mm-1071 === Subject: Re: Mechanical Turing Machine? In 1936, Turing would not have been limited to mechanics. The relay had been known 100 years, and telephone people were building quite elaborate systems with them, as well as laying the foundations of formal switching theory. (Which was in turn, the basis for the wartime machines Turning worked on). The Eccles-Jordan flipflop (bistable) had been described in 1919, so vacuum-tube logic was also known. :I'm looking for information on any previous designs :for a mechanical Turing Machine, i.e., a machine :fabricated in the technology that Turing would have :had available at the time of his thesis. : :I have conceived of a method for the (semi) infinite :tape using techniques from clock-making, but I'm interested :in seeing designs for a reprogrammable state-machine :in mechanics. : === Subject: Re: Polynomials > i want to show, that if r is a root in a splitting field of f=x^3+x^2-2x-1 > over Q, then r'=r^2-2 is another root. > I split f into factors f=(x-r)(x^2+(r+1)x+(r^2+r-2)), but how can I see, > that r^2-2 is a root of the quadratic factor? factorization of f(x) by just dividing it by (x - r). If that's the case, then you should have _no_ trouble finishing this -- just use polynomial long division to compute (x^2 + (r_1)x + (f^2+r-2))/(x - (r^2-2)) ... it'll work out nicely _because_ f(r) = 0 -- kinda neat, actually :-) (As a bonus, you'll also determine the third root of f in terms of r: it's r^2+r-1 ... ) > Johannes === Subject: Re: Uncountable many reals without Cantor >>there are often threads in this group concerning >>the cardinality of the set of real numbers. Some >>persons seem to have strong objections against the >>Cantor Proof of the fact that the set of real >>numbers is not denumerable by the naturals. >People may have strong objections, but nobody >has any _coherent_ objections - the people who >object seem to be unable to follow very simple >reasoning. Hence I doubt that they're going to >be able to follow complicated chains of reasoning... David is a lecturer in mathematics, his salary depends on the maths department maintaining its budget while I.T. is phased out of education. It doesn't really matter to David what is right, millions of mathematicians have fallen for a simple trick that you can generate a new infinite sequence when you can't, so he's not going to lose any argument. One side of his mind believes in the bogus maths, the other needs the money, there is no conflict within David to seek out the truth. But their arguments are clearly bogus, they CANNOT toss coins in a new sequence that infinite (in all other respects the same) other people have tried, it goes against simple probability, reasoning, range of functions and logic. Now these arguments are being formalised against this greatest monstrosity of miscomprehension of cardinal representations in human endevour, David is panicking. He is clearly commiting libel here. >nobody >has any _coherent_ objections - the people who >object seem to be unable to follow very simple >reasoning This is after David has discussed infinitessimal details of the proof over 1000s of posts with many, many people. Constructivists often paraphrase the exact arguments of the Cantorian followers like David to point out the error. Then again and again and again the only resonse is... assume the sequence is at n, therefore n=~n as we described, and the inevitable Y O U M U S T B E S T U P I D N O T T O F O L L O W I T We've all done the proof 1000 times, we know people with <90 iq cannot follow the proof at all, iq is not the issue here. If iq is relevant its at a higher level where constructivists can see the self defeating statements that you entwine your own arguments into. YOU CANT PROVE ME ~ this is another that you take for granted, the NEVER TO BE REVOKED famous Godel statement that puts the nature of the universe outside of the scope of denumerable I.T. what fuking hogwash Herc === Subject: Re: inverse modulos The Euclidean algorithm constructively produces coefficients a,b such that ax + bm = gcd(x,m). So given x,m with gcd(x,m) = 1, one easily finds the multiplicative inverse a of x modulo m. You are asking if a were known, could you find x (at least, up to modulo m), and the answer is yes. This is just turning the question around; x is again the multiplicative inverse of a modulo m. If you do a Google search for Euclidean algorithm, you'll see that it is a pretty iteration of the division algorithm. === Subject: Re: inverse modulos > Solving diophantine equations is easy... it uses the extended > Euclidean algorithm, for details see e.g.: > http://www.win.tue.nl/~ida/demo/c1s3f2.html >> makes use of the extended Euclidean algorithm, but it's not giving >> the result I want. I thought that the inverse modulo was supposed >> to be unique where it existed, but I must have misunderstood that. >> Apparently there are many solutions sometimes (or always?) > There are always infinitely many solutions. However, since: > ax - km = 1 > it follows that gcd(x, m) = 1, so there exists exactly one x which > satisfies the above equation and is less than m (and bigger than 0, > too). Indeed, suppose that we have two such x-es, x and x'(and two > respective k and k'). Then: > ax - k m = 1 > ax' - k'm = 1 > Subtracting the second equation from the first gives: > a(x - x') - m(k - k') = 0 > that is a(x-x') = m(k'-k), which yields x-x' = m(k'-k)/a. But also > gcd(a,m) = 1, which implies that a divides k'-k, so that (k'-k)/a is > an integer. That means x and x' differ by multiplicity of m, which is > possible only if one of them is bigger than m. Simpler: Inverses are always unique in any abelian group G. For if B has inverses A,C then A = A(BC) = (AB)C = C Here G = U(Z/m) = unit group of Z/m, the ring of integers (mod m), i.e. the units (invertibles) of the multiplicative monoid of Z/m. By the extended Euclidean/Bezout algorithm the units are simply the cosets n (mod m) := n + mZ such that n is coprime to m since then and only then does there exist integers j,k such that j n + k m = 1 <=> j n = 1 (mod m) <=> n in U(Z/m) Uniqueness theorems are powerful tools for proving equalities. For many further less trivial examples see my prior posts --Bill Dubuque === Subject: Re: inverse modulos >I thought that the inverse modulo was supposed >to be unique where it existed, but I must have misunderstood that. >Apparently there are many solutions sometimes (or always?) >>There are always infinitely many solutions. However, since: >> ax - km = 1 >>it follows that gcd(x, m) = 1, so there exists exactly one x which >>satisfies the above equation and is less than m (and bigger than 0, >>too). Indeed, suppose that we have two such x-es, x and x'(and two >>respective k and k'). Then: >> ax - k m = 1 >> ax' - k'm = 1 >>Subtracting the second equation from the first gives: >> a(x - x') - m(k - k') = 0 >>that is a(x-x') = m(k'-k), which yields x-x' = m(k'-k)/a. But also >>gcd(a,m) = 1, which implies that a divides k'-k, so that (k'-k)/a is >>an integer. That means x and x' differ by multiplicity of m, which is >>possible only if one of them is bigger than m. > Simpler: Inverses are always unique in any abelian group G. (...) Well, yeah, I know that much :-) But it seems that the person who asked that question didn't know much about groups, rings etc., so I was trying to explain the uniqueness of an inverse element in Z_m in the simplest possible terms... hope I haven't confused him too much. Sometimes one has to say difficult things, but one ought to say them as simply as one knows how - G. H. Hardy Pawel Gladki === Subject: Re: Unbounded Space >> What is this naive argument supposed to show? What does the actual > What all of them show, that you are an uneducated dummy. Learn how to converse first. I shall not respond to the rest of your swear words. Try to make a case like a civilized person, and I shall respond to your arguments. For this time only, I will neglect your improper attitude. === Subject: Re: need help in understanding Torkel's ZFC comment >>Summary of the discussion so far: >>Frege> http://au.metamath.org/mpegif/2p2e4.html >>Charlie-Boo> [6-line informal proof of 0'' + 0'' = 0''''] >>Charlie-Boo> I beat him by 19,725 steps. >>Ullrich> That's not a formal proof from the axioms of ZFC. >Neither is the MetaMath proof of 2+2=4. >>Yes it is. If you believe otherwise then point out an axiom that is used >>in the proof but is not part of ZFC. > Very few of the terminal nodes of the proof tree are ZFC axioms, > because ZFC has nothing to do with the fact that 2+2=4. (Only 1 set > is involved, so we need not consider the question of what sets exist > in general. 2+2=4 regardless of what ZFC says.) > At these nodes you will find other axioms, rules that allow > arbitrary wffs to be considered proven (unsound) and unverified > definitions (also unsound) which must be set up to verify new theorems > (a strict no-no in axiomatic systems.) There are also many nodes that > have no links or justification, such as ECOPRASS at > http://us.metamath.org/mpegif/ecoprass.html . What is the > justification for lines 23 and 24 (and the other lines labeled > ecoprass)? Looks like hypotheses of the theorem to me. > If you want to argue that certain rules don't count, that is not > productive [CapitalEth] the fact remains that each branch of Mathematics has its > own axioms. The assertion that everything comes from ZFC is a lie and > I would love to hear all sorts of professors claim that it is so. > MetaMath derives almost nothing from any axioms. It uses a number of > subterfuges to allow you to declare arbitrary expressions (even > meaningless sequences of symbols) to be theorems. > An axiomatic system is one that derives everything from a fixed set of > axioms and rules (and perhaps definitions.) After it is set up, (1) > you don't have to add anything to the system to develop a theorem, and > (2) only actual theorems can be constructed. (Agree?) > Otherwise you have no more than a word processor with a Mathematical > font. > C-B >Do you have any idea how the theorem was produced? How the proof was >produced? To what extent and how it was verified? >>Do you? It's only been explained to you two or three times by now. >>For the questions you ask, the web site really does >>explain it in detail. > Ok, then what are the possible justifications for the terminal nodes > in a proof tree? -- Replace Roman numerals with digits to reply by email === Subject: Re: need help in understanding Torkel's ZFC comment >>Ullrich> That's not a formal proof from the axioms of ZFC. >Neither is the MetaMath proof of 2+2=4. >>Yes it is. If you believe otherwise then point out an axiom that is used >>in the proof but is not part of ZFC. > Very few of the terminal nodes of the proof tree are ZFC axioms, > because ZFC has nothing to do with the fact that 2+2=4. (Only 1 set > is involved, so we need not consider the question of what sets exist > in general. 2+2=4 regardless of what ZFC says.) > At these nodes you will find other axioms, rules that allow > arbitrary wffs to be considered proven (unsound) and unverified > definitions (also unsound) which must be set up to verify new theorems > (a strict no-no in axiomatic systems.) There are also many nodes that > have no links or justification, such as ECOPRASS at > http://us.metamath.org/mpegif/ecoprass.html . What is the > justification for lines 23 and 24 (and the other lines labeled > ecoprass)? > Looks like hypotheses of the theorem to me. What are the hypotheses to 2+2=4? > An axiomatic system is one that derives everything from a fixed set of > axioms and rules (and perhaps definitions.) After it is set up, (1) > you don't have to add anything to the system to develop a theorem, and > (2) only actual theorems can be constructed. (Agree?) > Otherwise you have no more than a word processor with a Mathematical > font. > C-B You know, this is kind of like deja vue. All these people yelling and screaming and I'm the one who (as is often the case) puts out the formal definition to resolve technical issues - and they all go running for the hills until time for the next skirmish. What a life! === Subject: Re: dimension of free modules - looking for example > somewhere (probably in Lang's Algebra) I have read that there are rings R > and free modules R^n and R^m which are isomorphic although n!=m. > Does anybody know of an example? A search on invariant basis number (IBN) will turn up probably more than you wanted to know. --Bill Dubuque === Subject: Re: Kick classic in shins, win prizes Edward Green says... >A symbol v (bold-v) represents a particular infinitesimally long >displacement of the point P (script-P), which is the argument of the >function f = f(P). An inner product is formed and set equal to >a symbol @_v f (partial d subscript bold-v, italic-f), which is the >change of f in going from the tail of v to its tip. Mighty like a >differential, no? In fact we can identify the old notation with the >new term-by-term: > = @_v f == = df >Each slot on the rhs and lhs is occupied by the same object, but with >a different symbol (grad f is the gradient, dX the infinitesimal >displacement of the argument, df the infinitesimal change in the >function). Notice in particular that df and Df are _not_ mapped >to each other, but are distinct objects. I think the point is that in the *old* way of doing things, you need to pick a coordinate system in order to define grad f, you need a coordinate system in order to define dX, you need a coordinate a third time to define the inner product of grad f with dX. In MTW's way of doing things, the directional derivative of f with respect to v requires no coordinates for its definition. You write: >Heavens! This means that df is itself a variable or a function of >its argument (the change in P), as _well_ as an infinitesimal! What >are we to do! > But this is precisely what the exterior derivative Df represents... >Nope. Df represents an _engine_ to _calculate_ such a thing, given an >infinitesimal displacement as input, same as the gradient. In fact, >it _is_ the gradient, as they just finished telling us. Seems they >forgot. You agree that df is a *function* of the displacement. That's exactly what Df is. I don't understand your contrasting df is a variable or function of its argument and Df is an enging to calculate such a thing, given an infinitesimal displacement as input. Df is not the same thing as the gradient, even though it has the same information. The gradient is a vector, while Df is a 1-form (a linear function on vectors). Yes, if you have a metric, you can convert between 1-forms and vectors, but Df makes sense in the absence of any metric. Df is related to the usual gradient grad f via: (grad f)^i = (Df)_j g^{i,j} The main point of doing things the way MTW do it is to see what of your definitions require a metric, and what do not. Yes, in flat spacetime, when the metric is constant, you can ignore the difference between a 1-form and a vector, since they are always interconvertible via the constant metric. But when the metric itself varies from place to place, it is a little dangerous to ignore such implicit conversions. Where the metric is being used should be made explicit. -- Daryl McCullough Ithaca, NY === Subject: Re: Kick classic in shins, win prizes Hi Daryl. Nice to hear from you. Let me group your response under two headings: the behavior of grad f, and my allegation of fuzzy thinking by MTW. You write: > Df is not the same thing as the gradient, even though it has the same > information. The gradient is a vector, while Df is a 1-form (a linear > function on vectors). <... Df is related to the usual gradient grad f via: > (grad f)^i = (Df)_j g^{i,j} Ok. I take it that the usual gradient means the object whose components are @f/@x_i. One certainly must investigate whether this even defines a covector _or_ a vector, since a general rule for writing objects in terms of coordinates might define no geometric object at all (we call these things symbols). So you say in effect that the usual rule in fact gives a coordinate representation of a vector, not a covector. This would certainly be an important point. I haven't convinced myself whether this is true or not, but it's at least clear that if we are using the funny metric with the minus sign, we at least have to negate the x_0 component of grad f to use it as a covector, even in the nicest coordinate systems. However, I claim this is oblique to my criticism. You respond to: >This means that df is itself a variable or a function of >its argument (the change in P), as _well_ as an infinitesimal! > But this is precisely what the exterior derivative Df represents... >Nope. Df represents an _engine_ to _calculate_ such a thing, given an >infinitesimal displacement as input, same as the gradient. > You agree that df is a *function* of the displacement. Yes I do. > That's exactly what Df is. I find that a strange claim. Consider the ordinary scalar function: y = bx where y is a variable, x is a variable, b is a constant; x varies, y varies, b holds steady. The relation as a whole is a function, or y is a function of x. But we never call the constant b itself a function of x. Now consider the inner product: @_v f = where v is a variable, @_v f is a variable, and Df holds constant (considering a fixed point on the manifold). Df is not itself a function of the displacement. It is the constant in a linear function, like b above. > I don't understand your contrasting df is a variable or > function of its argument and Df is an enging to calculate such a > thing, given an infinitesimal displacement as input. Well, that's what I meant. I'm not saying there is something wrong with the new approach, or attacking the great brave new coordinate free geometric world. And you do remind us to be careful how we correctly understand the bad old coordinate shackled notations as representations of objects in the brave new world. But MTW stray into nonsense with their casual calumnies of the old world: There is nothing wrong with df being a function without explicitly mentioning them its arguments. It never bothered us with y -- if necessary we could always write y(x). And Df is not a more sophisticated geometrized version of df, despite the symbolic simularity; it is a more sophisticated geometrized version of grad f. And neither is Df a function of the displacement, any more than grad f was: they are constant linear operators waiting for input. Give them a differential input, they output a differential increment; give them a unit vector, they output a directional derivative. Stuff like this is a pet peave of mine. === Subject: Re: Compute plane with angles and vectors >I have two known vectors with the same origin and a unknown plain >which goes also through this origin. The two angles between the two >vectors and the normal of the plane are known, but not the normal >itself. How can I compute the normal? >Is there another way than this: >a1*n1 + a2*n2 + a3*n3 = cos(alpha1) >b1*n1 + b2*n2 + b3*n3 = cos(alpha2) >n1^2 + n2^2 + n3^2 = 1 >Because the result is really huge (over pages)... >There can be two results, right? >Can I fix this with a third vector (and a third angle), such that I >have only one result? >How would I do this? > Yes, with a third vector C it would be easy. You could compute both > N cross (A cross B) and N cross (A cross C) from your information, > each of which would be perpendicular to N, giving two vectors parallel > to the plane. Cross them again to get a normal. > --Lynn I'm sorry, but how would it look like? equationsystem? Because N is unknown. === Subject: Re: Compute plane with angles and vectors >> Yes, with a third vector C it would be easy. You could compute both >> N cross (A cross B) and N cross (A cross C) from your information, >> each of which would be perpendicular to N, giving two vectors parallel >> to the plane. Cross them again to get a normal. >> --Lynn >I'm sorry, but how would it look like? >equationsystem? Because N is unknown. Use the formula: N cross (A cross B) = (N dot B) A - (N dot A) B. Think of N as a unit vector, then N dot B = |B| cos(angle) which you know, and same for N dot A. You don't need a system of equations. --Lynn === Subject: Re: The Parageodetic Distance Formula (UBasic) Whoa!! Do you like to hear yourself or what? > 
 programmer has decided to first release the parageodetic (inverse
> problem) formulary core, before the more dressed, integrated
> quadratic-mean-spherical, parageodetic and geodetic inverse and
> direct GODx program core.
> But, rather than just spit out the program listing in numerical
> address order, however, it is being presented subroutine-with-execution,
> thereby providing another opportunity to disseminate this unique
> formulary approach (including all trigonometric quadrantic adjustments
> and division by zero bulletproofing) for mathematical study and
> discussion.
> As the concept of the parageodesic appears little acknowledged--if at
> all even recognized--it would seem prudent to comparatively define the
> geodesic and parageodesic from the conceptual approach used here (to
> the non-technical layman, the term graticular can be considered
> synonymous with spherical):
> Geodesic: The conformally delineated great-arc segment
> between two points;
> Parageodesic: The graticularly delineated great-arc segment
> between two points.
> Thus, with an ellipsoid, the traditional geodesic is the pulled
> string path and distance on an elliptically correct model, whereas
> the parageodesic is the pulled string path and distance, first
> DEFINED on a spherical model of the ellipsoid, then the sphere is
> reformed to its proper ellipsoidal shape, at which time the
> elliptically correct arcradius and angle/azimuth AT ANY GIVEN POINT
> ALONG THE DEFINED SEGMENT can be found (i.e., it is the local,
> elliptically correct Great-Circle valuation!).
> UBasic's system program can be found here:
> ftp://rkmath.rikkyo.ac.jp/pub/ubibm
jp/~kida/ubasic.htm
> (this web page is translated through Google, thus some of the wording is
off)
> UBasic's command dictionary can be found here:
> http://ed-thelen.org/bab/bab-ubhelp.html
> Of course, while it is presented here out of numerical address order,
> once the program code is cut and pasted into UBasic, it will revert to
> its natural arrangement.
> 
--------------------------------------------------------------------------
--------------------
> 3000 GoSub 58000
> 58000 '*** Radial Parameters ***
> 58010 a=6378.135:b=6356.75:b=b::GoTo 58200
> 58020 a=10000:b=5000
> 58200 Oz=2*Atan((a-b)^.5/(a+b)^.5)
> 10000 BLat=+43.66111111111111111111:BLong= -70.25583333333333333333
> 10010 DLat=+45.52305555555555555556:DLong=-122.67583333333333333333
> 1' Parageodetic = 4092.6024524649626 Km 
291.6521536Á/73.6370729Á
> 1' Geodetic = 4092.6016550196721 Km 
291.7142510Á/73.5743344Á
> 1' Q-M-Spherical = 4082.4708695785296 Km 
291.7214263Á/73.5858490Á
> 2200 Pi=#Pi:PH=.5*Pi:PD=2*Pi
> 2300 RF=Pi/180
> 55110 BLr=BLat*RF:DLr=DLat*RF:MD=(DLong-BLong)*RF:M=Abs(MD):If M>Pi Then
MD=MD-Sgn(MD)*PD
> 11000 GoSub 55000
> 1'############################################################
> 55000 '*** Spherical, ParaGeodetic Valuations ***
> 55100 ID%=0
> 55120 WBLr=DLr:WDLr=BLr:GoSub 55400:DAz(0)=WAz(0):TvL(5)=TvL(0)
> 55130 WBLr=BLr:WDLr=DLr:GoSub 55400:BAz(0)=WAz(0):TvL(1)=TvL(0)
> 1'##########
> 55400 '*** Spherical Elements ***
> 55410 GoSub 55420:GoSub 55440:Return
> 55420 SA=Cos(WDLr)*Sin(M):If SA=0 And Abs(WDLr)0 Then
SA=#Eps
> 55422 If Abs(WBLr/RF)>=90 Or Abs(WDLr/RF)>=90 Then SA=0
> 55425 SB=(Sin(WBLr+WDLr)*Sin(.5*M)^2)-(Sin(WBLr-WDLr)*Cos(.5*M)^2)
> 2150 IO=10^100
> 55430 WAz(ID%)=Abs(Atan(IO*SA/(IO*SB+1)))
> 55440 TvL(0)=Atan(IO*Sin(WBLr)/(Abs(IO*Cos(WBLr)*Cos(WAz(ID%)))+1))
> 55445 If SB<0 Then WAz(ID%)=Pi-WAz(ID%)
> 55439 Return
> 55449 Return
> 1'----------
> 55125 GoSub 55190:DAz(1)=WAz(1)
> 55135 GoSub 55190:BAz(1)=WAz(1)
> 1'##########
> 55190 ID%=1:SA=SA*.E1p(0,WBLr):SB=SB*.E0p(0,WBLr):GoSub 55430:GoSub
55445:ID%=0:Return
> 1'----------
> 2100 Dcm=Int(Log(#Eps)/Log(0.1)-5):Syota=0.1^Dcm
> 55140 If Cos(BAz(0))<0 Then TvL(1)=Sgn(IO*BLr+Syota)*Pi-TvL(1)
> 55142 If Cos(DAz(0))>0 Then TvL(5)=Sgn(IO*DLr+Syota)*Pi-TvL(5)
> 55145 If TvL(5) 55150
AD(0)=TvL(5)-TvL(1):ADq_0=0.25*AD(0):BTvL=TvL(1)/RF:DTvL=TvL(5)/RF:ADg=DTvL-

BTvL
> 55155 TvL(2)=TvL(1)+ADq_0:TvL(3)=TvL(1)+2*ADq_0:TvL(4)=TvL(1)+3*ADq_0
> 55147 GoSub 55450:APg=AP(0)/RF:GoSub 55485:ID%=1:GoSub 55485
> 1'##########
> 55450
AP(ID%)=Atan(Abs(IO*Cos(WBLr)*Sin(WAz(ID%)))/(IO*(Cos(WAz(ID%))^2+(Sin(WBLr)

*Sin(WAz(ID%)))^2)^.5+1)):Return
> 55485 If MD<0 Then BAz(ID%)=PD-BAz(ID%)
> 55487 If MD>0 Then DAz(ID%)=PD-DAz(ID%)
> 55489 BAz(ID%)=BAz(ID%)/RF:DAz(ID%)=DAz(ID%)/RF:Return
> 1'----------
> 55189 Return
> 1'------------------------------------------------------------
> 11100 GoSub 59100:KmDx(1)=a*EGDD*AD(0)
> 1'############################################################
> 59100 '*** PEOx ***
> 59000 '*** Integration ***
> 65100
.E2p(AP,TvL):Return((Cos(Oz)^2+(Sin(AP)*Sin(Oz))^2+(Cos(AP)*Cos(TvL)*Sin(Oz)

)^2)^.5)
> 1' = Elliptic Integrand Of The Second Kind
> 65110 .E1p(AP,TvL):Return(1/.E2p(AP,TvL))
> 1' = Elliptic Integrand Of The First Kind
> 1' x a = N (Normal) = Transverse Equatorial Arcradius
> 65120 .E0p(AP,TvL):Return(Cos(Oz)^2*.E1p(AP,TvL)^3)
> 1' x a = M (Meridian) = Vertical Meridional Arcradius
> 65130
.EGp(AP,TvL):Return((.E0p(AP,TvL)^2+Sin(AP)^2*(IO*(.E1p(AP,TvL)^2-.E0p(AP,Tv

L)^2)/(IO*(Cos(TvL)^2+(Sin(AP)*Sin(TvL))^2)+1)))^.5)
> 1' x a = O (Omniversal) = Transverse Meridional Arcradius
> 1'#########################
> 11020 XF=1:::For UL=2 To 20 Step 2::GoSub 60000
> 1' Integration Convergency for EGDD to .1^20 @ cos{Oz}, for XFxUL: 
cos{Oz}
= 1xUL/2xUL/3xUL
> 1' .9999 = 6/8/6; .995 = 8/12/15; .99 = 8/12/15; .9 = 11/16/18;
> 1' .75 = 13/18/24; .5 = 17/24/30; .25 = 27/36/45; .1 = 42/60/75;
> 1' .01 = --/--/231; (.001 ~=~ 20x94 = 1880; .0001 ~=~ 200x97 = 19400)
> 60000 '*** (Gaussian) Amplitudal Kernal Expansion ***
> 65000
.AE(NS,NX,XF):AE=2*Atan(((XF-NX+Sin(.5*AEg(NS))^2)/(NX-Sin(.5*AEg(NS))^2))^.

5):If AE>.5*Pi Then AE=Pi-AE
> 65009 Return(AE)
> 60100 DU=2*UL:For NS=1 To UL:CK=Cos(.5*Pi*(((4*NS)-1)/((4*UL)+1)))
> 60200 LP(1)=1:LP=1:For NP=1 To DU
> 60210
LP(2)=LP(1):LP(1)=LP:LP=((((2*NP)-1)*LP(1)*CK)-((NP-1)*LP(2)))/NP:Next NP
> 60220 DP=DU*(LP(1)-(LP*CK))/(1-CK^2)
> 60230 CK(2)=CK(1):CK(1)=CK:CK=CK(1)-(LP/DP)
> 60235 If Abs(CK-CK(1))>(10*Syota) And CK<>CK(2) Then 60200
> 60300 AEg(NS)=2*Atan((1-CK)^.5/(1+CK)^.5)
> 60310 OE(NS)=Sin(AEg(NS))^2/(DU*LP(1)^2):Next NS:Return:::::?NS;
;Using(,30),AEg(NS)/RF; ;OE(NS)/RF:Next NS:List 60010:End:Return
> 1'-------------------------
> 59110 EGDD=0:For NS=1 To UL:For NX=1 To XF
> 59120 AE=AEg(NS):If XF>1 Then AE=.AE(NS,NX,XF)
> 59130 TW=XF*UL:ABq=Cos(AE)*ADq_0:OW=OE(NS)/TW
> 59140 UUP=TvL(4)+ABq:UMP=TvL(4)-ABq:LMP=TvL(2)+ABq:LLP=TvL(2)-ABq
> 59150
EGDD=EGDD+0.25*(.EGp(AP(0),UUP)+.EGp(AP(0),UMP)+.EGp(AP(0),LMP)+.EGp(AP(0),L

LP))*OW
> 59160 Next NX:Next NS:Return
> 1'------------------------------------------------------------
> 10 ClS
> 1000 Dim
AEg(101),OE(101),LP(2),CK(2),AD(1),AP(1),TvL(5),KmDx(1),WAz(1),BAz(1),DAz(1)

> 2000 '*** Program Constants/Standard Values ***
> 2400 Deg$=Chr(248):'(Degree Sign)
> 58300 ?Tab(20);************************************:?Tab(19);/
Parageodetic Distance Formula Core :? *----------------* ( cos{Oz}
=;Using(,20),Cos(Oz); ) *-------------------*
> 58310 ?Using(,13), | a =;a;
Km;;:?Tab(55-Len(Str(Int(b))));Using(,13),b =;b; Km; |:?
*---------------------------------------------------------------------------

*:Return
> 11003 ?Tab(7);BTvL, DTvL: ;Using(,20),TvL(1)/RF;Deg$,TvL(5)/RF;Deg$
> 11005 ?Tab(25);ADg: ;Using(,20),ADg;Deg$
> 11007 ?Tab(25);APg: ;Using(,20),APg;Deg$
> 11010 ?Tab(5);BAz_g, DAz_g: ;Using(,20),BAz(0);Deg$,DAz(0);Deg$
> 11012 ?Tab(5);BAz_e, DAz_e: ;Using(,20),BAz(1);Deg$,DAz(1);Deg$
> 11015 ?Tab(5);APe_b, APe_d:
;Using(,20),.APe(AP(0),TvL(1))/RF;Deg$,.APe(AP(0),TvL(5))/RF;Deg$
> 65010
.ROz(APg):Return(Atan(IO*Cos(APg*RF)*Sin(Oz)/(IO*(1-(Cos(APg*RF)*Sin(Oz))^2)

^.5+1)))
> 1' = Reduced Oz: cos{ROz{0Á}} = cos{Oz}, 
cos{ROz{90Á}} = 1
> 65020
.APe(AP,TvL):Return(Atan(Sin(AP)*(.E2p(AP,TvL)^4*.EGp(AP,TvL)^2-Sin(AP)^2)^(

-.5)))
> 11017 GoSub 11990:?Tab(16);Cos{ROz{APg}} 
=;Using(,20),Cos(.Roz(APg)):?
> 11900 ?TW =;UL*XF;Tab(10);PEODx: ;Using(,20),KmDx(1); Km
(;Using(,25),EGDD; ):Next:?
> 11910 GoSub 11990
> 11990
?Tab(4);*-----------------------------------------------------------------
-
---*:Return
> 11980 ?:List 11020:List 58010-58030:List 10000-10010:End
> 64000 ' *** Functions ***
> 1 Point 7
> ' ~=~ .1^33
> 
--------------------------------------------------------------------------
--------------------
> This program is presented as a workbench specimen, intended for
> dissection and analysis.
> For just a no-frills output, the following modification can be
> made.
> -------------------------------------------------------
> Delete 11003-11900
> 4000 XF=1:UL=10:GoSub 60000
> 11100 GoSub 59100:KmDx(1)=A*EGDD*AD(0)
> 11200 ?:?Tab(12);PEODx: ;Using(,20),KmDx(1); Km:?
