not (but nowhere did I ask if you believed ininfinitesimals), you reply No, I don't believe in infinitesimals.Well. That's interesting. But nobody asked you and nobody implied youdid. And, no, they are not pure science fiction. Infinitesimals can bedefined just as rigorously as can the real numbers, the real line, andall the geometry that you are using in your little page. Theyconstitute a ->distinct but equally valid way<- of doingcalculus. Since it is clear that an actual textbook would be way overyour head, then I suggest that you read the following: Nonstandard Analysis, by Martin Davis and Reuben Hersh _Scientific American_, June 1972.> You think I'm part of the ignorant massesNo. I ->know<- you are ignorant because you have given ample evidencethat this is the case.> and I think you whorship false gods.Well, good for you. So far, your guesses as to what I supposedlybelieve have not even been in the right ballpark. Yet more evidenceof your ample ignorance on the subject what I accept as reality. --- Calvin (Calvin and === I'm not a mathematician. I don't know any mathematicians. > Mathematicians always delete my e-mails.if you are looking for attention, you are late!this ng already has a world-class crackpot/troll who practically getsall the attention. as far as your discoveries are concerned, they areextremely === prime counting> What I found when I talked to Ullrich's boss at Oklahoma State> University, is that he lies even to his university, as I heard a> rather fascinating tale, second-hand through his boss, justifying> Ullrich's odd posting behavior, which is a tale his boss apparently> believed.Just as cops lie in order to help prosecutors convict innocentpeople, just so DU distorts and misleads in order to discreditanyone who would cast doubt on principles under whose auspicesthose in his ÔHood transact their business. > Make no mistake, David Ullrich is a dedicated and persistent liar.> He also is a math professor at Oklahoma State University.> lies you have to makeyourself, and you can't be sure they are any good until you'veused them --- and then it's too late.John === you have to make> yourself, and you can't be sure they are any good until you've> used them --- and then it's too late. John SteinbeckYou lame bastard. That's *my* .sig.A Google search on the first four words brings up three occurrences.Someone gave a fuller quote of the same passage in 1999 on aSpringsteen newsgroup. I used it as a .sig on Nov 11, in<87vfpq60yf.fsf@phiwumbda.org> (alt.os.linux.slackware) and now youlamely use it.Of course, you're free to take it but golly, you look like an idiot.Anyway, when it comes up in my posts here on sci.math, I hope thereaders do not mistakenly believe that I'm copying you.I steal from better sources than that.-- I've been thinking about my problems with getting any kind ofadmission that my math arguments showing the core error in mathematicsare correct, so I've gone to marketing books. -- === Re: Numerical integration, prime countingX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>>> What I found when I talked to Ullrich's boss at Oklahoma State>> University, is that he lies even to his university, as I heard a>> rather fascinating tale, second-hand through his boss, justifying>> Ullrich's odd posting behavior, which is a tale his boss apparently>> believed.Just as cops lie in order to help prosecutors convict innocent>people, just so DU distorts and misleads in order to discredit>anyone who would cast doubt on principles under whose auspices>those in his ÔHood transact their business. As I explained to James, this would be less laughable if youtold us exactly what it was I said that was distorted andmisleading.>> Make no mistake, David Ullrich is a dedicated and persistent liar.>>> He also is a math professor at Oklahoma State University.>> lies you have to make>yourself, and you can't be sure they are any good until you've>used them --- and then it's too late.John Steinbeck*****************************John Correy says:Christian (what an oxymoron!): Degrade, demean, goad and bait meas Ullrich and the Boyz have done to JSH, and I won't triße withwriting your employer: I'll come after you with an === prime countingX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>> What I found when I talked to Ullrich's boss at Oklahoma State> University, is that he lies even to his university, as I heard a> rather fascinating tale, second-hand through his boss, justifying> Ullrich's odd posting behavior, which is a tale his boss apparently> believed.