> 11300 ?Tab(5);BAz_e, DAz_e: ;Using(,20),BAz(1);Deg$,DAz(1);Deg$
> 11980 ?:?:List 58010:?:?:?:List 10000:?:?:List 10010:End
> 58010 a=6378.135:b=6356.75
> 58020
> -------------------------------------------------------
> ~Kaimbridge M. GoldChild~
>
=== Subject: No Unique Initial Segment And No Characteristic Expansion N. U. I. S. A. N. C. E. No Unique Initial Segment And No Characteristic Expansion . . . . . . Infinite people each flip coins infinite times. Can you always find a different sequence of heads and tails? sci.math and sci.logic went quiet on this question for about a week, then against all logic, probability theory, and common sense they all agreed YES. Believers of hyperinfinities have no shame! How on Earth can you exhaust an infinite set? This is their solution, a standard application of Cantor's diagonalisation technique, the hailed method of proving infinity incomplete (infinite already means never ending or incomplete!) Take one of the people, whatever his 1st flip was, reverse it! If he flipped a head you select tail, if he flipped a tail, heads. That's your first outcome, cross him off and select someone else, whatever was their second flip, reverse it! Keep on going and you have an infinite sequence that is different to everyone's sequence in atleast one flip. THAT'S THE PROOF! That's hyperinfinity and cardinality theory 101. How many assumptions about infinity are they taking for granted here? What jumps straight out at most everyone is : Aren't all possible combinations of heads and tails for infinite flips already been done? This doesn't stir the Cantor supporter one bit. They think if the combination is on the list, it must be at some natural number position n, but the nth flip of person n is necessarily different! Voila a (somewhat contrived) contradiction. Hence the sequence does not appear anywhere on the list. Makes me reminiss to lectures on these types of problems. You decide for yourself no this is right. So the lecturer asks you for the further outcome of that conclusion, which goes around in a circle and proves what he wanted to. It's played out to everyone who studies theory, you can see the conversions taking place on sci.logic and sci.math each week. The other supporting argument Cantor followers have is demonstrated when you assume this infinite list: 0.222222.. 0.322222.. 0.332222.. 0.333222.. 0.333322.. 0.333332.. .. The diagonal is 0.22222.. If we modify every digit, we can get 0.33333.. Does 0.3.. occur on the list? 0.3333333 <--L--> There are unlimited 3s in sequence on the list! Although 0.333.. has No Unique Initial Segment, it does have a Characteristic Expansion. It ends in 333.., all members end in 222.. so no, 0.3.. does not occur on the list. Back to the infinite flippers list: httttt.. hhhttt.. hhhhhh.. tttttt.. hhhttt.. .. With probability 1, this list contains every initial segment possible of heads and tails sequences. Assume there is some initial segment that is not on the list. TTTHHH This sequence has a finite length L, there are 2^L possible sequences of length L, so with infinite amount of flippers that initial segment will be covered with probability 1. The diagonal sequence HHHTT.. inverted TTTHH.. has No Unique Initial Segment. That in itself does not prove it's on the list, remember 0.333..! But TTTHH.. has No Characteristic Expansion either. As far as can be determined the sequence appears on the list. So although we may not be able to disprove the hyperinfinity status of the diagonal yet, we can show that it is a N.U.I.S.A.N.C.E. Herc === Subject: Re: No Unique Initial Segment And No Characteristic Expansion H7> What jumps straight out at most everyone is : H7> Aren't all possible combinations of heads and tails for infinite flips H7> already been done? No, obviously; the anti-diagonal BY DEFINITION hasn't been done. H7> This doesn't stir the Cantor supporter one bit. They think if the H7> combination is on the list, it must be at some natural number position We DON'T just THINK this Herc: WE *KNOW* this for an absolute DEFINITION. That's what list MEANS if you are insisting that it is a list-as-we-know-it. Now, IF you want to start talking about lists of order-types LONGER THAN omega, lists where maybe an infinite number of things can come BEFORE something on the list as well as after, so that SOME things on the list are at positions that are NOT any particular natural number, WELL the, have at it. But at a bare minimum, if you are going to KEEP calling it a list, it is going to HAVE to be well-ordered, that is, it is going to have to have a first element and every element in it is going to have to have a unique next element. There are a great many infinities you can embed or permute the list into and still keep it countable; there are uncountably many countable ordinals. But the point remains that we are NOT believing something that MIGHT be mistaken when we say that everything on the list is on it at some natural number position: THAT is true BY DEFINITION, JUST as surely as it is true that every decimal place in the decimal expansion of a real is some NATURAL number of places to the right of the decimal point. THAT'S WHAT list MEANS, FOOL. Obviously we're sorry for you that you're so ignoran that you didn't know what a list was. If you want to talk about a list of things with MORE than w elements, well, have at it, but STATE YOUR AXIOMS and DEFINE YOUR TERMS. === Subject: Re: No Unique Initial Segment And No Characteristic Expansion H7> What jumps straight out at most everyone is : H7> Aren't all possible combinations of heads and tails for infinite flips H7> already been done? No, obviously; the anti-diagonal BY DEFINITION hasn't been done. H7> This doesn't stir the Cantor supporter one bit. They think if the H7> combination is on the list, it must be at some natural number position We DON'T just THINK this Herc: WE *KNOW* this for an absolute DEFINITION. That's what list MEANS if you are insisting that it is a list-as-we-know-it. Now, IF you want to start talking about lists of order-types LONGER THAN omega, lists where maybe an infinite number of things can come BEFORE something on the list as well as after, so that SOME things on the list are at positions that are NOT any particular natural number, WELL the, have at it. But at a bare minimum, if you are going to KEEP calling it a list, it is going to HAVE to be well-ordered, that is, it is going to have to have a first element and every element in it is going to have to have a unique next element. There are a great many infinities you can embed or permute the list into and still keep it countable; there are uncountably many countable ordinals. But the point remains that we are NOT believing something that MIGHT be mistaken when we say that everything on the list is on it at some natural number position: THAT is true BY DEFINITION, JUST as surely as it is true that every decimal place in the decimal expansion of a real is some NATURAL number of places to the right of the decimal point. THAT'S WHAT list MEANS, FOOL. Obviously we're sorry for you that you're so ignoran that you didn't know what a list was. If you want to talk about a list of things with MORE than w elements, well, have at it, but STATE YOUR AXIOMS and DEFINE YOUR TERMS. === Subject: Re: No Unique Initial Segment And No Characteristic Expansion That's my point, you don't show the member is unique, you just define it so. c1=!f1,1 & c2=!f2,2 &... Therefore c1!=f1,1 & c2!=f2,2 &.. WOW big conclusion. but your definition is flawed. your argument, assume sequence is at n, cn!=fn,n is self referential, its completely meaningless, no different to Russels paradox or many others. define a number that is not a number, what do you have? So how did you conclude the sequence is original? by changing digits? every digit in every digit position is saturated, its not going to run out of options. THINK this time, any 10 year old can work this out. Infinite people all toss coins infinite times each. Can you come up with a new sequence of heads and tails? The diagonal is not going to help here. All combinations to infinite length are done. Lets drop infinity and go by the book, only allow assertions about large objects as they tend to infinity. Many people toss coins many times each. How many coins do you have to toss in sequence, to eventually get a new sequence? log(number of people). As number of people -> oo, log(number of people) -> oo. All sequences have been done to infinite length, you need MORE THAN INFINITE flips to succeed. A sequence has a HEAD and a TAIL. The diagonal of a random infinite list has NO UNIQUE INITIAL SEGMENT in the head and NO CHARACTERISTIC EXPANSION i the tail. There is nothing to distinguish the diagonal from members of the list. The diagonal of UTM(X, Y) has NO UNIQUE INITIAL SEGMENT and NO CHARACTERISTIC EXPANSION to distinguish it from members of the list. Infinite means NEVER ENDING. Infinite means NEVER COMPLETED If you try to prove INFINITY is not complete you get TRUE. Herc === Subject: Re: No Unique Initial Segment And No Characteristic Expansion Herc, you just can't write down a list of the reals, and you can write down a list of the naturals. Get over it. === Subject: Re: No Unique Initial Segment And No Characteristic Expansion UTM(X,Y) the_diag_has_no_unique_digits! . . The following is clearly insufficient to base existence of a new sequence or a new infinity upon... Infinite people each flip coins infinite times. Can you always find a different sequence of heads and tails? . Take one of the people, whatever his 1st flip was, reverse it! If he flipped a head you select tail, if he flipped a tail, heads. That's your first outcome, cross him off and select someone else, whatever was their second flip, reverse it! Keep on going and you have an infinite sequence that is different to everyone's sequence in atleast one flip. X 0/10 WRONG F YOU'RE KIDDING! FAIL :( its really quite simple, infinite people all doing the same thing you are doing dispells any possibility of you being unique. Herc === Subject: Re: No Unique Initial Segment And No Characteristic Expansion Herc, you just can't write down a list of the reals, and you can write down a list of the naturals. Get over it. === Subject: Re: No Unique Initial Segment And No Characteristic Expansion > N. U. I. S. A. N. C. E. > No Unique Initial Segment And No Characteristic Expansion No Underwear Interdicting And Nonsense Crapitating Everywhere http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: No Unique Initial Segment And No Characteristic Expansion Uncle Al, the last remnant of pre Herc anarchy on the science web. (and a complete mental blank on any issues he can't popularise out of previous theorists work) (and the most obnoxious purile self obsessed fckwit mensa has every had to put up with) like nobody's seen a science student correct art students..... http://www.winternet.com/~mikelr/flame76.html Herc === Subject: Re: No Unique Initial Segment And No Characteristic Expansion > Uncle Al, the last remnant of pre Herc anarchy on the science web. > (and a complete mental blank on any issues he can't popularise out of > previous theorists work) > (and the most obnoxious purile self obsessed fckwit mensa has every had > to put up with) > like nobody's seen a science student correct art students..... > http://www.winternet.com/~mikelr/flame76.html > Herc Here is what you so desperately seek to subborn, steal, purchase, cheat, lie... anything but earn: The editors acknowledge receipt of the above manuscript on [date Uncle Al did a prdy good job, little boy, and now his work will be reviewed by Referees to stand or fall. The first of three experiments is happening in China as you read this. We then have four outcomes: 1) The paper is rejected and all three parity Eotvos experiments null within experimental error. Clever idea but empirically wrong. 2) The paper is accepted and all three parity Eotvos experiments null within experimental error. Clever idea but empirically wrong. Perhaps somebody else will look in a better way or extend quantitative parity divergence theory. 3) The paper is rejected and there is a reproducible non-null output. The paper will be resubmitted and quickly accepted. General Relativity and 3/5 of M-theory are falsified; quantum mechanics is in trouble (Lorentz invariance empirically fails). Weitzenboek and affine gravitation displace Einstein and metric gravitation. 4) The paper is accepted and there is a reproducible non-null output. General Relativity and 3/5 of M-theory are falsified; quantum mechanics is in trouble (Lorentz invariance empirically fails). Weitzenboek and affine gravitation displace Einstein and metric gravitation. (2), (3), or (4) will have you eating your liver. (3) or (4) will have Uncle Al sampling lutefisk 10 December. Pookie pookie little whining nothing. Uncle Al says, It's hard to make a comeback when you haven't been anywhere. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: No Unique Initial Segment And No Characteristic Expansion clap clap, and I just click the [confirm membership] button 20 times each morning for my crust. Herc === Subject: Re: Integral[Poly degree 4]= ? > Let f(x)=x^4 +2A*x^3 +B*x^2 +(1-3A)*x+ 4 with real coefficients, such that > f:[-1,1]-->R is non-decreasing on [-1,1] . > I want to compute I:= Integral_{x=-1 to x=1}f(x) dx. It's true that I = 3.4 ? No. One can show that (by examining f'(x) for x = 1, 1/2, 0, and -1) that for f to be non-decreasing on [-1,1], one must have A = 2B/3 + 1 and -2 <= B <= -1. Hence f'(x) = 2 (x + 1) (B + x + 1) (2 x - 1) with roots -1, -B-1, 1/2. We want f'(x) >= 0 on [-1,1] so we must choose B = -3/2. Hence f'(x) = (x + 1) (2 x - 1)^2 which is clearly non-decreasing on [-1,1] and f(x) = x^4 - 3/2 x^2 + x + 4 The integral over [-1,1] of f(x) is 37/5 = 7.4. Paul -- Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul@physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul === Subject: Re: Invariant Galilean Transformations (FAQ) On All Laws > : [snip lies] > [snip lies] > You back again, you ing imbecile? > Androcles Just admit it. You don't have a mathematical clue. Were there to be internal inconsistencies in SR (meaning inconsistencies of a purely mathematical logical nature) that would automatically lead to contradictions in number theory, itself, and arithmetic, since the mathematics of Minkowski geometry is equiconsistent with the theory of real numbers and with arithmetic. Eleaticus explicitly demonstrates that he is completely ignorant of multivariable calculus. He has no concept of the Chain Rule in multivariable calculus. Consider his Galilean Transformation goo and dribble: t' = t, x' = x - vt, y' = y, z' = z. His refusal to accept that t' must be introduced as a separate variable springs from a massive emprical stupidity re space and time are described as a four-dimensional manifold, with four coordinates instead of a time evolution of a three-dimensional manifold, and that the change of coordinate system should be a change of four coordinates, and not a time-dependent change of three coordinates. This is particularly vital when it comes to fields over space and time (electric and magnetic fields for example). The transformation law for the differential operators under the Galilean transformation is given by: d/dt' = d/dt + v d/dx, d/dx' = d/dx, d/dy' = d/dy, d/dz' = d/dz. This shows the necessity of introducing a new variable t', since partial differentiation with respect to t' (constant x', y', z') is a different operation to partial differentiation with respect to t (constant x, y, z). The above transformation law is determined by the Chain Rule: d/dt' = dt/dt' d/dt + dx/dt' d/dx + dy/dt' d/dy + dz/dt' d/dz, d/dx' = dt/dx' d/dt + dx/dx' d/dx + dy/dx' d/dy + dz/dx' d/dz, d/dy' = dt/dy' d/dt + dx/dy' d/dx + dy/dy' d/dy + dz/dy' d/dz, d/dz' = dt/dz' d/dt + dx/dz' d/dx + dy/dz' d/dy + dz/dz' d/dz. The presence of the term involving d/dx in the expression for d/dt' is indicative of the fact that x depends on t' (x', y', z', being held constant), as can be seen from the fact that the coefficient of d/dx in the expression for d/dt' is dx/dt'. Because of the now demonstrated fact that Eleaticus has no formal education in multivariable calculus, he has managed, somehow, to get it into his head that the presence of the term involving d/dx in the expression for d/dt' is indicative of t' depending on x (t, y, z, being held constant). Because of his stupidty Eleaticus cannot get the correct transformation law for the differential operators under the Galilean Transformation, and he cannot determine the invariance or otherwise of Maxwell's Equations under the Galilean Transformation. The first advice to Eleaticus is to learn multivariable calculus. Eleaticus should not pretend that he can understand how to determine invariance or otherwise of Maxwell's Equations under the Galilean Transformation, or under the Lorentz Transformation, until he understands the multivariable calculus which underlies such considerations. Eleaticus is a loud idiot. The homogeneous Maxwell equations are invariant under the Galilean Transformation, with transformation laws: E_x' = E_x, E_y' = E_y - v B_z, E_z' = E_z + v B_y, B_x' = B_x, B_y' = B_y, B_z' = B_z. The derivation of these transformation laws was determined using the transformation laws for the differential operators given above. These transformation laws have the additional advantage that they determine the correct transformation for the force law, thus providing further evidence in favour of the transformation law for the differential operators, as above. The inhomogeneous Maxwell equations are also invariant under the Galilean transformation, with transformation laws: E_x' = E_x, E_y' = E_y, E_z' = E_z, B_x' = B_x, B_y' = B_y + v/c^2 E_z, B_z' = B_z - v/c^2 E_y, rho' = rho, J_x' = J_x - v rho, J_y' = J_y, J_z' = J_z. Note the the transformation laws for the charge density and current density are as they should be under the Galilean transformation. Homogeneous equations are invariant under the Galilean Transformation, and inhomogeneous equations are invariant under the Galilean Transformation, but Maxwell's Equations as a whole are NOT invariant under the Galilean Transformation, since the transformation laws required for the EM field for the two cases are inconsistent with each other. The transformation law for the EM field which makes the homogeneous equations invariant will not also make the inhomogeneous equations invariant. The transformation law for the EM field which makes the inhomogeneous equations invariant will not also make the homogeneous equations invariant. On the other hand, all of Maxwell's equations are invariant under the Lorentz Transformation, with transformation laws: E_x' = E_x, E_y' = gamma (E_y - v B_z), E_z' = gamma (E_z + v B_y), B_x' = B_x, B_y' = gamma (B_y + v/c^2 E_z), B_z' = gamma (B_z - v/c^2 E_y), rho' = gamma (rho - v/c^2 J_x), J_x' = gamma (J_x - v rho), J_y' = J_y, J_z' = J_z, where gamma = 1/sqrt(1 - v^2/c^2). If you can't handle the straightforward math, and Androcles, you clearly can't, then stay the out of sci.physics.relativity and go bother [Hammond] === Subject: Re: Invariant Galilean Transformations (FAQ) On All Laws > If you can't handle the straightforward math, and Androcles, you > clearly can't, then stay the out of sci.physics.relativity and go > bother [Hammond] Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe and not make messes in the house. --Lazarus Long, Time Enough for Love, (Robert A. Heinlein) Bob Kolker === Subject: American cigarettes for $ 14.95 ! Marlboro :D American cigarettes for $ 14.95 ! Marlboro B&H $14.95 Camel $14.95 Shopcig - cigarettes shop online L&M $14.95! Marlboro $14.95 West $13.95 Free delivery around the world! www.shopcig.com *-----------------------* www.GroupSrv.com *-----------------------* === Subject: American cigarettes for $ 14.95 ! Marlboro :D American cigarettes for $ 14.95 ! Marlboro B&H $14.95 Camel $14.95 Shopcig - cigarettes shop online L&M $14.95! Marlboro $14.95 West $13.95 Free delivery around the world! www.shopcig.com *-----------------------* www.GroupSrv.com *-----------------------* === Subject: American cigarettes for $ 14.95 ! Marlboro :D American cigarettes for $ 14.95 ! Marlboro B&H $14.95 Camel $14.95 Shopcig - cigarettes shop online L&M $14.95! Marlboro $14.95 West $13.95 Free delivery around the world! www.shopcig.com *-----------------------* www.GroupSrv.com *-----------------------* === Subject: Root Finder vii. Root Finder vii. by Jon Giffen The polynomial a[0]+a[1]t+a[2]t^2+a[3]t^3+ ... + a[n] = 0 may be expressed as the dot product of the two vectors, T = (t,t^2,t^3,...,t^n) and N = (a[1],a[2],a[3],...,a[n]) then T*N+a[0]=0 is the nth degree polynomial Define, Q = (-a[0]/|N|^2)(a[1],a[2].a[3],..a[n]) Suppose the vectors D,S,C are defined, D=( 1 , 2 , 3 ,...,n ) S=(a[1],2a[2],3a[3],...,na[n]) C=(S*N)D+(D*N)S then it can be shown that, |Q| T = Q +/- --- C |C| t^3+2t^2+t-4=0 t=1 a[0]= -4 a[1]= 1 a[2]=2 a[3]=1 N=(1,2,1) |N|^2=6 Q=(2/3)(1,2,1) |Q|=(2/3)6^(1/2) D=(1,2,3) S=(1,4,3) S*N=(1,4,3)*(1,2,1)=1+8+3=12 D*N=(1,2,3)*(1,2,1)=1+4+3=8 C=12(1,2,3)+8(1,4,3)=(12+8,24+32,36+24)=(20,56,60)=4(5,14,15) |C|=(4)446^(1/2) (2/3)6^(1/2) (t,t^2,t^3)=(2/3)(1,2,1)+/- ------------ (5,14,15) 446^(1/2) (t,t^2,t^3)=(2/3)(1,2,1)+/- 0.07732(5,14,15) (t,t^2,t^3)=(0.6666,1.3333,0.6666)+/-(0.3866,1.08254,1.1599) Selecting the first component, t = 0.6666+0.3866 = 1.0532 ~ 1 then t^2 + 3t + 4 ------------------------------ t-1/ t^3 + 2t^2 + t - 4 t^3 - t^2 -------------------- 3t^2 + t - 4 3t^2 -3t ------------- 4t - 4 and the remaining roots are -3+/-{9-16|^(1/2) t = ----------------- = -1.5+/-1.3229i 2 Dividing t^3 by t^2, t^3 = 0.6666+1.1599 = 2.2656 ----------------------------- t^2 = 1.3333+1.0825 = 2.4158 or t = 0.9378 ~ 1 Jon Giffen === Subject: Re: Root Finder vii. That is, The polynomial a[0]+a[1]t+a[2]t^2+a[3]t^3+ ... + a[n] = 0 may be expressed as the dot product of the two vectors, T = (t,t^2,t^3,...,t^n) and N = (a[1],a[2],a[3],...,a[n]) or, (t,t^2,t^3,..,t^n)*(a[1],a[2],a[3],..,a[n])+a[0]=0 or T*N+a[0]=0 is the nth degree polynomial N may be considered the normal to the more general case of the plane, a[0] + a[1]x[1] + a[2]x[2] + ... + a[n]x[n] = 0 where x[1]=t x[2]=t^2 x[3]=t^3 ... x[n]=t^n Following T up from the origin, R is the variable vector parallel to N and orthogonal to T-R, as T varies along its curve. then R*(T-R)=0 taking the derivative with respect to t, R'*(T-R)+R*(T'-R')=0 or 2R'*R = R'*T + R*T' The projection of T' onto N/|N| is R' T'*N N ---- ---- = R' |N| |N| define Q = (T*N)N/|N|^2 = -a[0]N/|N|^2 Q = (-a[0]/|N|^2)(a[1],a[2].a[3],..a[n]) or Q = (q[1],q[2],q[3],...,q[4]) Since R' is in the same direction as Q, it holds that since Q and T'-(T'*N)N/|N|^2 are orthogonal, Q*(T'-(T'*N)N/|N|^2) = 0 where Q is the shortest vector from the origin to the plane. T = (t,t^2,t^3,...,t^n) T'= (1,2t,3t^2,..,nt^(n-1) ) T'*Q =q[1]+2q[2]t+3q[3]t^2+....+ nq[n]t^(n-1) =(1/t)T*(q[1],2q[2],3q[3],...,nq[n]) T'*N = a[1]+2a[2]t+3a[3]t^2+....+ na[n]t^(n-1) =(1/t)T*(a[1],2a[2],3a[3],...,na[n]) Suppose the vectors D,S,U are defined, D=( 1 , 2 , 3 ,...,n ) S=(a[1],2a[2],3a[3],...,na[n]) U=( 1 , 1 , 1 ,...,1 ) Then T'* U =(1/t)T*D T'* N =(1/t)T*S for instance, (1,2t,3t^2,..,nt^(n-1) )*(1,1,1,..) =(1/t)(t,t^2,t^3,...,t^n)*(1,2,3,..) and (1,2t,3t^2,..,nt^(n-1) )*(a[1],a[2],a[3],..) =(1/t)(t,t^2,t^3,...,t^n)*(a[1],2a[2],3a[3],..) dividing the below equations with each other, T'* U =(1/t)T*D T'* N =(1/t)T*S (T'*U)(T*S) - (T'*N)(T*D) = 0 factoring out T, T*((T'*U)S - (T'*N)D) = 0 so T is orthogonal with (T'*U)S - (T'*N)D or T is orthogonal with (1/t)((T*D)S - (T*S)D) or T is orthogonal with (T*D)S - (T*S)D this is true, since T*((T*D)S - (T*S)D) = 0 Let, G = (T*D)S - (T*S)D If m is some ratio of G such that, (mG-Q)*N = 0 mG*N - Q*N = 0 since Q*N=T*N, m[(T*D)(S*N)-(T*S)(D*N)]-(T*N) = 0 or T*[m((S*N)D-(D*N)S )-N] = 0 One solution for m, among others, can be found from m((S*N)D-(D*N)S - N = 0 or m((S*N)D-(D*N)S = N taking the dot product of both sides with D and solving for m, N*D m = --------------------- (S*N)(D*D)-(D*N)(S*D) Otherwise, T and m((S*N)D-(D*N)S )-N are orthogonal Notice that since N*((S*N)D-(D*N)S )= 0, ((S*N)D-(D*N)S ) is orthogonal to N. T is orthogonal with m((S*N)D-(D*N)S)-N N is orthogonal with (S*N)D-(D*N)S OY=m((S*N)D-(D*N)S)-N Y *------------ N T | / | / | / | / | / | / Z _________|/ |O OZ= | (S*N)D-(D*N)S | | | | -N T is orthogonal with OY N is orthogonal with OZ since angle YOZ = angle TON, OY*OZ T*N OY*N -------- = ------ = {1 - (-------)^2 }^(1/2) |OY||OZ| |T||N| |OY||N| (OY*OZ)^2 (OY*N)^2 ------------ = 1 - ----------- |OY|^2|OZ|^2 |OY|^2|N|^2 Multiplying both sides of this equation by |OY|^2|OZ|^2|N|^2, (OY*OZ)^2|N|^2 = |OY|^2|OZ|^2|N|^2 - (OY*N)^2|OZ|^2 eqn i. Letting (S*N)D-(D*N)S = C, OZ=C and OY=mC-N and (OY*OZ)^2=((mC-N)*C)^2|N|^2 = m^2|C|^2|N|^2 |OY|^2=(mC-N)*(mC-N) = m^2|C|^2+|N|^2 |OZ|^2=|C|^2 (OY*N)^2=((mC-N)*N)^2=-|N|^4 Substituting these in eqn i, (m^2|C|^2|N|^2)|N|^2 = (m^2|C|^2+|N|^2)|C|^2|N|^2 + |N|^4|C|^2 or m^2|N|^2 = m^2(|C|^2+|N|^2) + N^2 N^2 = m^2(|N|^2 - |C|^2 - |N|^2) |N| m = - ----- |C| So OY=m((S*N)D-(D*N)S)-N or |N| OY= - ---((S*N)D-(D*N)S)-N |C| or |N| OY= - ---------------((S*N)D-(D*N)S) - N |(S*N)D-(D*N)S| or |N| OY= - --- C - N |C| (qOY-Q)*N=0 qOY*N-Q*N=0 q(0-|N|^2) = Q*N q(-|N|^2) = -a[0] q = a[0]/|N|^2 angle ZOY OY*OZ T*N cos(ZOY)=-------- = ------ = cos(NOT) |OY||OZ| |T||N| T*N= -a[0] and a[0]|OY||OZ| |T| = - ------------- |N|(OY*OZ) (a[0])^2|OY|^2|OZ|^2 |T|^2 = --------------------- |N|^2(OY*OZ)^2 |OY|^2 = 2|N|^2 (OY*OZ)^2 = (|N||C|)^2 (a[0])^2 2|N|^2 |C|^2 |T|^2 = --------------------- = 2|Q|^2 |N|^2 |N|^2 |C|^2 T = Q + {|T|^2-|Q|^2}^(1/2)(-C/|C|) |Q| T = Q +/- ----- C |C| Where C = (S*N)D-(D*N)S D=( 1 , 2 , 3 ,...,n ) S=(a[1],2a[2],3a[3],...,na[n]) Alternatively, suppose T'*(-U)=(1/t)T*D T'* N =(1/t)T*S Then dividing the two equations, (T'*U)(T*S)+(T'*N)(T*D) =0 then C = (S*N)D+(D*N)S > Root Finder vii. > by Jon Giffen > The polynomial > a[0]+a[1]t+a[2]t^2+a[3]t^3+ ... + a[n] = 0 > may be expressed as the dot product of the two vectors, > T = (t,t^2,t^3,...,t^n) and > N = (a[1],a[2],a[3],...,a[n]) then > T*N+a[0]=0 is the nth degree polynomial > Define, > Q = (-a[0]/|N|^2)(a[1],a[2].a[3],..a[n]) > Suppose the vectors D,S,C are defined, > D=( 1 , 2 , 3 ,...,n ) > S=(a[1],2a[2],3a[3],...,na[n]) > C=(S*N)D+(D*N)S > then it can be shown that, > |Q| > T = Q +/- --- C > |C| > t^3+2t^2+t-4=0 t=1 > a[0]= -4 > a[1]= 1 a[2]=2 a[3]=1 N=(1,2,1) |N|^2=6 Q=(2/3)(1,2,1) > |Q|=(2/3)6^(1/2) > D=(1,2,3) S=(1,4,3) S*N=(1,4,3)*(1,2,1)=1+8+3=12 > D*N=(1,2,3)*(1,2,1)=1+4+3=8 > C=12(1,2,3)+8(1,4,3)=(12+8,24+32,36+24)=(20,56,60)=4(5,14,15) > |C|=(4)446^(1/2) > (2/3)6^(1/2) > (t,t^2,t^3)=(2/3)(1,2,1)+/- ------------ (5,14,15) > 446^(1/2) > (t,t^2,t^3)=(2/3)(1,2,1)+/- 0.07732(5,14,15) > (t,t^2,t^3)=(0.6666,1.3333,0.6666)+/-(0.3866,1.08254,1.1599) > Selecting the first component, > t = 0.6666+0.3866 = 1.0532 ~ 1 > then > t^2 + 3t + 4 > ------------------------------ > t-1/ t^3 + 2t^2 + t - 4 > t^3 - t^2 > -------------------- > 3t^2 + t - 4 > 3t^2 -3t > ------------- > 4t - 4 > and the remaining roots are > -3+/-{9-16|^(1/2) > t = ----------------- = -1.5+/-1.3229i > 2 > Dividing t^3 by t^2, > t^3 = 0.6666+1.1599 = 2.2656 > ----------------------------- > t^2 = 1.3333+1.0825 = 2.4158 > or > t = 0.9378 ~ 1 > Jon Giffen === Subject: Re: Generalizations of Weierstrass' approximation theorem > Hi Folks. > Weierstrass' theorem of approximations says (at least in the form I > learned it in an intro course on real analysis) that a function f(x) > defined on a <= x <= b, can be approximated with arbitrary presicion > by a certain polynomial p(x): > |f(x) - p(x)| < e, for some e > 0, a <= x <=b. [1] > Now, I have had the misfortune of getting myself mixed up with people > who claim that given some physical measurement g(t), there exist one > and only one physical (computational) model m(t) that explains the > measurement g(t). ... > Tell your friends that the temperature in my house is 21.2C and it was > 20.0 yesterday. Ask them where I live. Heh, I've tried that. My favourite example is based on a technique known as focalization which is very hot these days. The idea behind focalization is that one makes one measurement of a propagating wave in a medium where the speed of sound is unknown, and estimate *both* the speed of sound in the medium *and* the distance between the source and the reciever. This is equivalent to make a radar system where the speed of light, c, is unknown: The 2 way time of flight is 1 second. Find bot the slant range r and the speed of sound c. No, I'm not kidding. People actually publish such ideas and expect to be taken seriously: http://www.fysel.ntnu.no/~allnor/pdfs/paper1.pdf Note the first sentence under the heading I. Focalization: Focalization is a generalization of MFP in which both source parameters and environmental parameters are unknown or partially unknown. These guys don't waste any time in checking whether there are conditions to whether their general ideas work. The statement that the method works even if all relevant parameters are unknown is pretty unequivocal. Now, I am not clairvoyant so I don't know what the authors actually mean to say. I do know how lots of people interpret this paper, and, I guess, the particular statement about everything being unknown: Any measurement goes, and if inserted into this inversion scheme, the true model (i.e. the true environmental and source parameters) pop out in the other end as if by magic. It may be due to my mental abilities, but I am not capable of making sense of the focalization paper. Interestingly, it is registered with 81 citations in the literature, when I checked today. > Consider the sequence 1, 2, 3, 4, (not known). Offer several polynomials > that produce that sequence, each producing a different fifth member of > the sequence. (If you need help generating them, just ask.) Yeah, right. Have you ever tried to ask a professor to actually think through what he says? It's not recommended. When I tried, it cost me first a job and later a full year of sick leave. > What do they smoke? I don't have the faintest idea. I'm not sure I want to know. Rune === Subject: Re: Generalizations of Weierstrass' approximation theorem uncomfortable situation I'm in. I'll try to answer as much as poosible of all responses I've read so far, in this post. > Hi Folks. > Weierstrass' theorem of approximations says (at least in the form I > learned it in an intro course on real analysis) that a function f(x) > defined on a <= x <= b, can be approximated with arbitrary presicion > by a certain polynomial p(x): > |f(x) - p(x)| < e, for some e > 0, a <= x <=b. [1] > The slightly more abstract version is called Stone-Weierstrass for > general functions which satisfy some conditions, that polynomials > automatically satisfy. S-W is OK for proofs but is not a very practical > method of doing approximations. Consult a reasonable numerical analysis > source. I couldn't find the Stone-Weierstrass mentioned in any of my own books, so I guess I don't have a decent text on numerical analysis... or real analysis, for that matter. > Now, I have had the misfortune of getting myself mixed up with people > who claim that given some physical measurement g(t), there exist one > and only one physical (computational) model m(t) that explains the > measurement g(t). This self evident fact is then used as a basis > for remote sensing of just about anything. The actual algorithm > is based on searching over all possible models, and all possible > parametrizations of each model, where one claims that the true > Sounds like freshman rhetoric. Rather than waste time disproving > such nonsense one would be better off discovering what the true > statement should have been. Technical jargon, and its short cut > versions, can appear to be wildly improbable if it is taken out > of context. You sound like you are getting it out of context and > repeated, with editing for improvements and clarifications, > out of context yet again. Unfortunately, this is very in context. The technical jargon is model-based signal processing which is something of a buzzword in underwater acoustics, these days. The problem is that one does some sort of sonar survey at sea, where one locates objects partially or fully buried in the sea floor. Once such a buried object is detected and located (which can be done), the problem is to determine what kind of object this is. Is it a rock? A mine? A munition shell? A box containing munition shells? Something completely different? The problem is to answer these types of questions. Lots of people have done lots of research on developing forward models for acoustical scattering from buried and partially buried data. The forward problem is the problem where all aspects of the sea floor, the object and the sonar system is specified in advance, and the resulting measurements of the acoustic field are simulated. The inverse problem, the analysis problem, is generally seen as 'easy' once the forward modelling program is available: Just run all possible models with all possible parameters through the modelling routine, and see what fits best. The really disturbing part is that the people who suggest these kinds of things, are professors with 20, 30, 40 years of seniority. It's not just one who have lost it, it's a whole community who base their research on these methods. If you want me to, I can dig up some more recent published papers, but you could review my rant from a couple of years back, to get an impression of how these people think: > physical model is the model (with parameters) that minimizes > the residue > |g(t) - m(t)| < e, for some e > 0, a <= t <=b. [2] > No, I'm not kidding. There are people out there who actually > do these kinds of things, and expect to be taken seriously. > The fact that one basically searches a space of infinte x infinite > dimension [models x parameters] is not seen as a relevant hurdle. > Reckognizing the true model m(t) is seen as trivial, in that > it minimizes the residue e in [2]. Such trivial details as > observation and instrumentation noise are, of course, not > relevant in the measurement and are seen as insignificant > technical details. The task of actually dealing with such boring > technicalities are left as an excersise to the reader. > There is a major field of studies called inverse problems (with > applications like CAT scanning) that does quite well. But they tend > to have good models. In CAT scanning the source image may be all > possible images but material densities are bounded above and below, > objects have minimal sizes, etc, so all possible is less imposing > that it sounds. By the time the source function and the noise sources > are tamed (regularized in the jargon) and required to be self consistent > (satisfy the inverse problems conditions) it is surprising how little > information is lost. Getting it back is often not easy but that is > why there are inverse problem methods. Well, I have some experience from processing and analysis of seismic data. In that context, inverse problems are not automated at all, but are a highly interactive excersise. The best written account I have found for how seismologists work, is the first couple of paragraphs in Robinson: Model-driven predictive deconvolution Geophysics, v63 no2, March 1998, pp 713: INTRODUCTION In his perceptive and captivating book, Clay (1990) observes that geophysicists use essentially the same approach to finding oil that the fictional Sherlock Holmes used in solving mysteries. He points out that the subsurface structure of the earth represents and unknown, a mystery. The geophysicist acts like a detective in identifying a problem worthy of attention, collecting data, and then (like Holmes) deducing an explanation. Geophysicists identify a potential oil-bearing area, collect information in the form of reflected seismic waves, and then deduce the underground structure in order to delineate a possible oil reservoir. Some of the recorded seismic information is useful, and is called a signal. However, much of the recorded seismic information is made up of noise and interference that mask the useful signals. One of the most troublesome forms of interfernce is that caused by multiple reflections. Sheriff (1991) defines a model as a concept from which one can deduce effects for comparisions with observations. He states that a model can be used to develop a better understanding of the observations. However, he carefully points out that agreement between observations and effects derived from the model does not establish that the particular models represents the actual situation. Geophysical interpretation problems almost always lack uniqueness. Clay: 1990, Elementary Exploration Seismology, Prentice-Hall. Sheriff: 1991, Encyclopedic dictionary of exploration geophysics, 3rd. ed., Soc. Expl. Geophys. This is the tradition I'm trained in, this is the philosophy I apply to all my work. If you get as far as to come up with a base model that can be fed to a Levenberg-Marquart type of alhgorithm, all the hard work is done: 1) You have successfully(?) identified certain key effects in the data 2) You have used these effects to predict derived effects 3) You have found these secondary effects in the data and thus verified (or at least not contradicted) your first model 4) You have repeated the steps 2)-3) above until you either find a contradicting effect, or are comfortable with letting the computer take over the drudgery. One can use various types of inversion schemes for doing some of the drudgery, but all the detective work must still be done in advance. And whatever method is used to produca the end model, it can still be plain wrong. I have been involved with projects where three years of hard analysis work was abandoned because of the results of a small analysis job that took no more than half an hour to do. But the results contradicted the then prevailing hypothesis. > Of course, I don't agree with such methods. What I am concerned, > any data set can be approximated with arbitrary presicion of a > polynomial according to [1] above. A polynomial p needs not > have anything whatsoever to do with whatever physical process > dominated the measurement. Similarly, it is possible to find > a model m(t) with parameters, that approximates the data g(t) > with arbitrary presicion, without m(t) having any physical > significance whatsoever. > So my question is whether something like Weierstrass' approximation > theorem, dealing with polynomials in [1], can be adapted to the > case [2] where the approximation m(t) is a general model and not > a polynomial? > Read the literature on statistical fitting and on inverse problems > and you will find that poorly stated, otherwise known a silly, problems > provide many solutions like the ones you imagine. It is all the nasty > technical conditions to keep the problem well stated that get lost in > the retelling, editing and clarification of jargon filled claims by the > technical folks. These fields are already capable of producing curious > results when minor problems arise so it is hardly necessary to invent > new problems by miscommunication. Oh, how I wish mere miscommunication was the causes! It is not. People catually think data analysis is as simple as I described. I don't know what, if anything these people have smoked. What I do know, is that they have never been challenged on technical issues for decades. Thes guys can base an assumption on that the Earth is flat, and demand to be taken seriously. No one have ever had them stop and think twice about what they actually present under the cloak of science and research. Again, check out the thread I mentioned above, to get an impression of what's going on. Rune === Subject: Re: Generalizations of Weierstrass' approximation theorem Rune Allnor schrieb > Now, I have had the misfortune of getting myself mixed up with > people who claim that given some physical measurement g(t), > there exist one and only one physical (computational) model > m(t) that explains the measurement g(t). Err, isn't that true? There is only one physically true explanation of how the world works. We may not know the explanation, we may have only imperfect models explaining reality only partially, but nevertheless theoretically there is only one model. Anyway you may be interested in the Buckingham Pi theorem http://test.wikipedia.org/wiki/Buckingham_Pi_theorem or http://scienceworld.wolfram.com/physics/BuckinghamsPiTheorem.html which allows to draw up models of physics without necessarily knowing in-depth what's going on. Basically, you make up a list of variables, that you think are relevant to the problem and mingle them up to get dimensionless variables (what Buckingham called the (capital) Pi's, hence the name, it has nothing to do with 3.1415...). Then you can measurements, draw curves and approximate them with functions. Example of an application: Heat transfer variables: l = length of body [m] v = velocity of air [m/s] Ta = air temeperature [K] Tb = body temp [K] rho = body heat capacity [J/kgK] nu = viscosity of air [m2/s] ... and you end up with formulas like: Nu = c1 * (Re * Pr)^c2 where c1,c2 = empirically determined constants Nu = Nusselt number = dimensionless surface temperature gradient Re = Reynolds number = ratio of the inertia and viscous forces (dimensionless velocity) Pr = Prandtl number = ratio of momentum and thermal diffusivities This allows you to perform experiments on a model (e.g. a scaled down version of a Space Shuttle) and then apply the results to the real thing. You have to take care that the dimensionless parameters are similar in the model and in reality and you have to take care that you do not violate the underlying (perhaps unknown :-() physical model, e.g. normally you design such that Reynolds(model) = Reynolds(reality) but at higer velocities, you have to make sure that Mach(model) = Mach(reality) (Mach number = ratio of velocity to velocity of sound) because the effects of the compressibility of air are much more important than the others. I may be way off and this may be not what you are looking for, but I hope this helps. Martin === Subject: On the Euler phi values of pairwise coprime a,b, and c For a and b > 2, phi(a) and phi(b) are both even. For coprime a and b, there exist unique factorizations having no factor in common. Phi(n) calculates per http://mathworld.wolfram.com/TotientFunction.html according to a series of inverses of prime factors. Now, for coprime a and b, knowing phi even, are the values phi(a)/2 and phi(b)/2 distinct in every case? Without loss of generality, for the discussion, we may certainly take a max(phi(a)/2 phi(b)/2 phi(c)/2).... I need to prove just now, but it is intriguing that phi is so nearly universally even, and that it computes based on factorizations, which for gcd(a,b,c)=1 are distinct. To be blunt, I get a little nervous in a crowd, but there is some excitement to be had when crowding is present. :) Doug Goncz I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri, alphabetically. I drive: A double-step Thunderbolt with 657% range. I fight terrorism by: Using less gasoline. === Subject: Re: On the Euler phi values of pairwise coprime a,b, and c > If gcd(a,b,c) = 1, and a possibility that rather than phi(a)/2, phi(b)/2, and phi(c)/2 are > distinct. There are more number involved and so the condition is > closer. I'd like to read some opinions on this. Perhaps you wouldn't mind reading some facts, instead. phi(36391) = phi(36661) = phi(37231) = 36000, 36391 = 151 x 241, 36661 = 61 x 601, 37231 = 31 x 1201. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: On the Euler phi values of pairwise coprime a,b, and c > If gcd(a,b,c) = 1, and a possibility that rather than phi(a)/2, phi(b)/2, and phi(c)/2 are > distinct. There are more number involved and so the condition is > closer. I'd like to read some opinions on this. > Perhaps you wouldn't mind reading some facts, instead. > phi(36391) = phi(36661) = phi(37231) = 36000, > 36391 = 151 x 241, 36661 = 61 x 601, 37231 = 31 x 1201. A bit silly of me to post such large examples, when a few minutes scribbling on the back of an envelope produced phi(n) = 60 for n = 61, 77, and 93; phi(n) = 72 for n = 73, 91, 95, and 111. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: On the Euler phi values of pairwise coprime a,b, and c > If gcd(a,b,c) = 1, and a possibility that rather than phi(a)/2, phi(b)/2, and phi(c)/2 are > distinct. There are more number involved and so the condition is > closer. I'd like to read some opinions on this. > Perhaps you wouldn't mind reading some facts, instead. > phi(36391) = phi(36661) = phi(37231) = 36000, > 36391 = 151 x 241, 36661 = 61 x 601, 37231 = 31 x 1201. > A bit silly of me to post such large examples, when a few minutes > scribbling on the back of an envelope produced > phi(n) = 60 for n = 61, 77, and 93; > phi(n) = 72 for n = 73, 91, 95, and 111. It's funny - I was wondering quite why you were posting these numbers at all - I wondered how they were supposed to resolve Doug's query. I still wonder. However, that's because I simply cannot work out what Doug's query is actually supposed to mean! Isn't <5,8,10> an even smaller counter-example? Confused, Phil -- ... one Marine noticed one of the prisoners was still breathing. A Marine can be heard saying on the pool footage provided to Reuters Television: He's ing faking he's dead. He faking he's ing dead. The Marine then raises his rifle and fires into the man's head. The pictures are too graphic for us to broadcast, Sites said. === Subject: Re: On the Euler phi values of pairwise coprime a,b, and c > I still wonder. However, that's because I simply >cannot work out what Doug's query is actually supposed to mean! Well, I find in searches that the of the periods of the three dual exponential congruential sequences associated with FLT, and there are only three, *at least one* always has period two. This establishes that n is even, which in combination with n prime, shows that n=2, because they are purely periodic and the period of their conicidence must be the lcm of their individual periods. I thought that each of these sequences had period | phi of its corresponding base, but a,b,c = 5,6,7, phi(b)=3, t((c^n-a^n)mod b) = 2 shows that this is not universally true. So we can abandon this thread or if anyone would like, I can show sets of a,b,c for which ac - a != 0 (mod b)? a c^2 - a^2 == 0 mod b / c^2 - b^2 == 0 mod a in any conceivable way? a^2 < b^2 < c^2 < (a^2 + 2ab + b^2) --> c^2 - a^2 < b^2 + 2ab & c^2 - b^2 < a^2 + 2ab but that doesn't take *me* anywhere. All the searches are written in the obscure, unnamed, but typeset and highly readable programming language embedded in Mathcad, and since I am an original author and you are a potential customer for Mathsoft, the company has given me permission to distribute a few copies of my version 6 Pro, since they are on version 12 now and it's considered ancient and noncompetetive. I cling to using it because I don't need more power, and like to be able to give it away. I gave a copy to my bicycle mechanic friend, with my gear charts. I tolerance everything and tolerate everyone. I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri, alphabetically. I drive: A double-step Thunderbolt with 657% range. I fight terrorism by: Using less gasoline. === Subject: Re: On the Euler phi values of pairwise coprime a,b, and c > If gcd(a,b,c) = 1, and a possibility that rather than phi(a)/2, phi(b)/2, and phi(c)/2 are > distinct. There are more number involved and so the condition is > closer. I'd like to read some opinions on this. > Perhaps you wouldn't mind reading some facts, instead. > phi(36391) = phi(36661) = phi(37231) = 36000, > 36391 = 151 x 241, 36661 = 61 x 601, 37231 = 31 x 1201. > A bit silly of me to post such large examples, when a few minutes > scribbling on the back of an envelope produced > phi(n) = 60 for n = 61, 77, and 93; > phi(n) = 72 for n = 73, 91, 95, and 111. > It's funny - I was wondering quite why you were posting these > numbers at all - I wondered how they were supposed to resolve > Doug's query. I still wonder. However, that's because I simply > cannot work out what Doug's query is actually supposed to mean! > Isn't <5,8,10> an even smaller counter-example? Yes. I mistook gcd(a,b,c) = 1 for gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. Probably the reason I did that is that in a part of the original post that I left out of my response it said Think about it. a and b are not just coprime; there is a third variable, c, with which they are pairwise coprime. So it's not clear to me whether OP really meant the weaker or the stronger relative primality condition. My answer assumed the stronger. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: On the Euler phi values of pairwise coprime a,b, and c For positive integers a,b, and c, with gcd(a,b,c)=1, there exist six nontrivial pure exponential congurential generators with periods dividing phi(a), phi(b), and phi(c): a^n mod b a^n mod c b^n mod a b^n mod c c^n mod a, and c^n mod b. By the Euler Totient Theorem, which is for all a coprime to n, that a^(phi(n) == 1 (mod n) a^phi(c) == 1 mod c = a^0 a^phi(b) == 1 mod b = a^0 b^phi(c) == 1 mod c = b^0 b^phi(a) == 1 mod a = b^0 c^phi(b) == 1 mod b = c^0 c^phi(a) == 1 mod a = c^0 which is how we know the periods of these generators divide the phi of their corresponding bases. Ok, I got a that a little nonparallel but let's continue. phi(m) = m^b - m/p.1 - m/p.2 - ... m/p.b for a number m with b prime factors. That is phi(m) = m^b - r.1 - r.2 - ... r.b I think the loosest condition on phi(a)/2, phi(b)/2, and phi(c)/2 that can be of interest to me is that the values not all three the same, so let's forget for the moment any consideration of pairwise coprimality between these two. phi(n) < n- sqrt(n) so for a email Id of Frank Spodek - Undercover FBI Agent who monitored my Websurfing activities. You can do a quick investigative test by emailing to sfrank9@aol.com and asking Frank if he is an undercover FBI agent and I am absolutely positive he WONT respond and DENY he is an FBI agent because then it will be a LIE. With this test you can conclude Frank Spodek is indeed an undercover FBI agent. EBAY Ebay Id: cbsan, cidneybetz@aol.com, email Id of Cidney Betz - Undercover FBI agent who monitored my websurfing activities in 2002. Possible old phone numbers of Cidney Betz: 407-370-3043 or 352-291-2112 in the year 2002. 351 FTK (GA) Maroon Car, 7/28/04, 6:00 pm, 20ish Caucasian Male and Female, Metro Atlanta 131 YBP (GA) 35 yr old Caucasian Male, Silver colored car, 7/22/04, 10:45 pm, Hotel, Metro Atlanta APZ 7647 (GA) 35 yr old Caucasian Male, Truck, 5/25/04, 3:00pm, Ridgeview Hospital, Metro Atlanta YUU 243 (GA) Pickup truck, Caucasian male, 8/25/04, 4:40pm, Holcrombe Bridge Rd, Metro Altanta AQG 2185 (GA) 27 yr old Caucasian Female, Red Cherokee, 5/21/04, 11 Pep Boys, Cobb Parkway, Metro Atlanta WPS 578 (GA) SUV, One Caucasian Male and one female, 3/8/04, 8:30pm, Apartment Complex Coipel 770-577-0526 (5/2/04, 5:29pm), 4337 Lodgeview pl, Douglasville, GA 30315. undercover FBI agent in metro Altanta area. Please allow three years plus or minus to the ages of the undercover FBI agents I mentioned here. 2/29/04 Undercover FBI agent with a fictitious name Agent Johnson and IP address 81.132.210.89 said on internet that they are using me as a training target and have got a bet on as to when I'll do something really stupid and also they are taking bets on what it will be in alt.astrology Usenet newsgroup on 2/29/04. This FBI agent from Michigan field office might have posted it anonymously and hence the IP address might be a fake one. Url for Agent Johnson's post follows. 3/21/04 Undercover FBI agent by some fictitious internet name THOR and email Id: funk_a_del_ic2000@yahoo.com and IP address 24.94.125.55 in the Washington DC area monitored my web surfing activities and posted on rec.sport.cricket Usenet news group. 10/10/04 J NIcklebaker (Undercover FBI agent monitoring my websurfing activities) posted anonymously using Newsguy Usenet Service Provider in alt.liberalminded newsgroup and THREATENED ME. 10/6/04 Leonard (Undercover FBI agent monitoring my websurfing activities) with real IP address 24.209.79.66 (This IP address can be hacked) posted in rec.audio.car newsgroup 10/15/04 Els (Undercover FBI agent Els monitoring my websurfing activities) with probably fake IP address 82.169.229.173 posted in alt.test newsgroup I am absolutely confident all the above mentioned people are undercover FBI agents GUARANTEED. You can investigate them yourself and find out the truth. I request all you HACKERS on usenet to HACK these undercover FBI agents' IP addresses and trace their real REAL IDENTITIES and post the info here. FBI who is supposed to catch terrorists are WASTING humongous amount of tax payers money to ruin, wreck and destroy ordinary peoples lives and are MORE INTERESTED and spent three years to tell me how SMART and INFINITELY POWERFUL they are compared to me. FBI has even been eavesdropping, monitoring and recording my conversations with attorneys from pay phones when I was looking to hire an attorney to get help with this FBIs harassment, dehumanization, humiliation and mental torture. FBI agents have been entering my apartment regularly and torturing me by doing whatever they want because they know I am taking medication that puts me to sleep and I won't wake up until the next morning until the drug effect goes away. FBI has forced Google or used some technology at their disposal to some of my comments from getting posted on INTERNET newsgroups misc.legal, trial.misc.legal from a library in Farmington Hills, MI and denied and VIOLATED my right to FREEDOM OF SPEECH in 11/03 or 12/03. I moved in with another person (who advertised on a website for a room mate) on 5/24/04 and I went out in his car on the night of 5/25/04 and the next morning 5/26/04, at 9:30am the SADISTIC FBI called me and made this phone# 678-560-1322 and the caller name LOOKING FOR YOU appear on my caller id to tell me, Oh we were LOOKING for you last night, we did not know where you were because you went out in your room mates car. The sadistic and perverted FBI ILLEGALLY installed gps vehicle tracking devices in my room mates car the next day to TRACK ME, even though my room mate has nothing to do with me and we were COMPLETE STRANGERS until the previous day. The same day on 5/26/04 evening, I went out in my car and came back around 7:45 pm, parked my car in the parking lot and on the way I said nice car Within a few minutes at 8:44 pm, the SADISTIC FBI called and made phone# 770-690-8489 with caller name NEIGHBOR OUTREACH appear on my caller id to tell me, Oh we have seen and heard what you said to your neighbor in the parking lot. The FBI was able to see and hear my conversation because they installed VIDEO and AUDIO surveillance devices around my apartment. When I went to Best Buy in Feb 2002 to buy a cell phone, two caucasian FBI agents (possibly husband and wife) followed me into the store and observed me from a very close range and sent me a marketing fax for cell phones to my were WATCHING ME even in retail stores. SADISTIC FBI agents even PUNCTURED my room mates car tire and made it comments he made the previous day in our apartment. You might think this is crazy stuff, but I know it for a FACT that FBI SADISTIC MOTHERERS punctured my room mate's car tire INTENTIONALLY for the casual comments he made the previous day. FBI agents have terrorized me by calling my current unlisted phone# atleast asking for Roberto Fernandez (last name changed), knowing fully well that nobody by the name Roberto Fernandez lived in my apartment. I called undercover FBI agent, Kathleen Sue Taylor on 4/1/04 and told her I do not want to receive any HARASSING UNSOLICITED phone calls from the FBI and from next day onwards, the FBI had made my unlisted phone# appear on caller id of ordinary american people multiple # of times, where the owners of those phone#s listed below got upset and called me and left abusive messages in my answering machine. The owners of these phone numbers listed below called and asked me why I was calling them repeatedly, even though I DON'T KNOW who they are and I NEVER called them even once. 703-580-8992 4/2/04 Dale City, VA 540-309-1226 4/3/04 Roanoke VA 434-525-1448 4/3/04 Lynchburg, VA 757-404-1076 4/4/04 Harry Folsom left me an abusive message 703-820-6695 4/4/04 DC Suburb, VA 770-966-5236 4/4/04 Cooper P D, Woman left me an abusive message 757-483-0583 4/7/04 Norfolk, VA 540-463-7459 4/7/04 Lexington VA 770-253-6297 4/8/04 Hammond Stinson 703-541-0950 4/9/04 DC Suburb, VA FBI pretended to be me and made all these phone calls themselves from the FBI head office in Washington DC to those ordinary Americans and made my unlisted phone# appear on their phone caller ids, as if I am calling them. Please speak to, ADVISE and ENCOURAGE these people above to file lawsuits against the US government and FBI for violating their privacy and for phone harassment. (If you are RICH, you can use a PRIVATE EYE to get the PHONE CALL LISTS of COMMON PHONE# among the lists on the SPECIFIC DAYS mentioned above to zero in on the unlisted phone#) FBI agents have destroyed my livelihood by assasinating my character with potential employers by making anonymous calls to them when I sent resumes since FBI is privy to all the email addresses and phone numbers of the recruiters and potential employers. I do NOT think this is an EXCEPTION and imagine how many ordinary Americans have been suffering this FBI SADISM for years without knowing it. Once a Caucasian male FBI agent in a Comcast van threatened to ram into my car from behind and came within half an inch of hitting me from behind even though there was no traffic altercation between us and there is no traffica at all on a city street in Southfield (Detroit metro area) in Feb 2002. Two FBI agents once shouted at me and abused me verbally in a Sports Authority store parking lot approximately around Feb 2002 in Livonia, MI. FBI ILLEGALLY stole passwords of all my email and internet accounts with the help of bugs installed on my computer. FBI agents have followed me even when I went to see a doctor. There was one Caucasian FBI agent in a white truck with GA license plate APZ 7647 on 5/24/04, waiting for me in a truck in Ridgeview hospital parking lot listening to the conversation I was having with the doctor, using voice amplification devices. I am currently living in a two bedroom apartment sharing it with another person and the FBI sadists made anonymous calls to the apartment management office with fictitious complaints and tried to get me evicted from the FBI agents ILLEGALLY entered my apartment when I was not home in May stole a yellow highlighter and after I moved to a different Atlanta suburbs a month later entered my apartment again when I was not home and placed the yellow highlighter on the carpet in the middle of my bedroom by the time I came back from watching Fahrenheit 9/11 movie on 6/27/04. The FBI knew that I went to watch Fahrenheit 9/11 movie on 6/27/04 because they were watching me looking for directions and movie times on the internet and also with the help of gps vehicle tracking devices in my car. I spoke to an attorney from a pay phone on 6/24/04 and during the conversation I asked the attorney if we can get a court order to force the Internet Service Provider to reveal if my internet web surfing is being monitored by the FBI and lo and behold my internet account MYSTERIOUSLY DISAPPEARED from the ISP computers on 6/25/04. On inquiry my ISP told me which I never did. pasted a 8.5 X 11 page note on the inside of the apartment entrance door for my room mate to wake me up when he leaves so I can bolt the hotel lock from inside and go back to sleep. My room mate woke me up like I requested and I bolted the hotel lock from inside of my apartment and went back to sleep at 6 the morning of 7/24/04. The sadistic FBI agents STILL ENTERED my second 7/24/04 and REMOVED the 8.5X11 inch page pasted note on the inside of entrance door and LEFT IT on the ENTERTAINMENT CENTER in the living room just to TORTURE ME mentally and TERRORIZE me We entered your apartment even when you locked the hotel lock bolt and did this, but what can you do ? FBI agents ILLEGALLY entered my apartment on the night of 8/1/04 when I was asleep and stole prepaid calling card pin numbers from my trouser pockets. One of the prepaid calling cards in my trouser pocket was called I Conexion and the FBI next day called me on 8/2/04 and made the number 770-242-8775 with caller name Internet Conexion appear on my caller id to TERRORIZE me, that they entered my apartment the previous night and stole the prepaid calling numbers from my trouser pocket. The phone# above is NOT an FBI# but some other company. To protect myself from these SADISTIC FBI agents entering into my apartment and stealing things when I am asleep, I locked my bedroom door from the inside and put a small heavy box and a 19 inch TV on top of the box against the bed room door on the night of 8/7/04 so if the FBI agents try to open my bedroom door, the TV will fall off the box at the slightest movement of the door, make a big sound and wake me up. On the morning of 8/8/04 around 9:00 from the box and FBI agents speeding away by the time I realized what happened. FBI agents entered my apartment and opened my locked bedroom and pushed the the door. The FBI SADISTS are even willing to break my 19 inch TV to TORTURE and TERRORIZE me. FBI agents ILLEGALLY CHANGED three passwords (even though I used Kinkos to keep my PASSWORDS PRIVATE) of my three email ids that I was using to send this document to investigative reporters on www.ire.org between 8/26/04 and 8/28/04 to PREVENT ME from EXPOSING their SADISM, PERVERSION, TERRORISM, SEXUAL ABUSE and MENTAL TORTURE, by either ILLEGALLY installing key logging software on kinkos computers or by some other ILLEGAL MEANS unknown to me. So I created brand new email ids and sent this document and god knows when the SADISTIC FBI will change the passwords of my NEW email ids. SADISTIC FBI agents ILLEGALLY entered my apartment while I was taking SHOWER around 1:00pm and removed a shaving set scissors from the tote bag pocket in my bedroom and placed it on the iron board in my bedroom. FBI was able to EXECUTE this OPERATION WHILE I WAS TAKING SHOWER because they were watching me taking shower real time live with video devices installed in my bathroom and the rest of my apartment. To escape this FBI SADISM, I went and slept in an acquaintance's apartment came and entered that apartment, installed video surveillance devices in the acquaintance's apartment and drew a curvy line with a knife on the black colored apartment door of the acquaintance to TERRORIZE me. Sadistic FBI agents reacted to EACH and EVERY word that came out of my mouth for the last three years by making 2000 UNSOLICITED HARASSING phone calls at my unlisted private phone numbers. I know about registering with no call list websites but FBI PERVERTS and SADISTS just didn't care. To all of you who are gonna advise me to wear tinfoil or take drugs for schizophrenic paranoia, the SADISTIC FBI wants all of you to think exactly that way so they can CONTINUE to carry on their PERVERSE SADISM on ordinary american brains to treat the EVIL, SADISTIC, PERVERTED and DICTATORIAL FBI TERRORISTS like Gods. By now, you should have figured out the gist of what and how the FBI SADISTS and PERVERTS are TORTURING, TERRORIZING and SEXUALLY ABUSING me with their INFINITE POWER. You can read the rest of FBI SADISM below if you have time. **************************************************************************** I rented a car on 7/14/04, bought a cell phone and stayed in a hotel that night and by next morning 7/15/04, the FBI has figured out where I was staying and wire tapped my cell phone and tracked me by placing gps vehicle tracking devices even in the rental car. FBI agents ILLGEALLY entered my apartment when I was not home on 7/16/04 and LOCKED the bathroom door from inside, by the time I came back in the evening. FBI agents ILLEGALLY entered my apartment when I was not home between 7/15/04 and 7/18/04 and logged on to my computer and REMOVED a stock list from my Yahoo account. FBI agents ILLEGALLY entered my apartment when I was not home on 7/18/04 in the morning and turned the Air Conditioner ON intentionally by the time I came back to TORTURE me. FBI agents ILLEGALLY entered my apartment on 7/18/04 when I was not home and placed a razor blade and a nickel (which they STOLE from my apartment in WHAT CAN YOU DO ? FBI agents ILLEGALLY entered my apartment in the early hours of 7/22/04 (while I was still sleeping in bed) and opened a tote bag I kept next to the pillow and unzipped a small bag (zip which I closed the previous night) inside, just to TERRORIZE ME as if to say, we can and will do anything to you and you can't do anything to us because we FBI are GODS. FBI agents ILLEGALLY entered my apartment when I was not home on 7/22/04 during the day time and placed the TV remote this time hanging on the edge of the bed to TERRORIZE me that they entered my apartment. FBI agents ILLEGALLY entered my apartment sometimes when I was not home, pretending to be Wall Street Journal delivery personnel and delivered sometimes MUTIPLE copies of the Journal on the same day and also sometimes delivered multiple chinese restaurant take out menus to avoid the neighbors noticing and suspecting them. FBI agents ILLEGALLY entered my apartment when I was not home in the evening on 7/24/04 and again opened the tote bag and unzipped a small bag which is inside the tote bag and also STOLE a one time use Kodak camera from my bedroom which I bought on 7/23/04 in Wal Mart. FBI agents have REGULARLY entered my apartment ILLEGALLY with laptops when I was not home and copied information from my computer and floppy disks and messed them up. Sadistic FBI agents called me at least 30 times repeatedly in the space of an hour and a half between 7:00 and 8:30 pm on 7/27/04 and asked for Roberto Fernandez knowing fully well that nobody by that name lived in my apartment and infact said You are so funny and made WEIRD SOUNDS like aye, uye, ick, ihe etc while watching me real time live all the time with the video surveillance devices installed in the ceiling of my bedroom when I answered the phone. FBI agents ILLEGALLY entered my apartment while I was asleep on 8/2/04 night even though the hotel lock was bolted from inside, and placed the tv remote face down on the end table of my bedroom, folded a prepaid calling card in my trouser pocket and formed a crease on the card, removed a library map from the file folder (which was inside a tote bag) and placed it outside the file folder and unzipped the back pocket of the tote bag half way down and left it there to torture me. FBI agents have been alerting the security departments of WalMart, Best Buy, JC Penney stores etc with fictitious complaints when I went there for shopping, so the security cameras in those stores are constantly focused on me and watching me while I was in those stores. I rented a car on 8/5/04 and stayed in a hotel to get some privacy and the sadistic FBI agents still tracked me down that night and left the rental car doors unlocked by next morning on 8/6/04. I changed the car rented the previous day on 8/6/04 and traveled a couple of hours on a highway, took an exit, parked my car in a strip mall parking lot and spent about 20 minutes in a retail store and by the time I got back to the car, FBI agents followed me and put two quarters on a plastic bag containing donuts in the back seat of the car just to taunt and terrorize me they are still following me even when I rented a car and drove out of the city. FBI agents STOLE a pilot pen pack (which I bought from CVS Pharmacy on 8/8/04) from my car on the night of 8/8/04 and entered my apartment on pen pack under a chair on the carpet in the living room to torture me. FBI agents ILLEGALLY entered my apartment on 8/10/04 between 1:00pm and 1:15 pm (just 15 min) when I went to Publix and turned the TV On in my bedroom by the time I came back to mentally torture me. FBI agents ILLEGALLY entered my apartment even when the hotel lock was bolted from inside in the early hours of 8/16/04 while I was asleep and removed the middle row donuts in a 12 pack krispy kreme donut box in the refrigerator and placed them in the empty left row and placed the concord grape jelly bottle horizontally in the refrigerator and unplugged the tv cable from the cable outlet in my bedroom to TORTURE me that they are still entering my apartment anytime they want at their will. FBI agents ILLEGALLY entered my apartment while I was asleep in the early hours of 8/20/04 even when I put a coffee table against the apartment entrance door and a small heavy box and a 19in TV against my bedroom door from the inside and searched the locked tote bag and unzipped the back pocket of the tote bag half way down and left it there to TERRORIZE ME that they are still entering my apartment no matter what precautions I take. FBI agents ILLEGALLY entered my apartment on 8/22/04 around 7:30pm while I was taking a nap and turned the Air Conditioner ON to TAUNT me that they were entering my apartment EVEN WHEN I WAS still in my apartment. FBI agents stole a Gillette disposable razor on the night of 8/25/04 from my duffel bag and placed it between the drivers seat and driver side door near the trunk opening lever of my car. FBI agents ILLEGALLY opened the rental car I rented, in a hotel premises on the night of 8/27/04 and removed the cup holder and placed it infront of the parking brake of the car to HARASS ME that they are still following me even when I rented a car and stayed in a hotel. FBI agents ILLEGALLY even tapped MetroPCS cell phone numbers instantaneously as soon as I bought them on the same day. in my bedroom, ON and OFF by using remote control devices, sometimes while I was browsing the web. FBI agents ILLEGALLY entered the hotel rooms even when locked from inside where I stayed in August and September and installed video surveillance devices and moved things around from the tote bag's zipped side pocket into the tote bag to TERRORIZE me that they will INVADE my privacy and MENTALLY TORTURE me even when I slept in hotel rooms. Sometimes when I spoke to my roommate in non-English language, the psycho-analysis OBSESSED SADISTIC FBI agents called me on the phone immediately and said komo-isthaaaa in a SARCASTIC way and asked for Roberto Fernandez knowing fully well that nobody by that name lived in my apartment. === Subject: Re: Functional Analysis: Equivalent of Taylor series for operators >Hi all, >is there an equivalent of Taylor series for operators between Banach >Spaces ?- if yes, what is the name of this equivalent, and what are >the conditions on this operators ? I can think of at least two things the question might mean (does an operator have a Taylor series/can we apply a Taylor series to an operator). Try to be a little more specific about the question... ************************ David C. Ullrich === Subject: how to calculate the volume of cap? Assumed that an outer convex body formed by Minkowski sum between sphere (radius is r) and an convex body with any shape. The volume, surface area, mean breadth of innner convex body is known. The part between the outer convex body and the inner convex body is defined as convex ring. Now let a plane touch the outer surface of the inner convex body and move along the tangent direction of inner convex ring, then we call the intercept part between plane and convex ring with smaller volume as the intercepting cap. Now would you please tell me how to obtain the statistical average ratio between the volume of small cap and the volume of inner convex body, or the ratio between the volume of small cap and the volume of convex ring? If there is no general solution for the convex body with any shape. How about the convex body with regular shape, suck as ellipsoid with axis lengt of a, b,c, or the cuboid with side length of a, b, c? thank you very much in advance! === Subject: Re: how to calculate the volume of cap? > Assumed that an outer convex body formed by Minkowski sum between > sphere (radius is r) and an convex body with any shape. The volume, > surface area, mean breadth of innner convex body is known. The part > between the outer convex body and the inner convex body is defined as > convex ring. Now let a plane touch the outer surface of the inner > convex body and move along the tangent direction of inner convex ring, > then we call the intercept part between plane and convex ring with > smaller volume as the intercepting cap. Now would you please tell me > how to obtain the statistical average ratio between the volume of > small cap and the volume of inner convex body, or the ratio between > the volume of small cap and the volume of convex ring? > If there is no general solution for the convex body with any shape. > How about the convex body with regular shape, suck as ellipsoid with > axis lengt of a, b,c, or the cuboid with side length of a, b, c? It seems to me that even for polytopes the volume you ask for is a *very complicated* expression in terms of r, the volumes of each facet, and all interior angles at all faces of the polytope. The value seems to depend on much more than just volume, surface area, and mean breadth of innner convex body. On the other hand, I am not an expert in convex geometry... === Subject: Re: how to calculate the volume of cap? > Assumed that an outer convex body formed by Minkowski sum between > sphere (radius is r) and an convex body with any shape. The volume, > surface area, mean breadth of innner convex body is known. The part > between the outer convex body and the inner convex body is defined as > convex ring. Now let a plane touch the outer surface of the inner > convex body and move along the tangent direction of inner convex ring, > then we call the intercept part between plane and convex ring with > smaller volume as the intercepting cap. Now would you please tell me > how to obtain the statistical average ratio between the volume of > small cap and the volume of inner convex body, or the ratio between > the volume of small cap and the volume of convex ring? > If there is no general solution for the convex body with any shape. > How about the convex body with regular shape, suck as ellipsoid with > axis lengt of a, b,c, or the cuboid with side length of a, b, c? > It seems to me that even for polytopes > the volume you ask for is a *very complicated* expression > in terms of r, the volumes of each facet, and all > interior angles at all faces of the polytope. > The value seems to depend on much more than just > volume, surface area, and mean breadth of innner convex body. > On the other hand, I am not an expert in convex geometry... The textbook of geometrical probability give the quantitative relationship of volume, surface area between inner convex body and the outer convex body. But I didn't find the way on how to calculate the cap volume fraction. Obviously, for sphere, it is easy. then how about ellipsoid, cube....? === Subject: Non-split sequences Hello. Let M,N, and P be modules over some k-algebra, and let 0 --> M --> N --> P --> 0 be an exact sequence that doesn't split. Let Q be an indcomposable summand of N. Why is Hom(M,Q)=0, and Hom(Q,P)=0? -- Michael Knudsen === Subject: Re: Non-split sequences > Let M,N, and P be modules over some k-algebra, and let > 0 --> M --> N --> P --> 0 > be an exact sequence that doesn't split. Let Q be an indcomposable summand > of N. Why is Hom(M,Q)=0, and Hom(Q,P)=0? Argh! Both equalities should be neq (not equal to)! Sorry! My question is: Why must the sequence split if for example Hom(M,Q)=0? -- Michael === Subject: Re: Non-split sequences >> Let M,N, and P be modules over some k-algebra, and let >> 0 --> M --> N --> P --> 0 >> be an exact sequence that doesn't split. Let Q be an indcomposable >> summand of N. Why is Hom(M,Q)=0, and Hom(Q,P)=0? > Argh! Both equalities should be neq (not equal to)! Sorry! My question > is: Why must the sequence split if for example Hom(M,Q)=0? I don't think it has to. If we take a non-split sequence 0 -> M -> N -> P -> 0 and form 0 -> M -> N (+) Q -> P (+) Q -> 0 where Hom(M,Q) = Hom(Q,M) 0, that's non-split surely ... I still don't think you have the right question. :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Non-split sequences > Hello. > Let M,N, and P be modules over some k-algebra, and let > 0 --> M --> N --> P --> 0 > be an exact sequence that doesn't split. Let Q be an indcomposable summand > of N. Why is Hom(M,Q)=0, and Hom(Q,P)=0? It isn't necessarily. What if N is indecomposable, but not simple? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Non-split sequences > It isn't necessarily. What if N is indecomposable, but not simple? Oh, I forgot to require that M and P are indecomposable. Is it true then? -- Michael Knudsen === Subject: Re: Non-split sequences >>It isn't necessarily. What if N is indecomposable, but not simple? > Oh, I forgot to require that M and P are indecomposable. Is it true then? No. That doesn't rule out the possibility Q=N (if M and P are two non-isomorphic simple modules with a non-trivial Ext?). In that case we trivially have the inclusion of M into N and the projection from N onto P as non-zero homomorphisms. Did you want Hom(P,Q) and Hom(Q,M) to be zero by any chance? That does rule out the above counterexample anyway: if Hom(P,Q) is non-zero, and P is simple, then we have a copy of P in the socle of N, and consequently we have Poplus M inside N, so the original sequence would split. It's been 10 years plus since I used the theory of blocks at all, so I'm a bit rusty there, sorry: I can't immediately tell, whether this modification of the original claim is correct. My guess would be that it isn't: couldn't the socle of M be isomorphic to the head of P for example? Say, if there are two irreducible modules X and X', and M has socle isomorphic to X, M/X is isomorphic to X'(so M isn't semisimple) and similarly P has X' in the socle, P/X'=X. Say, if we then can locate a module N with socle series levels X, X'oplus X' and X. Then N would be indecomposable, but we would have non-trivial homomorphisms in both Hom(P,N) and Hom(N,M). I don't have a suitable k-algebra in mind, though. Jyrki Lahtonen, Turku, Finland === Subject: Re: Non-split sequences >> It isn't necessarily. What if N is indecomposable, but not simple? > Oh, I forgot to require that M and P are indecomposable. Is it true then? No. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Need Help Understanding Conjugacy Classes in A_5 (take 2) I came across the following approach on showing why the elements of order 5 in A_5 must break into two conjugacy classes. However, I do not fully understand the logic. Can someone elaborate the logic here? ... The Sylow 5-subgroup of A_5 is Z_5. Hence the conjugacy classes of elements of order 5 in A_5 correspond to the orbits of N(Z_5) -- the normalizer of Z_5. (1) 24 = 60 / |N(Z_5)| * 4. Thus |N(Z_5)|=10 and hence N(Z_5) has two orbits, which correspond to two conjugacy classes. Ok, I see why Sylow 5-subgroup of A_5 is Z_5 since Z_5 is the only group of order 5. Unfortunately for me, this is the only thing I understood. How do the conjugacy classes correspond to the orbits of N(Z_5)? What action is implied by the orbits? By conjugation? I understand that 24 is the # of elements of order 5, 60 is the # of elements in A_5, but how is (1) formed? How does |N(Z_5)|=10 implies N(Z_5) has 2 orbits? Help... Kira. === Subject: Re: Need Help Understanding Conjugacy Classes in A_5 (take 2) no@mail.com >I came across the following approach on showing why the elements of >order 5 in A_5 must break into two conjugacy classes. However, I do not >fully understand the logic. Can someone elaborate the logic here? >... The Sylow 5-subgroup of A_5 is Z_5. Hence the conjugacy classes of >elements of order 5 in A_5 correspond to the orbits of N(Z_5) -- the >normalizer of Z_5. >(1) 24 = 60 / |N(Z_5)| * 4. >Thus |N(Z_5)|=10 and hence N(Z_5) has two orbits, which correspond to >two conjugacy classes. >Ok, I see why Sylow 5-subgroup of A_5 is Z_5 since Z_5 is the only group >of order 5. Unfortunately for me, this is the only thing I understood. First you need to prove the following result. Lemma If the finite group G has an abelian Sylow p-subgroup P, and g,h in P are conjugate in G, then they are conjugate in N(P). Sketch proof. If g^x = h with x in G, then P and P^x are Sylow p-subgroups of C(h), so are conjugate in C(h). Hence P = P^(xy) with y in C(h), and so g^(xy) = h with xy in N(P). Now we can take P = < g >, and N(P) = < g,h > with g = (1,2,3,4,5), h = (2,5)(3,4), and you can calculate directly that {g,g^-1} and {g^2,g^-2} are the N(P) conjuagcy classes of 5-elements. Hence there are also two classes in A_5. Since x and x^2 are conjugate in S_5, the two classes have the same size. Derek Holt. >How do the conjugacy classes correspond to the orbits of N(Z_5)? >What action is implied by the orbits? By conjugation? >I understand that 24 is the # of elements of order 5, 60 is the # of >elements in A_5, but how is (1) formed? >How does |N(Z_5)|=10 implies N(Z_5) has 2 orbits? >Help... >Kira. === Subject: Re: Need Help Understanding Conjugacy Classes in A_5 (take 2) I came across the following approach on showing why the elements of order 5 in A_5 must break into two conjugacy classes. However, I do not fully understand the logic. Can someone elaborate the logic here? ... The Sylow 5-subgroup of A_5 is Z_5. Hence the conjugacy classes of elements of order 5 in A_5 correspond to the orbits of N(Z_5) -- the normalizer of Z_5. (1) 24 = 60 / |N(Z_5)| * 4. Thus |N(Z_5)|=10 and hence N(Z_5) has two orbits, which correspond to two conjugacy classes. Ok, I see why Sylow 5-subgroup of A_5 is Z_5 since Z_5 is the only group of order 5. Unfortunately for me, this is the only thing I understood. How do the conjugacy classes correspond to the orbits of N(Z_5)? What action is implied by the orbits? By conjugation? I understand that 24 is the # of elements of order 5, 60 is the # of elements in A_5, but how is (1) formed? How does |N(Z_5)|=10 implies N(Z_5) has 2 orbits? Help... Kira. === Subject: Re: sparse matrices and eigenvalue computation >I am in the position where I may need to implement an eigenvalue solver >myself. The objective is to obtain only the lower n eigenvalues (in >magnitude). I do the eigenvalue computation at present (in C++ using TNT >few of the lower n eigenvalues will be much faster. >In MATLAB, I have seen some people use sparse matrices to compute >eigenvalues (usually when the input matrix, X, is quite large). >First, where do I find information on what sparse matrices are and why, >where and how to use them? I would like to go from easy to more detailed >explanations. a matrix whose majority of entries is zero is called a sparse matrix. the number of nonzeros divided by n^2 gives an impression of sparsity and often this is below 0.1%. sparse matrices can be stored and handled in different format >Second, are there any C++ programs that already compute only a subset of >eigenvalues of a given input matrix? >->HS >-- arpack++ (it corresponds to eigs in matlab) http://www.ime.unicamp.br/~chico/arpack++ http://www.caam.rice.edu/software/ARPACK if you presently are able to get all eigenvalues this gives me the idea that you might have dense and not really large n<1000 say, matrices. if you have a symmetric case, then you could use the bisection technique applied to the tridiagonal form of the matrix in order to restrict yourself to the computation of only some eigenvalues. or translate f.l.bauers ratqr which determines the eigenvalues ordered by magnitude. ratqr is the handbook of linear algebra (wilkinson-reinsch) in algol60, but tanslation to c++ should be easy. hth peter === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals : Before I begin I would make three request, you can grant them or not. : : 1) Please set your e-mail program (if it allows you to) to wrap at 72 : characters as is the standard. It becomes increasingly difficult to : respond ot messages that are as off centered as yours. : 2) Please refrain from personal attacks, I have made none on you. : 3) Please maintain a civil tone. I enjoy spirited debate but will not : have a discussion if you can not remain civil. : : message : > : > : > : > : > : > : No, I am a poor High School teacher. : > : > : : > : > : However, I found a few who have. : > : > : : > : > : http://www.physlink.com/Education/AskExperts/ae433.cfm -- This is : > : > : what I have heard before that clocks in airplanes moving fast have shown : > : > : the effects. Also the muon counting, although I believe I read in : > : > : previous part of the thread that you disagree with that. : > : > : By contrast http://www2.rideau.net/gaasbeek/spap5.html -- Is an doesn't offer and : > : > : counter evidence, just his theory, which I don't really think is right. : > : > : Perhaps I don't fully understand what a helical wave electron, but neither following web : > : > : page, http://tinyurl.com/5dhh6, which points out the flaws in the previous that : > : > : website hasn't been updated since 1998, and the author's credentials : > : > : aren't listed.) : > : > : : > : > : So, if I weigh the data: : > : > : : > : > : Pro Time Dilation : > : > : ----------------- : > : > : Every textbook I have ever read (alone this is not a good source I : > : > : know) : > : > : Experimental Data (Although not accepted by all, but all of my : > : > : college professors believe it.) : > : > : Theory that can logically explain it. : > : > : > : > Uh huh. Faith is a powerful ally, but logic is even more powerful. : > : : > : The Force of course if my strongest ally! :) : > : : > : Seriously, I don't see how this answers anything. Did you look at any of : > : these sites, or is your only line going to be that scientists just : > : justify each other to look smart. I don't buy that. : > There are many other scientists that deny Einstein's relativity. : > Did you look into any of those, or is your line going to be that : > Einstein was right by majority opinion? If so, you'd be joining : > the Church against Galileo and supporting Ptolemy in a different era. : : To be clear, I *NEVER* said anything even remotely like Einstein is : correct by majority opinion. Please do not misquote or skew my words. : You will not convince anyone that you are correct with those tactics. If : you want to engage in a serious discussion, then do so. : : You simply choose to ignore my reasoning which for accepting relativity : which is that it is based in fact (pr oven by experiments) and experts : have explained it and agree with it. I have not found in my searches : (they have not been exhaustive as I only have limited to spend on this) : nearly as many credible sources that oppose relativity. If you have any : credible sources, please share them, you have yet to do that. : : > : Do you think that none of these people are like you and question it. : > No. : > : You don't even attempt to controvert any of the evidence that these reports : > : present. : > Bull, I've been quite prolific and controversial of any so-called : > evidence that the disciples of Einstein have presented. Do you think : > you are the only person I've ever conversed with? : : Yet when given a list of several experiments you haven't explain to me : how they are wrong. You seem to believe that every experiment ever : fabricated was a lie. : : > : People went on an airplane and did this and you still don't : > : believe it. What do you want? : > Proof. : : Please define proof. This happened. If you want to say that every : experiment is a lie, I guess you can, but then I don't know what you : will ever believe, unless you have the means to recreate every : experiment ever performed. : : Let me be specific. In the experiment performed by Haefele and Keating, : (Haefele and Keating, Nature 227 (1970), pg 270 (Proposal); Science Vol. : 177 pg 166--170 (1972)) in which two clocks were flown in opposites : directions with a control clock on ground and the two clock came back to : the Earth with different times that were within the predictions of : relativity, what was wrong with that experiment? Why isn't that proof? It is a statement. Where is your data? It is not a proof without data, and did you subject the clocks to the same buffeting of turbulence and acceleration inherent with a commercial flight, the changing magnetic field of the Earth, log the altitude and duration at that altitude throughout the experiment? Without a control clock on the ground being subjected to the same environment, all you have is a couple of clocks that do not record the time correctly when tossed around. : : > : Nothing in the above explanation was based on faith. Perhaps you were : > : implying that me believing my professors is faith, but these people are : > : *EXPERTS* in physics, who else will I believe? : > Ptolemy, of course. He was an *EXPERT* for 1400 years. Forget : > Copernicus, he was just a dumb preacher, what did he know? As long : > as YOU *know* the Earth cannot move, Ptolemy has to be right. : > That is an example of faith, the kind of faith you display. : : Many scientists that were marveled at have been proven wrong. Aristotle, : Freud, Hook, the list goes on. Ptolemy, had a theory, this was based on : experiments and he was believed to be correct until someone was able to : prove him wrong. This required a telescope! So, of course others will be : able to prove him wrong once that is made. : : Please see my discussion of the Ptolemy Fraud issue below. : : Let me repeat: I do not base my beliefs on faith. I have look at : experiments and at theories and drawn a conclusion on the information I : have available to me. : : > : Someone who doesn't explain why the data is wrong other than to claim it is biased & : > : doesn't offer any credentials (at least I can't find them on your wed site.) : > My anonimity is my personal choice. Take what I say based on : > what I say, not on who I am. : : That is your right, you will then have to understand that when I compare : your theory versus the theory of experts in physics, they will get a : slight edge. I have to consider my source. That is your right. I have to consider mathematics and logic without regard to the source. : : > : I don't know what the answer is, but Tom Roberts listed several : > : experiments that seem to justify SR and it is not logical to presume : > : them all to be fabricated. : > Bill Clinton did Not have sex with that woman. His credentials : > are President of the United States of America, so it is illogical : > to presume he fabricated that statement. : : I agree, one should presume that President would tell truth under oath. : I don't know if we can anymore, and I think that is a shame. Based upon : your theory, I can't believe a single law or experiment in physics : unless I try it. Is it safe to believe anyone? No, it isn't. : : > :You have not refuted a single experiment that : > : I have presented to you? : > You haven't presented any data, just a bold assertion that someone : > told you someone flew in a plane. You don't even know the : > name of the indvidual that did so, but are willing to believe in : > time dilation. How am I supposed to communicate with you on : > the subject when you haven't a clue what you are talking about? : : See the previous reference above. : : > : > : Anti Time Dilation : > : > : ------------------ : > : > : A few people whose reputations I don't know who just try to poke : > : > : holes that don't make a lot of sense in tested theories and have very few : > : > : reputable supporters. : > : > : : > : > : I guess that leads me to believe in time dilation. : > : > : : > : > : Sorry. :( : > : > : > : > Ah... I see. You judge before hearing all the evidence, basing your : > : > opinion on the beliefs of your college professors, who base their beliefs on : > : > each other's opinions, not one of whom could stand against me in a logical : > : > debate on the subject. : > : : > : What evidence have you presented? : > The burden of proof is upon the claimant. I deny time dilation. You : > prove it. : : This is very convienent for you, you can always say that it is my burden : and never to prove it wrong. Yes is and yes I can. That's one of the rules. I claim bright green flying elephants lay eggs. Would I be justified in asking you to prove they do not? Of course not. Its my claim, it is up to me to prove it. I cannot. Therefore I have no proof. You claim that time dilation exists. I ask you for proof and your reply has been Experts say so. I happen to be an expert in bright green flying elephant's nests, so why would you not believe me? Absurd, you say. Well, I say time dilation is absurd. Prove it. I'll listen. And I'll challenge any and all assumptions you make. : Why don't you point to experimental data by : one of the many other scientists that don't believe in relativity. : That would perhaps end this debate right now. Michelson and Morley disproves Einstein's relativity. You'll find that on your list of experiments that claim it as proof of relativity. If it is a disproof you seek, learn how the experiment was conducted, note that Michelson never once accepted Einstein's relativity, and Michelson was an expert. : : > : You haven't yet proved that any of these experiments are wrong. : > You haven't so much as named one. : : See above. : : > : My professors are the sources for this. : > Name one experiment that you consider to be absolute proof of time : > dilation. I'm not wasting my time on all of them, pick the one you like the : > best, present the data and we'll analyse it. : : See above. H&K? That's it? Even my wristwatch will not keep correct time when buffeted or subjected to changing magnetic fields and temperatures. : : > : That along with the data you have been presented and ignored is how I : > : draw my conclusions. *NOT ON FAITH ALONE!