>>Just as cops lie in order to help prosecutors convict innocent>>people, just so DU distorts and misleads in order to discredit>>anyone who would cast doubt on principles under whose auspices>>those in his ÔHood transact their business. As I explained to James, this would be less laughable if you>told us exactly what it was I said that was distorted and>misleading.another thread. There I'd pointed out that you habitually refuseto answer simple questions and you said I should give an example.Fair enough. But I have better rephrase the above as a question:Exactly what distortions of mine are you referring to? (Thequestion is regarding James statement that I lie to thedepartment head here - exactly what curious tale is it thatI told him?)> Make no mistake, David Ullrich is a dedicated and persistent liar.> He also is a math professor at Oklahoma State University.> but lies you have to make>>yourself, and you can't be sure they are any good until you've>>used them --- and then it's too late.>>John Steinbeck*****************************>John Correy says:Christian (what an oxymoron!): Degrade, demean, goad and bait me>as Ullrich and the Boyz have done to JSH, and I won't triße with>writing your employer: I'll come after you === Numerical integration, prime counting>[...]Answer my question: [...]> Sorry - you habitually ignore repeated direct questions that> you don't feel like answering, so you're in no position to> demand answers from others.Then it should be easy for you to cite such a question.But you can no more do this than you coulddeduce Ex~(x=x) from MKC3,MKC4 (with the deductiveapparus of FOL), or formalize MKC3,MKC4 in ZF. MKC3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)MKC4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]--John...my belief is that (2) is just something you made up (or somethingyou misunderstood based on something you read.)2) AxAy[Az(z in x <-> z in y) -> ((set x & set y) <-> === V9oKxQZcQeRpazRI350mQUszx1sW6Il04tcvde9O-9ZO-qcKDDH5gv (2) AxAy[Az(z in x <-> z in y) -> ((set x & set y) <-> x = y)]> ...my belief is that (2) is just something you made up. (David Ullrich)>But John, actually this i s *true*. (2) is something that YOU made up.Hence the resulting theory is properly called === counting> Tee-hee. Yeah, that's exactly right. Of course you're not going toNothing makes a pig happier than a roll in the wallow, as this tittering porker knows! http://www.shop4egifts.com/target.asp?item=/jpg/25112.jpg& width=314&height=400#http://www.fetchfido.co.uk/sound_[Capital Thorn]les/ === counting> Giggle.Nothing makes a pig happier than a roll in the wallow, as this giggling porker knows! http://www.shop4egifts.com/target.asp?item=/jpg/25112.jpg& width=314&height=400#http://www.fetchfido.co.uk/sound_[Capital Thorn]les/ === counting> Giggle.Nothing makes a pig happier than a roll in the wallow, as this giggling porker knows! http://www.shop4egifts.com/target.asp?item=/jpg/25112.jpg& width=314&height=400#http://www.fetchfido.co.uk/sound_[Capital Thorn]les/ === counting>[...]Answer my question: [...]> Sorry - you habitually ignore repeated direct questions that> you don't feel like answering, so you're in no position to> demand answers from others.Aren't my questions ones you would prefer others not see?> Why should JSH have zero credibility, while--> after your blunders, you, a Ph.D. in mathematics and professor> of same--retain yours?> Don't *all* of the following indicate that you had *no idea* (and> *still* have no idea) wh Ex~(x=x) follows from x <-> z in y) -> Az(x in z <-> y in z)] C3 EyAx[x in y <-> Et(x in t) & A] (with y not free inA)ClassificationC4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (WeakExtensionality)David Ullrich:If you meant NGB set theory then no, C1-C4 arenot inconsistent with set theory. It does _not_follow that C1-C4 give an example of somethingwhich is not equal to itself, or an example ofsomething which does not exist. It is correct that I have no idea why Ex~(x=x)follows from C1-C4. This is because (assumingthat NBG is consistent) NBG has a model inFOL=. In that model everything is equal Correy:You have no idea why Ex~(x=x) follows from C1-C4because you are brain-dead analysis teacher whocan only work with the routines he has memorized.David Ullrich:Could be. Now show us why Ex~(x=x) _does_ followfrom Ullrich:Exhibit of proof of Ex~(x=x) from C1-C4 and someone will pointout the Chapman, Jesse(Dogberry) Hughes, or Arturo (Gradgrind) Magidin === integration, prime counting>[...]Answer my question: [...]> > Sorry - you habitually ignore repeated direct questions that> you don't feel like answering, so you're in no position to> demand answers from others.> Aren't my questions ones you would prefer others not see?> Why should JSH have zero credibility, while--> after your blunders, you, a Ph.D. in mathematics and professor> of same--retain yours?> Don't *all* of the following indicate that you had *no idea* (and> *still* have no idea) wh Ex~(x=x) follows from C1-C4?