* : > What data have you presented? You draw your conclusions on faith : > alone. : > : > Well, I'm not here to change your beliefs. Only you can do that. Nor : > : > am I seeking renown. If your mind is open, read on. If it be closed, : > : > stop now. : > : : > : My mind is open. However, my mind works like any scientists mind does. : > Does it indeed... : > : I have a theory or belief, and then I test it with experiment. : > Hmm... and that is what you mean by an open mind, is it? Start with : > a belief and then try to justify it. This is what you think science is : > all about. I don't. : : Again, you are warping words. In science you develop a theory or belief : (generally referred to as a hypothesis) and then you test it. I have a : hypothesis right now that time dilation exists (I developed this after : doing background research). I want to test it. I can not, because I can : not move fast enough, so I went to find other tests. They have yielded : results. I have analyzed them, and I now agree that time dilation exists. Show your analysis then. : : If you prefer, I could have started with your hypothesis, Time Dilation : doesn't exist. Fine with me. It never did until Einstein read H.G. Wells' Time Machine. : Followed the same procedure and drawn the conclusion : that my hypothesis is wrong. It's your hypothesis. I say time dilation doesn't exist. : Either way I arrive at the same end point. : Although, in your world I wouldn't be allowed to reference the works of : other physicists. Of course you are allowed to reference them. I'm not preventing you, but naming names isn't a proof. : > : Lacking the ability to do that myself, I turn to others who have done : > : experiments or analyzed this topic. I weight them an draw a conclusion. : > In other words, Nature works by popular opinion. I don't think that : > Nature is a democrat. : : Again, you warp my words. I didn't say popular opinion. You could 2,000 : flawed experiments that disprove it and 1 good one that proves it, and I : would follow the good one. Oh, ok. Well, in my experiment with bright green flying elephant's eggs, I've been trying to hatch one. Unfortunately, like you not having enough speed, I don't have the right incubator. When I do, I'll prove it to you. : Fortunately, I found several good ones that : prove it and a handful that kind of disprove it. Give me your best good one. : : > : You spend a lot of time on a lot of websites and newsgroups as far back : > : as at least '02 trying to prove Einstein wrong. So, if your goal isn't : > : to change my beliefs, what is your goal? : > To communicate with people that share a desire to understand Nature, : > less so with those that prefer to take on faith what they want to : > hear. : : Yet, those people who have done experiments with & about nature you call : liars because you think Ptolemy was a liar. Newton thought Ptolemy was a liar. He should know, he's an expert. : : > : If you goal is to prove Einstein wrong, you better start addressing some : > : of the issues I present as well presenting some of your own. : > My goal isn't to prove Einstein wrong, that is a secondary issue. : > : > The most recent accusations of forgery made against Ptolemy came from : > : > Newton in [12]. He begins this book by stating clearly his views:- : > : > : > : > This is the story of a scientific crime. ... I mean a crime committed : > : > by a scientist against fellow scientists and scholars, a betrayal of : > : > the ethics and integrity of his profession that has forever deprived : > : > mankind of fundamental information about an important area of : > : > astronomy and history. : > : > : > : > Towards the end Newton, having claimed to prove every observation [12]:- : > : > : > : > [Ptolemy] developed certain astronomical theories and discovered that : > : > they were not consistent with observation. Instead of abandoning the : > : > theories, he deliberately fabricated observations from the theories so : > : > that he could claim that the observations prove the validity of his : > : > theories. In every scientific or scholarly setting known, this : > : > practice is called fraud, and it is a crime against science and : > : > scholarship. : > : > http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html : > : : > : I do not know if this if founded or not, but it is irrelevant. Because : > : Ptolemy did this, does that mean that every scientific theory out there : > : is wrong because people lied? Do you believe any progressive theory : > : besides Newton's Laws? : > I don't believe charlatans like Einstein, if that's what you want. : : What you fail to mention is that the next paragraph on this web-sites : clears Ptolemy of most of these charges. Are you saying Ptolemy was right then? : : Source: IBID : Although the evidence produced by Brahe, Delambre, Newton and others : certainly do show that Ptolemy's errors are not random, this last quote : from [12] is, I [EFR] believe, a crime against Ptolemy (to use Newton's : own words). The book [8] is written to study validity of these : accusations and it is a work which I strongly believe gives the correct : : ... one has to assume that a substantial proportion of the : Ptolemaic star catalogue is grounded on those Hipparchan observations : which Hipparchus already used for the compilation of the second part of : his Commentary on Aratus. Although it cannot be ruled out that : coordinates resulting from genuine Ptolemaic observations are included : in the catalogue, they could not amount to more than half the catalogue. : : ... the assimilation of Hipparchan observations can no longer be : discussed under the aspect of plagiarism. Ptolemy, whose intention was : to develop a comprehensive theory of celestial phenomena, had no access : to the methods of data evaluation using arithmetical means with which : modern astronomers can derive from a set of varying measurement results, : the one representative value needed to test a hypothesis. For : methodological reason, then, Ptolemy was forced to choose from a set of : measurements the one value corresponding best to what he had to consider : as the most reliable data. When an intuitive selection among the data : was no longer possible ... Ptolemy had to consider those values as : 'observed' which could be confirmed by theoretical predictions. : : I think that it really hurts you credibility to only present one side of : argument when you know there is a counter just to make a point. I quoted the source for you. Besides, we are not discussing my credibility, but Einstein's. It is : dishonest and makes not want to believe anything else you say. 2) Please refrain from personal attacks, I have made none on you. What the is that then, calling me dishonest? You are a two-faced jackass. : : Above I provided a link to experiments that claim to prove and disprove : time dilation and analyzed both. You took a quote out of context and : took on *FAITH* that Newton was right, when he was in the process of : justifying his beleifs, not the best source in this case. : : > : > I accuse Einstein of the same crime, fraud. : > : > I accuse his disciples of the fabrication of observation, including : > : > Hafele and Keating, Algevar et. al. and many others. : > : > I accuse those responsible for providing the FAQ's to this newsgroup : > : > of bluster and nonsense, and the claim that the Michelson Morley Experiment : > : > supports relativity is a lie. MMX disproves SR. : > : : > : It disproved it because it was a bad experiment. They were basing things : > : on aether, which we know now doesn't exist. : > Ah... I see. A bad experiment. Since it succeeded in disproving : > aether, I think you are wrong. I consider it a highly successful experiment. : > What did you say above? : > However, my mind works like any scientists mind does. I : > have a theory or belief, and then I test it with experiment. : : It was good for proving that aether didn't exist. Bad for dealing with : relativity because there assumptions about aether skewed there data. : Please see the above reference for more details. (The Roberts site.) MMX disproves SR as well as the non-existance of aether. : : > Michelson was a real scientist. He had a theory or belief, and he : > tested it with experiment. You call that bad. Contradictory, are you not? : > I don't think you even think, let alone think like a scientist. : : It is unfortunate that you have to result to personal attacks to try to : win your point. :( : Win my point? All you've done is told me your beliefs. One one shred of evidence, not one iota of data, and you claim to think like a scientist. You ramble on about bad experiments... What do you WANT? : The experiment was poorly constructed as a test of relativity, because : of the assumptions made about aether. Holy ! Michelson first conducted his experiment in 1881, it supposed to be a test of SR? : So, yes, the experiment disproved relativity, but it was not a well designed experiment for that purpose, so I don't view that as a defeat of relativity. It is clear now that you know absolutely NOTHING about it. How old are you, 18? : If that is the only : experimental basis upon which you are laying your claim, I fear you are : grasping at straws. Take your blind faith and shove it, little boy who claims to think like a scientist! : : > : > To discover why Einstein was a fraud takes a careful analysis of his : > : > paper. : > : > : > : > For quotations following, reference: : > : > http://www.fourmilab.ch/etexts/einstein/specrel/www/ : > : > (On the Electrodynamics of Moving Bodies by Albert Einstein) : > : > : > : > 2) In agreement with experience we further assume the quantity : > : > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in : > : > empty space., : > : > an admitted assumption that is quite worthless when there is any : > : > relative motion between A and B, yet essential to the derivation of : > : > the remainder of Einstein's nonsense. : > : : > : He says it is in agreement with experience, not just an assumption. You : > : need to read all of the words. : > It is not in agreement with my experience, and you have none. : It is becoming ever clearer that you have none. : : Our experience, and the experience of our experimental data is that the : speed of light is constant in a vacuum. : : > : Of course it changes when things move. : > So does the ü in the equation that follows. : > : He makes this claim when the objects are in the same FofR. : > So SR is quite valid when there is no motion. I'll accept that. : : Things can be moving and be in the same frame of reference. The key is : to not be accelerating. Increasingly clearer. : : > : This is the first postulate of one his : > : papers (i get the mixed up), that all laws of physics are the same in : > : non-accelerating frames of reference. : > So you are mixed up, but take what you are told on faith alone, : > and at the same time claim to think like a scientist. : > Einstein's thought experiment is a bad experiment, does that help? : : Why is bad? Can you defend any of your claims with anything besides math? I don't need to, sonny. Math is what it is all about. : : The only thing I was referring to about being mixed up is forgetting : which is GR & which is SR. One the difference in this discussion is that : I am willing to admit when I don't know something, where as you will : continue to say things and make personal attacks when you don't know : what to say. All you've shown is your faith. Why waste my time? : : > : > 3) The equation : > : > ü[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = : > : > tau(x',0,0,t+x'/(c-v)) , : > : > the ü of which is derived from 2) above and is tantamount to saying : > : > (1/3 + 2/3)/2 = 1/3. : > : : > : 3 - 7) I will be honest here that I don't know enough about math to : > : discuss this, but if it is wrong, why are you the only person saying, : > : and why do so many other physicists believe it? : > Who said or is your only line going to be that scientists just justify each : > other to look smart. I don't buy that. : > Yet you've fallen for them looking smart, hook, line and sinker, : > haven't you? It was really easy to hoodwink you, you have no math. : : You have not hoodwinked anyone. Einstein has. : : : : > : > And last but really first, : > : > 1) light is always propagated in empty space with a definite velocity : > : > c which is independent of the state of motion of the emitting body, : > : > a totally unproven assumption without any evidence to support it, a : > : > great deal of evidence to deny it. : : Still waiting for that evidence that says light depends on its source, : since there is a great deal to deny it. http://www.androc1es.pwp.blueyonder.co.uk/actual_data.htm : : > : : > : Based upon your analysis then, the speed of light emitted from a source : > : on the north pole, should be different than the speed of light emitted : > : from a source at the equator, since the equator moves faster? : > Yes, at the top of the atmosphere. : : Why does being at the top of the atmosphere matter? It is pointed out that the extinction theorem of dispersion theory, for which for which the elementary derivation is given, shows that an incident light wave is extinguished at the surface of a dielectric. J.G, Fox, Experimental Evidence for the Second Postulate of Special Relativity. Am J. Phys. Vol 30 1962. That's why. My point was that a : light source on the equator would move faster than a light source on the : North Pole, so given your beliefs, the light at the equator should move : faster. I disagree with that claim. : : > The way I first proposed to test : > it (do a bad experiment) was to shoot the moon with a laser from : > the ISS. Since then I've revised my opinion. Using Roemer's method : > and the Cassini probe currently in orbit about Saturn, we can : > determine the speed of light as Cassini approaches and recedes from the Earth. : > It will be carrying an adequately accurate clock. Radio transmissions : > are acceptable. : : Didn't we basically accomplish your experiment a long time ago using the : moons of Saturn (or which ever planet Roehmer and Cassini were watching, : I don't recall right now)? Why would we repeat it? How do you know it : won't be faked? : : Additionally, we already know the speed of light, why would we test it : again? : : > : So, if we were to deduce the length of a meter (which is defined by the : > : speed of light) a meter in Ecuador would be longer than a meter on the : > : north pole? This is absurd. : > Yes, I'll agree with you that xi = (x-vt)/sqrt(1-v^2/c^2) : > is absurd. Well done, you've just given up length contraction. : > Now all you need to give up is time dilation tau = (t - vx/c^2) / sqrt (1-v^2/c^2) : > which is equally as absurd. : : Stop twisting my words. Obviously I wouldn't have said that xi = : (x-vt)/sqrt(1-v^2/c^2) is absurd considering the whole rest of my advocacy. I am not twisting your words. You've clearly stated that if lengths measured at the pole and Equator differ, that is absurd. I agree. Since that is a prediction of relativity, then relativity is absurd. : : My point was not that length contraction was absurd. My point, which I : will explain again, is that the definition of a meter is in terms of the : speed of light in a vacuum. : According to your theory, using this method : a meter on the North Pole would have to be shorter than a meter on the : equator, because light moved slower on the North Pole. The fact that you : would propose such a theory is absurd to me. Ok then, that IS what Lorentz proposed and Einstein took up, based on MMX. It IS absurd. : : > : This bit of logic should be enough to make you realize that Einstein's : > : derivation and proof along with the many experiments in agreement with : > : him are correct. : > I thought you'd just said length contraction was absurd? It is. But then, I'm not so dumb as to change the length of a metre based on where I happen to be. : : Yes, that is what you thought, you were wrong. :( Since you have no idea what you are discussing, you clearly haven't clue, you do not have the right to tell me I'm wrong, little boy. : : > : You still offer no evidence of an experiment that disagrees with the : > : theory or relativity or any analysis of the many that do. : > If you want to present the data, we'll analyze it. : : I've shown my hand, will you ever show an experiment that supports you? : I have been asking since the start. You haven't done that. I don't have the burden of proof and you have NOT presented any data. Androcles. : : > : If you are going to convince anyone that Einstein is wrong, you will have to do : > : this, because, if you are right and science is self fulfilling itself, : > : you will have to convince young people like, and trying to poke little : > : whole in math ain't gonna cut it. : > As I said, I can't overcome blind faith. I can only reason with people : > of intelligence. Have a nice day. : > Androcles. : : Then reason away. : : Sorry the reply took so long. I have been busy, and while replying I : have had to consistently correct your warping of my words that it has : added to my typing time. : : Have a nice day, : : Peter === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals > : Before I begin I would make three request, you can grant them or not. > : 1) Please set your e-mail program (if it allows you to) to wrap at 72 > : characters as is the standard. It becomes increasingly difficult to > : respond ot messages that are as off centered as yours. > : 2) Please refrain from personal attacks, I have made none on you. > : 3) Please maintain a civil tone. I enjoy spirited debate but will not > : have a discussion if you can not remain civil. I see you ignored my requests. (I will address you allegation of hypocrisy below.) That is unfortunate, although not surprising. You will have a hard time convincing others of your ideas if you can not present them in a professional manner. You may be a brilliant scientists, I don't know, but you may want to take some time to study argument and persuasion before you try to convince the rest of the world of your ideas. Just trying to be helpful. > : > : > : > : : > : > : No, I am a poor High School teacher. > : > : > : > : > : > : However, I found a few who have. > : > : > : : > : So, if I weigh the data: > : > : > : > : > : > : Pro Time Dilation > : > : > : ----------------- > : > : > : Every textbook I have ever read (alone this is not a good source I > : > : > : know) > : > : > : Experimental Data (Although not accepted by all, but all of my > : > : > : college professors believe it.) > : > : > : Theory that can logically explain it. > : > : : > : > Uh huh. Faith is a powerful ally, but logic is even more powerful. > : > : > : > : The Force of course if my strongest ally! :) > : > : > : > : Seriously, I don't see how this answers anything. Did you look at any of > : > : these sites, or is your only line going to be that scientists just > : > : justify each other to look smart. I don't buy that. > : : > There are many other scientists that deny Einstein's relativity. > : > Did you look into any of those, or is your line going to be that > : > Einstein was right by majority opinion? If so, you'd be joining > : > the Church against Galileo and supporting Ptolemy in a different era. > : To be clear, I *NEVER* said anything even remotely like Einstein is > : correct by majority opinion. Please do not misquote or skew my words. > : You will not convince anyone that you are correct with those tactics. If > : you want to engage in a serious discussion, then do so. > : You simply choose to ignore my reasoning which for accepting relativity > : which is that it is based in fact (proven by experiments) and experts > : have explained it and agree with it. I have not found in my searches > : (they have not been exhaustive as I only have limited to spend on this) > : nearly as many credible sources that oppose relativity. If you have any > : credible sources, please share them, you have yet to do that. > : > : Do you think that none of these people are like you and question it. > : : > No. > : : > : You don't even attempt to controvert any of the evidence that these reports > : > : present. > : : > Bull, I've been quite prolific and controversial of any so-called > : > evidence that the disciples of Einstein have presented. Do you think > : > you are the only person I've ever conversed with? > : Yet when given a list of several experiments you haven't explain to me > : how they are wrong. You seem to believe that every experiment ever > : fabricated was a lie. > : > : People went on an airplane and did this and you still don't > : > : believe it. What do you want? > : : > Proof. > : Please define proof. This happened. If you want to say that every > : experiment is a lie, I guess you can, but then I don't know what you > : will ever believe, unless you have the means to recreate every > : experiment ever performed. > : Let me be specific. In the experiment performed by Haefele and Keating, > : (Haefele and Keating, Nature 227 (1970), pg 270 (Proposal); Science Vol. > : 177 pg 166--170 (1972)) in which two clocks were flown in opposites > : directions with a control clock on ground and the two clock came back to > : the Earth with different times that were within the predictions of > : relativity, what was wrong with that experiment? Why isn't that proof? > It is a statement. Where is your data? > It is not a proof without data, and did you subject the clocks to the same > buffeting of turbulence and acceleration inherent with a commercial > flight, the changing magnetic field of the Earth, log the altitude and > duration at that altitude throughout the experiment? Without a control clock on the > ground being subjected to the same environment, all you have is a couple of > clocks that do not record the time correctly when tossed around. Yes, in the experiment those factors are taken into account. You can read the paper and many different analysis of the report if you want to. In the experiment They flew atomic clocks on commercial airliners around the world in both directions, and compared the time elapsed on the airborne clocks with the time elapsed on an earthbound clock (USNO). Their eastbound clock lost 59 ns on the USNO clock; their westbound clock gained 273 ns; these agree with GR predictions to well within their experimental resolution and uncertainties (which total about 25 ns). (http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#5.%20T win%20paradox) For analysis on your other concerns one place I found was http://www.phys.unsw.edu.au/~jw/time.html. Sorry I didn't include the data before, I assumed you were familiar with it. > : > : Nothing in the above explanation was based on faith. Perhaps you were > : > : implying that me believing my professors is faith, but these people are > : > : *EXPERTS* in physics, who else will I believe? > : : > Ptolemy, of course. He was an *EXPERT* for 1400 years. Forget > : > Copernicus, he was just a dumb preacher, what did he know? As long > : > as YOU *know* the Earth cannot move, Ptolemy has to be right. > : > That is an example of faith, the kind of faith you display. > : Many scientists that were marveled at have been proven wrong. Aristotle, > : Freud, Hook, the list goes on. Ptolemy, had a theory, this was based on > : experiments and he was believed to be correct until someone was able to > : prove him wrong. This required a telescope! So, of course others will be > : able to prove him wrong once that is made. > : Please see my discussion of the Ptolemy Fraud issue below. > : Let me repeat: I do not base my beliefs on faith. I have look at > : experiments and at theories and drawn a conclusion on the information I > : have available to me. : I don't know what the answer is, but Tom Roberts listed several > : > : experiments that seem to justify SR and it is not logical to presume > : > : them all to be fabricated. > : : > Bill Clinton did Not have sex with that woman. His credentials > : > are President of the United States of America, so it is illogical > : > to presume he fabricated that statement. > : I agree, one should presume that President would tell truth under oath. > : I don't know if we can anymore, and I think that is a shame. Based upon > : your theory, I can't believe a single law or experiment in physics > : unless I try it. Is it safe to believe anyone? > No, it isn't. I disagree with this claim. We have systems in the scientific community to prevent the things that may have happened in the past. We have many, many journals into which people can submit there findings and they can be verified by the community. This is what scientists do when they have a claim or a theory, they publish it so others can critique, if it is wrong, we will know. This has been a tradition since the days of Newton when he and Hooke had many of their famous exchanges. The community monitors itself. If you have a belief you should get it published so others can either agree or disagree with you. If you have reference so that I may read it. : > : Anti Time Dilation > : > : > : ------------------ > : > : > : A few people whose reputations I don't know who just try to poke > : > : > : holes that don't make a lot of sense in tested theories and have very few > : > : > : reputable supporters. > : > : > : > : > : > : I guess that leads me to believe in time dilation. > : > : > : > : > : > : Sorry. :( > : > : : > : > Ah... I see. You judge before hearing all the evidence, basing your > : > : > opinion on the beliefs of your college professors, who base their beliefs on > : > : > each other's opinions, not one of whom could stand against me in a logical > : > : > debate on the subject. > : > : > : > : What evidence have you presented? > : : > The burden of proof is upon the claimant. I deny time dilation. You > : > prove it. > : This is very convienent for you, you can always say that it is my burden > : and never to prove it wrong. > Yes is and yes I can. That's one of the rules. I claim bright green flying elephants lay eggs. Would I be justified > in asking you to prove they do not? Of course not. Its my claim, it is up to me to prove it. > I cannot. Therefore I have no proof. You claim that time dilation exists. I ask you for proof and your > reply has been Experts say so. I happen to be an expert in bright green > flying elephant's nests, so why would you not believe me? Absurd, you say. Well, I say time dilation is absurd. Prove it. > I'll listen. And I'll challenge any and all assumptions you make. I will try to respond to this analogy, but I think it would be more useful if you could provide a topic that could actually be tested by experiment. You see, because my 1st question would be for you to show me and experiment that proves you correct. This is what you rightly asked me for. I provided you one. Then, in turn, I would provide and experiment that was designed to test your elephant theory that disproves your elephant theory. This we would move forward. It is all about well designed experiments with testable and reproducible results. I can concede that I have to justify that time dilation exists, but you have to justify that it doesn't exists. We each have a claim to prove. I have provided evidence for my claim that has been published in a journal and accepted by the community as being correct. You have *still* provided no experiment that justifies your claim. You have been asked in several threads to do this. In fact in the very start of this thread before I was involved it was said that you that no experiment has ever been done that doesn't support the claims or relativity. You would think that if you point is so clear, that you could easily provide, and yet you don't. Why is that? > : Why don't you point to experimental data by > : one of the many other scientists that don't believe in relativity. > : That would perhaps end this debate right now. > Michelson and Morley disproves Einstein's relativity. You'll find > that on your list of experiments that claim it as proof of relativity. > If it is a disproof you seek, learn how the experiment was conducted, > note that Michelson never once accepted Einstein's relativity, and > Michelson was an expert. Michelson also never accepted the results of his own experiment. (http://www.ncsu.edu/felder-public/kenny/papers/relativity.html) So, I don't know if that is a good frame of reference. Do you agree with MMX's conclusion that the speed of light is the same relativity which would imply to me that you accept MMX, but in other cases you say that the speed of light depends on the source (when you discuss the north pole and the equator.) > : > : You haven't yet proved that any of these experiments are wrong. > : : > You haven't so much as named one. > : See above. > : > : My professors are the sources for this. > : : > Name one experiment that you consider to be absolute proof of time > : > dilation. I'm not wasting my time on all of them, pick the one you like the > : > best, present the data and we'll analyse it. > : See above. > H&K? That's it? Even my wristwatch will not keep correct time when > buffeted or subjected to changing magnetic fields and temperatures. Fortunately they used more advanced clocks than a wristwatch. :) All of the sources of error you mention, like gravitational fields, were accounted for. (See above source.) > : > : That along with the data you have been presented and ignored is how I > : > : draw my conclusions. *NOT ON FAITH ALONE!* > : : > What data have you presented? You draw your conclusions on faith > : > alone. > : : > : > Well, I'm not here to change your beliefs. Only you can do that. Nor > : > : > am I seeking renown. If your mind is open, read on. If it be closed, > : > : > stop now. > : > : > : > : My mind is open. However, my mind works like any scientists mind does. > : : > Does it indeed... > : : > : I have a theory or belief, and then I test it with experiment. > : : > Hmm... and that is what you mean by an open mind, is it? Start with > : > a belief and then try to justify it. This is what you think science is > : > all about. I don't. > : Again, you are warping words. In science you develop a theory or belief > : (generally referred to as a hypothesis) and then you test it. I have a > : hypothesis right now that time dilation exists (I developed this after > : doing background research). I want to test it. I can not, because I can > : not move fast enough, so I went to find other tests. They have yielded > : results. I have analyzed them, and I now agree that time dilation exists. > Show your analysis then. I showed you my analysis before in my second post and it is still above in quotes where I compare the pro time dilation arguments and the con time dilation arguments. Every published, reputable experiment that I have found supports relativity. None disprove it. No one but you has found the math loop hole you present. I know that the speed of light is always the same. This is never addressed or explained in a different way by you, so there has to be an explanation of how the laws of physics can co-exist. Given all of the evidence in support of relativity, I am left with only one conclusion. > : If you prefer, I could have started with your hypothesis, Time Dilation > : doesn't exist. > Fine with me. It never did until Einstein read H.G. Wells' Time > Machine. Source on that? Honestly... > : Followed the same procedure and drawn the conclusion > : that my hypothesis is wrong. > It's your hypothesis. I say time dilation doesn't exist. And following the scientific method at some point you should test your hypothesis and publish your results. Where would I find that? > : Either way I arrive at the same end point. > : Although, in your world I wouldn't be allowed to reference the works of > : other physicists. > Of course you are allowed to reference them. I'm not preventing you, > but naming names isn't a proof. But your contention is that they are all liars. (This I feel is unsubstantiated.) So, I may as well not mention them at all. > : > : Lacking the ability to do that myself, I turn to others who have done > : > : experiments or analyzed this topic. I weight them an draw a conclusion. > : : > In other words, Nature works by popular opinion. I don't think that > : > Nature is a democrat. > : Again, you warp my words. I didn't say popular opinion. You could 2,000 > : flawed experiments that disprove it and 1 good one that proves it, and I > : would follow the good one. > Oh, ok. Well, in my experiment with bright green flying elephant's eggs, > I've been trying to hatch one. Unfortunately, like you not having enough > speed, I don't have the right incubator. When I do, I'll prove it to > you. Then, you have to maintain your elephant predictions as hypothesis, I suppose. I hope you get your incubators soon. Fortunately for me, there are people that actually have been able to do experiments that show that relativity, more specifically time dilation, exists. : You spend a lot of time on a lot of websites and newsgroups as far back > : > : as at least '02 trying to prove Einstein wrong. So, if your goal isn't > : > : to change my beliefs, what is your goal? > : : > To communicate with people that share a desire to understand Nature, > : > less so with those that prefer to take on faith what they want to > : > hear. > : Yet, those people who have done experiments with & about nature you call > : liars because you think Ptolemy was a liar. > Newton thought Ptolemy was a liar. He should know, he's an expert. You have taken my claim that we should listen to experts to a whole level that I never said. Perhaps I should clarify. I didn't say that we should listen to every expert every time. Sometimes they will be wrong or draw a bad conclusion. The beauty of the system is that even an expert can be wrong and still be an expert. (This would be analogous to Michael Jordan, back in the day at least, having a bad game.) The other great thing about the scientific community is that they police each he would publish it and if people couldn't reproduce his claims then we would know they were wrong. This is why people haven't accepted string theory yet, we can't test it. You will see where I discuss the Newton/Ptolemy issue below. : > The most recent accusations of forgery made against Ptolemy came from > : > : > Newton in [12]. He begins this book by stating clearly his views:- > : > : : > : > This is the story of a scientific crime. ... I mean a crime committed > : > : > by a scientist against fellow scientists and scholars, a betrayal of > : > : > the ethics and integrity of his profession that has forever deprived > : > : > mankind of fundamental information about an important area of > : > : > astronomy and history. > : > : : > : > Towards the end Newton, having claimed to prove every observation > : > : : > : > [Ptolemy] developed certain astronomical theories and discovered that > : > : > they were not consistent with observation. Instead of abandoning the > : > : > theories, he deliberately fabricated observations from the theories so > : > : > that he could claim that the observations prove the validity of his > : > : > theories. In every scientific or scholarly setting known, this > : > : > practice is called fraud, and it is a crime against science and > : > : > scholarship. > : > : http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html > : > : > : > : I do not know if this if founded or not, but it is irrelevant. Because > : > : Ptolemy did this, does that mean that every scientific theory out there > : > : is wrong because people lied? Do you believe any progressive theory > : > : besides Newton's Laws? > : : > I don't believe charlatans like Einstein, if that's what you want. > : What you fail to mention is that the next paragraph on this web-site > : clears Ptolemy of most of these charges. > Are you saying Ptolemy was right then? No, of course not. My point was that it is not as cut and dry as you claim that Ptolemy was a liar. He drew a conclusion from the best information he had at the time. Would you call everyone who thought light was instantaneous a liar because we now know it is wrong. Even Galileo couldn't prove it. No. People make the best conclusion they can at a given time. Additionally, my point was that there is another side to the issue that you deliberately excluded. This would be understandable if it was hard to find, but it is in the *NEXT* paragraph. > : Source: IBID > : Although the evidence produced by Brahe, Delambre, Newton and others > : certainly do show that Ptolemy's errors are not random, this last quote > : from [12] is, I [EFR] believe, a crime against Ptolemy (to use Newton's > : own words). The book [8] is written to study validity of these > : accusations and it is a work which I strongly believe gives the correct > : ... one has to assume that a substantial proportion of the > : Ptolemaic star catalogue is grounded on those Hipparchan observations > : which Hipparchus already used for the compilation of the second part of > : his Commentary on Aratus. Although it cannot be ruled out that > : coordinates resulting from genuine Ptolemaic observations are included > : in the catalogue, they could not amount to more than half the catalogue. > : ... the assimilation of Hipparchan observations can no longer be > : discussed under the aspect of plagiarism. Ptolemy, whose intention was > : to develop a comprehensive theory of celestial phenomena, had no access > : to the methods of data evaluation using arithmetical means with which > : modern astronomers can derive from a set of varying measurement results, > : the one representative value needed to test a hypothesis. For > : methodological reason, then, Ptolemy was forced to choose from a set of > : measurements the one value corresponding best to what he had to consider > : as the most reliable data. When an intuitive selection among the data > : was no longer possible ... Ptolemy had to consider those values as > : 'observed' which could be confirmed by theoretical predictions. > : I think that it really hurts you credibility to only present one side of > : argument when you know there is a counter just to make a point. > I quoted the source for you. Besides, we are not discussing my > credibility, but Einstein's. You credibility is paramount in this discussion because you are making a lot of claims without evidence. So, I have to trust that you are giving a fair and balanced view of the situation and that you are not introducing your own bias. I have made every effort to do this, I don't know if the same is true for you. Furthermore, as you have taken your right to anonymity in this discussion, you credibility is even more important because I know nothing else about you besides your other posts that I have read and what you say here. This is the only frame of reference I have about your character, knowledge, or believability. If you show that you are not presenting a balanced picture, I can only conclude that you are letting your bias control your thought and beliefs. In argumentation terms, I guess I am just questioning how much ethos you would have in this forum when you intentionally exclude counter arguments. > : It is dishonest and makes not want to believe anything else you say. > 2) Please refrain from personal attacks, I have made none on you. > What the is that then, calling me dishonest? > You are a two-faced jackass. There is no inconsistency between what I said and my criticism of your argument. I am not sure what part of that statement you are referring to as a personal attack? What you did is intellectually dishonest. I did not refer to you personally in any way. I only made reference to your actions in this discussion. > : Above I provided a link to experiments that claim to prove and disprove > : time dilation and analyzed both. You took a quote out of context and > : took on *FAITH* that Newton was right, when he was in the process of > : justifying his beleifs, not the best source in this case. While you are addressing contradictions, could you please address this one? > : > : > I accuse Einstein of the same crime, fraud. > : > : > I accuse his disciples of the fabrication of observation, including > : > : > Hafele and Keating, Algevar et. al. and many others. > : > : > I accuse those responsible for providing the FAQ's to this newsgroup > : > : > of bluster and nonsense, and the claim that the Michelson Morley Experiment > : > : > supports relativity is a lie. MMX disproves SR. > : > : > : > : It disproved it because it was a bad experiment. They were basing things > : > : on aether, which we know now doesn't exist. > : : > Ah... I see. A bad experiment. Since it succeeded in disproving > : > aether, I think you are wrong. I consider it a highly successful experiment. > : > What did you say above? > : > However, my mind works like any scientists mind does. I > : > have a theory or belief, and then I test it with experiment. > : It was good for proving that aether didn't exist. Bad for dealing with > : relativity because there assumptions about aether skewed there data. > : Please see the above reference for more details. (The Roberts site.) > MMX disproves SR as well as the non-existance of aether. I guess I must have misread something about the MMX. It seems that the experiment shows that light moves at the same speed at all times. How, exactly does this disprove relativity? > : > Michelson was a real scientist. He had a theory or belief, and he > : > tested it with experiment. You call that bad. Contradictory, are you not? > : > I don't think you even think, let alone think like a scientist. > : It is unfortunate that you have to result to personal attacks to try to > : win your point. :( > Win my point? All you've done is told me your beliefs. One one shred > of evidence, not one iota of data, and you claim to think like a > scientist. You ramble on about bad experiments... What do you WANT? I want a reputable, published experiment that was designed to disproved relativity, or time dilation, and succeeds in doing so. If you are correct this should be easy to do. > : The experiment was poorly constructed as a test of relativity, because > : of the assumptions made about aether. > Holy ! Michelson first conducted his experiment in 1881, > it supposed to be a test of SR? Then, how can it disprove it. Einstein took its conclusions and used them to write his papers. So, if the MMX is the basis of Einstein's claims, how can it disprove it? > : So, yes, the experiment disproved relativity, but it was not a well > designed experiment for that purpose, so I don't view that as a defeat > of relativity. Having re-read what I typed here, I would like to rescind it, as I do not know what I meant. It was late. Let me try again. I don't see how an experiment can disprove relativity if relativity was: a) developed long after the MMX b) based upon the results of the MMX > It is clear now that you know absolutely NOTHING about it. > How old are you, 18? Not sure why you would ask this when you have made it clear that personal background, in your opinion, is irrelevant. But, I am 24. > : If that is the only experimental basis upon which you are laying your claim, I fear you are > : grasping at straws. > Take your blind faith and shove it, little boy who claims to think > like a scientist! Perhaps you are unclear as to what blind faith is. That would be if I was told time dilation didn't exists and I just said, okay. This has not happened, so I would appreciate it if you would stop saying un-true things (please see my discussion of ethos above). > : > : > To discover why Einstein was a fraud takes a careful analysis of his > : > : > paper. > : > : : > : > For quotations following, reference: > : > : > http://www.fourmilab.ch/etexts/einstein/specrel/www/ > : > : > (On the Electrodynamics of Moving Bodies by Albert Einstein) > : > : : > : > 2) In agreement with experience we further assume the quantity > : > : > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in > : > : > empty space., > : > : > an admitted assumption that is quite worthless when there is any > : > : > relative motion between A and B, yet essential to the derivation of > : > : > the remainder of Einstein's nonsense. > : > : > : > : He says it is in agreement with experience, not just an assumption. You > : > : need to read all of the words. > : : > It is not in agreement with my experience, and you have none. > It is becoming ever clearer that you have none. What experience do you have that the speed of light in empty space is not a universal constant? : This is the first postulate of one his > : > : papers (i get the mixed up), that all laws of physics are the same in > : > : non-accelerating frames of reference. > : : > So you are mixed up, but take what you are told on faith alone, > : > and at the same time claim to think like a scientist. > : > Einstein's thought experiment is a bad experiment, does that help? > : Why is bad? Can you defend any of your claims with anything besides math? > I don't need to, sonny. Math is what it is all about. I will ask again, if you are correct, and Einstein's math is wrong, why has no one disproved him and why are you not on the cover of every magazine in the world? > : The only thing I was referring to about being mixed up is forgetting > : which is GR & which is SR. One the difference in this discussion is that > : I am willing to admit when I don't know something, where as you will > : continue to say things and make personal attacks when you don't know > : what to say. > All you've shown is your faith. Why waste my time? How does anything in this block of text show faith? > : > : > 3) The equation > : > : > ü[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = > : > : > tau(x',0,0,t+x'/(c-v)) , > : > : > the ü of which is derived from 2) above and is tantamount to saying > : > : > (1/3 + 2/3)/2 = 1/3. > : > : > : > : 3 - 7) I will be honest here that I don't know enough about math to > : > : discuss this, but if it is wrong, why are you the only person saying, > : > : and why do so many other physicists believe it? > : : > Who said or is your only line going to be that scientists just justify each > : > other to look smart. I don't buy that. > : > Yet you've fallen for them looking smart, hook, line and sinker, > : > haven't you? It was really easy to hoodwink you, you have no math. > : You have not hoodwinked anyone. > Einstein has. You didn't answer the question. I repeat, if it is wrong, why are you the only person saying, and why do so many other physicists believe it? : > And last but really first, > : > : > 1) light is always propagated in empty space with a definite velocity > : > : > c which is independent of the state of motion of the emitting body, > : > : > a totally unproven assumption without any evidence to support it, a > : > : > great deal of evidence to deny it. > : Still waiting for that evidence that says light depends on its source, > : since there is a great deal to deny it. > http://www.androc1es.pwp.blueyonder.co.uk/actual_data.htm I am not sure what this proves. You don't really explain anything here. What am I supposed to get out of these pictures? What was the experiment? > : > : Based upon your analysis then, the speed of light emitted from a source > : > : on the north pole, should be different than the speed of light emitted > : > : from a source at the equator, since the equator moves faster? > : : > Yes, at the top of the atmosphere. > : Why does being at the top of the atmosphere matter? > It is pointed out that the extinction theorem of dispersion theory, > for which for which the elementary derivation is given, shows that > an incident light wave is extinguished at the surface of a > dielectric. > J.G, Fox, Experimental Evidence for the Second Postulate of Special > Relativity. Am J. Phys. Vol 30 1962. > That's why. This paragraph taken out of context doesn't really mean anything to me. Could you explain what it is saying. What di-electric are you referring to? > : My point was that a light source on the equator would move faster than a light source on the > : North Pole, so given your beliefs, the light at the equator should move > : faster. I disagree with that claim. > : > The way I first proposed to test > : > it (do a bad experiment) was to shoot the moon with a laser from > : > the ISS. Since then I've revised my opinion. Using Roemer's method > : > and the Cassini probe currently in orbit about Saturn, we can > : > determine the speed of light as Cassini approaches and recedes from the Earth. > : > It will be carrying an adequately accurate clock. Radio transmissions > : > are acceptable. > : Didn't we basically accomplish your experiment a long time ago using the > : moons of Saturn (or which ever planet Roehmer and Cassini were watching, > : I don't recall right now)? Why would we repeat it? How do you know it > : won't be faked? > : Additionally, we already know the speed of light, why would we test it > : again? ? > : > : So, if we were to deduce the length of a meter (which is defined by the > : > : speed of light) a meter in Ecuador would be longer than a meter on the > : > : north pole? This is absurd. > : : > Yes, I'll agree with you that xi = (x-vt)/sqrt(1-v^2/c^2) > : > is absurd. Well done, you've just given up length contraction. > : > Now all you need to give up is time dilation tau = (t - vx/c^2) / sqrt (1-v^2/c^2) > : > which is equally as absurd. > : Stop twisting my words. Obviously I wouldn't have said that xi = > : (x-vt)/sqrt(1-v^2/c^2) is absurd considering the whole rest of my advocacy. > I am not twisting your words. You've clearly stated that if lengths > measured at the pole and Equator differ, that is absurd. I agree. Since that is a > prediction of relativity, then relativity is absurd. Let me try again to explain this in a way which you can not warp my words. In the real world length is define in terms of how far light travels in a given time. Specifically it is how far light travels in a vacuum in 1/299,792,458 of a second. This measurement to find the length of a meter should be the same in places. Which, if you look at relativity it is, because it shows us that a light source at the equator and at the north pole will emit light that is still traveling at c. In your world the light source at the north pole would emit a light beam at c (if we assume the north pole has a velocity of 0) and at the equator it will emit a light beam with speed c + 1000 mi/hr. Hence in your world the standard unit of a meter would not be the same. It would be a twitch shorter at the equator. In the real world experiments have shown that c is a constant, so the define length of a meter would be the same in all places. Since I have never actually picked up an object with mass in any part of this analysis, length contraction is irrelevant to this. > : My point was not that length contraction was absurd. My point, which I > : will explain again, is that the definition of a meter is in terms of the > : speed of light in a vacuum. > : According to your theory, using this method > : a meter on the North Pole would have to be shorter than a meter on the > : equator, because light moved slower on the North Pole. The fact that you > : would propose such a theory is absurd to me. > Ok then, that IS what Lorentz proposed and Einstein took up, based on MMX. > It IS absurd. I am still only referring the distance light travels, it has nothing to do with length contraction which refers to a massive object. : You still offer no evidence of an experiment that disagrees with the > : > : theory or relativity or any analysis of the many that do. > : : > If you want to present the data, we'll analyze it. > : I've shown my hand, will you ever show an experiment that supports you? > : I have been asking since the start. You haven't done that. > I don't have the burden of proof and you have NOT presented any data. In the end this is true: 1) There is evidence that time dilation and relativity are true. It has been published and verified by thousands of scientists and mathematicians over the past 100 years. 2) There has not been any published and verified experiments presented in opposition to time dilation and relativity. Until that happens we can not engage in a discussion to the contrary. 3) The speed of light is a universal constant. This was demonstrated by MMX, Cassini & Rohmer, and others. Einstein has put forth a logical, testable, and verified explanation. You seem to ignore this reality. If you don't then, at a minimum, you have not presented an alternative explanation for it. Have a good day, Peter === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals >> : Before I begin I would make three request, you can grant them or >> not. >> : >> : 1) Please set your e-mail program (if it allows you to) to wrap at >> 72 >> : characters as is the standard. It becomes increasingly difficult to >> : respond ot messages that are as off centered as yours. >> : 2) Please refrain from personal attacks, I have made none on you. >> : 3) Please maintain a civil tone. I enjoy spirited debate but will >> not >> : have a discussion if you can not remain civil. >> : > I see you ignored my requests. Yes, I have. You alleged I was dishonest, so who gives a damn? > (I will address you allegation of hypocrisy below.) That is > unfortunate, although not surprising. You will have a hard time > convincing others of your ideas if you can not present them in a > professional manner. You may be a brilliant scientists, I don't know, > but you may want to take some time to study argument and persuasion > before you try to convince the rest of the world of your ideas. Just > trying to be helpful. At my age, sonny, it's damnly my frank, I don't give a dear. I'm not seeking fame and fortune. I'm seeking honesty and reason, traits you seem to lack. >> message >> : > : > : > : > >> : > : > : No, I am a poor High School teacher. I didn't write that. I've never been a poor high school teacher. In fact, I've never been inside a poor High School. >> : > : > : >> : > : > : However, I found a few who have. >> : > : > : > relativity to save space.> Anyone can look up http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#5.%20Tw in%20paradox and find lists of experiments purporting to support relativity. I've analyzed quite a few of them. You are a great Haefele & Keating supporter, I believe. Henri Wilson (who can think) suggested a plane went around the Earth directly underneath a GPS satellite. Do you still think the clock it carries would disagree with the GPS clock? And don't give me any bull about it being possible, either. Concorde (scrapped after 30 years) could have managed it. The period of the GPS constellation is 12 hours. 25,000 miles around the Earth, 12 hours, 2083 mph. Mach 3... well, almost manage it. Maybe we'd need a Blackbird. >> : > : > : So, if I weigh the data: >> : > : > : >> : > : > : Pro Time Dilation >> : > : > : ----------------- >> : > : > : Every textbook I have ever read (alone this is not a good >> source I >> : > : > : know) >> : > : > : Experimental Data (Although not accepted by all, but all of >> my >> : > : > : college professors believe it.) >> : > : > : Theory that can logically explain it. >> : > : > : > : > Uh huh. Faith is a powerful ally, but logic is even more >> powerful. >> : > : >> : > : The Force of course if my strongest ally! :) >> : > : >> : > : Seriously, I don't see how this answers anything. Did you look >> at any of >> : > : these sites, or is your only line going to be that scientists >> just >> : > : justify each other to look smart. I don't buy that. >> : > : > There are many other scientists that deny Einstein's relativity. >> : > Did you look into any of those, or is your line going to be that >> : > Einstein was right by majority opinion? If so, you'd be joining >> : > the Church against Galileo and supporting Ptolemy in a different >> era. >> : >> : To be clear, I *NEVER* said anything even remotely like Einstein >> is >> : correct by majority opinion. Please do not misquote or skew my >> words. >> : You will not convince anyone that you are correct with those >> tactics. If >> : you want to engage in a serious discussion, then do so. >> : >> : You simply choose to ignore my reasoning which for accepting >> relativity >> : which is that it is based in fact (proven by experiments) and >> experts >> : have explained it and agree with it. I have not found in my >> searches >> : (they have not been exhaustive as I only have limited to spend on >> this) >> : nearly as many credible sources that oppose relativity. If you have >> any >> : credible sources, please share them, you have yet to do that. >> : >> : > : Do you think that none of these people are like you and >> question it. >> : > : > No. >> : > : > : You don't even attempt to controvert any of the evidence that >> these reports >> : > : present. >> : > : > Bull, I've been quite prolific and controversial of any >> so-called >> : > evidence that the disciples of Einstein have presented. Do you >> think >> : > you are the only person I've ever conversed with? >> : >> : Yet when given a list of several experiments you haven't explain to >> me >> : how they are wrong. You seem to believe that every experiment ever >> : fabricated was a lie. >> : >> : > : People went on an airplane and did this and you still don't >> : > : believe it. What do you want? >> : > : > Proof. >> : >> : Please define proof. This happened. If you want to say that every >> : experiment is a lie, I guess you can, but then I don't know what >> you >> : will ever believe, unless you have the means to recreate every >> : experiment ever performed. >> : >> : Let me be specific. In the experiment performed by Haefele and >> Keating, >> : (Haefele and Keating, Nature 227 (1970), pg 270 (Proposal); Science >> Vol. >> : 177 pg 166--170 (1972)) in which two clocks were flown in opposites >> : directions with a control clock on ground and the two clock came >> back to >> : the Earth with different times that were within the predictions of >> : relativity, what was wrong with that experiment? Why isn't that >> proof? >> It is a statement. Where is your data? >> It is not a proof without data, and did you subject the clocks to the >> same >> buffeting of turbulence and acceleration inherent with a commercial >> flight, the changing magnetic field of the Earth, log the altitude >> and duration at that altitude throughout the experiment? Without a >> control clock on the ground being subjected to the same environment, >> all you have is a couple of clocks that do not record the time >> correctly when tossed around. > Yes, in the experiment those factors are taken into account. Show me where it says the movement through the Earth's magnetic field was taken into account, I can't find it. > You can read the paper and many different analysis of the report if > you want to. In the experiment They flew atomic clocks on commercial > airliners around the world in both directions, and compared the time > elapsed on the airborne clocks with the time elapsed on an earthbound > clock (USNO). Their eastbound clock lost 59 ns on the USNO clock; > their westbound clock gained 273 ns; these agree with GR predictions > to well within their experimental resolution and uncertainties (which > total about 25 ns). > (http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#5.%20T w in%20paradox) Ever considered the uncertainties might be more that 25 ns? No, of course not. You take it on faith. How about reading this: 8. Experiments Which Apparently are NOT Consistent with SR/GR It is clear that most if not all of these experiments have difficulties which are unrelated to SR. In some cases the inconsistent experiment has been carefully repeated and been shown to be in error. Did anyone ever bother to repeat Haefele and Keating to see if it was in error? Well? Did they? No, of course not. It's whoopee! We get the result we like. Anyone else performs an experiment that disproves Einstein and it HAS to be in error. Yeah, right. > For analysis on your other concerns one place I found was > http://www.phys.unsw.edu.au/~jw/time.html. That's a neat web page with the animation. Shows time dilation nicely. Here's my version. http://www.androc1es.pwp.blueyonder.co.uk/gardner.htm It shows time compression just be changing the angle of the beam. > Sorry I didn't include the data before, I assumed you were familiar > with it. You still haven't produced any data. You've produced Baez's bias. I want first-hand data, not second-hand. >> : > : Nothing in the above explanation was based on faith. Perhaps >> you were >> : > : implying that me believing my professors is faith, but these >> people are >> : > : *EXPERTS* in physics, who else will I believe? >> : > : > Ptolemy, of course. He was an *EXPERT* for 1400 years. Forget >> : > Copernicus, he was just a dumb preacher, what did he know? As >> long >> : > as YOU *know* the Earth cannot move, Ptolemy has to be right. >> : > That is an example of faith, the kind of faith you display. >> : >> : Many scientists that were marveled at have been proven wrong. >> Aristotle, >> : Freud, Hook, the list goes on. Ptolemy, had a theory, this was >> based on >> : experiments and he was believed to be correct until someone was >> able to >> : prove him wrong. This required a telescope! So, of course others >> will be >> : able to prove him wrong once that is made. >> : >> : Please see my discussion of the Ptolemy Fraud issue below. >> : >> : Let me repeat: I do not base my beliefs on faith. I have look at >> : experiments and at theories and drawn a conclusion on the >> information I >> : have available to me. > : > : I don't know what the answer is, but Tom Roberts listed several >> : > : experiments that seem to justify SR and it is not logical to >> presume >> : > : them all to be fabricated. >> : > : > Bill Clinton did Not have sex with that woman. His credentials >> : > are President of the United States of America, so it is >> illogical >> : > to presume he fabricated that statement. >> : >> : I agree, one should presume that President would tell truth under >> oath. >> : I don't know if we can anymore, and I think that is a shame. Based >> upon >> : your theory, I can't believe a single law or experiment in physics >> : unless I try it. Is it safe to believe anyone? >> No, it isn't. > I disagree with this claim. Do as you please. > We have systems in the scientific community to prevent the things that > may have happened in the past. We have many, many journals into which > people can submit there findings and they can be verified by the > community. I'm part of the community. > This is what scientists do when they have a claim or a theory, they > publish it so others can critique, if it is wrong, we will know. Yep. And I know. That you are incompetent in mathematics isn't my fault. That I wasn't born in 1905 isn't my fault either. Einstein was a VERY competent huckster. So competent he deceived the community for 100 years. > This has been a tradition since the days of Newton when he and Hooke > had many of their famous exchanges. > The community monitors itself. If you have a belief you should get it > published so others can either agree or disagree with you. If you have > reference so that I may read it. Frankly, my damn, I don't give a dear. I've already published the below, and you are not competent to understand it. The Seven Deadly Sins of Special Relativity. For quotations following, reference: http://www.fourmilab.ch/etexts/einstein/specrel/www/ (On the Electrodynamics of Moving Bodies by Albert Einstein) 1) light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body, a totally unproven assumption without any evidence to support it. 2) In agreement with experience we further assume the quantity 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in empty space., an admitted assumption that is quite worthless when there is any relative motion between A and B, yet essential to the derivation of the remainder of Einstein's nonsense. 3) The equation ü[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = tau(x',0,0,t+x'/(c-v)) , the ü of which is derived from 2) above and is tantamount to saying (1/3 + 2/3)/2 = 1/3. 4) The missing 0' from that equation, since x' = x-vt, hence 0' = 0-vt, and the equation should be ü[tau(-vt,0,0,t)+tau(-vt,0,0,t+x'/(c-v)+x'/(c+v))] = tau(x',0,0,t+x'/(c-v)) at the very least. 5) The further assumption IF we place x' = x-vt ... without considering IF we place x' = x+vt, from which we derive (using Einstein's method) tau = (t+xv/c^2)/sqrt(1-v^2/c^2) xi = (x + vt)/sqrt(1-v^2/c^2) -Paul B. Andersen 6) The statements But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c-v... and It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light. For this case we obtain V = (c+w)/(1+w/c) = c. which are contradictory, the first being Galilean, the second being contrary to the vector addition of velocities, an axiom of a vector space. 7) The lack of a check to verify the theory is self-consistent by feeding the new PoR given in 6) into the equation given in 3) and finding a total failure. Check: (t1-t)/(t2-t)*[tau(-vt,0,0,t)+tau(-vt,0,0,t+x'/V+x'/V)] = tau(x',0,0,t+x'/V) where V = (c+v)/(1+v/c) as required by the redefined PoR. > : > : > : Anti Time Dilation >> : > : > : ------------------ >> : > : > : A few people whose reputations I don't know who just try to >> poke >> : > : > : holes that don't make a lot of sense in tested theories and >> have very few >> : > : > : reputable supporters. >> : > : > : >> : > : > : I guess that leads me to believe in time dilation. >> : > : > : >> : > : > : Sorry. :( >> : > : > : > : > Ah... I see. You judge before hearing all the evidence, >> basing your >> : > : > opinion on the beliefs of your college professors, who base >> their beliefs on >> : > : > each other's opinions, not one of whom could stand against me >> in a logical >> : > : > debate on the subject. >> : > : >> : > : What evidence have you presented? >> : > : > The burden of proof is upon the claimant. I deny time dilation. >> You >> : > prove it. >> : >> : This is very convienent for you, you can always say that it is my >> burden >> : and never to prove it wrong. >> Yes is and yes I can. That's one of the rules. I claim bright green >> flying elephants lay eggs. Would I be justified in asking you to >> prove they do not? Of course not. Its my claim, it is up to me to >> prove it. >> I cannot. Therefore I have no proof. You claim that time dilation >> exists. I ask you for proof and your reply has been Experts say so. >> I happen to be an expert in bright green >> flying elephant's nests, so why would you not believe me? Absurd, you >> say. Well, I say time dilation is absurd. Prove it. >> I'll listen. And I'll challenge any and all assumptions you make. > I will try to respond to this analogy, but I think it would be more > useful if you could provide a topic that could actually be tested by > experiment. You see, because my 1st question would be for you to show > me and experiment that proves you correct. This is what you rightly > asked me for. I provided you one. No you haven't. You produced hearsay, written by Baez. > Then, in turn, I would provide and experiment that was designed to > test your elephant theory that disproves your elephant theory. This we > would move forward. It is all about well designed experiments with > testable and reproducible results. Well designed? Throw a clock on a plane and fly it around? I call that uncontrolled. > I can concede that I have to justify that time dilation exists, but > you have to justify that it doesn't exists. No I do not, but I have anyway. See the Seven Deadly Sins above. If you want more, then I'll oblige. One orbit of a planet around the sun is a measure of time. In the case of the Earth, this defines one year exactly. Feel free to subdivide the year into nanoseconds if you wish. Mercury, Venus, Mars, Jupiter, Saturn, Neptune, Uranus and Pluto will all circumnavigate the Sun in their own period of time, and we can express any pair of them as a ratio. For Mercury, the ratio will be (to the nearest integer) 365/88. So Mercury will make about 4 orbits of the sun in a year, or one every three months. The ratio between Mercury and Venus is about 225/88. Mercury will make 2.5 orbits of the Sun for each orbit of Venus. We can use the ratios to predict where Ganymede, Io, Europa and Callisto will be relative to Jupiter on April 30th, 2005 by making use of universal time, which is given by the table of ratios between any two oscillators Mercury Venus Earth * Mercury 1 225/88 365/88 x/88 Venus 88/225 1 365/225 x/225 Earth 88/365 225/365 1 x/365 * 88/x 225/x 365/x 1 If * changes according to some fanciful function of speed, ALL its entries in the table change, but the others do not. Feel free to measure time by any of the instruments in my list. When a body makes less than or more than one orbit per orbit, so that it's own entry in the table differs from exactly 1, let me know. You could change the orbit of any one of them, of course, as will happen to GPS satellites as their orbits are perturbed by the moon. Just change all its row and column entries or it will show the right time in the wrong place and wont work properly. The best way to do this is upload its position and time on a regular basis automatically, which is the role of the ground stations. > We each have a claim to prove. Done mine, it disproves yours. > I have provided evidence for my claim that has been published in a > journal and accepted by the community as being correct. You've provided hearsay. Being part of the community, I disagree with your conclusion. Therefore you are engaging in the majority verdict. > You have *still* provided no experiment that justifies your claim. You > have been asked in several threads to do this. I already said I can't prove bright green flying elephants lay eggs. The pity of it all is, you, as a High School teacher, will tell young minds lies that you believe, and they'll believe you. Instead of encouraging them to think for themselves, you'll indoctrinate them with nonsense, they'll get older and do the same to the next generation, and so it continues. How to break the cycle? > In fact in the very start of this thread before I was involved it was > said that you that no experiment has ever been done that doesn't > support the claims or relativity. You would think that if you point is > so clear, that you could easily provide, and yet you don't. Why is > that? No experiment has ever been done that DOES prove the claims of relativity. NOT ONE. The reason is that relativity is nonsense. ALL experiments show experimenter bias. >> : Why don't you point to experimental data by >> : one of the many other scientists that don't believe in >> relativity. >> : That would perhaps end this debate right now. >> Michelson and Morley disproves Einstein's relativity. You'll find >> that on your list of experiments that claim it as proof of >> relativity. >> If it is a disproof you seek, learn how the experiment was >> conducted, >> note that Michelson never once accepted Einstein's relativity, and >> Michelson was an expert. > Michelson also never accepted the results of his own experiment. Nonsense. > (http://www.ncsu.edu/felder-public/kenny/papers/relativity.html) So, I > don't know if that is a good frame of reference. A rather poor web page, but at least it explains Galilean relativity. > Do you agree with MMX's conclusion that the speed of light is the same > at all times? No. > which would imply to me that you accept MMX, Of course I accept the null result of MMX. I could perform MMX on a plane. I can talk to the flight attendant in a normal voice, I can hear her in a normal voice, the speed of sound is added to the speed of the plane. If I perform MMX on a plane, the speed of light is added to the speed of the plane. That is only possible if the speed of light is NOT observer dependent. but in other > cases you say that the speed of light depends on the source (when you > discuss the north pole and the equator.) In the vacuum of space, there is no medium for light's speed to be relative to. There are only three choices, 1) SoL is medium dependent. 2) SoL is observer dependent. 3) SoL is source dependent. Aetherialism selects the first. It is correct in the presence of a medium. Relativity selects the second. We all carry a personal aether round with us. Emission theory selects the third. It is Galilean relativity, and is supported by observation. The difficulty is in interpreting the observation. Here it is. http://www.britastro.org/vss/ >> : > : You haven't yet proved that any of these experiments are wrong. >> : > : > You haven't so much as named one. >> : >> : See above. >> : >> : > : My professors are the sources for this. >> : > : > Name one experiment that you consider to be absolute proof of >> time >> : > dilation. I'm not wasting my time on all of them, pick the one >> you like the >> : > best, present the data and we'll analyse it. >> : >> : See above. >> H&K? That's it? Even my wristwatch will not keep correct time when >> buffeted or subjected to changing magnetic fields and temperatures. > Fortunately they used more advanced clocks than a wristwatch. :) All > of the sources of error you mention, like gravitational fields, were > accounted for. (See above source.) You want to nitpick nanoseconds. I want to nitpick Hulse and Taylor, Nobel prize winners. http://www.androc1es.pwp.blueyonder.co.uk/SekerinTime.htm >> : > : That along with the data you have been presented and ignored is >> how I >> : > : draw my conclusions. *NOT ON FAITH ALONE!* >> : > : > What data have you presented? You draw your conclusions on faith >> : > alone. >> : > : > : > Well, I'm not here to change your beliefs. Only you can do >> that. Nor >> : > : > am I seeking renown. If your mind is open, read on. If it be >> closed, >> : > : > stop now. >> : > : >> : > : My mind is open. However, my mind works like any scientists >> mind does. >> : > : > Does it indeed... >> : > : > : I have a theory or belief, and then I test it with experiment. >> : > : > Hmm... and that is what you mean by an open mind, is it? Start >> with >> : > a belief and then try to justify it. This is what you think >> science is >> : > all about. I don't. >> : >> : Again, you are warping words. In science you develop a theory or >> belief >> : (generally referred to as a hypothesis) and then you test it. I >> have a >> : hypothesis right now that time dilation exists (I developed this >> after >> : doing background research). I want to test it. I can not, because I >> can >> : not move fast enough, so I went to find other tests. They have >> yielded >> : results. I have analyzed them, and I now agree that time dilation >> exists. >> Show your analysis then. > I showed you my analysis before in my second post and it is still > above in quotes where I compare the pro time dilation arguments and > the con time dilation arguments. You demonstrated your belief. You have yet to provide first-hand data. > Every published, reputable experiment that I have found supports > relativity. None disprove it. No one but you has found the math loop > hole you present. Well, I KNOW that. Someone has to be first! It happens to be me. > I know that the speed of light is always the same. Of course you do. You've been indoctrinated. That's why you are a poor High School teacher and I'm a scientist. You don't know how to question. Faith is powerful. > This is never addressed or explained in a different way by you, so > there has to be an explanation of how the laws of physics can > co-exist. Given all of the evidence in support of relativity, I am > left with only one conclusion. Right. Go on believing whatever you wish. I say you are wrong, but you are not about to believe me anyway. >> : If you prefer, I could have started with your hypothesis, Time >> Dilation >> : doesn't exist. >> Fine with me. It never did until Einstein read H.G. Wells' Time >> Machine. > Source on that? Honestly... Yes, honestly. Imagination is more important than knowledge. -- Albert Einstein If we knew what it was we were doing, it would not be called research, would it? - Albert Einstein Common sense is the collection of prejudices acquired by age eighteen. - Albert Einstein Albert read Time Machine as a teenager. It was his greatest influence. I read it myself at fourteen, and was fascinated by it. I was expecting to see my older self suddenly appear and tell me how to build one. That is better known today as the Grandfather Paradox, but I had my own version. I was a kid then, today I know better. >> : Followed the same procedure and drawn the conclusion >> : that my hypothesis is wrong. >> It's your hypothesis. I say time dilation doesn't exist. > And following the scientific method at some point you should test your > hypothesis and publish your results. Where would I find that? http://www.androc1es.pwp.blueyonder.co.uk/ Other results are found in the data of many astronomical texts, and include cepheids, flare stars, eclipsing variables and recurrent novae, the light curves of which I can reproduce. >> : Either way I arrive at the same end point. >> : Although, in your world I wouldn't be allowed to reference the >> works of >> : other physicists. >> Of course you are allowed to reference them. I'm not preventing you, >> but naming names isn't a proof. > But your contention is that they are all liars. (This I feel is > unsubstantiated.) So, I may as well not mention them at all. >> : > : Lacking the ability to do that myself, I turn to others who >> have done >> : > : experiments or analyzed this topic. I weight them an draw a >> conclusion. >> : > : > In other words, Nature works by popular opinion. I don't think >> that >> : > Nature is a democrat. >> : >> : Again, you warp my words. I didn't say popular opinion. You could >> 2,000 >> : flawed experiments that disprove it and 1 good one that proves it, >> and I >> : would follow the good one. >> Oh, ok. Well, in my experiment with bright green flying elephant's >> eggs, >> I've been trying to hatch one. Unfortunately, like you not having >> enough >> speed, I don't have the right incubator. When I do, I'll prove it >> to you. > Then, you have to maintain your elephant predictions as hypothesis, I > suppose. I hope you get your incubators soon. > Fortunately for me, there are people that actually have been able to > do experiments that show that relativity, more specifically time > dilation, exists. > : > : You spend a lot of time on a lot of websites and newsgroups as >> far back >> : > : as at least '02 trying to prove Einstein wrong. So, if your >> goal isn't >> : > : to change my beliefs, what is your goal? >> : > : > To communicate with people that share a desire to understand >> Nature, >> : > less so with those that prefer to take on faith what they want to >> : > hear. >> : >> : Yet, those people who have done experiments with & about nature you >> call >> : liars because you think Ptolemy was a liar. >> Newton thought Ptolemy was a liar. He should know, he's an expert. > You have taken my claim that we should listen to experts to a whole > level that I never said. Perhaps I should clarify. I didn't say that > we should listen to every expert every time. Sometimes they will be > wrong or draw a bad conclusion. The beauty of the system is that even > an expert can be wrong and still be an expert. (This would be > analogous to Michael Jordan, back in the day at least, having a bad > game.) The other great thing about the scientific community is that > they police each other. If Stephen Hawking were to publish a bad > reproduce his claims then we would know they were wrong. This is why > people haven't accepted string theory yet, we can't test it. > You will see where I discuss the Newton/Ptolemy issue below. > : > : > The most recent accusations of forgery made against Ptolemy >> came from >> : > : > Newton in [12]. He begins this book by stating clearly his >> views:- >> : > : > : > : > This is the story of a scientific crime. ... I mean a crime >> committed >> : > : > by a scientist against fellow scientists and scholars, a >> betrayal of >> : > : > the ethics and integrity of his profession that has forever >> deprived >> : > : > mankind of fundamental information about an important area of >> : > : > astronomy and history. >> : > : > : > : > Towards the end Newton, having claimed to prove every >> observation >> [12]:- >> : > : > : > : > [Ptolemy] developed certain astronomical theories and >> discovered that >> : > : > they were not consistent with observation. Instead of >> abandoning the >> : > : > theories, he deliberately fabricated observations from the >> theories so >> : > : > that he could claim that the observations prove the validity >> of his >> : > : > theories. In every scientific or scholarly setting known, >> this >> : > : > practice is called fraud, and it is a crime against science >> and >> : > : > scholarship. >> : > : > >> http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html >> : > : >> : > : I do not know if this if founded or not, but it is irrelevant. >> Because >> : > : Ptolemy did this, does that mean that every scientific theory >> out there >> : > : is wrong because people lied? Do you believe any progressive >> theory >> : > : besides Newton's Laws? >> : > : > I don't believe charlatans like Einstein, if that's what you >> want. >> : >> : What you fail to mention is that the next paragraph on this >> web-site >> : clears Ptolemy of most of these charges. >> Are you saying Ptolemy was right then? > No, of course not. > My point was that it is not as cut and dry as you claim that Ptolemy > was a liar. He drew a conclusion from the best information he had at > the time. Would you call everyone who thought light was instantaneous > a liar because we now know it is wrong. Even Galileo couldn't prove > it. No. People make the best conclusion they can at a given time. > Additionally, my point was that there is another side to the issue > that you deliberately excluded. This would be understandable if it was > hard to find, but it is in the *NEXT* paragraph. >> : Source: IBID >> : Although the evidence produced by Brahe, Delambre, Newton and >> others >> : certainly do show that Ptolemy's errors are not random, this last >> quote >> : from [12] is, I [EFR] believe, a crime against Ptolemy (to use >> Newton's >> : own words). The book [8] is written to study validity of these >> : accusations and it is a work which I strongly believe gives the >> correct >> : >> : ... one has to assume that a substantial proportion of the >> : Ptolemaic star catalogue is grounded on those Hipparchan >> observations >> : which Hipparchus already used for the compilation of the second >> part of >> : his Commentary on Aratus. Although it cannot be ruled out that >> : coordinates resulting from genuine Ptolemaic observations are >> included >> : in the catalogue, they could not amount to more than half the >> catalogue. >> : >> : ... the assimilation of Hipparchan observations can no longer >> be >> : discussed under the aspect of plagiarism. Ptolemy, whose intention >> was >> : to develop a comprehensive theory of celestial phenomena, had no >> access >> : to the methods of data evaluation using arithmetical means with >> which >> : modern astronomers can derive from a set of varying measurement >> results, >> : the one representative value needed to test a hypothesis. For >> : methodological reason, then, Ptolemy was forced to choose from a >> set of >> : measurements the one value corresponding best to what he had to >> consider >> : as the most reliable data. When an intuitive selection among the >> data >> : was no longer possible ... Ptolemy had to consider those values as >> : 'observed' which could be confirmed by theoretical predictions. >> : >> : I think that it really hurts you credibility to only present one >> side of >> : argument when you know there is a counter just to make a point. >> I quoted the source for you. Besides, we are not discussing my >> credibility, but Einstein's. > You credibility is paramount in this discussion because you are making > a lot of claims without evidence. So, I have to trust that you are > giving a fair and balanced view of the situation and that you are not > introducing your own bias. I have made every effort to do this, I > don't know if the same is true for you. > Furthermore, as you have taken your right to anonymity in this > discussion, you credibility is even more important because I know > nothing else about you besides your other posts that I have read and > what you say here. This is the only frame of reference I have about > your character, knowledge, or believability. If you show that you are > not presenting a balanced picture, I can only conclude that you are > letting your bias control your thought and beliefs. > In argumentation terms, I guess I am just questioning how much ethos > you would have in this forum when you intentionally exclude counter > arguments. >> : It is dishonest and makes not want to believe anything else you >> say. >> 2) Please refrain from personal attacks, I have made none on you. >> What the is that then, calling me dishonest? >> You are a two-faced jackass. > There is no inconsistency between what I said and my criticism of your > argument. > I am not sure what part of that statement you are referring to as a > personal attack? > What you did is intellectually dishonest. I did not refer to you > personally in any way. I only made reference to your actions in this > discussion. >> : Above I provided a link to experiments that claim to prove and >> disprove >> : time dilation and analyzed both. You took a quote out of context >> and >> : took on *FAITH* that Newton was right, when he was in the process >> of >> : justifying his beleifs, not the best source in this case. > While you are addressing contradictions, could you please address this > one? >> : > : > I accuse Einstein of the same crime, fraud. >> : > : > I accuse his disciples of the fabrication of observation, >> including >> : > : > Hafele and Keating, Algevar et. al. and many others. >> : > : > I accuse those responsible for providing the FAQ's to this >> newsgroup >> : > : > of bluster and nonsense, and the claim that the Michelson >> Morley Experiment >> : > : > supports relativity is a lie. MMX disproves SR. >> : > : >> : > : It disproved it because it was a bad experiment. They were >> basing things >> : > : on aether, which we know now doesn't exist. >> : > : > Ah... I see. A bad experiment. Since it succeeded in disproving >> : > aether, I think you are wrong. I consider it a highly successful >> experiment. >> : > What did you say above? >> : > However, my mind works like any scientists mind does. I >> : > have a theory or belief, and then I test it with experiment. >> : >> : It was good for proving that aether didn't exist. Bad for dealing >> with >> : relativity because there assumptions about aether skewed there >> data. >> : Please see the above reference for more details. (The Roberts >> site.) >> MMX disproves SR as well as the non-existance of aether. > I guess I must have misread something about the MMX. It seems that the > experiment shows that light moves at the same speed at all times. How, > exactly does this disprove relativity? >> : > Michelson was a real scientist. He had a theory or belief, and >> he >> : > tested it with experiment. You call that bad. Contradictory, are >> you not? >> : > I don't think you even think, let alone think like a scientist. >> : >> : It is unfortunate that you have to result to personal attacks to >> try to >> : win your point. :( >> : >> Win my point? All you've done is told me your beliefs. One one shred >> of evidence, not one iota of data, and you claim to think like a >> scientist. You ramble on about bad experiments... What do you WANT? > I want a reputable, published experiment that was designed to > disproved relativity, or time dilation, and succeeds in doing so. If > you are correct this should be easy to do. >> : The experiment was poorly constructed as a test of relativity, >> because >> : of the assumptions made about aether. >> Holy ! Michelson first conducted his experiment in 1881, >> it supposed to be a test of SR? > Then, how can it disprove it. Einstein took its conclusions and used > them to write his papers. So, if the MMX is the basis of Einstein's > claims, how can it disprove it? >> : So, yes, the experiment disproved relativity, but it was not a well >> designed experiment for that purpose, so I don't view that as a >> defeat of relativity. > Having re-read what I typed here, I would like to rescind it, as I do > not know what I meant. It was late. Let me try again. > I don't see how an experiment can disprove relativity if relativity > was: > a) developed long after the MMX > b) based upon the results of the MMX >> It is clear now that you know absolutely NOTHING about it. >> How old are you, 18? > Not sure why you would ask this when you have made it clear that > personal background, in your opinion, is irrelevant. But, I am 24. >> : If that is the only experimental basis upon which you are laying >> your claim, I fear you are >> : grasping at straws. >> Take your blind faith and shove it, little boy who claims to think >> like a scientist! > Perhaps you are unclear as to what blind faith is. That would be if I > was told time dilation didn't exists and I just said, okay. This has > not happened, so I would appreciate it if you would stop saying > un-true things (please see my discussion of ethos above). >> : > : > To discover why Einstein was a fraud takes a careful analysis >> of his >> : > : > paper. >> : > : > : > : > For quotations following, reference: >> : > : > http://www.fourmilab.ch/etexts/einstein/specrel/www/ >> : > : > (On the Electrodynamics of Moving Bodies by Albert >> Einstein) >> : > : > : > : > 2) In agreement with experience we further assume the >> quantity >> : > : > 2AB/(t'A-tA) = c to be a universal constant- the velocity of >> light in >> : > : > empty space., >> : > : > an admitted assumption that is quite worthless when there is >> any >> : > : > relative motion between A and B, yet essential to the >> derivation of >> : > : > the remainder of Einstein's nonsense. >> : > : >> : > : He says it is in agreement with experience, not just an >> assumption. You >> : > : need to read all of the words. >> : > : > It is not in agreement with my experience, and you have none. >> : >> It is becoming ever clearer that you have none. > What experience do you have that the speed of light in empty space is > not a universal constant? > : > : This is the first postulate of one his >> : > : papers (i get the mixed up), that all laws of physics are the >> same in >> : > : non-accelerating frames of reference. >> : > : > So you are mixed up, but take what you are told on faith alone, >> : > and at the same time claim to think like a scientist. >> : > Einstein's thought experiment is a bad experiment, does that >> help? >> : >> : Why is bad? Can you defend any of your claims with anything besides >> math? >> I don't need to, sonny. Math is what it is all about. > I will ask again, if you are correct, and Einstein's math is wrong, > why has no one disproved him and why are you not on the cover of every > magazine in the world? I told you. I'm not seeking fame. >> : The only thing I was referring to about being mixed up is >> forgetting >> : which is GR & which is SR. One the difference in this discussion is >> that >> : I am willing to admit when I don't know something, where as you >> will >> : continue to say things and make personal attacks when you don't >> know >> : what to say. >> All you've shown is your faith. Why waste my time? > How does anything in this block of text show faith? >> : > : > 3) The equation >> : > : > ü[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = >> : > : > tau(x',0,0,t+x'/(c-v)) , >> : > : > the ü of which is derived from 2) above and is tantamount to >> saying >> : > : > (1/3 + 2/3)/2 = 1/3. >> : > : >> : > : 3 - 7) I will be honest here that I don't know enough about >> math to >> : > : discuss this, but if it is wrong, why are you the only person >> saying, >> : > : and why do so many other physicists believe it? >> : > : > Who said or is your only line going to be that scientists just >> justify each >> : > other to look smart. I don't buy that. >> : > Yet you've fallen for them looking smart, hook, line and sinker, >> : > haven't you? It was really easy to hoodwink you, you have no >> math. >> : >> : You have not hoodwinked anyone. >> Einstein has. > You didn't answer the question. I repeat, if it is wrong, why are you > the only person saying, and why do so many other physicists believe > it? Faith, of course. > : > : > And last but really first, >> : > : > 1) light is always propagated in empty space with a definite >> velocity >> : > : > c which is independent of the state of motion of the emitting >> body, >> : > : > a totally unproven assumption without any evidence to support >> it, a >> : > : > great deal of evidence to deny it. >> : >> : Still waiting for that evidence that says light depends on its >> source, >> : since there is a great deal to deny it. >> http://www.androc1es.pwp.blueyonder.co.uk/actual_data.htm > I am not sure what this proves. You don't really explain anything > here. What am I supposed to get out of these pictures? What was the > experiment? >> : > : Based upon your analysis then, the speed of light emitted from >> a source >> : > : on the north pole, should be different than the speed of light >> emitted >> : > : from a source at the equator, since the equator moves faster? >> : > : > Yes, at the top of the atmosphere. >> : >> : Why does being at the top of the atmosphere matter? >> It is pointed out that the extinction theorem of dispersion theory, >> for which for which the elementary derivation is given, shows that >> an incident light wave is extinguished at the surface of a >> dielectric. >> J.G, Fox, Experimental Evidence for the Second Postulate of Special >> Relativity. Am J. Phys. Vol 30 1962. >> That's why. > This paragraph taken out of context doesn't really mean anything to > me. Could you explain what it is saying. What di-electric are you > referring to? Sigh... look it up. >> : My point was that a light source on the equator would move faster >> than a light source on the >> : North Pole, so given your beliefs, the light at the equator should >> move >> : faster. I disagree with that claim. >> : >> : > The way I first proposed to test >> : > it (do a bad experiment) was to shoot the moon with a laser from >> : > the ISS. Since then I've revised my opinion. Using Roemer's >> method >> : > and the Cassini probe currently in orbit about Saturn, we can >> : > determine the speed of light as Cassini approaches and recedes >> from the Earth. >> : > It will be carrying an adequately accurate clock. Radio >> transmissions >> : > are acceptable. >> : >> : Didn't we basically accomplish your experiment a long time ago >> using the >> : moons of Saturn (or which ever planet Roehmer and Cassini were >> watching, >> : I don't recall right now)? Why would we repeat it? How do you know >> it >> : won't be faked? >> : >> : Additionally, we already know the speed of light, why would we test >> it >> : again? >> : > : So, if we were to deduce the length of a meter (which is >> defined by the >> : > : speed of light) a meter in Ecuador would be longer than a meter >> on the >> : > : north pole? This is absurd. >> : > : > Yes, I'll agree with you that xi = (x-vt)/sqrt(1-v^2/c^2) >> : > is absurd. Well done, you've just given up length contraction. >> : > Now all you need to give up is time dilation tau = (t - vx/c^2) / >> sqrt (1-v^2/c^2) >> : > which is equally as absurd. >> : >> : Stop twisting my words. Obviously I wouldn't have said that xi = >> : (x-vt)/sqrt(1-v^2/c^2) is absurd considering the whole rest of my >> advocacy. >> I am not twisting your words. You've clearly stated that if lengths >> measured at the pole and Equator differ, that is absurd. I agree. >> Since that is a >> prediction of relativity, then relativity is absurd. > Let me try again to explain this in a way which you can not warp my > words. > In the real world length is define in terms of how far light travels > in a given time. Specifically it is how far light travels in a vacuum > in 1/299,792,458 of a second. This measurement to find the length of a > meter should be the same in places. Which, if you look at relativity > it is, because it shows us that a light source at the equator and at > the north pole will emit light that is still traveling at c. > In your world the light source at the north pole would emit a light > beam at c (if we assume the north pole has a velocity of 0) and at the > equator it will emit a light beam with speed c + 1000 mi/hr. Hence in > your world the standard unit of a meter would not be the same. It > would be a twitch shorter at the equator. > In the real world experiments have shown that c is a constant, so the > define length of a meter would be the same in all places. It is constant with respect to the source. > Since I have never actually picked up an object with mass in any part > of this analysis, length contraction is irrelevant to this. >> : My point was not that length contraction was absurd. My point, >> which I >> : will explain again, is that the definition of a meter is in terms >> of the >> : speed of light in a vacuum. >> : According to your theory, using this method >> : a meter on the North Pole would have to be shorter than a meter on >> the >> : equator, because light moved slower on the North Pole. The fact >> that you >> : would propose such a theory is absurd to me. >> Ok then, that IS what Lorentz proposed and Einstein took up, based on >> MMX. >> It IS absurd. > I am still only referring the distance light travels, it has nothing > to do with length contraction which refers to a massive object. > : > : You still offer no evidence of an experiment that disagrees >> with the >> : > : theory or relativity or any analysis of the many that do. >> : > : > If you want to present the data, we'll analyze it. >> : >> : I've shown my hand, will you ever show an experiment that supports >> you? >> : I have been asking since the start. You haven't done that. >> I don't have the burden of proof and you have NOT presented any data. > In the end this is true: > 1) There is evidence that time dilation and relativity are true. It > has been published and verified by thousands of scientists and > mathematicians over the past 100 years. > 2) There has not been any published and verified experiments presented > in opposition to time dilation and relativity. Until that happens we > can not engage in a discussion to the contrary. > 3) The speed of light is a universal constant. This was demonstrated > by MMX, Cassini & Rohmer, and others. Einstein has put forth a > logical, testable, and verified explanation. You seem to ignore this > reality. If you don't then, at a minimum, you have not presented an > alternative explanation for it. > Have a good day, > Peter There are more things in heaven and earth, Peter, Than are dreamt of in your philosophy. William Shakespeare, Hamlet. Androcles. === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals > : > : > : Given three clocks, A, B, and C, where A is moving > inertially > and > : > : > : clocks B and C move in a circle in opposite directions (as > : > measured > : > by > : > : > : A, specifically where the rate at which clock B ticks > relative > : > to A > : > is > : > : > : independant of where in the orbit clock B is, and where the > rate > at > : > : > : which clock C ticks relative to A is independant of where in > the > orbit > : > : > : clock C is) around A, SR predicts that clock B SEES clock A > tick > : > at > : > a > : > : > : faster rate than clock B, and that clock C SEES clock A tick > at > a > : > : > : faster rate than clock C, and that clock A SEES both clock B > and > clock > : > : > : C tick slower than clock A. SR further predicts that clock > B > SEES > : > : > : clock C tick first slower and then faster in such a way as > that > after > : > : > : a full period it has seen as many ticks from C as it has > seen > from > : > : > : clock B. SR further predicts that clock C SEES clock B tick > first > : > : > : slower and then faster in such a way as that after a full > period > it > : > : > : has seen as many ticks from B as it has seen from clock C. > : > : > : > : > : > : How can we reconcile these valid SR predictions with > Androcles's > : > : > : assertions on > http://www.androc1es.pwp.blueyonder.co.uk/Theorem.htm > : > : > : that SR actually predicts some jibberish when Androcles uses > it, > even > : > : > : though this only happens to Androcles, and everyone (else?) > who > : > : > : understands the modern SR theory agrees with my claim about > SR's > : > : > : predictions which contradict Androcles' claims? So far the > only > : > : > : explanations I could come up with were: > : > : > : > : > : > : (1) Androcles is dishonest, > : > : > : (2) Androcles doesn't understand