> <-> z in y) -> Az(x in z <-> y in z)] > C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in> A)Classification> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak> Extensionality)> David Ullrich:> If you meant NGB set theory then no, C1-C4 are> not inconsistent with set theory. It does _not_> follow that C1-C4 give an example of something> which is not equal to itself, or an example of> something which does not exist. > It is correct that I have no idea why Ex~(x=x)> follows from C1-C4. This is because (assuming> that NBG is consistent) NBG has a model in> FOL=. In that model everything is equal to> itself.> Correy:> You have no idea why Ex~(x=x) follows from C1-C4> because you are brain-dead analysis teacher who> can only work with the routines he has memorized.> David Ullrich:> Could be. Now show us why Ex~(x=x) _does_ follow> from C1-C4.> Ullrich:> Exhibit of proof of Ex~(x=x) from C1-C4 and someone will point> out the error.> --John> Jesse> (Dogberry) Hughes, or Arturo (Gradgrind) Magidin will === Numerical integration, prime countingX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>>>[...]>>Answer my question: [...]>>> Sorry - you habitually ignore repeated direct questions that>> you don't feel like answering, so you're in no position to>> demand answers from others.Aren't my questions ones you would prefer others not see?Chortle. Guffaw. Your estimate of your importance, um, gimmea second to figure out how to phrase this...The idea that I'm concerned about what people are going to thinkof me on the basis of what they read in posts from, uh, peoplelike you on usenet is hilarious. You seem to have the idea thatthe things we say here matter. I suppose it's understandable howyou might get that impression, since your only intellectuallife appears to be on usenet, but news ßash, it's not so.>> Why should JSH have zero credibility, while-->> after your blunders, you, a Ph.D. in mathematics and professor>> of same--retain yours?Ok, I'll explain. JSH has zero credibility because he's been wrongover and over and over, for years.On the other hand, (i) the blunders of mine you refer to consistof at most one blunder, which you've been citing repeatedly for,I haven't been keeping track, maybe more than a year now (youshould _really_ find a few more examples of blunders I've made -you look silly harping on this one) (ii) as I've explained severaltimes, I don't regard anything below as a blunder - to make it looklike a blunder one has to ignore the context. _As_ has been explained_many_ times: You said we were talking about standard set theory._If_ we're talking about standard set theory then _if_ Ex~(x=x)follows from C1-C4 then that's profoundly uninteresting, because itsimply shows that the system we're talking about is inconsistent,since Ax(x=x) is a theorem of standard set theory.Yes, it may well be whatever you said that made me think youwere talking about deductions within standard set theory wasnot meant to imply that. We've seen recently that it appearsto imply that to people other than me. The _blunder_ I'vecommitted here consisted of not being able to read yourmind.>> Don't *all* of the following indicate that you had *no idea* (and>> *still* have no idea) wh Ex~(x=x) follows from in x <-> z in y) -> Az(x in z <-> y in z)] >C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in>A)Classification>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak>Extensionality)David Ullrich:>If you meant NGB set theory then no, C1-C4 are>not inconsistent with set theory. It does _not_>follow that C1-C4 give an example of something>which is not equal to itself, or an example of>something which does not exist. It is correct that I have no idea why Ex~(x=x)>follows from C1-C4. This is because (assuming>that NBG is consistent) NBG has a model in>FOL=. In that model everything is equal John Correy:>You have no idea why Ex~(x=x) follows from C1-C4>because you are brain-dead analysis teacher who>can only work with the routines he has memorized.David Ullrich:>Could be. Now show us why Ex~(x=x) _does_ follow>from Ullrich:>Exhibit of proof of Ex~(x=x) from C1-C4 and someone will point>out the (Moloch) Chapman, Jesse>(Dogberry) Hughes, or Arturo (Gradgrind) Magidin will try to bail>you out...--John*****************************John Correy says:Christian (what an oxymoron!): Degrade, demean, goad and bait meas Ullrich and the Boyz have done to JSH, and I won't triße withwriting your employer: I'll come after you with an === prime counting>>>[...]>>Answer my question: [...]>>> Sorry - you habitually ignore repeated direct questions that>> you don't feel like answering, so you're in no position to>> demand answers from others.Aren't my questions ones you would prefer others not see?> Chortle. Guffaw. Your estimate of your importance, um, gimme> a second to figure out how to phrase this...> The idea that I'm concerned about what people are going to think> of me on the basis of what they read in posts from, uh, people> like you on usenet is hilarious. You seem to have the idea that> the things we say here matter. I suppose it's understandable how> you might get that impression, since your only intellectual> life appears to be on usenet, but news ßash, it's not so.> You're damn straight! And that's not all. For the Ôintellectual life'of you and your homies, I harbour the same profound contempt thatworking people everywhere harbour for the sterile posturing ofindolent and spoiled four intellectuals atthe most, but it seemed to him that he had never known one ofthem who did not seem to be talking to himself, or, which wasthe same thing, exclusively for the benefit of each other.The more intellectual they became the less communication theyhad with life and people. They revolved in increasingly smallerand smaller circles, without any real interest in anything,sustained only by a specialized vocabulary that was utterlyincomprehensible to outsiders and little more than a standardizedyet ever-changing ritual to themselves, not unlike the newestslang phrases of the slick advertising world that are droppedalmost as soon as they become known and quickly replaced byothers even newer. (Charles Jackson, _The Outer Edges_(New York: Reinhart, 1948), p. 232; cited in Leo Gurko,_Heroes, Highbrows and the Popular === Numerical integration, prime counting> You're damn straight! And that's not all. For the Ôintellectual> life' of you and your homies, I harbour the same profound contempt> that working people everywhere harbour for the sterile posturing of> indolent and spoiled parasites.This is a poorly informed (and rather sad) view ofacademics. Professional academics, whether they work at universities,community colleges or your local grade school, work hard every daytrying to teach others. They don't spend all their time exchangingjargon with their fellow academics in some narrow research specialty.I can see why someone dead-set on avoiding education would think === EepOyTZDQeln6WNDWUfwzqHpg2d-x3o2fZW5-0FrIGV-mWWigSIkQ7> Yes, it may well be whatever you said that made me think you> were talking about deductions within standard set theory was> not meant to imply that. We've seen recently that it appears> to imply that to people other than me. [...]>Indeed Correy proposed a *variation* of MK set theory. He's calling thistheory MKC n o w - which is a rather good idea. For this way furtherconfusion is avoided.In h i s theory we actually have ~Ax(x = x).(Since equality in his theory only holds between /sets/, but not for/proper classes/. And -as we all know- there ARE proper classes in MK, and hence in === countingX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>[...]Answer my question: [...]Sorry - you habitually ignore repeated direct questions thatyou don't feel like answering, so you're in no position todemand answers from others.*****************************John Correy says:Christian (what an oxymoron!): Degrade, demean, goad and bait meas Ullrich and the Boyz have done to JSH, and I won't triße withwriting your employer: I'll come after you with an === bend in a carI'm looking for a some help on the friction forces acting on a car as itgoes round a bend.perhaps can offer an explanation.To be a bit more specific, I'm considering a car travelling in a strightline at a constant speed (neglecting friction at wheel bearings, windresistance, road gradiant, etc.). So what happens when the front wheeldirection is chnaged that causes the car to drive the round the === bend.Subject: Re: Friction - driving round a bend in a to impar an angularacceleration to the car until it is rotating at the rate its going round thecircle. If you turn the wheel to fast, the angular acc of the car will beless than that of the required change in direction and you will get a skid.If you wath the incar camara on a racing car, this is happening all thetime. only on high speed bends does the balance between friction andcventrifugal force at back and Harveybruce@bearsoft.co.ukThe Alternative Physics Sitehttp://users.powernet.co.uk/bearsoft> I'm looking for a some help on the friction forces acting on a car as it> goes round a bend. perhaps can offer an explanation. To be a bit more specific, I'm considering a car travelling in a stright> line at a constant speed (neglecting friction at wheel bearings, wind> resistance, road gradiant, etc.). So what happens when the front wheel> direction is chnaged that causes === driving round a bend in a carIn sci.math, Jim Scott:> I'm looking for a some help on the friction forces acting on a car as it> goes round a bend.> perhaps can offer an explanation.> To be a bit more specific, I'm considering a car travelling in a stright> line at a constant speed (neglecting friction at wheel bearings, wind> resistance, road gradiant, etc.). So what happens when the front wheel> direction is chnaged that causes the car to drive the round the bend.> Qualitatively, your tires get scuffed. Quantitatively, one canprobably estimate the acceleration by simply parameterizing theproblem (we assume a 2-D Universe for simplicity):P(t) = (vt,0) for t < 0 = (r * sin(vt/(2*pi*r)), r - r cos(vt/(2*pi*r))), 0 <= t < pi*r/(2v) = (r, vt) for t >= pi*r/(2v)which models the car going around a perfectly circular 90degree curve at constant speed v (which isn't all thatrealistic, admittedly, but it makes the math easy --note that speed != velocity).The acceleration A(t) = P''(t). Note that the accelerationis *inward* -- centripetal force.If this is overly simplistic you can try to compute thecar's path given the steering wheel's position at time t,and working out the path that way; the car in this casewill not describe a perfect circle. There are some issuesregarding tire contact, as the tires will scuff (sincethe rear wheels are not describing a straight track, forexample), although in a pinch one could model the tiresas perfect discs, with infinite sliding friction and zerorolling friction, perhaps.I'm not entirely sure how to describe the road force.Obviously, the weight of the car is a factor thereon (andso is the tire inßation, tire composition, road surface,oil/water/ice thereon, etc.), but it's clear that theroad can only provide so much centripetal force; if theconditions are right and the steering wheel is turned toofar, the car will skid and depending on conditions it willeither plough (understeer, push) or fishtail (oversteer,loose/get loose). If one is lucky one will only hit thesand barrels at the gore point, spewing sand everywhere... :-)If one models the axle in a certain way the front wheelswill bend as they try to negotiate the corner. However,I'm not sure that's all that significant for this problemunless one is modeling very high speeds (e.g. SSC).-- #191, ewill3@earthlink.netIt's still legal to go === while and never really thought itwould be useful when I had the best opportunity to learn (ie. atschool). I now find that I was wrong... to cut a long story short, I'mshort of the tools required to solve a specific problem, and I neednudged in the right direction so I can figure it out for myself.Basically, it's a chemical reaction of the variety:c+d ---> e+fIt also produces a byproduct (I'll call the byproduct b1), formed ina quantity dependent on various parameters, which I've given arbitrarynames:b1=1-(b/m(1+a/q)+(2b^2)/mn)*(1+a/p+b/m(1+a/q+b/ n))^-1(the letters have replaced equilibrium constants and ionconcentrations... doesn't really matter what they are as this isapplicable for a number of reactions.)one of the parameters, b, that b1 is dependent on is of the samesubstance as b1.My problem is that b1 gets smaller the larger b gets... and Iwould like to be able to calculate what the end b is if a certainnumber of c+d is converted to e+f where all other parameters are knownDoes that make sense?I suspect I need to === derivativeFor interger b and n the nth derivative of a(z^b) is equal to(a(b!))/((b-n)!)z^(b-n) The factorial can be extended to real andcomplex arguments by the gamma function thus giving the first and ahalf derivative or the ith derivative. Has any one done a paper onthis? I might point out that when a mathematician thinks a branch ofmathematics has no applications to the real world a physicist findsthat is === derivativeX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>For interger b and n the nth derivative of a(z^b) is equal to>(a(b!))/((b-n)!)z^(b-n) The factorial can be extended to real and>complex arguments by the gamma function thus giving the first and a>half derivative or the ith derivative. Has any one done a paper on>this? I might point out that when a mathematician thinks a branch of>mathematics has no applications to the real world a physicist finds>that is does.Yes, you might point that out. You'd look a little less sillypointing that out if you were talking about something thatmathematicains had not studied for a long time.David C. === interger b and n the nth derivative of a(z^b) is equal to>>(a(b!))/((b-n)!)z^(b-n) The factorial can be extended to real and>>complex arguments by the gamma function thus giving the first and a>>half derivative or the ith derivative. Has any one done a paper on>>this? I might point out that when a mathematician thinks a branch of>>mathematics has no applications to the real world a physicist finds>>that is does.>>>Yes, you might point that out. You'd look a little less silly>pointing that out if you were talking about something that>mathematicains had not studied for a long time.>David C. Ullrich>What a helpful comment to share with the OP and this forum!-- Stephen J. Herschkorn === derivative> For interger b and n the nth derivative of a(z^b) is equal to> (a(b!))/((b-n)!)z^(b-n) The factorial can be extended to real and> complex arguments by the gamma function thus giving the first and a> half derivative or the ith derivative. Has any one done a paper on> this?'Fraid so. In the MSC indexhttp://www.ams.org/msc/we find:26A33 Fractional === derivative> For interger b and n the nth derivative of a(z^b) is equal to> (a(b!))/((b-n)!)z^(b-n) The factorial can be extended to real and> complex arguments by the gamma function thus giving the first and a> half derivative or the ith derivative. Has any one done a paper on> this? I might point out that when a mathematician thinks a branch of> mathematics has no applications to the real world a physicist finds> that is does.Look up fractional derivative...For example === Subject: Re: The first and a half derivativeInteresting that there is a well developed theory for this type of operator,when the far simpler operator that takes a function f(x) and makes g(x) suchthat f(x)=g(g(x)) - a sort of square root operator for functions - is muchless well defined. As I recall from a previous thread, not a whole lot isknown about this - unless you have some links for this as well? For interger b and n the nth derivative of a(z^b) is equal to> (a(b!))/((b-n)!)z^(b-n) The factorial can be extended to real and> complex arguments by the gamma function thus giving the first and a> half derivative or the ith derivative. Has any one done a paper on> this? I might point out that when a mathematician thinks a branch of> mathematics has no applications to the real world a physicist finds> that is does. Look up fractional derivative... For example === Subject: Summing an Uncountably Infinite No. of Terms - Meaningless?I assume the following are meaningless. Please confirm.A product consisting of an uncountably infinite number of factors.A sum consisting of an uncountably infinite number === of Terms - Meaningless?Leonard M. Wapner says...I assume the following are meaningless. Please confirm.A product consisting of an uncountably infinite number of factors.A sum consisting of an uncountably infinite number of terms.I was discussing this subject with a friend years ago when my mathprofessor at Northwestern, Mark Pinsky, burst into the room to setus straight.Suppose that you have an infinite collect C of positive realnumbers, and suppose that they sum to a finite number. Then Cmust be countable.Proof: Split C up into an infinite collection of subsets,C0 = those elements of C that are greater than 1C1 = those elements of C that are greater than 1/2C2 = those elements of C that are greater than 1/4...Cn = those elements of C that are greater than 1/2^n...Obviously, each set is finite (otherwise it would sum to aninfinite number). So C is equal to the union of countablymany finite sets, which is countable.--Daryl === of Terms - Meaningless?>Suppose that you have an infinite collect C of positive real>numbers, and suppose that they sum to a finite number. Then C>must be countable.Infinite collect? I have some issues with the assumptions given.>Proof: Split C up into an infinite collection of subsets,>C0 = those elements of C that are greater than 1>C1 = those elements of C that are greater than 1/2>C2 = those elements of C that are greater than 1/4>...>Cn = those elements of C that are greater than 1/2^nObviously, each set is finite (otherwise it would sum to an>infinite number).Your starting assumption was that there exists an infinite (possiblyuncountably so) set of reals that sum to a finite limit. You can'tturn around and say such sets can't exist when you start from theassumption that they do exist _without_limitations_.Of course any series (countably infinite sum) only converges if|an|->0 as n->oo, but since your proof speaks of possibly uncountablecollects, not series, this === an Uncountably Infinite No. of Terms - Meaningless?Toni Lassila says...>>Suppose that you have an infinite collect C of positive real>>numbers, and suppose that they sum to a finite number. Then C>>must be countable.Infinite collect?Whoops! Infinite *collection*.>>Proof: Split C up into an infinite collection of subsets,>>C0 = those elements of C that are greater than 1>>C1 = those elements of C that are greater than 1/2>>C2 = those elements of C that are greater than 1/4>>...>>Cn = those elements of C that are greater than 1/2^n>>Obviously, each set is finite (otherwise it would sum to an>>infinite number).Your starting assumption was that there exists an infinite (possibly>uncountably so) set of reals that sum to a finite limit. You can't>turn around and say such sets can't exist when you start from the>assumption that they do exist _without_limitations_.I started with the assumptions 1. C is infinite. 2. The reals in C have a finite sum.Fact 2 is a limitation on C, and I showed that that limitationimplies that C is countable. I didn't show that there are noinfinite collections that sum to a finite number, I showed thatthere are no *uncountable* collections.Perhaps you are confused by the fact that for each n,Cn has to be finite? That's a different case, becauseeach Cn has a *minimum* size for its elements, namely1/2^n. If the sum of all the elements in Cn is K (forinstance) the size of Cn can be no greater than2^n K.>Of course any series (countably infinite sum) only converges if >|an|->0 as n->oo, but since your proof speaks of possibly uncountable >collects, not series, this requirement isn't immediately obvious.Changing the order of summation cannot affect the sum in the casewhere all the reals are === an Uncountably Infinite No. of Terms - Meaningless?> I assume the following are meaningless. Please confirm.> A product consisting of an uncountably infinite number of factors.> A sum consisting of an uncountably infinite number of terms.> > L> You can have such product or sums provided all but countably many factors of such a product equal 1 and all but countably many of the terms of such a sum are zero, so that, in effect only countably many factors/terms can inßuence the === replies to this message constitute permission for an emailed You can have such product or sums provided all but countably many > factors of such a product equal 1 and all but countably many of the > terms of such a sum are zero, so that, in effect only countably many > factors/terms can inßuence the result.Provided that the terms are reals, but the original didn't ask that.In set theory, you can (and do) entertain such uncountable sums andproducts. It's defined of course in the expected way.The factors/terms are indexed by an arbitrary ordinal, and you cansum/multiply them by either ordinal or cardinal arithmetic. Theresult is of course the limit of the === Re: Summing an Uncountably Infinite No. of Terms - Meaningless?> I assume the following are meaningless. Please confirm. A product consisting of an uncountably infinite number of factors.Consider the product of all reals 0 A sum consisting of an uncountably infinite number of terms.>Again, consider the sum of all reals -1I assume the following are meaningless. Please confirm.>>A product consisting of an uncountably infinite number of factors.>>>Consider the product of all reals 0This makes *some* sense.>>A sum consisting of an uncountably infinite number of terms.>>>Again, consider the sum of all reals -1Oh, really? Why isn't this sum the sum of {x - x^2: 0 < x <1} = + infinity?Others have posted correct answers-- Stephen J. Herschkorn === Uncountably Infinite No. of Terms - Meaningless?>Again, consider the sum of all reals -1 infinity?His set contains negative numbers, for a start. I think IE vulnerability: HTTP error handler Local Zone XSSDescription: HTML/Script injection in the Local ZoneReference: http://sec.greymagic.com/adv/gm014-ie/Exploit: === an Uncountably Infinite No. of Terms - Meaningless?>>Again, consider the sum of all reals -1> Oh, really? Why isn't this sum the sum of {x - x^2: 0 < x <1} = +>> infinity?> His set contains negative numbers, for a no?If you pair up +x with -x^2 instead of +x with -x what do === algorithm>What is the fastest algorithm for computing factorial, for very large >numbers (e.g. 10000!)?Normally this would take n-2 multiplications, by multiplying out each >term n(n-1)(n-2)(n-3)...3.2. Is a better way known? > ln (n!)= (ln(n)-1)*n+(1/2)*ln((2*pi*n)) + 1/(12*n) - 1/(360*n^3)> more terms can be added when the n is smaller> hjs.> a people that don't remember their past are doomed to repeat it.but this is not exact, is it? You may as well approximate === algorithm> Just to compare. and show that 10000! is not that big> On my Mac: 10000! took .05 seconds to run> 100000! took 1.88 seconds> 1000000! took 57.32 secondsCan you print === factorial algorithm> What is the fastest algorithm for computing factorial, for very large > numbers (e.g. 10000!)?> > Normally this would take n-2 multiplications, by multiplying out each > term n(n-1)(n-2)(n-3)...3.2. Is a better way known?I don't know much about that, but I would surmise there should is.Someone said O(log n), that sounds about right.I just wanted to write this post in case you awnted to write your ownfactorial multiplier or whatnot, to offer some ideas I had afterreading your post.Every time you go down by 3 numbers, you accumulate a factor of three,because one of the numbers was divisible by 3. Going further, youaccumulate a factor of 4, or so on. Or you may just want to deal withprime factors. The trick: to make use of these factors and collectthem instead of multiplying the numbers. That should speed up yourprocess hopefully. Although I have to say, this is probably too wackya solution, since computers are able to handle large multiplicationsanyway, and going beyond 100! isn't very feasible in your regularprogarms === What is the fastest algorithm for computing factorial, for very large > numbers (e.g. 10000!)?> Normally this would take n-2 multiplications, by multiplying out each > term n(n-1)(n-2)(n-3)...3.2. Is a better way known?> I don't know much about that, but I would surmise there should is.> Someone said O(log n), that sounds about right.This assumes multiplication is constant time, regardless of the sizeof the numbers. One implication of this assumption is that factoringcan be done in O(log n) time, so it is hardly a realistic measure.> I just wanted to write this post in case you awnted to write your own> factorial multiplier or whatnot, to offer some ideas I had after> reading your post.> Every time you go down by 3 numbers, you accumulate a factor of three,> because one of the numbers was divisible by 3. Going further, you> accumulate a factor of 4, or so on. Or you may just want to deal with> prime factors. The trick: to make use of these factors and collect> them instead of multiplying the numbers. That should speed up your> process hopefully.Indeed. n! = 2^(n//2)*3^(n//3)*5^(n//5)*...*p^(n//p), where // isinteger division and p the largest prime less than or equal to p. a^bcan be calculated in O(log b) multiplications, so you save some.Unlike the O(log n) factorial, all the numbers involved are smallerthan the result, so the saving is more realistic.> Although I have to say, this is probably too wacky a solution, since> computers are able to handle large multiplications anyway,Define large. Once you go beyond the size of a machine integer(typically 32 or 64 bits), multiplications aren't very fast.Typically, a multi-word multiplication takes time proportional to theproduct of the number of words in the two numbers. Faster methodsexist (multiplying two N-bit numbers in O(N*log N), but they needfairly large numbers to be worthwhile.> and going beyond 100! isn't very feasible in your regular progarms> anyway.If you stay within the range of machine integers, I agree, but manylanguages support unbounded inetgers. Example: Typing product [1..1000]into Hugs (a Haskell interpreter) produces (with no appreciable delay): 402387260077093773543702433923003985719374864210714632543799910 429938512398629020592044208486969404800479988610197196058631666 872994808558901323829669944590997424504087073759918823627727188 732519779505950995276120874975462497043601418278094646496291056 393887437886487337119181045825783647849977012476632889835955735 432513185323958463075557409114262417474349347553428646576611667 797396668820291207379143853719588249808126867838374559731746136 085379534524221586593201928090878297308431392844403281231558611 036976801357304216168747609675871348312025478589320767169132448 426236131412508780208000261683151027341827977704784635868170164 365024153691398281264810213092761244896359928705114964975419909 342221566832572080821333186116811553615836546984046708975602900 950537616475847728421889679646244945160765353408198901385442487 984959953319101723355556602139450399736280750137837615307127761 926849034352625200015888535147331611702103968175921510907788019 393178114194545257223865541461062892187960223838971476088506276 862967146674697562911234082439208160153780889893964518263243671 616762179168909779911903754031274622289988005195444414282012187 361745992642956581746628302955570299024324153181617210465832036 786906117260158783520751516284225540265170483304226143974286933 061690897968482590125458327168226458066526769958652682272807075 781391858178889652208164348344825993266043367660176999612831860 788386150279465955131156552036093988180612138558600301435694527 224206344631797460594682573103790084024432438465657245014402821 885252470935190620929023136493273497565513958720559654228749774 011413346962715422845862377387538230483865688976461927383814900 140767310446640259899490222221765904339901886018566526485061799 702356193897017860040811889729918311021171229845901641921068884 387121855646124960798722908519296819372388642614839657382291123 125024186649353143970137428531926649875337218940694281434118520 158014123344828015051399694290153483077644569099073152433278288 269864602789864321139083506217095002597389863554277196742822248 757586765752344220207573630569498825087968928162753848863396909 959826280956121450994871701244516461260379029309120889086942028 510640182154399457156805941872748998094254742173582401063677404 595741785160829230135358081840096996372524230560855903700624271 243416909004153690105933983835777939410970027753472000000000000 000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 === the fastest algorithm for computing factorial, for very large C> numbers (e.g. 10000!)? C> Normally this would take n-2 multiplications, by multiplying out each C> term n(n-1)(n-2)(n-3)...3.2. Is a better way known?The fastest algorithm known is: A. Shamir. Factoring numbers in O(log n) arithmetic steps. Information Processing Letters,1:28-31, 1979.As far as I know, the question was first posed in my PhD thesis, ``On SomeOptimal Algorithms,'' Department of Computer Science, Cornell University,January, 1971.-- Professor Edward M. Reingold Email: reingold@iit.eduChairman, Department of Computer Science Voice: (312) 567-3309Illinois Institute of Technology Assistant: (312) 567-5152Stuart Building Fax: (312) 567-506710 West 31st Street, Suite 236Chicago, IL 60616-3729 ===