SR, or > : > : > : (3) Androcles is very bad at math and/or logic. > : > : > : > : > : > : But maybe there is another explanation, anyone have any > ideas? > : > : > : > : > : > : P.S. If someone can explain to me how the modern SR theory > is > : > : > : internally inconsistent in a rigorous way (i.e. such that I > can > use > : > : > : the proof to show that set theory is false and thus collect > the > : > : > : US$7,000,000.00 bounty on that result), then I will gladly > bestow > : > : > : US$500,000 upon that person, provided that (1) It is legal > for > me to > : > : > : give that person money and (2) That that person has not > first > : > : > : collected that bounty themselves or in some other manner > interfered > : > : > : with my collection of said bounty. > : > : : > : : > : > Try this. > : > : > (4) J.E. is dishonest AND > : > : > (5) J.E. doesn't understand SR AND > : > : > (6) J.E. wouldn't understand math/ logic if it bit him in the > arse. > : > : : > : > : Given three clocks, A, B, and C, where A is moving > inertially > and > : > : > : clocks B and C move in a circle > : > : : > : > B and C are not moving inertially. Inconsistency #1. > : > : > : > : I agree that clock B and clock C don't move inertially, and I > never > : > : said they were, rest of raving about Androcles responsing to > voices > probably > : > : read it and respond later, I don't think I can actually sink to > : > : Androcles level and standards, it's just too hard.) > : : > Chew on this, you dumb jerk. > : the > : > points A and B of K there are stationary clocks which, viewed in > the > : > stationary system, are synchronous; and if the clock at A is moved > : > with the velocity v along the line AB to B, then on its arrival at > B > : > the two clocks no longer synchronize, but the clock moved from A > to B > : > lags behind the other which has remained at B by > : : > (1/2) t.v^2/c^2 > : : > (up to magnitudes of fourth and higher order), t being the time > : > occupied in the journey from A to B. > : : > It is at once apparent that this result still holds good if the > clock > : > moves from A to B in any polygonal line, and also when the points > A > : > and B coincide. > : > If we assume that the result proved for a polygonal line is also > valid > : > for a continuously curved line, we arrive at this result: If one > of > : > two synchronous clocks at A is moved in a closed curve with > constant > : > velocity until it returns to A, the journey lasting t seconds, > then by > : > the clock which has remained at rest the travelled clock on its > : > arrival at A will be > : : > (1/2) t.v^2/c^2 > : > second slow. Thence we conclude that a balance-clock at the > equator > : > must go more slowly, by a very small amount, than a precisely > similar > : > clock situated at one of the poles under otherwise identical > : > conditions. - Albert Einstein > : Wow A.E. agrees that the clock that moves in a curved path moves > : slower than the one that goes in a straight path, amazing, that > means > : HE AGRESS WITH ME. > Ok, you dumb cluck, obviously you are such a jerk you cannot > even spell agrees in capital letters. > Actually A.E does NOT agree with a moron like you that claims > SR further predicts that clock C SEES clock B tick first > slower and then faster in such a way as that after a full period it > has seen as many ticks from B as it has seen from clock C. > Why don't you just off, imbecile? > This newsgroup is for beating up idiots and you are such an > easy target it's getting boring as shooting fish in a barrel. > Tell you what... I'll help you. > *plonk* > Androcles. > PS... you are so stupid I doubt you can understand the meaning > of *plonk*. It means I can no longer read your noise. You are > kill filed. Unfortunately Androcles does not know what killfiled means any more than he understands mathematics or relativity theory. By the way, moron, there is no such thing as 0'. Zero is zero. Speaking of which, your mathematical ability is zero in all frames of reference. === Subject: Re: final proof Einstein 1905 is crap. >> One more obvious proof that Einstein's derivation in 1905 is just as much >> crap as his 1916/1962 derivation. >SR is an excercise in the algebra of Lorentz-Poincare transformations as >applied to electromagnetic fields. If it is mathematically inconsistent >then so is the theory real variables and linear algebra. That takes down >ALL of physics. You haven't been paying attention, Bob. He has issues with quantum mechanics and thermodynamics, too. What's left? -- I'm giving you the chance to look fate in those pretty eyes of hers and say, 'Step off, bitch. This is my party and you're not invited.' -- Chris Shugart, _Testosterone Magazine_ === Subject: Re: final proof Einstein 1905 is crap. >> One more obvious proof that Einstein's derivation in 1905 is just as much >> crap as his 1916/1962 derivation. >SR is an excercise in the algebra of Lorentz-Poincare transformations as >applied to electromagnetic fields. If it is mathematically inconsistent >then so is the theory real variables and linear algebra. That takes down >ALL of physics. > You haven't been paying attention, Bob. He has issues with quantum > mechanics and thermodynamics, too. What's left? I do? Would you mind terrible if I ask for a quote for either of those? eleaticus === Subject: Re: final proof Einstein 1905 is crap. > You haven't been paying attention, Bob. He has issues with quantum > mechanics and thermodynamics, too. What's left? Aristotle's philosophy? Bob Kolker === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals > By the way, moron, there is no such thing as 0'. Zero is zero. Idiot. A zero is no different than a letter that describes a particular coordinate. Both have a different values in any second system (unless a rotation only). It would not be beyond the intellectual capacity of a moral person to accept the obvious usage. eleaticus === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals : : : : > By the way, moron, there is no such thing as 0'. Zero is zero. : : Idiot. A zero is no different than a letter that describes a particular : coordinate. Both have a different values in any second system (unless a : rotation only). : : It would not be beyond the intellectual capacity of a moral person to accept : the obvious usage. : : eleaticus : Being on my list of idiots not worth bothering with, Tit Pot would never have succeed in reaching me without your reply. I would like to point out that there is no such thing as x' . x is x (particularly when x takes the value 0). Androcles. === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals > By the way, moron, there is no such thing as 0'. Zero is zero. > Idiot. A zero is no different than a letter that describes a particular > coordinate. Both have a different values in any second system (unless a > rotation only). > It would not be beyond the intellectual capacity of a moral person to accept > the obvious usage. Obvious usage like in O' = 0 + vt? http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/LooksGood.html By the way, Oren Coward Webster, I never got a reply to No balls? Dirk Vdm === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals > By the way, moron, there is no such thing as 0'. Zero is zero. > Idiot. A zero is no different than a letter that describes a particular > coordinate. Both have a different values in any second system (unless a > rotation only). > It would not be beyond the intellectual capacity of a moral person to accept > the obvious usage. > Obvious usage like in > O' = 0 + vt? Only by typo would someone say O' = 0 + vt. 0' = 0 +- vt, yes. O' = O +- vt, yes. eleaticus === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals : : > By the way, moron, there is no such thing as 0'. Zero is zero. : : > Idiot. A zero is no different than a letter that describes a particular : > coordinate. Both have a different values in any second system (unless a : > rotation only). : : > It would not be beyond the intellectual capacity of a moral person to : accept : > the obvious usage. : : > Obvious usage like in : > O' = 0 + vt? : : Only by typo would someone say O' = 0 + vt. : : 0' = 0 +- vt, yes. O' = O +- vt, yes. : : eleaticus Let's face it, it is beyond the intellectual capacity of moortel to understand how the sheer genius of Einstein hoodwinked the community for 100 years. Recall that the light leaves A and reflects at B in 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in empty space. Hence ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) because it is a coordinate system, and that the velocity of light in our theory plays the part, physically, of an infinitely great velocity. There will always be morontels. Androcles === Subject: final proof Einstein 1905 is crap. > Let's face it, it is beyond the intellectual capacity of moortel to > understand > how the sheer genius of Einstein hoodwinked the community for 100 > years. > Recall that the light leaves A and reflects at B in > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in > empty > space. > Hence > ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) > because it is a coordinate system, and > that the velocity of light in our theory plays the part, physically, > of an infinitely great velocity. x'/(c+v)). Hmmm. Never thought of working that out before. Better show my work so far. AE's AB means 'the distance from A to B' in his usage, and that is x' later The total distance involved in his 2AB/(t'A-tA) = c is thus 2x'. AE's (t'A-tA) is the elapsed time for the light round trip, which is, per his final tau, (x' /(c-v)+ x'/(c+v)). Thus, 2AB/(t'A-tA) = c becomes 2x'/(x' /(c-v)+ x'/(c+v)) = c. c = 2x'/(x' /(c-v)+ x'/(c+v)) c(x' /(c-v)+ x'/(c+v)) = 2x' cx'(c+v)/(c-v) + cx' = 2x'(c+v) cx'(c+v) + cx'(c-v) = 2x'(c+v)(c-v) cx'(2c) = 2x'(c^2 - v^2) c^2 = (c^2 - v^2) v=0, the only value for which his derivation could possibly be valid (except for t=0). One more obvious proof that Einstein's derivation in 1905 is just as much crap as his 1916/1962 derivation. eleaticus === Subject: Re: final proof Einstein 1905 is crap. > Let's face it, it is beyond the intellectual capacity of moortel to > understand > how the sheer genius of Einstein hoodwinked the community for 100 > years. > Recall that the light leaves A and reflects at B in > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in > empty > space. > Hence > ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) > because it is a coordinate system, and > that the velocity of light in our theory plays the part, physically, > of an infinitely great velocity. > x'/(c+v)). > Hmmm. Never thought of working that out before. > Better show my work so far. Ha, his work... let's have a look. > AE's AB means 'the distance from A to B' in his usage, and that is x' later > The total distance involved in his 2AB/(t'A-tA) = c is thus 2x'. HAHAHAHAHA, Oren Coward Webster http://users.pandora.be/vdmoortel/dirk/Stuff/WebsterAtSchool.gif imitating John Androcles Parker http://users.pandora.be/vdmoortel/dirk/Stuff/AndrolesAtSchool.gif Nice :-) By the way, coward, how come you never replied to No balls? Dirk Vdm === Subject: Re: final proof Einstein 1905 is crap. : : > Let's face it, it is beyond the intellectual capacity of moortel to : > understand : > how the sheer genius of Einstein hoodwinked the community for 100 : > years. : > Recall that the light leaves A and reflects at B in : > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in : > empty : > space. : > Hence : > ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) : > because it is a coordinate system, and : > that the velocity of light in our theory plays the part, physically, : > of an infinitely great velocity. out. : /(c-v)+ : x'/(c+v)). : : Hmmm. Never thought of working that out before. : : Better show my work so far. : : AE's AB means 'the distance from A to B' in his usage, and that is x' later : : The total distance involved in his 2AB/(t'A-tA) = c is thus 2x'. : : AE's (t'A-tA) is the elapsed time for the light round trip, which is, per : his final tau, (x' /(c-v)+ x'/(c+v)). : : Thus, 2AB/(t'A-tA) = c becomes 2x'/(x' /(c-v)+ x'/(c+v)) = c. : : c = 2x'/(x' /(c-v)+ x'/(c+v)) : : c(x' /(c-v)+ x'/(c+v)) = 2x' : : cx'(c+v)/(c-v) + cx' = 2x'(c+v) : : cx'(c+v) + cx'(c-v) = 2x'(c+v)(c-v) : : cx'(2c) = 2x'(c^2 - v^2) : : c^2 = (c^2 - v^2) : : v=0, the only value for which his derivation could possibly be valid (except : for t=0). Correct. Right on. Bravo! The ray leaves the coincident origins of K and k, travels at an infinite speed to x', reflects and returns at an infinite speed to the coincident origins of K and k. They have not moved in the time it takes the light to make the round trip and are still coincident. Speed = distance/time. Time = 0. We have a divide-by-zero. v is actually undefined. However, we also have a zero distance from A to B because c is actually finite. The one case where divide-by-zero is permitted is 0/0. Oddly enough, and whilst not knowing why, the idiot moortel is correct. There is no 0'. x' = x-vt so 0' = 0-vt and vt = 0. x' = x, 0' - 0. : : One more obvious proof that Einstein's derivation in 1905 is just as much : crap as his 1916/1962 derivation. Err... he died in 1956... I recall reading about it in a newspaper at the time. The 1905 paper demonstrates that Einstein was a huckster. It is a careful plot drawn up on paper that has hoodwinked millions for nearly 100 years. Many, such as yourself, have realized that something is wrong with it, but I think I alone have pinpointed exactly how the trick was performed. I'm not boasting, I'm stating a fact. By the careful use of primes and the expedient of using real numbers, he was able to make the prime vanish right before anyone's eyes, and on paper at that. Like all of us, he would have scribbled his first thoughts on the back of an envelope, then written a draft, then polished his work before presenting it as a paper. The introduction would have been written after the equations, then such things as doppler shift would have been added to round out the paper for completeness. The way to make the prime vanish was to use the number 0 instead of an algebraic expression like this. ü[tau(A,t)+tau(A',t+(B-A) /(c-v)+ (A'-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) while leaving it in place at x'. The ü is plucked out of nowhere. It should be (t1-t)/(t2-t) or just plain t1/t2. We need A' to show A' = A -vt. Making A = (0,0,0) disguises the missing prime exceeding well. You can stare at ü[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = tau(x',0,0,t+x'/(c-v)) for a long time and never see it, because it is NOT there! Neither is the return distance, A'-B. That is still x'. To me, it is the cleverest piece of misdirection imaginable. Sheer genius. If he had been caught, he could have always pleaded stupidity. He knew what he was doing, he was not stupid and he succeeded in gaining exactly what he sought. Fame and fortune. To be fair, I think in later life he wanted to be caught, it shows in his writings. The secret to creativity is knowing how to hide your sources. = I hid that prime really well. Weakness of attitude becomes weakness of character. = I'm really tough and can fool anyone. The only thing that interferes with my learning is my education. = I really don't know about mathematics (snigger). God does not care about our mathematical difficulties. He integrates empirically. = God made me do it. If A is a success in life, then A equals x plus y plus z. Work is x; y is play; and z is keeping your mouth shut. = I'm not telling how I hid that prime, suckers. Not everything that counts can be counted, and not everything that can be counted counts. (Sign hanging in Einstein's office at Princeton) = Never forget the prime really counts, but if you cant see it, you can't count it. Only I know it is there. Keep mouth shut. A man's ethical behavior should be based effectually on sympathy, education, and social ties; no religious basis is necessary. Man would indeed be in a poor way if he had to be restrained by fear of punishment and hope of reward after death. = I was unethical, but I have a tiger by the tail. I'd be slaughtered if I let it go. I'll leave the analysis to the soft-hearted sociologists. My duty is to prosecute him for the crime he committed, and that's just what I'm doing here. Welcome aboard. Poke around inside his head by reading his paper in a new light, now that you know he was a fraud. Androcles. : eleaticus : : : : : === Subject: Re: final proof Einstein 1905 is crap. You are unqualified. My comments stand. > One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. Einstein died in 1955. > eleaticus === Subject: Re: final proof Einstein 1905 is crap. > One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. > Einstein died in 1955. And so you slavishly replicate the idiocy of other cretins postings here. Were you knowledgeable you would know that one Einstein-authored book was published in 1962, but his intro was dated 1916. What is not so funny is that your are so corrupt a jackass that the juxtaposition of 1916/1962 didn't push any intellectual curiousity buttons. What a 1905 jackass you are. eleaticus === Subject: Re: final proof Einstein 1905 is crap. > One more obvious proof that Einstein's derivation in 1905 is just as > much > crap as his 1916/1962 derivation. > Einstein died in 1955. Mark your snips, coward. > And so you slavishly replicate the idiocy of other cretins postings here. My idiocy is outshined by your idiocy. > Were you knowledgeable you would know that one Einstein-authored book was > published in 1962, but his intro was dated 1916. Like most other cranks, you have this unwavering belief that Einstein and his theories are inseperable. I do not need Einstein's books to learn his theories. > What is not so funny is that your are so corrupt a jackass that the > juxtaposition of 1916/1962 didn't push any intellectual curiousity buttons. > What a 1905 jackass you are. Have you managed to learn limits yet? > eleaticus === Subject: Re: final proof Einstein 1905 is crap. > One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. SR is an excercise in the algebra of Lorentz-Poincare transformations as applied to electromagnetic fields. If it is mathematically inconsistent then so is the theory real variables and linear algebra. That takes down ALL of physics. Produce a single vetted, reproduced experiment that refutes a single prediction made by either SR or GTR. Please provide citations in a refereed journal. Bob Kolker === Subject: SR consistency is crap. > One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. > SR is an excercise in the algebra of Lorentz-Poincare transformations as > applied to electromagnetic fields. If it is mathematically inconsistent > then so is the theory real variables and linear algebra. That takes down > ALL of physics. Reliance on the concept of mathematical consistency is nonsense. Just consider the inconsistency as to which of two identically manufactured and maintained objects is shorter/longer when in relative motion. You're shorter! No! YOU are! Wrong, asshole, I can clearly see you are much shorter than me! Idiot! Are you insane?! etc. No inconsistency there? LOL. You assholes! . Just consider the inconsistency of deriving spatial coorinate transformations and not being able to actually use them with EM quantities like E,B, D, H, etc. Or having 'cook the books' when you actually do try to apply them to an equation. The restort to 'mathematical consistency' reminds me of Chritian resort to the Books of Moses to exclude their violence: immaterial, irrelevant, and incompetent. eleaticus === Subject: Re: SR consistency is crap. > Reliance on the concept of mathematical consistency is nonsense. Consistency does not prove empirical truth. However mathematical inconsistency is absolute proof of nonsense and balderdash. We know SR is mathematically consistent being an excercise in the algebra of Lorentz-Poincare transforms. I also asked you to produce a single experiment that has ever falsified a prediction of SR. Please only vetted reproducable experiments need apply. Can you produce such an experiment or not? If you can, then do and if not, shut the up. Bob Kolker === Subject: Re: final proof Einstein 1905 is crap. > Let's face it, it is beyond the intellectual capacity of moortel to > understand > how the sheer genius of Einstein hoodwinked the community for 100 > years. > Recall that the light leaves A and reflects at B in > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in > empty > space. > Hence > ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) > because it is a coordinate system, and > that the velocity of light in our theory plays the part, physically, > of an infinitely great velocity. > x'/(c+v)). > Hmmm. Never thought of working that out before. > Better show my work so far. > AE's AB means 'the distance from A to B' in his usage, and that is x' later > The total distance involved in his 2AB/(t'A-tA) = c is thus 2x'. > AE's (t'A-tA) is the elapsed time for the light round trip, which is, per > his final tau, (x' /(c-v)+ x'/(c+v)). > Thus, 2AB/(t'A-tA) = c becomes 2x'/(x' /(c-v)+ x'/(c+v)) = c. > c = 2x'/(x' /(c-v)+ x'/(c+v)) > c(x' /(c-v)+ x'/(c+v)) = 2x' > cx'(c+v)/(c-v) + cx' = 2x'(c+v) > cx'(c+v) + cx'(c-v) = 2x'(c+v)(c-v) > cx'(2c) = 2x'(c^2 - v^2) > c^2 = (c^2 - v^2) > v=0, the only value for which his derivation could possibly be valid (except > for t=0). > One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. Hmmm. Well, I have thought of an 'out'. > eleaticus === Subject: Re: final proof Einstein 1905 is crap. > Let's face it, it is beyond the intellectual capacity of moortel to > understand > how the sheer genius of Einstein hoodwinked the community for 100 > years. > One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. > Hmmm. Well, I have thought of an 'out'. > eleaticus What can you say relatively of the following experimental discovery in the domain of a gravitation of the solar system?: .82THE GRAVITATION SYMMETRY INSIDE THE SOLAR SYSTEMé I maintain, that my discovery is direct experimental refutation of GTR. Aleksandr === Subject: Re: final proof Einstein 1905 is crap. > I maintain, that my discovery is direct experimental refutation of > GTR. Has your discovery been vetted or verified by other worker's in the field. If so, you may get a trip to Stockholm sooner than you think. Bob Kolker === Subject: Re: final proof Einstein 1905 is crap. > Let's face it, it is beyond the intellectual capacity of moortel to > understand > how the sheer genius of Einstein hoodwinked the community for 100 > years. > Recall that the light leaves A and reflects at B in > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in > empty > space. > Hence > ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) > because it is a coordinate system, and > that the velocity of light in our theory plays the part, physically, > of an infinitely great velocity. > x'/(c+v)). > Hmmm. Never thought of working that out before. > Better show my work so far. > AE's AB means 'the distance from A to B' in his usage, and that is x' > later > The total distance involved in his 2AB/(t'A-tA) = c is thus 2x'. > AE's (t'A-tA) is the elapsed time for the light round trip, which is, per > his final tau, (x' /(c-v)+ x'/(c+v)). > Thus, 2AB/(t'A-tA) = c becomes 2x'/(x' /(c-v)+ x'/(c+v)) = c. > c = 2x'/(x' /(c-v)+ x'/(c+v)) > c(x' /(c-v)+ x'/(c+v)) = 2x' > cx'(c+v)/(c-v) + cx' = 2x'(c+v) > cx'(c+v) + cx'(c-v) = 2x'(c+v)(c-v) > cx'(2c) = 2x'(c^2 - v^2) > c^2 = (c^2 - v^2) > v=0, the only value for which his derivation could possibly be valid > (except > for t=0). > One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. > Hmmm. Well, I have thought of an 'out'. > eleaticus Yes, you've just proven that the speed of light is the same between two inertial frames of reference. Well done! === Subject: Re: final proof Einstein 1905 is crap. Let's face it, it is beyond the intellectual capacity of moortel to > understand > how the sheer genius of Einstein hoodwinked the community for 100 > years. > Recall that the light leaves A and reflects at B in > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light in > empty > space. > Hence > ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = tau(B,t+(B-A)/(c-v)) > because it is a coordinate system, and > that the velocity of light in our theory plays the part, physically, > of an infinitely great velocity. out. > /(c-v)+ > x'/(c+v)). Hmmm. Never thought of working that out before. Better show my work so far. AE's AB means 'the distance from A to B' in his usage, and that is x' > later The total distance involved in his 2AB/(t'A-tA) = c is thus 2x'. AE's (t'A-tA) is the elapsed time for the light round trip, which is, per > his final tau, (x' /(c-v)+ x'/(c+v)). Thus, 2AB/(t'A-tA) = c becomes 2x'/(x' /(c-v)+ x'/(c+v)) = c. c = 2x'/(x' /(c-v)+ x'/(c+v)) c(x' /(c-v)+ x'/(c+v)) = 2x' cx'(c+v)/(c-v) + cx' = 2x'(c+v) cx'(c+v) + cx'(c-v) = 2x'(c+v)(c-v) cx'(2c) = 2x'(c^2 - v^2) c^2 = (c^2 - v^2) v=0, the only value for which his derivation could possibly be valid > (except > for t=0). One more obvious proof that Einstein's derivation in 1905 is just as much > crap as his 1916/1962 derivation. > Hmmm. Well, I have thought of an 'out'. Wrong!? > Yes, you've just proven that the speed of light is the same between > two inertial frames of reference. Well done! Wrong! (As usual.) What I showed is that AE's stationary system expression for elapsed time of the light transit is valid only if v=0. eleaticus === Subject: Re: final proof Einstein 1905 is crap. Let's face it, it is beyond the intellectual capacity of moortel to > understand > how the sheer genius of Einstein hoodwinked the community for 100 > years. > Recall that the light leaves A and reflects at B in > 2AB/(t'A-tA) = c to be a universal constant- the velocity of light > in > empty > space. > Hence > ü[tau(A,t)+tau(A,t+(B-A) /(c-v)+ (A-B) /(c+v))] = > tau(B,t+(B-A)/(c-v)) > because it is a coordinate system, and > that the velocity of light in our theory plays the part, > physically, > of an infinitely great velocity. out. /(c-v)+ > x'/(c+v)). Hmmm. Never thought of working that out before. Better show my work so far. AE's AB means 'the distance from A to B' in his usage, and that is x' > later The total distance involved in his 2AB/(t'A-tA) = c is thus 2x'. AE's (t'A-tA) is the elapsed time for the light round trip, which is, > per > his final tau, (x' /(c-v)+ x'/(c+v)). Thus, 2AB/(t'A-tA) = c becomes 2x'/(x' /(c-v)+ x'/(c+v)) = c. c = 2x'/(x' /(c-v)+ x'/(c+v)) c(x' /(c-v)+ x'/(c+v)) = 2x' cx'(c+v)/(c-v) + cx' = 2x'(c+v) cx'(c+v) + cx'(c-v) = 2x'(c+v)(c-v) cx'(2c) = 2x'(c^2 - v^2) c^2 = (c^2 - v^2) v=0, the only value for which his derivation could possibly be valid > (except > for t=0). One more obvious proof that Einstein's derivation in 1905 is just as > much > crap as his 1916/1962 derivation. Hmmm. Well, I have thought of an 'out'. > Wrong!? You're contradicting yourself, doofus. > Yes, you've just proven that the speed of light is the same between > two inertial frames of reference. Well done! > Wrong! (As usual.) > What I showed is that AE's stationary system expression for elapsed time of > the light transit is valid only if v=0. Nope. You've demonstrated that there is no elapsed time for photons and therefore all inertial frames measure the same speed of light. === Subject: Re: Uncle assAl: (SR) Lorentz t', x' = Intervals > By the way, moron, there is no such thing as 0'. Zero is zero. > Idiot. A zero is no different than a letter that describes a particular > coordinate. Both have a different values in any second system (unless a > rotation only). > It would not be beyond the intellectual capacity of a moral person to > accept > the obvious usage. > Obvious usage like in > O' = 0 + vt? > Only by typo would someone say O' = 0 + vt. > 0' = 0 +- vt, yes. O' = O +- vt, yes. Moron, I meant 0' = 0 + vt. By the way, Oren Coward Webster, I never got a reply to No balls? Dirk Vdm === Subject: Re: pi + e one of pi + e and pi e is irrational. >>Isn't it _known_ that _pi_ is transcendental? >Yes, it is. >Why do you ask? Probably because you used a space for a multiplication operator, which can be confusing. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: pi + e By the way, http://mathworld.wolfram.com/GelfondsTheorem.html states that Gelfond's Theorem establishes the fact the e*pi is irrational. I don't see it. Anyone care to elaborate? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: pi + e > By the way, http://mathworld.wolfram.com/GelfondsTheorem.html states > that Gelfond's Theorem establishes the fact the e*pi is irrational. I > don't see it. Anyone care to elaborate? One of the many errors on Math World. This one has been pointed out in this newsgroup before: 3ud5mnk02%40sneakemail.com&rnum=2&prev=/groups%3Fq%3Dirrational%2Bpi%2Bg 02%25253ud5mnk02%2540sneakemail.com%26rnum%3D2> -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: pi + e >By the way, http://mathworld.wolfram.com/GelfondsTheorem.html states >that Gelfond's Theorem establishes the fact the e*pi is irrational. I >don't see it. Anyone care to elaborate? It's a misprint for e^pi, to which Gelfond-Schneider applies since (e^pi)^i = -1. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: pi + e >>By the way, http://mathworld.wolfram.com/GelfondsTheorem.html states >>that Gelfond's Theorem establishes the fact the e*pi is irrational. I >>don't see it. Anyone care to elaborate? >> >It's a misprint for e^pi, to which Gelfond-Schneider >applies since (e^pi)^i = -1. Actually, it mentions both e^pi and e pi. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: pi + e > By the way, http://mathworld.wolfram.com/GelfondsTheorem.html states > that Gelfond's Theorem establishes the fact the e*pi is irrational. I > don't see it. Anyone care to elaborate? >> It's a misprint for e^pi, to which Gelfond-Schneider >> applies since (e^pi)^i = -1. > Actually, it mentions both e^pi and e pi. We know the source of Eric Weisstein's error. It's from Hardy and Wright page 39. http://mathworld.wolfram.com/IrrationalNumber.html See: === Subject: Re: pi + e >> >By the way, http://mathworld.wolfram.com/GelfondsTheorem.html states >that Gelfond's Theorem establishes the fact the e*pi is irrational. I >don't see it. Anyone care to elaborate? >>It's a misprint for e^pi, to which Gelfond-Schneider >>applies since (e^pi)^i = -1. >Actually, it mentions both e^pi and e pi. OK, it's a misprint in or misreading of whatever source Weisstein used. In any case, I'm pretty sure it's not known that e pi is irrational. It's certainly not an obvious consequence of Gelfond-Schneider. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: pi + e >> Proofs of irrationality (for numbers that are not known to be >> algebraic) tend to be very difficult. >> On the other hand, it is known (and not difficult) that at least >> one of pi + e and pi e is irrational. >Isn't it _known_ that _pi_ is transcendental? > Yes, it is. > Why do you ask? Because I misunderstood what pi e meant -- I hadn't realized until reading another answer that you meant the _product_ of the two quantities, which is usually designated as pi x e or pi * e or pi . e or ... Norm === Subject: Re: pi + e > Because I misunderstood what pi e meant -- I hadn't realized until > reading another answer that you meant the _product_ of the two quantities, > which is > usually designated as pi x e or pi * e or pi . e or ... No, it's usually designated as pi e. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: pi + e >> Because I misunderstood what pi e meant -- I hadn't realized until >> reading another answer that you meant the _product_ of the two quantities, >> which is >> usually designated as pi x e or pi * e or pi . e or ... >No, it's usually designated as pi e. In the mail I get that means something else... === Subject: B,uy SECRET Appl;e pi e rec1pes NOW!!! to en'larg,e you're ... === Subject: Re: pi + e > Because I misunderstood what pi e meant -- I hadn't realized until > reading another answer that you meant the _product_ of the two quantities, > which is > usually designated as pi x e or pi * e or pi . e or ... > No, it's usually designated as pi e. Not in most ASCII USENET posts. Norm === Subject: Re: pi + e >> Because I misunderstood what pi e meant -- I hadn't realized until >> reading another answer that you meant the _product_ of the two > quantities, >> which is >> usually designated as pi x e or pi * e or pi . e or ... >> No, it's usually designated as pi e. > Not in most ASCII USENET posts. Really? How come I've never seen pi x e, pi * e or pi . e save in this thread? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: pi + e Because I misunderstood what pi e meant -- I hadn't realized until > reading another answer that you meant the _product_ of the two >> quantities, > which is > usually designated as pi x e or pi * e or pi . e or ... > No, it's usually designated as pi e. >> Not in most ASCII USENET posts. >Really? How come I've never seen pi x e, pi * e or pi . e >save in this thread? Because it's usually written e*pi? Rich === Subject: Re: pi + e >> Because I misunderstood what pi e meant -- I hadn't realized until >> reading another answer that you meant the _product_ of the two > quantities, >> which is >> usually designated as pi x e or pi * e or pi . e or ... >> No, it's usually designated as pi e. >> Not in most ASCII USENET posts. >>Really? How come I've never seen pi x e, pi * e or pi . e >>save in this thread? > Because it's usually written e*pi? Only in this post? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: pi + e Given the above, what's the difference between irrational and >transcendal numbers? I read, only a few days ago, e is transcendal, pi >is transcendal, but it's unknown if pi + e is. Are these two concepts >related? The word is transcendental, not transcendal. A number x is > transcendental if there is no polynomial of degree at least 1 > with rational coefficients that has x as a root. All transcendental > numbers are irrational, but not all irrational numbers are > transcendental. Also, 100% of real numbers are transcendental. > Almost all != all. Isn't that a bit like trying to argue that 0.999... isn't quite 1 :) === Subject: Re: pi + e > Isn't that a bit like trying to argue that 0.999... isn't quite 1 :) No. A set of measure zero need not be empty. Bob Kolker === Subject: simple probabilities ( but confusing) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBC4tM731729; I would apprciate your input with this problem; it seems like perfectly correct ways of looking at it lead to different answers. Any way, we are choosing three points in a circle at random, and we join every pair of points to form a triangle.We want to know the probability that the triangle includes the center of the circle. My answer was that the triangle will include the center iff no 2 of the pairs of points are more than pi radians appart. So, we choose the first point x_1. We draw a diameter thru x_1 , and then we have to choose x_2 x_3. There are 4 ways of choosing x_2 and x_3 so that they are either all in the same side of the diameter or not, and in exactly two of these ways the three points are of the same side,out of 4 possibilities so the answer is 1/2. My friend says it is 1/4. Who's right? Now we must === Subject: Re: simple probabilities ( but confusing) > Any way, we are choosing three points in a circle > at random, and we join every pair of points to form > a triangle.We want to know the probability that the > triangle includes the center of the circle. Let the circle have circumference 1. Choose one of the points randomly, and let its position be the origin for measurements around the circle, taken in a fixed direction. Then the distances of the other two points from the chosen point are independent Uniform(0,1) random variables, say X1 and X2. The events min(X1,X2) > 1/2, max(X1,X2)-min(X1,X2) > 1/2, and 1 - max(X1,X2) > 1/2 are mutually exclusive. Pr(the triangle does not include the center of the circle) = Pr( min(X1,X2) > 1/2 OR max(X1,X2)-min(X1,X2) > 1/2 OR 1-max(X1,X2) > 1/2 ). The random variables min(X1,X2), max(X1,X2)-min(X1,X2), and 1-max(X1,X2) are identically distributed. Pr(the triangle does not include the center of the circle) = 3 Pr(max(X1,X2) < 1/2) = 3/4. Pr(the triangle includes the center of the circle) = 1/4. === Subject: Re: simple probabilities ( but confusing) >>Any way, we are choosing three points in a circle >>at random, and we join every pair of points to form >>a triangle.We want to know the probability that the >>triangle includes the center of the circle. >> As I indicated in an earlier post, here is an easier way than Ray's: Let T be the length of the smaller arc between the first two points. Note that the center lies in the triangle's interior iff the third point lies in the opposite arc (determined by the diameters from the first two points). Let A be this event. I assume a unit radius. PA = EP(A|T) = E[T/(2 pi)] = ET / (2 pi) = (pi / 2) / (2 pi) = 1/4. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: simple probabilities ( but confusing) http://mygate.mailgate.org/mynews/sci/sci.math/35664ae5716193d30039c6e8cbb03 a 8c.48257%40mygate.mailgate.org > So, we choose the first point x_1. We draw a > diameter thru x_1 , and then we have to choose x_2 > x_3. There are 4 ways of choosing x_2 and x_3 so that > they are either all in the same side of the diameter or not, > and in exactly two of these ways the three points are > of the same side,out of 4 possibilities so the answer is > 1/2. My friend says it is 1/4. Who's right? But having the 2nd and 3rd points fall on opposite sides of the diameter established by the first isn't sufficient to guarantee that the triangle they establish includes the circle's center. xanthian. -- === Subject: Re: simple probabilities ( but confusing) >> So, we choose the first point x_1. We draw a >> diameter thru x_1 , and then we have to choose x_2 >> x_3. There are 4 ways of choosing x_2 and x_3 so that >> they are either all in the same side of the diameter or not, >> and in exactly two of these ways the three points are >> of the same side,out of 4 possibilities so the answer is >> 1/2. My friend says it is 1/4. Who's right? >> >But having the 2nd and 3rd points fall on opposite sides of the >diameter established by the first isn't sufficient to guarantee >that the triangle they establish includes the circle's center. Correct; I was wrong in my previous post. I would approach this by conditioning on the length of the smaller arc between the first two points. In his First Course book, Ross has an exercise with a nifty approach of determining the probability that the center lies within the convex hull of n randomly selected points on the circle. The answer is 1 - n / 2^(n-1). -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: simple probabilities ( but confusing) >we are choosing three points in a circle > at random, and we join every pair of points to form > a triangle.We want to know the probability that the > triangle includes the center of the circle. > My answer was that the triangle will include the > center iff no 2 of the pairs of points are more than > pi radians appart. > So, we choose the first point x_1. We draw a > diameter thru x_1 , and then we have to choose x_2 > x_3. There are 4 ways of choosing x_2 and x_3 so that > they are either all in the same side of the diameter or not, > and in exactly two of these ways the three points are > of the same side,out of 4 possibilities so the answer is > 1/2. My friend says it is 1/4. Who's right? First, I assume that the process by which you choose the points is to independently choose uniformly with respect to one-dimensional on the circle. (Or did you mean in the disc?) I agree with your (Carlos's) assessement. Draw the diameter ending in the first point. In two out of four equally likely pairs of positions on semicircles, the triangle contains the center. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Google [was: Jesus christ What's going on?] Is this working? Andrew Usher === Subject: Re: Google [was: Jesus christ What's going on?] <31majcF3bo1lbU1@individual.net> <0rhjr0thhcujrgrikmhhq9p2i4i1l2kn33@4ax.com> Well, I can't post with the old version of Google anymore, apparently because all the message IDs have changed. I'm using the new version to post now, and I see that my name is gone again. Andrew Usher === Subject: Re: Impossible sampling theory! > A number of texts suggest that sampling can be modelled > by multiplying the incoming waveform by a comb of > Diracian Delta Functions. > How can this be? > 1. The samples that you get are measured in the order > of single volts whereas the Diracian is infinitely tall. Surely, > if something of the order of unity were to be multiplied by > something of the order of infinity, the result would > be of the order of infinity? Yes, it would, but it is also multiplied by something on the order of reciprocal infinity...the dt in the integral. > 3. If you are one of those who claim that the sampled signal is a short > spike of zero width, then it is zero-integrable and not analysable by > any process involving Laplace Transforms. It's wrong to say zero width. It's infinitely close to zero width. -- different MP3 every day! http://gweep.net/~shifty/snackmaster . . . . . . . . ... . . . . . . Anything moving in the zone, even a three-year- | Niente old, needs to be killed. -Captain R. | shifty@gweep.net === Subject: opportunity is knocking Hey Here's an opportunity to get a free desktop computer that's actually not a scam. The catch is that you have to sign up for one of the offers, most of which are free trials and things like that, that have to refer ten other people for the deal too. That's what I'm doing now. I already got one free computer and I'm trying to get another one for my Mother. If you click the link below and then complete and offer we'll both be on our way to a new desktop computer. Here's the Link: http://www.FreeDesktopPC.com/?r=8662679 === Subject: functional derivatives hi, This one is on basic maths w.r.t poisson brackets. I'm not sure if I've fully understood functional derivatives, which might be the root cause. I've these two constraints 1) E1(x): d_i A^i(x) = 0 2) E2(x): d_i PI^i(x) + J^0(x) = 0 where PI^i is the conjugate momenta of A^i and d_i is partial derivative. Now, calculating the poisson brackets [E1(x),E2(y)]_p is where my confusion is. To calculate the poisson brackets, I need to find the functional derivative of E1 and E2 w.r.t A^i and PI^i Neither E1 nor E2 contain A^i or PI^i; rather, they contain derivatives of A^i and PI^i. So, shouldnt the derivative of either E1 or E2 w.r.t say A^i be zero always?? But the result of [E1(x),E2(y)]_p = div grad delta^2(x-y) Can someone enlighten me on this elementary aspect?? ganesh === Subject: Re: Counterexamples to FLT Do you have a different posting address each day or something? Sure sign of a troll or a loon - morphing all over the place. Phil -- The gun is good. The penis is evil... Go forth and kill. === Subject: Re: Reviewing errors === >>Subject: Re: Reviewing errors >>Message-id: >Now we know! I wonder if a set of onions is not a set either? > I got a TV set. > And I've got a chess set! the ancient egyptians had a god set Phil -- The gun is good. The penis is evil... Go forth and kill. === Subject: Re: Reviewing errors > === >>Subject: Re: Reviewing errors >>Message-id: >Now we know! I wonder if a set of onions is not a set either? > I got a TV set. > And I've got a chess set! > the ancient egyptians had a god set I think I'll put this thread into TeX and have it typeset. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Help needed with diagonalization of a matrix expression Hello everybody I am trying to diagonalize a matrix expression, but I have not been able to solve it. The expression is: WZ[W]=WBD[WB]-WBE[WC]-WCF[WB]+WCG[WC] where [X] = the transposed of X I am trying to diagonalize WZ[W] such that WZ[W]=PD[P] where P is a unitary matrix My question is: How do I find an expression for P? Some helpful properties: W is a whitening matrix WB is a unitary matrix WC(G^ü) is a unitary matrix where an entry of the diagonal of G^ü is equal to the squareroot of the corresponding entry in G D,E,F and G are diagonal matrices I really appreciate any help. I know that WBD[WB] is a diagonalization WC(G^ü)[WC(G^ü)] is also a diagonalization Jakob === Subject: matlab problem -martix if I have that P * X = X P is 8 * 8 and X is a vector (8*1) and the values of P is given. How to find X in matlab? Any command will do? === Subject: Re: matlab problem -martix > if I have that P * X = X > P is 8 * 8 and X is a vector (8*1) > and the values of P is given. > How to find X in matlab? > Any command will do? In general, the newsgroup comp.softsys-matlab is a good place for matlab questions. However, I'll answer it here. X is an eigenvector with eigenvalue 1; the function EIG finds eigenvalues and eigenvectors: Here's the help description of eig: >> help eig EIG Eigenvalues and eigenvectors. E = EIG(X) is a vector containing the eigenvalues of a square matrix X. [V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D. [V,D] = EIG(X,'nobalance') performs the computation with balancing disabled, which sometimes gives more accurate results for certain problems with unusual scaling. If X is symmetric, EIG(X,'nobalance') is ignored since X is already balanced. E = EIG(A,B) is a vector containing the generalized eigenvalues of square matrices A and B. [V,D] = EIG(A,B) produces a diagonal matrix D of generalized eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that A*V = B*V*D. EIG(A,B,'chol') is the same as EIG(A,B) for symmetric A and symmetric positive definite B. It computes the generalized eigenvalues of A and B using the Cholesky factorization of B. EIG(A,B,'qz') ignores the symmetry of A and B and uses the QZ algorithm. In general, the two algorithms return the same result, however using the QZ algorithm may be more stable for certain problems. The flag is ignored when A and B are not symmetric. See also CONDEIG, EIGS. For other options, do this: >> lookfor('eigenvector') === Subject: Re: matlab problem -martix but I tried with eig(P) to calculate. The answer doesn't seem to right. === Subject: Re: matlab problem -martix > but I tried with eig(P) to calculate. > The answer doesn't seem to right. You had the correct form? That is, [V,D] = EIG(X) and found the entry of the diagonal matrix D that was equal to 1 (remember, the notion of equal needs to be taken with a bit of slack, because Matlab uses 8-byte values for doubles, and will usually have random stuff on the order of 10^(-10) or so). The corresponding column of V should do the trick. Post your matrix P and what you got from Matlab. Could be your equation doesn't have the solution you think it does. Dale. === Subject: Re: matlab problem -martix For finding Eigenvectors one starts with M*X= c*X c any real number. Mathcat asked for c=1. Have fun Hero === Subject: Re: matlab problem -martix > For finding Eigenvectors one starts with > M*X= c*X > c any real number. > Mathcat asked for c=1. > Have fun > Hero ... and ....? The Matlab function that I mentioned computes (and returns) approximate (since Matlab works numerically) eigenvalues and eigenvectors, provided the calling you've responded to, I suggested that the OP makes sure the calling sequence is correct, and to check the eigen- vector corresponding to the eigenvalue 1. I'm wondering how this is different from what you're saying, with the possible exception of being a bit more complete in terms of the details of using Matlab's EIG function. Dale. === Subject: Re: matlab problem -martix I guess the X should be a vector of 1. So that P X = X p=[1/6 0 0 1/6 1/6 1/6 1/6 1/6; 0 1/6 0 1/6 1/6 1/6 1/6 1/6; 0 0 1/6 1/6 1/6 1/6 1/6 1/6; 1/4 1/4 1/4 1/4 0 0 0 0; 1/4 1/4 1/4 0 1/4 0 0 0; 1/4 1/4 1/4 0 0 1/4 0 0; 1/4 1/4 1/4 0 0 0 1/4 0; 1/4 1/4 1/4 0 0 0 0 1/4] I tested it in matlab, it is fine. But i don't know how did I get this answer. Can anyone help me to back track the steps? so that it is reasonable? I don't really think [V D] = eig(p) will help. === Subject: Re: matlab problem -martix I guess the X should be a vector of 1. So that P X = X Let P be the matrix with ones in the main diagonal and zero else, than X can be... Let P be another matrix than this and with determinant not zero, than X must be... Can You do this step ? Hero === Subject: Re: matlab problem -martix I guess the X should be a vector of 1. So that P X = X p=[1/6 0 0 1/6 1/6 1/6 1/6 1/6; 0 1/6 0 1/6 1/6 1/6 1/6 1/6; 0 0 1/6 1/6 1/6 1/6 1/6 1/6; 1/4 1/4 1/4 1/4 0 0 0 0; 1/4 1/4 1/4 0 1/4 0 0 0; 1/4 1/4 1/4 0 0 1/4 0 0; 1/4 1/4 1/4 0 0 0 1/4 0; 1/4 1/4 1/4 0 0 0 0 1/4] I tested it in matlab, it is fine. But i don't know how did I get this answer. Can anyone help me to back track the steps? so that it is reasonable? I don't really think [V D] = eig(p) will help. === Subject: Re: matlab problem -martix > I guess the X should be a vector of 1. > I don't really think [V D] = eig(p) will help. I just ran [V D] = eig(p) and got the answer. Of course, you might not recognize it as the answer. What I got associated with eigenvalue 1 was a vector whose elements are all -0.3536. What you need to know about eigenvectors is that they can be multiplied by any constant and still be eigenvectors. If AX = cX, then AY = cY where Y = kX, k = any constant. So a vector of identical nonzero values is an eigenvector if and only if a vector of all 1's is an eigenvector of the same eigenvalue. - Randy === Subject: Re: matlab problem -martix > I guess the X should be a vector of 1. > So that P X = X > p=[1/6 0 0 1/6 1/6 1/6 1/6 1/6; 0 1/6 0 1/6 1/6 1/6 1/6 1/6; 0 0 1/6 > 1/6 1/6 1/6 1/6 1/6; 1/4 1/4 1/4 1/4 0 0 0 0; 1/4 1/4 1/4 0 1/4 0 0 0; > 1/4 1/4 1/4 0 0 1/4 0 0; 1/4 1/4 1/4 0 0 0 1/4 0; 1/4 1/4 1/4 0 0 0 0 > 1/4] > I tested it in matlab, it is fine. > But i don't know how did I get this answer. A good guess? You noticed that the sum of the elements in each row = 1, and realized that P*X where X is a vector of 1's sums each row? > Can anyone help me to back track the steps? so that it is reasonable? Of course not. We don't know your thought process, especially if it was an intuitive leap of some sort. > I don't really think [V D] = eig(p) will help. I'm not sure why you think that. People have told you EXACTLY how this call will give you the solution you want. Why is it that getting the answer to your question is not helpful? - Randy === Subject: Re: matlab problem -martix saying,.. It's the same, but expressed in different form. I think mathcad got stuck, thinks too difficult, where there's a simple solution, isn't it ? Hero === Subject: Re: matlab problem -martix You're too deep in it. Take a step back and look from more distance. In this case try 2d first: ( a b// c d ) times (x,y) =(x,y). Still clueless? Try in 1D. Be happy Hero === Subject: Lorentz transformations, Maxwell Eq., and Special Relativity imply the Cosmos is a huge sphere such as the inside of an atom Today I heard in conversation the formula E = mc^2 and Special Relativity. And I am familar to the old puzzle of why gravity and Coulomb forces are inverse square laws is because they are spherical. But not enough has been said as to the spherical foundation that is the Maxwell Equations and Special Relativity and Lorentz transformation invariance. The reason you have Special Relativity is because if you did not then you would destroy the symmetry of Maxwell Eq that a moving magnet causes current in a wire loop is the same as a moving wire loop to a stationary magnet. The inverse themselves. Again in Special Relativity in order for the speed of light to be constant then distance or time have to increase or decrease to keep c the same. What it all says is that the Cosmos is one huge sphere and it has some constants such as the speed of light and that if you change distance by increasing it then you decrease time. If you move a magnet yet keep the wire loop stationary is the same as move the loop and keep the magnet stationary. It all means that the Cosmos is a huge sphere in that if you increase parameter X you decrease parameter Y. This all makes sense in a AtomTotality because the cosmos is one big atom. But it does not make sense in a BigBang theory. In the BigBang theory Lorentz transformations would not be invariant in Maxwell theory, nor would Special Relativity be invariant to the speed of light because the cosmos would be open flat Euclidean with c going to infinity. Nor would the BigBang theory favor inverse square laws for gravity and Coulomb because there is no sense of a permanent spherical cosmos except in the AtomTotality theory where all atoms are spherical in geometry. I think I said all of the above at various times in the past 11 years of posting to the sci newsgroups but it is fun to watch how my mind through the years develops a sharper clarity over many various topics. And although I am sure I said all of the above in the 1990s, I may not have said them as clear as the above musings. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: R^n and R^m are not homeomorphic Does anyone have the solution for that problem? I know S^n is not homeomorphic to S^m but I cannot prove with those things removed 1 point. Khoa === Subject: Re: R^n and R^m are not homeomorphic === Subject: Re: R^n and R^m are not homeomorphic > Does anyone have the solution for that problem? I know S^n is not > homeomorphic to S^m but I cannot prove with those things removed 1 point. > Khoa Suppose f:R^n->R^m is a homeomorphism. Show that as |x|->infinity, |f(x)|->infinity as well. This would allow you to extend f to a homeomorphism f':S^n->S^m. === Subject: Re: R^n and R^m are not homeomorphic > Does anyone have the solution for that problem? I know S^n is not > homeomorphic to S^m but I cannot prove with those things removed 1 point. Let n1: take 0 out of real line and image of 0 out of R^m. Consider points -1 and +1 and their images. There's no path between -1 and 1 in R^1 but certainly there is path between images of -1 and 1 in R^m. Now You can do the rest :-) sirix. === Subject: Re: R^n and R^m are not homeomorphic > Does anyone have the solution for that problem? Suppose that m > n and that f:R^m ---> R^n is a homeomorphism. Let i:R^n ---> R^m the function defined by i(x_1,...,x_n) = (x_1,...,x_n,0,...,0) then i o f would be a continous injective function from R^m into itself whose image would not be an open set. This is impossible, by the domain invariance theorem. Jose Carlos Santos === Subject: Re: R^n and R^m are not homeomorphic > Does anyone have the solution for that problem? I know S^n is not > homeomorphic to S^m but I cannot prove with those things removed 1 point. > Khoa If R^n and R^m were homeomorphic, their one-point compactifications would be. So S^n and S^m would be homeomorphic. But they're not. === Subject: Re: OT: Re: Google Groups is much better now http://mygate.mailgate.org/mynews/sci/sci.math/838408d54a1fe64eaa134498f5352 9 7a.48257%40mygate.mailgate.org > No, sir! I do not like it! Well, if you want your complaints to receive some attention, send them to and ignore the auto-response; they do, eventually, let a human read email to that address. HTH xanthian. -- === Subject: Re: OT: Re: Google Groups is much better now >[...] >Well, if you want your complaints to >receive some attention, send them to >and ignore the auto-response; they do, >eventually, let a human read email to >that address. True. To Google's great credit, you do usually get a human response. In addition to the partial switch to monospaced fonts, one of my other gripes (that a groups search terms didn't automatically carry over to a Web search, unless JavaScript was on) was addressed a few days after the complaint. So it isn't just that they read the mail, they actually do something about it. But there are still lots of problems. Since Google can't make everyone happy, they should have an optional groups- -basic version of the archive, without all the new bells and whistles, font/scripting/email mangling nonsense, and with archival URLs that would allow one to refer to threads and individual posts for years to come. -- Eric Behr | NIU Mathematical Sciences | (815) 753 6727 behr@math.niu.edu | http://www.math.niu.edu/~behr/ | fax: 753 1112 === Subject: Re: OT: Re: Google Groups is much better now Discussion, linux) > When was the last time you looked? Sci.math and several others > (comp.programming for one) are now defaulting to monospaced fonts. > The default sorting has been changed to most recently posted instead > of most recently created, so I don't have to click on the goddamn link > every time. The tree view was not working when they first launched it. > Sorting the tree by date is still inactive, but maybe you should just > give them some more time. I can't stand that they obfuscate email addresses, so that if you want to reply by email, you have to have a Google account. I have often my question months ago and emailed him to hear if he's found a solution not reported on Google. If they are going to maintain an archive, they ought to present posts exactly as they were posted. I didn't mangle my email address and I didn't ask them to. -- So I speak before a crowd of the damned, cursed to be unloved throughout time, with only their hatred and bile to comfort them now, having betrayed what should have been their one true lover: Mathematics. -- James Harris reaches a bit === Subject: Re: GCH vs. Axiom of Choice. ZF+AC that are about L inconsistent with the therorems of ZF+~AC that > are about L? Clearly the theorems of ZF+AC about V are different than > the theorems of ZF+~AC about V, but I want to know about L. I have just realized if ZF+~AC could prove a certain proposition is true in L, then it would be consistent with ZFC that that proposition was true in L. However, you might like to try the other example. Suppose ZF+Con(ZF) is consistent. Then ZF+Con(ZF) is consistent and proves Con(ZF) is true in L, whereas ZF+~Con(ZF) is consistent and proves ~Con(ZF) is true in L. === Subject: Re: GCH vs. Axiom of Choice. I was asking about FO sentances being true in L. Neither Con(ZF) not ~Con(ZF) is a FO sentance is it? I was wondering if for a FO sentance X, such that Con(A)=>Con(A&X) where A is the conjuction of axioms of ZF, whether the sentances TX that are of the form S[x] & x in L that are proved by the conjuction of axioms of ZF with each other and with X will be consistent with the sentances TY that are of the form S[x] & x in L that are proved by the conjuction of axioms of ZF with each other and with ~X given that Con(A) => Con (A&~X). I don't even know if the question is a real question because I don't know if x in L is a FO sentance, if it isn't, then TX and TY are collections of no sentances, and my question is vacuously answered. I don't understand what L is well enough to know if L is FO expressable. I don't know if adding a discussion of L automatically makes one talking outside ZF or whether if can be done inside ZF. Talking with Keith Ramsey makes me worry that L is well defined, that it's existance isn't proveable maybe not even by assuming Con(ZF), and since Con(ZF) isn't a FO sentance to my knowledge, I don't know the first thing about how to prove things given Con(ZF), hence why I'm asking on Usenet. === Subject: Re: GCH vs. Axiom of Choice. > I was asking about FO sentances being true in L. Neither Con(ZF) not > ~Con(ZF) is a FO sentance is it? What on earth is a FO sentence? === Subject: Re: GCH vs. Axiom of Choice. > I was asking about FO sentances being true in L. Neither Con(ZF) not > ~Con(ZF) is a FO sentance is it? > What on earth is a FO sentence? first order === Subject: Re: GCH vs. Axiom of Choice. >first order And what does that mean? === Subject: Re: GCH vs. Axiom of Choice. I don't think the question is meaningless. He means, is there proposition relativized to L that you can prove in ZFC but disprove in ZF+~AC. I don't know whether this is the case or not. === Subject: Re: GCH vs. Axiom of Choice. > He means, is there > proposition relativized to L that you can prove in ZFC but disprove in > ZF+~AC. I don't know whether this is the case or not. If ZF proves AC->A_L, ZF proves AC_L->A_L_L, and so proves A_L_L, which implies A_L. So ZF can't prove -AC->-A_L. === Subject: Re: GCH vs. Axiom of Choice. > False axioms can be consistent with ZF. You could add a false > axiom > to > ZF and the results that you could prove to be true in L with this > axiom system might not in fact be true in L. Could you please be so kind as to explain what a false axiom is? > I'm > familiar that somone has proven (in ZF?) that Con(ZF) => Con(ZFC) > as > well as Con(ZF) => Con(ZF+~C), so I didn't think C or ~C was > considered false, just relatively independant axioms. What is an > example of a false axiom that is (relativily?) consistent with ZF? Well, I believe that every sentence in the first-order language of > set > theory is either true or false. ~AC would be an example of a false > axiom consistent with ZF. > If xey is fixed as either true or false in a domain of discourse > for every x and y in the domain of discourse then yes every FOL > sentance is either true or false of that same domain of discourse, it's > just a fact of combinatorics. If you start with an unknown domain of > discourse that satisifies ZF and then consider a sub domain that is L > then you can assume AC or ~AC in the original domain and I don't see > how it can effect the subdomain's truth. But you might be able to > prove more. > If you don't believe this, you shouldn't really use the notion of > truth at all, just provability in this or that formal system. > I'm not sure what you are saying that there is to believe. Why is AC > true and ~AC false? You didn't explain that. I just happen to believe it. Another example you might like better: Con(ZF) is true and ~Con(ZF) is false. However ~Con(ZF) is consistent with ZF. > But assuming one only talked about provability, are the theorems of > ZF+AC that are about L inconsistent with the therorems of ZF+~AC that > are about L? Clearly the theorems of ZF+AC about V are different than > the theorems of ZF+~AC about V, but I want to know about L. I'm not sure. === Subject: Re: GCH vs. Axiom of Choice. |I don't even understand what L even is, I gave you one definition, that it's the smallest class of sets *that contains all ordinals* and satisfies ZF. For a more concrete definition and treatment you can read about it in Cohen's book about the axiom of choice. L is a hierarchy indexed by ordinals but with only one set being added at each step. If alpha is a limit ordinal, L_{alpha+n} for integers n are the subsets of {L_beta : beta < alpha} that are definable by formulas in the language of ZF with all the quantifiers restricted to {L_beta : beta < alpha}. L is the proper class consisting of all the sets that occur as L_alpha for some ordinal alpha. L satisfies AC because everything in it is well-ordered by the order in which the elements appear in the L_alpha. |but some people |seem to think it's well defined enough to have every oFOL sentenance |with one binary relation be either true or false of it, which is pretty |well-defined in my book if that's the case. I doubt you'll find anyone who believes that about L who doesn't believe it because they also believe that about V, the cumulative hierarchy. At first glance, the hierarchical definition of L is less problematic because the sets at each step are being defined by formulas quantifying over previous steps. But we still have to start with a belief in the well-definedness of the sequence of all ordinals, and once one believes in the ordinals, belief in arbitrary sets is not usually seen as a problem. | My naive understanding is |that it's the smallest intended universe that satisfies the ZF axioms. No. I don't know what you mean by intended, but if you mean what I would mean by intended, there's just one intended universe, V. V_0 is {{}}. For each ordinal alpha, V_alpha+1 is the power set of V_alpha. If alpha is a limit ordinal, V_alpha is the union of the V_beta for beta < alpha. Many sets are added at each successor ordinal. V is the class of sets that appear at some stage V_alpha for an ordinal alpha. Keith Ramsay === Subject: Re: GCH vs. Axiom of Choice. > |but some people > |seem to think it's well defined enough to have every oFOL sentenance > |with one binary relation be either true or false of it, which is pretty > |well-defined in my book if that's the case. > I doubt you'll find anyone who believes that about L who doesn't > believe it because they also believe that about V, the cumulative > hierarchy. At first glance, the hierarchical definition of L is less > problematic because the sets at each step are being defined by > formulas quantifying over previous steps. But we still have to start > with a belief in the well-definedness of the sequence of all ordinals, > and once one believes in the ordinals, belief in arbitrary sets is > not usually seen as a problem. I think there are probably people (by which I mean people who know what they're talking about) who might take the position that any sentence of LST relativized to L is either true or false, but not any sentence period. One reason this might be attractive is that you can't, by forcing, change the truth value of a sentence relativized to L. It's certainly not *my* position. I think the honest way of describing the position would be to say you think the powerset axiom is really false--that there isn't any completed whole that contains, say, all the reals. I guess that's not directly inconsistent with thinking that you *can* collect (say) all the countable ordinals into a completed whole. === Subject: Re: GCH vs. Axiom of Choice. <3228t8F3eejn4U1@individual.net> >It's certainly not *my* position. I think the honest >way of describing the position would be to say you think >the powerset axiom is really false--that there isn't any completed >whole that contains, say, all the reals. I guess that's not >directly inconsistent with thinking that you *can* collect >(say) all the countable ordinals into a completed whole. I think that's it. As a physicist I don't like saying that a set exists that contains all the reals, because it's not strongly falsifiable in the way that I'd want it to be, not even well defined enough. If there are any questions about what is or isn't a set, then it's unclear to have all the subsets, because we haven't described what is or isn't a set. It's better to prove things about an arbitrary set of reals than a set of all reals, and with an arbitrary set that's surely enough for anything practical. Whereas if one is clear that every ordinal is a set, then considering all the ordinals might be easier, then again I don't get why one can't interpret P(x) as a successor operation, and define P-ordinals the same way as xU{x} ordinals, and get a collection isomorphic in structure to the ordinals. If one interpreted P(x) as the successor of x, then separation, for instance, would be more complicated if one made S[z] act on P(x) the same way as S[x] acts on xU{x}. So I'm not sure this changes anything. But right now understanding what L is and what true in L is and whether L exists and if L can be complete in only one way is enough for me now. === Subject: Re: GCH vs. Axiom of Choice. |I'm |familiar that somone has proven (in ZF?) Only finitistic reasoning is actually required. | that Con(ZF) => Con(ZFC) That someone was Goedel. |as |well as Con(ZF) => Con(ZF+~C), That someone was Cohen. | so I didn't think C or ~C was |considered false, just relatively independant axioms. Are you assuming that every statement X for which we can prove Con(ZF)=>Con(ZF+X) and Con(ZF)=>Con(ZF+~X) is neither true nor false? I don't mean it as a rhetorical question. I don't assume anything like that, and to me this seems like a nonsequitur. If you believe such a thing, and you don't know why you believe it, it's liable to be a source of some confusion. Keith Ramsay === Subject: Re: GCH vs. Axiom of Choice. >Are you assuming that every statement X for which we can prove >Con(ZF)=>Con(ZF+X) and Con(ZF)=>Con(ZF+~X) is neither true >nor false? I don't mean it as a rhetorical question. I think of them as sentance such that (~ZF)v(X) cannot be validities. As opposed to theorems T of ZF which I consider to be sentances such that (~ZF)v(T) is a validity. I define the truth of a sentance based on the truth of atomic sentances, z in x for elements x,z of a domain of discourse, then a sentance is true or false based on the existence of a winning strategy for one side in a particular game played where atomic sentances give winners to the game and choices of elements are from a fixed domain of discourse. A validity is a sentence that is has a winning strategy (i.e. is true) in any domain of discourse. If the game is the usual E and A game, then this a standard view of truth and falseness, if the game is an IF-logic game with more moves like Ex//y instead of just Ex, then it's less popular, but not very different IMO. The concept of a complex sentance being true or false without reference to a domain of discourse just doesn't make sense to me, it could be valid (is true in all possible worlds with well defined true/false for the atomic sentances) inconsistent (is false in all possible worlds with will defined true/false for atomic sentances) or consistent (true or false depending on which possible world or equivalently where it's truth depends). Examples: Ex Ey x in y is consistent. (Ax Ay ~ x in y )v(Ex Ey x in y) is a validity for non-empty domains of discourse (and for empty domains I wouldn't know how to play the game). While Ex Ey ((x in y)&(~x in y)) is inconsistent for non-empty domains of discourse. >I don't assume anything like that, and to me this seems like a >nonsequitur. If you believe such a thing, and you don't know >why you believe it, it's liable to be a source of some confusion. I don't even know what it MEANS to say a sentance X is true without reference to a domain of discourse. Saying it's valid I get. Saying it's true, I don't get. Saying it's false, I don't get. Saying for any nonempty domain of discourse, it is true or it is false I get and if the game is that perfect information A and E game, then I even believe that it's true. That was the basis of boot straping, that I take sentances like A is a validity where A is formed from composite sentance B and C, to get binary relations B in C that can be interpreted as true when A is a validity, and false otherwise and thus get a concrete domain of discourse from considering the case of arbitrary domains of discourse. It's really a kind of meta-quantification, but if's a possible world that I can understand that's better than a world that I can't. So L contains all the ordinals that can be proven and any set generated by a cumulative application of, for instance, separation? That sounds a bit vague since we need both all and ordinal before we could know what was in L. === Subject: Re: GCH vs. Axiom of Choice. |But assuming one only talked about provability, are the theorems of |ZF+AC that are about L inconsistent with the therorems of ZF+~AC that |are about L? What do you mean by about L? It's a theorem of ZF+~AC that there exist sets not in L. Is that about L? In another posting I outlined a proof that for any sentence X in the language of ZF, we have ZF|-(L|=X) if and only if ZF+(V=L)|-(L|=X), and since AC is a consequence of V=L, these two are also equivalent to ZFC|-(L|=X). I believe it's also the case that for every such X, ZF+~AC|-(L|=X) if and only if ZF|-(L|=X). I don't think that would be too hard to prove, but might require something like redoing Cohen's proof of the relative consistency of ~AC. If I'm correct, then ZFC|-(L|=X) if and only if ZF+~AC|-(L|=X). Is that the kind of result you want? What's the motivation for the line of inquiry? Keith Ramsay === Subject: Internal Set Theory Uniqueness Principle Throughout, I will use the notation st for standard, fin for finite, and for sets C and D, C inter D denotes the intersection of C and D. I will denote there exists by TE and for all by FA. In Edward Nelson's original paper on Internal Set Theory, he gave a proof that if (1) P(x) is a formula in Internal Set Theory with one free variable x, (2) P(x) is relativized to a standard set V (i.e. all quantifiers are of the form FA x in V, TE x in V, FA st x in V, TE st x in V), (3) there is a unique value of x in V for which P(x) holds, then the unique value of x in V, such that P(x) holds, is standard. It is reasonably simple to generalize this to the theorem that if (1) P(u,v_1,...,v_k) is a formula in Internal Set Theory, with free variables u, v_1, ..., v_k, (2) y_1, ..., y_k are standard sets, (3) the only occurrences of the predicate 'standard' in the formula are in quantifiers of the form FA st z in V for some standard set V or TE st z in V for some standard set V (the standard set V being allowed to vary from quantifier to quantifier). (4) there is a unique value of x such that P(x,y_1,..,y_k) holds, then the unique value of x, such that P(x,y_1,...,y_k) holds, is standard. For a formula P(u,v_1,...,v_k) in Internal Set Theory, take the formula FA st y_1 ... FA st y_k FA x (FA w [P(w,y_1,...,y_k) <=> w = x] => st(x)). (*) Is (*) a theorem in Internal Set Theory for all formulae P(u,v_1,...,v_k)? Is there a formula P(u,v_1,....,v_k) for which the negation of (*) is a theorem in Internal Set Theory? That is, is there a formula P such that TE st y_1 ... TE st y_k TE x (FA w [P(w,y_1,...,y_k) <=> w = x] and not st(x)) is a theorem of Internal Set Theory? Do there exist formulae such that (*) is undecideable in Internal Set Theory? Equivalently, my question is that if (1) P(u,v_1,...,v_k) is a formula in Internal Set Theory, with free variables u, v_1, ..., v_k, (2) y_1, ..., y_k are standard sets, (3) there is a unique value of x such that P(x,y_1,..,y_k) holds, then is it necessarily true that the unique value of x specified in (3) is standard? Or does there exist a formula P(u,v_1,...,v_k) in Internal Set Theory with free variables u, v_1, ..., v_k, and standard values of y_1, ..., y_k, such that (1) there is a unique value of x such that P(x,y_1,..,y_k) holds, (2) the unique value of x such that P(x,y_1,...,y_k) holds is nonstandard? Or is the question undecideable? If there is a proof in the general case that x must be standard, I would be grateful to see it. If it is not true in general that x must be David ----- P.S. The proof of the generalization of Nelson's result, as given above, is as follows. The proof follows similar lines to those used by Nelson. If P(u,v_1,...,v_k) is a formula in Internal Set Theory with free variables u, v_1, ..., v_k, such that (1) the only occurrences of the predicate 'standard' in the formula are in quantifiers of the form FA st z in V for some standard set V or TE st z in V for some standard set V (the standard set V being allowed to vary from quantifier to quantifier), then P(u,v_1,...,v_k) is equivalent to the formula TE st s in A FA st t in B Q(s,t,u,v_1,...,v_k) for some internal formula Q(s,t,u,v_1,...,v_k), and for some nonempty standard sets A and B. The presumption is that y_1, ..., y_k have standard values, and that there is a unique value of x for which P(x,y_1,...,y_k) holds. Since P(x,y_1,...,y_k) holds, then TE st s in A FA st t in B Q(s,t,x,y_1,...,y_k). (**) Since x is the unique value for which P(x,y_1,...,y_k) holds, then FA w (P(w,y_1,...,y_k) => w = x), so that FA w (TE st s in A FA st t in B Q(s,t,w,y_1,...,y_k) => w = x). It follows that FA w FA st s in A TE st t in B (Q(s,t,w,y_1,...,y_k) => w = x), so that FA st s in A FA w TE st t in B (Q(s,t,w,y_1,...,y_k) => w = x), and so FA st s in A TE st fin t' FA w TE t in t' inter B (Q(s,t,w,y_1,...,y_k) => w = x), by idealization. So FA st s in A TE st fin t' FA w (FA t in t' inter B Q(s,t,w,y_1,...,y_k) => w = x). (***) By (**), there exists standard S in A such that FA st t in B Q(S,t,x,y_1,...,y_k). TE st fin t' FA w (FA t in t' inter B Q(S,t,w,y_1,...,y_k) => w = x), so there exists a standard finite set, T, such that FA w (FA t in T inter B Q(S,t,w,y_1,...,y_k) => w = x). (+) Suppose t is an element of T inter B. Since T is standard and finite, then t is standard, and so t is a standard element of B, and so Q(S,t,x,y_1,...,y_k). Since Q(S,t,x,y_1,...,y_k) for all t in T inter B, then FA t in T inter B Q(S,t,x,y_1,...,y_k), and so TE w FA t in T inter B Q(S,t,w,y_1,...,y_k). Since Q(s,t,u,v_1,...,v_k) is an internal formula, and T, B, S, y_1, ..., y_k, are all standard, then by transfer, TE st w FA t in T inter B Q(S,t,w,y_1,...,y_k). So there exists W standard such that FA t in T inter B Q(S,t,W,y_1,...,y_k). (++) As a consequence of (+), FA t in T inter B Q(S,t,W,y_1,...,y_k) => W = x. (+++) As a consequence of (++) and (+++), then W = x. Since W is standard, and x = W, then x is standard. ----- === Subject: Primitive recursive sets I have a basic question: Is every primitive recursive subset of N^p definable with a Sigma_0 formula (over the langage of arithmetic) ? === Subject: Re: Primitive recursive sets > Is every primitive recursive subset of N^p definable with a Sigma_0 formula > (over the langage of arithmetic) ? No, since the set of true Sigma_0 formulas is primitive recursive. === Subject: Intersection of two surfaces in 3-space Related to the recent thread :Curve of intersection of two surfaces (in 3D). The same is re-worded here for a possible parametric link. Common or 3D coordinates obtained by intersection of two surfaces s and S are : (x,y,z) = f(u,v) = F(U,V). what combined PDE/DE/intrinsic/vector relations may be there between u,v,U and V ? i.e., if linef(u,v)=0 is given for curve of intersection on one surface, how is LINEF(U,V)=0 obtained on the other (or vice-versa)? The First Fundamental form derivables arc length and tangent rotation between corresponding parameter limits is same for the stand alone 3D Space curve. Generalized Frenet-Serret relations with scalars kg,kn,taug may be involved. TIA for further thoughts. Narasimham === Subject: Condition of a binary polynomial with no real roots? Hi ! Under which condition does a binary polynomial have no real roots? it means: Let f(x,,y) be a binary polynomial, Under which condition does $f(x,y)neq 0$, for any x, y in R holds? === Subject: Re: Condition of a binary polynomial with no real roots? >Hi ! >Under which condition does a binary polynomial have no real roots? >it means: >Let f(x,,y) be a binary polynomial, Under which condition does >$f(x,y)neq 0$, for any x, y in R holds? If the polynomial has no zeroes in the xy-plane, it must have no zeroes in any line in the xy-plane. For any line, you can collapse the binary polynomial into a single-variable polynomial. Then you can apply the things you know about ordinary polynomials, for example it must be of even degree in order for the function value not to cross 0. Start with lines x=0 and y=0. Then y=mx+b, choosing your m and b artfully to prove whatever point you need to prove. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Condition of a binary polynomial with no real roots? >Under which condition does a binary polynomial have no real roots? >it means: >Let f(x,,y) be a binary polynomial, Under which condition does >$f(x,y)neq 0$, for any x, y in R holds? By binary polynomial do you mean a polynomial with coefficients in {0,1}, or just a polynomial in two variables with real coefficients? I don't know if there's a useful necessary and sufficient condition, but there are obvious sufficient conditions: f(x,y) is a sum of squares of polynomials f_j such that the intersection of the sets f_j^(-1)(0) is empty. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Condition of a binary polynomial with no real roots? >>Under which condition does a binary polynomial have no real roots? >>it means: >>Let f(x,,y) be a binary polynomial, Under which condition does >>$f(x,y)neq 0$, for any x, y in R holds? >I don't know if there's a useful necessary and sufficient condition, >but there are obvious sufficient conditions: f(x,y) is a sum of squares >of polynomials f_j such that the intersection of the sets f_j^(-1)(0) >is empty. That's not necessary, but a similar condition is necessary and sufficient: a polynomial is non-negative iff it is the sum of squares of some rational functions. This was Hilbert's 17th problem. There are many variants you could ask about. For some discussions, see http://www.math-atlas.org/96/hilb17 There are some fairly simple polynomial inequalities which cannot be proved by expressing (something)-(something) as a sum of squares of _polynomials_; I think the arithmetic-geometric mean inequality for three variables is one of them. (That's good to know when confronted by those inequalities challenges, which often involve 3 variables arranged symmetrically -- one recently appeared in this newsgroup, for example. To prove them you'll either need the AGM inequality or the sum-of-squares trick.) dave === Subject: Re: Condition of a binary polynomial with no real roots? [reply address is baloglouAToswego.edu] >>Under which condition does a binary polynomial have no real roots? >>it means: >>Let f(x,,y) be a binary polynomial, Under which condition does >>$f(x,y)neq 0$, for any x, y in R holds? >By binary polynomial do you mean a polynomial with coefficients >in {0,1}, or just a polynomial in two variables with real coefficients? >I don't know if there's a useful necessary and sufficient condition, >but there are obvious sufficient conditions: f(x,y) is a sum of squares >of polynomials f_j such that the intersection of the sets f_j^(-1)(0) >is empty. A recent expoerience with a certain homogeneous 3-variable inequality made me think of the necessity mentioned above: if the inequality can be proven to hold for any choice of variables, does it follow that the corresponding n-variable polynomial can be written as a sum of squares? [I felt that this might be an Algebraic Geometry issue ... and withdrew the question!] baloglouAToswego.edu === Subject: Re: Condition of a binary polynomial with no real roots? When You ask for x square + u *x +v =0 and x Elemt R - then start with ( (x,0) - (a,b) )*( ( (x,0) - (c,d) ) =(0,0) and b and d not zero. Have fun Hero === Subject: Re: Proof of Sum_{i=1...n} i^k is a polynomial expression over n First of all, Ignacio, sorry for no answer to you in a supreme answer to the my question of Touchard congruence. I wanted to answer but then I have to occupy my time in other _obligatory_ things. It's not excuse.... sorry. > Xan escribi.97: >> Say S_k(n) = Sum_{i=1...n} i^k = 1^k + 2^k + .... + n^k. >> I want to prove that S_k(n) is a polynomial expression over n, that >> is, that there exists a polynomial p_k(x) in R[x] such that S_k(n) = >> p_k(n) for all n (and p_k only depends of k). >> I prefer proofs by induction and elementals. I know that it could be >> proved using the Bernoulli polynomials, but I want a proof without >> that (I want more elemental proof). Could it be?. I tried but I did >> not get it. >> Xan. > Let > S(n, k) = 1^k + 2^k + ... + n^k > be the sum of the k-powers of the first n positive integers. > Then > (1+1)^(k+1) = 1^(k+1)+(k+1)*1^k + Comb(k+1, 2)*1^(k-1) + ... + 1 > (2+1)^(k+1) = 2^(k+1)+(k+1)*2^k + Comb(k+1, 2)*2^(k-1) + ... + 1 > ..... > (n+1)^(k+1) = n^(k+1)+(k+1)*n^k + Comb(k+1, 2)*n^(k-2) + ... + 1 > (n+1)^(k+1) = 1 + (k+1)Sum(n, k) + Comb(k+1, 2)Sum(n, k-1) + ... + n > and > S(n, k)=((n+1)^(k+1) - (n+1) - Sum(Comb(k+1, k+1 - i)S(n, i), i, 1, > k-1))/(k+1) This is a ingenious recursion. Until here, all good. > Sou, you can determine S(n, k) from S(n, 1), S(n, 2), ..., S(n, k-1). > As S(n, 0) = n is a poynomial, also S(n, k) is for k > 0. You can deduce > also that degree of S(n, k), with repect to n, is k +1. This is the dark part. How can I determine that S(n,k) is a polynomial over n?. By induction of n?. Really, for proving that S(n,k) is a polynomial over n, you need to prove here that: Theorem 1 If p(x) is a polynomial, then P(n) = Sum_{i=1...n} p(i) is a polynomial expression over n, that is, there is q(x) a polynomial such that P(n) = q(n) but for proving this, you need (at least, I no see alternatives) to prove a previous lema that S(n,k) is a polynomial over n. So there is a perversion circle. I did not say, but really I want to prove Theorem 1 and I saw that I need a lema (S(n,k) is a polynomial over n). So if now, I need to prove Theorem 1 for proving lema.... ;-) Surely, I'm wrong, but I don't see where. For the other hand, it's well-known that this results are true. Xan. the proofs of Ignacio type. More simple as it could be. === Subject: Re: Proof of Sum_{i=1...n} i^k is a polynomial expression over n Xan escribi.97: > First of all, Ignacio, sorry for no answer to you in a supreme answer > to the my question of Touchard congruence. I wanted to answer but > then I have to occupy my time in other _obligatory_ things. It's not > excuse.... sorry. It isn't matter ... >> Xan escribi.97: > Say S_k(n) = Sum_{i=1...n} i^k = 1^k + 2^k + .... + n^k. > I want to prove that S_k(n) is a polynomial expression over n, that > is, that there exists a polynomial p_k(x) in R[x] such that S_k(n) = > p_k(n) for all n (and p_k only depends of k). > I prefer proofs by induction and elementals. I know that it could be > proved using the Bernoulli polynomials, but I want a proof without > that (I want more elemental proof). Could it be?. I tried but I did > not get it. > Xan. >> Let >> S(n, k) = 1^k + 2^k + ... + n^k >> be the sum of the k-powers of the first n positive integers. >> Then >> (1+1)^(k+1) = 1^(k+1)+(k+1)*1^k + Comb(k+1, 2)*1^(k-1) + ... + 1 >> (2+1)^(k+1) = 2^(k+1)+(k+1)*2^k + Comb(k+1, 2)*2^(k-1) + ... + 1 >> ..... >> (n+1)^(k+1) = n^(k+1)+(k+1)*n^k + Comb(k+1, 2)*n^(k-2) + ... + 1 >> (n+1)^(k+1) = 1 + (k+1)Sum(n, k) + Comb(k+1, 2)Sum(n, k-1) + ... + n >> and >> S(n, k)=((n+1)^(k+1) - (n+1) - Sum(Comb(k+1, k+1 - i)S(n, i), i, 1, >> k-1))/(k+1) > This is a ingenious recursion. Until here, all good. >> Sou, you can determine S(n, k) from S(n, 1), S(n, 2), ..., S(n, k-1). >> As S(n, 0) = n is a poynomial, also S(n, k) is for k > 0. You can >> deduce also that degree of S(n, k), with repect to n, is k +1. > This is the dark part. How can I determine that S(n,k) is a polynomial > over n?. By induction of n?. Really, for proving that S(n,k) is a > polynomial over n, you need to prove here that: By induction on k, of course: S(n, 0) = n is a polynomial. Then suppose that all S(n, k), for k < m, is a polynomial. Then S(n, m) is sum/difference of With respect to the degree of S(n, k), the unique sumand of degree (k +1) is the first: (n+1)^(k+1) (newly it can be very easily show by induction in k). Also is easy to determine some coefficients of S(n, k), as the principal coefficient: 1/(k+1). But also some others. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Science, Philosophy, Mysticism, Art, Mathematics, and Physics > --------------- > Allow me to summarize certain lines of reasoning relating to the > definition of science, philosophy, and mysticism so as to make these > ideas explicit. > Science argues that A is C because A is B and B is C. Very amusing.. I'll try some.... Science argues that A is C because of a A=C grant. B is window dressing. Mysticism just postulates that A is C because C is unknowable and makes us afraid and so all things including B must be subject to it. Art just argues that A is anything I want it to be.. just watch and listen! Physics argues that A is C cuz it just got a new A=C scope. Philosophy argues that A is A, B is B and C is C. They are all part of an didactic letternomic set with has been assigned arbitrary cultural and cognitive importance... at least that's what it says here... in this book. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> --------------- >> Allow me to summarize certain lines of reasoning relating to the >> definition of science, philosophy, and mysticism so as to make these >> ideas explicit. >> Science argues that A is C because A is B and B is C. >Very amusing.. I'll try some.... >Science argues that A is C because of a A=C grant. B is window dressing. >Mysticism just postulates that A is C because C is unknowable and makes us >afraid and so all things including B must be subject to it. > Art just argues that A is anything I want it to be.. just watch and listen! > Physics argues that A is C cuz it just got a new A=C scope. > Philosophy argues that A is A, B is B and C is C. They are all part of an >didactic letternomic set with has been assigned arbitrary cultural and >cognitive importance... at least that's what it says here... in this book. All very amusing as well, ZerkanX. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >Science argues that A is C because of a A=C grant. B is window dressing. >Mysticism just postulates that A is C because C is unknowable and makes us >afraid and so all things including B must be subject to it. > Art just argues that A is anything I want it to be.. just watch and listen! > Physics argues that A is C cuz it just got a new A=C scope. > Philosophy argues that A is A, B is B and C is C. They are all part of an >didactic letternomic set with has been assigned arbitrary cultural and >cognitive importance... at least that's what it says here... in this book. I argue that = is an emergant generalization of distinguishing dissimilarity between A and B, a conceptual positive to a physical negative. -:|:- AngleWyrm === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >>Science argues that A is C because of a A=C grant. B is window dressing. >>Mysticism just postulates that A is C because C is unknowable and makes us >>afraid and so all things including B must be subject to it. >> Art just argues that A is anything I want it to be.. just watch and listen! >> Physics argues that A is C cuz it just got a new A=C scope. >> Philosophy argues that A is A, B is B and C is C. They are all part of an >>didactic letternomic set with has been assigned arbitrary cultural and >>cognitive importance... at least that's what it says here... in this book. >I argue that = is an emergant generalization of distinguishing dissimilarity >between A and B, a conceptual positive to a physical negative. I believe I've argued myself along similar lines. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41bd1833.17476667@netnews.att.net> <41bd9f23.21123889@netnews.att.net> In physics = as in A=B, means A is directly proportional to B multiplied by something very close to 1 most of the time. In computer engineering it means, A is now B, until I tell you otherwise. The main point is, nothing is identical to anything else, or may be so, for such a short time as turns out to be not observable. But, we wouldnt get anywhere is we couldnt identify things with names in systems of knowledge (or belief). It is important for assigning meaning. So there is the approximation of the adjective, EQUALS, used in many systems, with no general definition (a definition that holds true all of the time) === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > In physics = as in A=B, means A is directly proportional to B > multiplied by something very close to 1 most of the time. In computer > engineering it means, A is now B, until I tell you otherwise. Computer science might also describe '=' as A is copied from B. How about: '=' describes an proximity relationship between two measurements, where the distance between A and B approaches zero to within some arbitrary threshhold? Just by changing the threshhold, we can say that two things are equal. For instance, my mom and dad live in different cities, but if I move the distance threshhold up: They still live in different states, but they live in the same country/continent/planet/galaxy. I would continue with same universe, but I don't have the distance between two universes to work with in this implied measurement of length. I can therefore say that '=' is a partial description of a distance delta less than a threshhold, which assumes the reader will know the scale of the threshhold, usually based on S.O.P. for units of measure employed. Seems like it might even be more appropriate to call it less than e away from, but that's just hard to pronounce. Maybe a better symbol would include the distance threshold under which we choose to ignore differences: A =e= B If we were free to designate the unit of measure for a given application, I would propose using length e as unit length, resulting in integer math. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >In physics = as in A=B, means A is directly proportional to B >multiplied by something very close to 1 most of the time. In computer >engineering it means, A is now B, until I tell you otherwise. The main >point is, nothing is identical to anything else, or may be so, for such >a short time as turns out to be not observable. But, we wouldnt get >anywhere is we couldnt identify things with names in systems of >knowledge (or belief). It is important for assigning meaning. So there >is the approximation of the adjective, EQUALS, used in many systems, >with no general definition (a definition that holds true all of the >time) I'm mostly interested in definitions that hold true all the time. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Science argues that A is C because A is B and B is C. >I'd say science argues A appears to correlate with C because A appears >to correlate with B and B appears to correlate with C and the chain is >not so long so if the probability of good enough correlation is high >enough for each link we still have a high probablity of correlation >between A and C for the whole chain. Ooooooooooooooooooooookay. === Subject: Re: Special Relativity for Mathematicians > It's impossible to understand special relativity with words. > You must be mathematically inclined. You think mathematics uses things that can't be described with words? What might those things be? > Here is the bare > minimum that's required for understanding special relativity > correctly. > http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Special Relativity for Mathematicians > It's impossible to understand special relativity with words. > You must be mathematically inclined. > You think mathematics uses things that can't be described with words? > What might those things be? The seemingly endless cycle of wasted words in trying to explain the alleged riddle of why two inertial observers see each other's clocks running slow is a waste of time. My whole approach to special relativity proves that anyone who wastes his or her time on this alleged riddle doesn't understand SR. http://www.everythingimportant.org/viewtopic.php?p=2091#2091 http://www.everythingimportant.org/relativity/special.pdf Eugene Shubert === Subject: Re: Special Relativity for Mathematicians Why should mathematicians need a special-special relativity? AFAIK the physicists aren't staking any claim of ownership. (or taking much blame ) It's a little like these emerging democracies. Why should they be writing brand new constitutions when the U.S. has one that they don't use anymore. ;-) Sue... === Subject: Re: Special Relativity for Mathematicians > Why should mathematicians need a special-special > relativity? AFAIK the physicists aren't staking any > claim of ownership. (or taking much blame ) > It's a little like these emerging democracies. Why > should they be writing brand new constitutions when > the U.S. has one that they don't use anymore. ;-) > Sue... What do you mean, anymore? Was there ever a time when they did? Ever notice they have an organisation called Veterans of Foreign Wars? That's because all the veterans of domestic wars are dead. The constitution of the Caucasian States of America doesn't extend to indigenous peoples (Navaho, Cherokee etc) and never did. We hold these truths to be self-evident. Androcles === Subject: Re: Special Relativity for Mathematicians > It's impossible to understand special relativity with words. > You must be mathematically inclined. Here is the bare > minimum that's required for understanding special relativity > correctly. > http://www.everythingimportant.org/relativity/special.pdf > Eugene Shubert Since when is your derivation the standard for understanding SR? Even if your paper had merit, which it does not, what makes you think that every other resource for understanding SR should be abandoned for your paper? === Subject: Re: Special Relativity for Mathematicians >> It's impossible to understand special relativity with words. >> You must be mathematically inclined. Here is the bare >> minimum that's required for understanding special relativity >> correctly. >> http://www.everythingimportant.org/relativity/special.pdf >> Eugene Shubert > Since when is your derivation the standard for understanding SR? > Even if your paper had merit,..., what makes you think that every > other resource for understanding SR should be abandoned for > your paper? I never said that every other resource for understanding SR should be abandoned. However, I did say that Dr. Tom Roberts needs to remove the misleading mathematical errors in his paper on SR. http://www.everythingimportant.org/relativity/special.pdf Eugene Shubert === Subject: Re: Special Relativity for Mathematicians <41bc48c9$1@sys13.hou.wt.net> >> It's impossible to understand special relativity with words. >> You must be mathematically inclined. Here is the bare >> minimum that's required for understanding special relativity >> correctly. >> http://www.everythingimportant.org/relativity/special.pdf >> Eugene Shubert > Since when is your derivation the standard for understanding SR? The most generous support I see for your paper is that it is mathematically correct - no more. The concept is flawed because it doesn't work in higher dimensions and it destroys the beautiful geometry behind SR. Your derivation of SR contributes nothing to the field. > Even if your paper had merit,..., what makes you think that every > other resource for understanding SR should be abandoned for > your paper? More intellectual dishonesty from Eugene Shubert. If you don't like what I say, you should respond instead of editing what I said to make it appear something I said wasn't. > I never said that every other resource for understanding SR should > be abandoned. However, I did say that Dr. Tom Roberts needs to > remove the misleading mathematical errors in his paper on SR. This is not about Tom Roberts! This is about you posting crap that is being ignored in the real world and laughed at in USENET. > http://www.everythingimportant.org/relativity/special.pdf ...still unfit for publishing! > Eugene Shubert Congratulations Eugene. I am firmly in the habit of checking my post before I reply to yours because you so often edit my posts then respond to them. === Subject: Re: Special Relativity for Mathematicians <41bc48c9$1@sys13.hou.wt.net> >> http://www.everythingimportant.org/relativity/special.pdf > The most generous support I see for your paper is that it is > mathematically correct may be due their mathematical incompetence. Since you are on that list of incompetent whiners, there is no reason for me to respond to any more of your ignorant tripe. http://www.everythingimportant.org/viewtopic.php?t=451 http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: Special Relativity for Mathematicians <41bc48c9$1@sys13.hou.wt.net> >> http://www.everythingimportant.org/relativity/special.pdf > The most generous support I see for your paper is that it is > mathematically correct archives, > may be due their mathematical incompetence. No. That is not what I admitted. Reading is fundamental, Eugene. Your derivation is mathematically correct *BUT* it is the same way of being correct that saying if a = b then if a = PV and b = nRT, you have derived the ideal gas law. The derivation is mathematically correct but offers no physical insight and is wrong as a physical theory. > Since you are on that list of incompetent whiners, there is no reason > for me to respond to any more of your ignorant tripe. Oh well. > http://www.everythingimportant.org/viewtopic.php?t=451 > http://www.everythingimportant.org/relativity/special.pdf === Subject: The State-of-the-Art in Mathematics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBCDJRk09922; MATHEMATICAL UPDATE AND CLARIFICATION THIS IS part of my occasional update of mathematics and physics. It refers to the resolution of mathematical issues that have appeared in several journals recently. (1) The complex number i is nonsense since i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. Dividing by i, I obtain 1 = -1 or 1 = 0 and the real number system goes down the drain.. If, instead of dividing, I add i on both sides, I get 2i = 0 or i = 0 and the complex number system vanishes in thin air. This nonsense reduces both mathematical systems to triviality. The remedy is offered in Nonlinear Studies, Vol. 5, 1998 where I replaced the nonsense i by the operator j that rotates a plane vector counterclockwise by pi/2 radians. (2) Another issue is the Ullrich-Israel equation, 1 = 0.99á; both the Ullrich-Israel equation and the symbol 0.99á are nonsense, the real number system itself being nonsense since its trichotomy and completeness axioms are false. Brouwer and Banach-Tarski counterexamples to them. I reconstructed the real number system without these axioms into the new real number system, R*, +, x, under simple axioms: Axiom 1: R* contains the basic integers 0, 1, á, 9; Axioms 1 & 2: the operations + and x are defined by the addition and multiplication tables of elementary arithmetic. I enlarged the basic integers to decimals by defining 10 = 1+9 and a decimal, say, abc.defg = a102 + b10 + c + d/10 + e/102 + f/103 +g/1104. The integral part left of the decimal point is called integer and the number on the right is called the decimal part; it is less than 1. When the decimal part is unending it is called nonterminating. It is well-defined when every digit is known or computable, that is there! is a rule for finding its digits. A well-defined nonterminating decimal N.abcá is a sequence, N.a, N.ab, á, N.ab,á, s(k), á, where there is a rule for finding the kth term s(k). Thus a periodic decimal is well-defined. A well-defined nonperiodic decimal is called irrational. R* has natural ordering < called lexicographic ordering where 0.99á < 1 (Applied Mathematics and Computation, Vol. 130, 2001) which demolishes the Ullrich-Israel equation that I now call Ullrich-Israel nonsense. Note that the well-defined real numbers are now embedded in the new real number system. (4) The dark number d* = 1 .9a 0.99á is the smallest positive new real number; R* is finite; unbounded; free from contradiction; enriched by d*, a new integer, and unbounded number u*, the largest new real number; has natural ordering (the real numbers have none) and. adequate for scientific and practical purposes. The new real numbers: N.99á, N = 0, 1, 2, á, behave like integers; therefore, I call them new integers. In particular, d* and 0.99á behave like 0 and 1, respectively. One-one correspondence (isomorphism) between the integers and new integers (N .9a 1).99á, N = 1, 2, á is established in the above paper. (4) The new integer d* satisfies these properties: given new real number x > 0 that is not a new integer, x + d* = x, (d*)^n = d*, n = 1, 2, á, and xd* = d*. The exception is: N.99á + d* = N+1, N= 0, 1, 2, á (5) The irrationals pi and the nth root of a prime, n = 2, 3, á, are well-defined because their digits are computable. (6) The introduction of qualitative mathematics, the complement of computation, extends the breadth and reach of computation; it is the principal tool of dynamic modeling. Items (2) and (5) are appearing, for the first time, in the Journal of Nonlinear Analysis and Phenomena, January 2005. With only three axioms students need not bother to memorize the 12 axioms of the real number system to learn R*. Moreover, the redundant fractions can be discarded since fractions are decimals anyway. MathForge.net provided me the forum to criticize publicly Dillon.89s and Wiles.89 supposed solutions of FLT. FLT is not only nonsense but false. This is shown by the counterexamples below. FLT says, for n > 2, the equation xn + yn = zn has no solution in nonzero integers. Let x = (0.99á)10T, y = d*, z = 10T, T = 1, 2, á Then xn + yn = (0.99á)10nT + d* = 10nT = znT. Let k = 1, 2, á, then (kx)n + (ky)n = kn(xn + yn) = knzn = (kz)n. http://www.users.bigpond.com/pidro/home.htm http://home.iprimus.com.au/pidro/ Wiles.89 proof and J. C. Dillon.89s supposedly simpler proof of FLT turn out to be wrong. No one disputed my findings and the threads are now retired in the archives of MathForge.net. E. E. Escultura === Subject: Re: The State-of-the-Art in Mathematics > MATHEMATICAL UPDATE AND CLARIFICATION > THIS IS part of my occasional update of mathematics and physics. It > refers to the resolution of mathematical issues that have appeared in > several journals recently. (1) The complex number i is nonsense since > i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. Wrong. SQRT is not a single valued function. The so-called imaginary i is one of two rootes of the equation x^2 + 1 = 0. It is no more mysterious than adjoining a formal root to the polynomial x^2 - 2 = 0 over the field of rational numbers. Bob Kolker === Subject: Re: The State-of-the-Art in Mathematics >> MATHEMATICAL UPDATE AND CLARIFICATION >> THIS IS part of my occasional update of mathematics and physics. It >> refers to the resolution of mathematical issues that have appeared in >> several journals recently. (1) The complex number i is nonsense since >> i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. -i = 1/i -i = 1/sqrt(-1) i =/= -1 i = sqrt(-1) Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: The State-of-the-Art in Mathematics > MATHEMATICAL UPDATE AND CLARIFICATION THIS IS part of my occasional update of mathematics and physics. It > refers to the resolution of mathematical issues that have appeared in > several journals recently. (1) The complex number i is nonsense since > i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. > -i = 1/i > -i = 1/sqrt(-1) typo; i =/= -i > i =/= -1 > i = sqrt(-1) Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: The State-of-the-Art in Mathematics >> MATHEMATICAL UPDATE AND CLARIFICATION >> >> THIS IS part of my occasional update of mathematics and physics. It >> refers to the resolution of mathematical issues that have appeared in >> several journals recently. (1) The complex number i is nonsense since >> i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. >> -i = 1/i >> -i = 1/sqrt(-1) >typo; i =/= -i >> i =/= -1 >> i = sqrt(-1) i = sqrt( -1) i^2 = 1 i^3 = -i Let's see if the reciprocal of 1/i really is -i. It looks like to me, the reciprocal of i is, i^-1. But, let's check, sqrt(-1)/sqrt(-1) = 1 i^1 * i^-1 = i^(1-1) = i^0 = 1 If the reciprocal of i was -1 then i^1 * (-i^1) = -i^(1+1) = -i^2 = 1 It looks like it gives the same answer in this case. But what about this ? (i^-1) * (i^-1) = i^(-2) = 1/i^2 = 1 compare, (-i^1) * (-i^1) = i^2 = 1 It looks like to me, the reciprocal of i can equal -i or i^-1, because you get the same results for either way you express it. But does, i^3 = i^-1 i^3 = -i It was shown that, -i = i^-1 So it does. And, i =/= -i But when you raise this to the power of 2, two negatives turns into positive, THEN they are equal to 1. i^2 = 1 -i^2 = 1 Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: The State-of-the-Art in Mathematics In sci.math, S. Enterprize Company refers to the resolution of mathematical issues that have appeared in > several journals recently. (1) The complex number i is nonsense since > i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. > -i = 1/i > -i = 1/sqrt(-1) >>typo; i =/= -i > i =/= -1 > i = sqrt(-1) > i = sqrt( -1) > i^2 = 1 > i^3 = -i Be *VERY* careful here. sqrt(4) = +2 or -2. sqrt(-1) is traditionally called i, but again sqrt(-1) = +i or -i. (There's at least one joke proof running around that deliberately confuses the issue by inappropriately taking the square root and proving a nonsense result.) > Let's see if the reciprocal of 1/i really is -i. Well, if b is the reciprocal of a, then a * b = +1. Since i^2 = -1, i * i * (-1) = +1, and one can take i * (-1) = -i to be the reciprocal of i. QED. > It looks like to me, the reciprocal of i is, > i^-1. > But, let's check, > sqrt(-1)/sqrt(-1) = 1 > i^1 * i^-1 = i^(1-1) = i^0 = 1 > If the reciprocal of i was -1 then Did you mean -i here? > i^1 * (-i^1) = -i^(1+1) = -i^2 = 1 > It looks like it gives the same answer in this case. > But what about this ? > (i^-1) * (i^-1) = i^(-2) = 1/i^2 = 1 Eh? 1/i^2 = 1/(-1) = -1. > compare, > (-i^1) * (-i^1) = i^2 = 1 > It looks like to me, the reciprocal of i can equal -i or i^-1, > because you get the same results for either way you express it. And the two are in fact equal. > But does, > i^3 = i^-1 > i^3 = -i > It was shown that, > -i = i^-1 > So it does. > And, > i =/= -i > But when you raise this to the power of 2, two negatives turns into > positive, THEN they are equal to 1. > i^2 = 1 > -i^2 = 1 I think you mean (-i)^2 = 1, which is also true. -(i^2) = 1 is also true. However, i^2 = -1. [.sigsnip] -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: The State-of-the-Art in Physics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBCDJRI09927; THE STATE-OF-THE-ART IN PHYSICS There is similarity and distinction between mathematics and physics. They are similar in the sense that they both study the structure, properties and behavior of their respective subject matter. The difference lies in their subject matter: that of mathematics is the representation of thought, making it a language, and that of physics is nature, making it a science. Both suffer from similar defects: mathematical spaces and their concepts are ill-defined and nature and physical concepts are also ill-defined. Until my recent critique-rectification of foundations, number theory, the real number system and physics this was true of both mathematics and physics. A physical concept is well-defined if its existence, structure, behavior, properties, etc., are specified by the laws of nature. Physics has another problem. Its present methodology of mathematical modeling that describes nature in terms of mathematical spaces is both flawed and inadequate because two distinct spaces are involved which can only be well-defined by their axioms and the laws of nature, respectively. Therefore, solving a physical problem in some mathematical space is flawed and this is the reason mathematical physics has long standing problems such as the gravitational n-body problem, finding the basic constituent of matter and what the structure of the electron is. This inadequacy of methodology leads to error. This occurs, for instance in the search for the basic constituents of matter. The main difficulty is the ambiguity on the requirements they must satisfy. There are two: (1) they must be indestructible otherwise the first law of thermodynamics would not hold and (2) some basic constituents must comprise every piece if matter. It follows from (2) that there can only be one basic con! stituent aside from form. Immediately, the physicists.89 candidate for basic constituent, the quark, does not qualify because it is destructible. Moreover, in view of (2) the electron must consist of quark, too, and they do not seem to be looking for it there. Another error in physics lies in the physical concept .8bblack hole..8a First of all, it is ill-defined. I think physicists agree that a black hole is dark matter and yet it is supposed to suck visible matter around it. This is a contradiction since dark matter does not interact with visible matter. The remedy is a new methodology, dynamic modeling that EXPLAINS nature in terms of its laws. This methodology was first applied to solve the gravitational n-body problem in 1997 (Nonlinear Analysis, Vol. 30, No. 8). The crucial step was to find the appropriate laws of nature to well-define the key physical concepts in this problem, .8bbody.8a and .8bgravity..8a It took 11 natural laws to do this; they constitute the initial laws of nature that well-define gravity and establish the flux theory of gravitation. To well-define .8bbody.8a it was necessary to well-define the basic constituent of dark matter, the superstring. Then natural laws were discovered that convert dark matter to visible matter. These eleven laws of nature yielded the solution of the gravitational n-body problem. Moreover, every time this theory is applied to solve new problems new laws of nature are required. To-date, 42 natural laws including biological laws and laws of learning, mathematical principles and a general! law have been discovered that anchor the theories of turbulence, intelligence and learning and explain the Columbia Space Shuttle disaster. They anchor the full development of the flux theory of gravitation and applications (published in a series of over a dozen papers in Nonlinear Analysis, Nonlinear Analysis and Phenomena, Applied Mathematics and Computation, Indian Journal of Pure and Applied Mathematics and proceedings of several international conferences). For full listing of references visit my websites: http://www.users.bigpond.com/pidro/home.htm http://home.iprimus.com.au/pidro/ E. E. Escultura === Subject: Re: Counterexamples to FLT by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBCDJRn09940; >Do you have a different posting address each day or something? >Sure sign of a troll or a loon - morphing all over the place. >Phil >-- >The gun is good. The penis is evil... Go forth and kill. pointing them out. Your post inadvertently reveals the absence of worthwhile ideas. E. E. Escultura === Subject: Re: Counterexamples to FLT by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBCDJSQ10019; >Do you have a different posting address each day or something? >Sure sign of a troll or a loon - morphing all over the place. >Phil >-- >The gun is good. The penis is evil... Go forth and kill. I now post again the messages with the corrected posting addresses. E. E. Escultura === Subject: Re: Change in meaning of inertial motion in Newton to Einstein > Well he did a poetic thing, so how does one respond? > I am the universe > What universe will prevail upon your senses beyond > that one? > That's GR! Everybody has a different universe, and > even a termite has a universe. Different systems. > I enjoy a philosophical to the precepts of GR, that's > ok. I think everyone will have one universe in their > existence, as a summation of experience in space > time and maybe a few more dimensions. > Furthermore, since all humans share one thing in > common, and that is, they each and only have one > universe relatively, then we are all relatively equal. Systems are not separate, but infinitely embedded. > That statement is very important in the Consitution > of the United States, and I think it wise. > Take a retarded kid, he has one universe, take > a genious he has one universe. The number of > universes is invariant as intelligence varies, and > will always equal one, furthermore, the fathers's > of our constitution set forth a policy of equality, > by stating all men's (and later womens) universes > are equal. > In my mind the General Principle of Relativity, > is a practical way of underwriting democracy, and > I understood better the meaning of equality as I > came to appreciate GR. In a very abstract sense I would agree with this. It's important to consider the different contexts each inhabit. This is analogous to coevolution where systems produce positive sum relationships as opposed to conflict. But complexity science shows so much more. The power structure of society must be modeled after nature. By following a power-law distribution. In a pyramid of layers of equal thickness, each level must have the same total power as all the others. So the bottom layer....the people...must have equal power to the one at the peak. And so on. Also, a society must follow nature in that the structure must be a balance between the static, dynamic and chaotic attractors. So law, competition and freedom must have equal weight. > So if Jonathan needs to fuse GR and poetry, with > something as precious as human rights, I'll help. Complexity science is universal, interdisciplinary, so it works as well with physics as it does with human rights, clouds even emotions. Works everywhere in fact. Jonathan > Ken S. Tucker === Subject: Re: Change in meaning of inertial motion in Newton to Einstein > I am the universe > I see that you'll have to wait for your granddaughter > to explain to you how the universe really works > ...pity. > And the fact that a simple frame of reference > problem is the source of classical ignorance > makes this all the more tragic. > Jonathan > I am the universe > Is GR in your poetry, PERCEPTION of an Object costs Precise the Object's loss. Perception in itself a gain Replying to its price; The Object Absolute is nought, Perception sets it fair, And then upbraids a Perfectness That situates so far > it's true for all, I was > hoping you might have thought that through for > yourself. The poem shows why your frame is backwards. There is inherent uncertainty or incompleteness with observing when reducing to the smallest to derive fundamental laws. You expect that the physical universe will display these laws so that we might comprehend the more complex such as life. That is backwards. It is through life that we understand the nature of the physical universe. Our fundamental laws must be derived from evolving systems....or complex adaptive is the realm from which a unified theory must emanate. Or more specifically, from Darwin...evolution...the science of self organizing systems, originally called chaos theory, now called complexity science. Life is a cyclic attractor solution. So is the universe. As shown by the Quintessence paper from the physics dept at Princeton. http://wwwphy.princeton.edu/~steinh/ A complete online book is below from a graduate level course taught at MIT. If you dare to exit the Dark Age that is. http://www.necsi.org/publications/dcs/ A brief intro into the 'integral' of the Enlightenment is here from Oxford. http://users.ox.ac.uk/~quee0818/complexity/complexity.html And another nice site from the Univ of Mich http://www.pscs.umich.edu/ Univ of Md http://www-chaos.umd.edu/ Faq http://www.calresco.org/sos/sosfaq.htm Jonathan FAITH is a fine invention For gentlemen who see; But microscopes are prudent In an emergency! Poems by E Dickinson s > BTW Jonathan, how fast are you moving and how > fast are you accelerating? > Ken === Subject: Re: Change in meaning of inertial motion in Newton to Einstein <41b7d775$1_4@127.0.0.1> <41b8ec21$1_3@127.0.0.1> <41bc44e7$1_2@127.0.0.1> Hi Jonathan et al... IMO, the existentialist philosophy is complimentary to the Theory of Relativity., except the the existentialist philosophy does integrate experience, including genetic into the preception. Memory and experience are certainly beyond the immediate spacetime. But consider our current solar-system, it to is a result of many inputs of energy over time, suppose I were to conject that our solar system was like a photograph. Turn our attention to the moon's surface, if anything is like a photographic plate that *statically* accumulates knowledge over the billion year period, that qualifies., we see a surface - like a photographic plate - accumulating energy data for billions of years, by impact craters. In comparison, Earths dynamic, subsumes, and revitalizes it's surface so everything there is more recent. Too my point, while we humans have so far a brief glimpse of spacetime and the accumulation of that knowledge in some organized manner, we have the benefit to understand our immediate solar system has done that for billions of years for our experience. So I will argue, the moon portrays it's memory in an existential fashion, unhibited and apparent, of it's personal experience within the solar system. Silly Dr. Dolittle speaks to the animals. Ken S. Tucker === Subject: Re: Change in meaning of inertial motion in Newton to Einstein <41b7d775$1_4@127.0.0.1> <41b8ec21$1_3@127.0.0.1> So what you're saying Mike is to refrain from discussing GR with poetic philosophers on their terms? I'll need to give up talking to myself! Minkowski to Einstein, I raise you one time, and that is spacetime, Einstein to Minkowski, with a sneer replies you've had one to many beer. Ok, I won't quit my day job. Ken === Subject: Re: The State-of-the-Art in Mathematics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBCEDiM14611; >i = sqrt(-1) = sqrt(1/-1) = 1/i am I missing some state-of-the-art? The operation sqrt(-1)=sqrt(1/-1)=sqrt(1)/sqrt(-1) is only valid if the product is positive. This is clearly not the case now.. Indeed in general sqrt(uw)=sqrt(u)sqrt(w) is not generally true in the complex plane. >MATHEMATICAL UPDATE AND CLARIFICATION >THIS IS part of my occasional update of mathematics and physics. It refers to the resolution of mathematical issues that have appeared in several journals recently. > (1) The complex number i is nonsense since i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. Dividing by i, I obtain 1 = -1 or 1 = 0 and the real number system goes down the drain.. If, instead of dividing, I add i on both sides, I get 2i = 0 or i = 0 and the complex number system vanishes in thin air. This nonsense reduces both mathematical systems to triviality. The remedy is offered in Nonlinear Studies, Vol. 5, 1998 where I replaced the nonsense i by the operator j that rotates a plane vector counterclockwise by pi/2 radians. > (2) Another issue is the Ullrich-Israel equation, 1 = 0.99á; both the Ullrich-Israel equation and the symbol 0.99á are nonsense, the real number system itself being nonsense since its trichotomy and completeness axioms are false. Brouwer and Banach-Tarski counterexamples to them. I reconstructed the real number system without these axioms into the new real number system, R*, +, x, under simple axioms: Axiom 1: R* contains the basic integers 0, 1, á, 9; Axioms 1 & 2: the operations + and x are defined by the addition and multiplication tables of elementary arithmetic. I enlarged the basic integers to decimals by defining 10 = 1+9 and a decimal, say, abc.defg = a102 + b10 + c + d/10 + e/102 + f/103 +g/1104. The integral part left of the decimal point is called integer and the number on the right is called the decimal part; it is less than 1. When the decimal part is unending it is called nonterminating. It is well-defined when every digit is known or computable, that is ther! e! > is a rule for finding its digits. A well-defined nonterminating decimal N.abcá is a sequence, N.a, N.ab, á, N.ab,á, s(k), á, where there is a rule for finding the kth term s(k). Thus a periodic decimal is well-defined. A well-defined nonperiodic decimal is called irrational. R* has natural ordering < called lexicographic ordering where 0.99á < 1 (Applied Mathematics and Computation, Vol. 130, 2001) which demolishes the Ullrich-Israel equation that I now call Ullrich-Israel nonsense. Note that the well-defined real numbers are now embedded in the new real number system. > (4) The dark number d* = 1 .9a 0.99á is the smallest positive new real number; R* is finite; unbounded; free from contradiction; enriched by d*, a new integer, and unbounded number u*, the largest new real number; has natural ordering (the real numbers have none) and. adequate for scientific and practical purposes. The new real numbers: N.99á, N = 0, 1, 2, á, behave like integers; therefore, I call them new integers. In particular, d* and 0.99á behave like 0 and 1, respectively. One-one correspondence (isomorphism) between the integers and new integers (N .9a 1).99á, N = 1, 2, á is established in the above paper. > (4) The new integer d* satisfies these properties: given new real number x > 0 that is not a new integer, x + d* = x, (d*)^n = d*, n = 1, 2, á, and xd* = d*. The exception is: N.99á + d* = N+1, N= 0, 1, 2, á > (5) The irrationals pi and the nth root of a prime, n = 2, 3, á, are well-defined because their digits are computable. > (6) The introduction of qualitative mathematics, the complement of computation, extends the breadth and reach of computation; it is the principal tool of dynamic modeling. > Items (2) and (5) are appearing, for the first time, in the Journal of Nonlinear Analysis and Phenomena, January 2005. With only three axioms students need not bother to memorize the 12 axioms of the real number system to learn R*. Moreover, the redundant fractions can be discarded since fractions are decimals anyway. > MathForge.net provided me the forum to criticize publicly Dillon.89s and Wiles.89 supposed solutions of FLT. FLT is not only nonsense but false. This is shown by the counterexamples below. FLT says, for n > 2, the equation xn + yn = zn has no solution in nonzero integers. Let x = (0.99á)10T, y = d*, z = 10T, T = 1, 2, á Then xn + yn = (0.99á)10nT + d* = 10nT = znT. Let k = 1, 2, á, then (kx)n + (ky)n = kn(xn + yn) = knzn = (kz)n. >http://www.users.bigpond .com/pidro/home.htmhttp://home.iprimus.com.au/pidro/< /a Wiles.89 proof and J. C. Dillon.89s supposedly simpler proof of FLT turn out to be wrong. No one disputed my findings and the threads are now retired in the archives of MathForge.net. >E. E. Escultura === Subject: Basic factoring algorithm? Yesterday, I made a post mentioning this basic factoring idea I had a couple of days ago. Now I'm going to step it out into an algorithm. To see the derivation of the formulas that follow see my previous thread. The base equation is yx^2 + Ax - k = T where T is the target, and A is picked. y = n + T, where n is given by n = (2k +/- sqrt((2k + T)^2 + A^2 - T^2)) where you use the factors of A^2 - T^2 to determine k, so that n will be an integer. That's done simply enough as if you have factors 4f_1 f_2 = A^2 - T^2, and 2k + T = f_1 - f_2 then (2k+T)^2 + A^2 - T^2 = f_1^2 - 2f_1 f_2 + f_2^2 + 4f_1 f_2 so (2k+T)^2 + A^2 - T^2 = f_1^2 + 2f_1 f_2 + f_2^2 = (f_1 + f_2)^2 and you have simply enough that n = (2k +/- (f_1 + f_2)). Here to minimize that value of A^2 - T^2 you can just let A = floor(sqrt(T)) or A = floor(sqrt(T)) + 1 where the first gives a negative number and the second a positive one. Now you have y and k, so you can go back to yx^2 + Ax - k = T and substitute those values in, which allows you to solve for x, and x will be rational, but it will be fraction. Now you also need to factor the left side, to get the factors of T. One way is to introduce z, where yz^2 + Az - k = 0, and solve for z. It will come out rational and a fraction, and you make sure that the numerators and denominators of its solutions are coprime, so you have z_1 = -b_1/a_1, and z_2 = -b_2/a_2 and that gives (a_1 x + b_1)(a_2 x + b_2) = yx^2 + Ax - k so now you have all the values you need. To finish you just substitute in the value of x, from earlier as you have (a_1 x + b_1)(a_2 x + b_2) = T and simplify, and you will necessarily have factors of T, but I don't know if they'll be trivial or not. If T is prime, of course, you'll get T(1), or (-T)(-1) or something like that, but at this point, mathematically I don't see why you wouldn't get non-trivial factors of T if it's not prime. But I've STILL not tested this idea, only worked it out theoretically and now given the algorithm. Talking it out helps me. Hopefully if there's some flaw in it, I'll see it sooner than later, versus investing a lot of time and effort in an idea that doesn't work. For me talking it out is part of a process that usually has ended in failure. But hey, it's also an engaging process for me, which I enjoy, so I keep at it! James Harris === Subject: S_n, transposition Is it true that any transposition and any n-cycle wiill generate S_n? Tony === Subject: Re: S_n, transposition > Is it true that any transposition and any n-cycle wiill generate S_n? > Tony Not true in general, e.g. if you number the vertices of a square for 1 to 4, the symmetry group of that square will become a subgroup of S_4 (of order 8) that contains both a transposition (diagonal reflection) and a 4-cycle (rotation by 90Á). However, an n-cycle c=(a_1 ... a_n) and a transposition t=(a_i a_(i+1)) of *successive* points in c *will* generate S_n. (Hint: wlog assume c=(12...n) and t=(12), then repeatedly conjugate t by c.) LD === Subject: Re: S_n, transposition > Is it true that any transposition and any n-cycle wiill generate S_n? It is not true. For a counterexample, consider S_4, let p = ( 1 2 3 4 ; 2 3 4 1 ), q = ( 1 2 3 4 ; 3 2 1 4 ). p is an 4-cycle and q is a transposition. Let's look at the parities: (E for even, O for odd) p will map ( O E O E) to ( E O E O ), and (E O E O ) to (O E O E). q will map ( O E O E ) to (O E O E), and (E O E O) to (E O E O). So every combination of p and q will map (O E O E) to (O E O E ) or to (E O E O). So it will not map (1 2 3 4), which is (O E O E), to (1 3 2 4), which is (O O E E). That is, (1 2 3 4; 1 3 2 4) will not be generated by p and q. -- Dae-jung Yoo (IGNJSA YOO) (Delete DELETE to reply by e-mail) === Subject: Galois groups of quartics help What's the easiest way to compute Galois groups of quartics over Q? For example, f(x) = x^4 + x^2 - 6 I could always take the easy way out and calculate the discriminant and resolvent cubic and just figure it out from there. But what if I do not have access to discriminant formula (say, during a test)? Is there an easier way to calculate the Galois group of something like this? For a cubic it's much easier because the discriminant is easier to memorize and all one has to check is if it's the square of a rational number. To start, by rational roots theorem, the above f(x) has no rational roots. Taking the derivative, we get 4x^3 + 2x, which equals zero when x = 0 or when x^2 = -1. So there is only one real valued critical number. Taking derivatives again, we get 12x^2 + 2, which is always positive and which equals zero never (on the real line). So we have 2 real roots and 2 complex roots since f(0) < 0 and as x approaches negative and positive infinity, we go towards infinity. Thus, 2 real roots and 2 complex roots implies that Gal(f) contains a transposition. Now if I was able to tell that f(x) was irreducible (which I haven't shown whether f can or can't split into quadratics), then Gal(f) must contain a 4-cycle since adjoining a root gives a degree 4 extension and thus 4 divides |Gal(f)|, thus contains an element of order 4 which can only be a 4-cycle. Thus if it contained a transposition and 4-cycle, it would be all of S_4. Is there a different way to solve this problem, and in general, a different way of solving Galois groups of quartics? I ask because the method above I believe won't always work...Also, is f irreducible? Tony === Subject: Re: Galois groups of quartics help You do seem to have most of the tools needed to handle many of the quickie special cases. We'll stick with characteristic zero stuff for the moment. A general principle that I don't think you explicitly mentioned is that the Galois group for an irreducible polynomial would be transitive, in the sense that any two roots can be mapped one to another by suitable choice of automorphism. To drop back, though, is x^4 + x^2 - 6 irreducible? Think of factoring a quadratic in x^2. === Subject: Re: Galois groups of quartics help === Subject: Re: 2 difficult groups of order 16 > * The group G_{4,4} with presentation > I haven't looked at his, and all I see from C_4 acting on > C_2 x C_2 is interchanging the generators, since the action must have > order 2, and A(C_2 x C_2) = S_3. > So I will also have to work more on this. > Van This has errors, and I started a new thread titled 2 different presentation of D_n with n = 2^m where I would prefer to continue this. Van === Subject: Re: 2 difficult groups of order 16 > Anyway, The remaining 2 which I have not studied or seen > before are * The group with presentation subgroups of order 2. That is, if x and y are the central elements > of order 2 in D_4 and C_4, respectively, then D_4 o C_4 is the > quotient group (D_4 x C_4) / <(x,y)>. > I have looked into the central problem, which I haven't come across > yet, and its interesting, especially as it seems to give a completely > different presentation and way of looking at > ; as > (D_4 x C_4) / <(x,y)>. > In the notation I was using, which is neither of the above, > D_4 = C_4 x| C_2 = where x^4 = 1 = y^2 is the usual presentation > of D_4. Let (the other) C_4 = , and let > G = D_4 x C_4 = (C_4 x| C_2) x C_4 = ( x| ) x , and |G| = 32. > Then A == <(x^2,z^2)> = C_2, and D_4 o C_4 = G/A and |G/A| = 16. > Let g_ijk = x^i y^j z^k ; where i,k in Z_4 and j in Z_2. Then > the elements of D_4 o C_4 = G/A are (g_ijk, g_(i+2)j(k+2) ). > e.g. a == (g_111,g_313) = (xyz,x^3yz^3), and a^2 = (x^4 z^2, x^4 z^2) > = (z^2,z^2) ==> a^4 = (1,1) = 1 so |a| = 4, as above. > b = (g_101,g_303) = (xz, x^3 z^3), and b^2 = x^2 z^2 (1,1) ==> b^4 = 1, > c = (g_100,g_302) = (x, x^3z^2) ; c^2 = x^2 (1,1) ==> c^4 = 1 > Well, I was trying to find the other element(s) of order 2, and > show that we have and iso between this and > I haven't looked at his, and all I see from C_4 acting on > C_2 x C_2 is interchanging the generators, since the action must have > order 2, and A(C_2 x C_2) = S_3. What I meant here is that if (C_2 x C_2) x| C_4 has generators x,y,z with z^4 = 1 = x^2 = y^2, then z acts on by interchanging x and y, but does the semi-direct product alway refer to action by inner auto?. Let N = = (1,x,y,xy) is abelian. I am not sure how z acts on Z_2 x Z_2. I mean, is it by the inner auto so that zxz^-1 = y; zyz^-1 = x ; z(xy)z^-1 = xy = yx ? Then I though consider (C_2 x C_2) x| C_2 = G with |G| = 8, and see what group I get. But if I use the above method, all elements x,y,z have order 2, so I should get Z_2^3. zxz = y ; zyz = x ; z(xy)z = xy = yx I had a hard time proceeding from here. I have probably overlooked something obvious. Any comments are welcome. Van === Subject: Presentations of D_n when n = 2^m I have found (from Zundark) an alternative presentation of D_4 which should be useful for alternative presentations of groups of order 8, 16, and I think for 2^m, m > 4. First D_n = Z_n x| Z_2 = where x^n = 1 = z^2 and zxz^(-1) = x^(-1). This is the usual presentation most are familiar with. For n = 2^m = 4 say, consider the groups of order 8; I show that Z_4 x| Z_2 = (Z_2 x Z_2) x| Z_2 = D_4. Z_4 x| Z_2 appears to have 2 generators, and (Z_2 x Z_2) x| Z_2 appears to have 3, but they are both D_4. I will discuss the following group of order 16 later; > * The group G_{4,4} with presentation > This is just (C_2 x C_2) x| C_4. Note that G_{4,4} has only 2 generators, while (C_2 x C_2) x| C_4 appear to have 3, though one considers the groups as generated by elements of different orders in the 2 cases. This led me to consider (C_2 x C_2) x| C_2 which turned out to be D_4. C_2 x C_2 = (1,x,y,xy = yx). Let C_2 = act on C_2 x C_2 by an inner auto as follows: zxz = y ; zyz = x ; z(xy)z = xy = yx Then let a = xz, a^2 = xzxz = xy, a^3 = yxxz = yz, a^4 = xzyz = x^2 = 1 and let z act on C_4 = ; zaz = zxzz = zx = yz = a^3 ==> D_4. Van === Subject: Re: Why (or not) use single letter variable names? >> Why do mathematicians use single letter variable names (even when >their >> work is intended to be read by others) when programmers are typically >> taught to use descriptive variable names? >Because mathematicians write big symbols on blackboards, which are slow >input devices. >Then, mathematicians name things in a common abstraction. Tolerance >becomes Epsilon, Angle becomes Theta, etc. These are arbitrary, >and some mathematicians even take the time to define each one at the >start of their writings. Knowing this arbitrary mapping becomes a >secret handshake for mathdom. >> Meanwhile in programming land people are happy to use long variables >> names and insist that code is unreadable without. >Right. We make up variable names much faster and more frequently than >mathematicians. Each math proof pushes the value of previous proofs by >one tiny increment, so nearly all the letters get re-used from the last >proof. Programmers, by contrast, must write entirely new functions, >from scratch, with _specific_ angles and tolerances in them. It could be done much more efficiently and produce better code. Computer terminology seems to be deliberately intended for confusion with most of mathematics terminology beyond arithmetic and simple algebra; symbols with many mathematical meanings are misused, and simple ways of doing things well are replaced by SUPPOSEDLY idiot-proof methods which nobody with intelligence can use reasonably well. Also, as Einstein said, nature will produce better idiots. We need to get away from ASCII and have longer codes, which can avoid writing epsilon, but just have the one Greek letter appear on the screen or the print, without print spacing. Editing becomes otherwise a laborious task. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Why (or not) use single letter variable names? >> Note, this isn't a rhetorical question. I'm asking quite literally what >> the difference between these fields is that results in quite different >> naming conventions? >Mostly, it's because mathematicians are so much smarter than >programmers that they don't need to be clubbed over the head >repeatedly to be reminded what a variable stands for. >(Even more, mathematicians _don't care_ what the variable >stands for.) This is correct. A mathematical argument (or a program) does not depend on the meaning of its variables to be valid. I find the use of long names for variables, locations, functions, etc., to be annoying and also a cause for typing errors. They stand out better with short names. I also produce code which needs goto's to be efficient. Mathematics is intended to be understood by people who know the background; frankly, we need good WYSIWYG means of writing it, with fixed spacing, so it is easy to edit and produce. It seems difficult to do now. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Why (or not) use single letter variable names? > I find the use of long names for variables, locations, functions, > etc., to be annoying and also a cause for typing errors. Please, learn how to use word completion in your editor, or get a new editor, if your current one can't do this. For example both vim and emacs have this capability (CTRL-N and CTRL-P in vim). Tomasz === Subject: Re: Why (or not) use single letter variable names? |In mathematical notation juxtaposition denotes multiplication: |ab is a times b. If 'ab' were also allowed as a variable name |then there would be no way to tell whether ab was a single variable |or the product of two variables. | |In the typical programming language, otoh, ab must be a single |variable name because the product of a and b is a*b. It would be possible for mathematicians to drop this notational convention, which leaves us with the question of why we don't do so in order to have more mnemonic variable names. I'd say that it's partly that in mathematics we tend to have more multiplications appearing in formulas than in programming. Once in awhile we wind up with some slightly tedious formula in a program; someone might write a[0]*a[0]+a[1]*a[1]+a[2]*a[2], say. But in some areas of mathematics at least, the formulas are liberally sprinkled with multiplications. Exponents are used in programming infrequently enough that a lot of languages don't have a build-in exponential function. We can get some idea of the awkwardness of using * or ^ in mathematics by the way it is to write formulas in ASCII form. We still can usually do without *s for multiplication, but we wind up with enough ^ characters for exponentiation to make it feel a little clunky at times to me. We also wind up writing out Greek letters, which leads to our needing some *s after all, as in 2*pi*i, which just looks so much less elegant than the printed form. Some versions of what would in a program be a function application are also denoted in mathematics with slick position notation like subscripts on series. In ASCII form, or TeX, having to type the _s sometimes seems clunky to me. One ends up writing a_{n+1} and the like, which is just plain less elegant. One reason for longer names is to cope with a larger number of identifiers that need to be kept distinct. One of the main sources of large numbers of identifiers that need to be kept distinct is the presence of many things that need to be referred to globally. Programs often have a lot of them. There is some locality; a subroutine usually has its own stash of variables that don't need to be differently named from variables used elsewhere, for example. It seems as though those variables are often the ones with the shortest names, however. Variables that are to be used right away in a small loop and then not used again are more likely to be given single-letter names than the average identifier is. At least some of the identifiers used in proofs, especially the more global ones, are given longer names. Subroutines in programs are somewhat analogous to lemmas in proofs, and lemmas are sometimes provided with longer and more descriptive names. I do sometimes see formulas written as rate * time, at least in textbooks. There are other notations that are essentially departures from the convention of using single-letter variable names. In a notation like GL(n,R), the GL is taken as a single unit and not a product of G and L. Likewise for the trig functions sin, etc., and tor, ext, rad, spec and so on. (I have heard of students who are initially confused by thinking sin is s*i*n.) To some extent decorated variables such as u'' or V^{^} count as names, and they are more than just single letters. Mathematical logic is awash in alphabet soup, with all the acronyms for different kinds of systems. I know at least one programmer who thinks it would be kind of cool if we could use unicode characters for variable names. Occasionally, for instance, we use variables with names like theta, and he thinks it would be cute if we could just have a little theta instead. The inconveniences that go with using non-ASCII character sets in programming are worse, however, which also adds to the incentive to use longer names there. Keith Ramsay === Subject: Re: Why (or not) use single letter variable names? > I know at least one programmer who thinks it would be kind of cool > if we could use unicode characters for variable names. Occasionally, > for instance, we use variables with names like theta, and he thinks > it would be cute if we could just have a little theta instead. The > inconveniences that go with using non-ASCII character sets in > programming are worse, however, which also adds to the incentive > to use longer names there. > Keith Ramsay You can, for example in Java where identifiers are in Unicode. Most programming editors and IDE's (integrated development environments) are not geared towards Unicode. Many are. You might consider the Intentional Programming concept where the input is a DSL (Domain Specific Language) that may be iconic or graphical, or in the form of a mathematical formula, eg via the MathML content layer. Ha ha ha, Simonyi: Intentional Programming relates to a Programming Language as a powerset relates to a set. What a nerd. Basically Intentional Programming is the use of a domain specific language, sometimes with a specialized input editor. It is propped in Czarnecki's survey Generative Programming, he mentions spending a summer working on it. Intentional Programming doesn't hold much of a candle to Automatic Programming. What you would probably want at some level is Literate Programming, built upon TeX by Knuth. Me, I basically promote cross-platform or portable programming via code generation and specification reuse. See http://www.tiki-lounge.com/~raf/BlandDesignDocument.html, Bland: a system and a programming language. The ASCII-fication as we see on sci.math, , or as Leroy Quet calls it linear mode, for what it may be cumbersome, it is at least legible in most cases. Ross Finlayson === Subject: Re: Why (or not) use single letter variable names? linear mode, for what it may be cumbersome, it is at >least legible in most cases. More imporantly, it doesn't take a GB's worth of running software to read it. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Why (or not) use single letter variable names? if we could use unicode characters for variable names. Occasionally, >for instance, we use variables with names like theta, and he thinks >it would be cute if we could just have a little theta instead. Your programmer needs to consider a lot more things than cute. Some of the programmers I worked with (a long time ago) could type 60 words/minute. Adding more characters on the keyboard would slow them down. Yes, children, code is still typed in using keyboard-flavored interfaces by Real Programmers. Another thing that he needs to consider very seriously is backwards compatibility. Now, I know that this is not a PC thing but dismissing this as a goal is going to byte everybody's butt soon. I could on with other problems with being cute. :-) > ..The >inconveniences that go with using non-ASCII character sets in >programming are worse, however, which also adds to the incentive >to use longer names there. People are using longer names because they can. I would have never allowed the functional specs to pass if I'd been in those development groups. The reason people use these things is to avoid documenting what they do. Take a good look at the mess we have on certain PCs and you'll figure out why documenting thoroughly in human-readable form is a basic necessity in the computing biz. A programmer who knows his biz will not have to read the purpose of a variable every time he needs to reference it in his code. This is wasting valuable time and resources. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Why (or not) use single letter variable names? >Why do mathematicians use single letter variable names (even when > their >work is intended to be read by others) when programmers are typically >taught to use descriptive variable names? ... > If I am trying to program the Black Scholes formula (for example), > it is much better for me to > define a function (in Fortran, for illustration) > real function > call_price(spot,strike,volatility,time_to_exp,rate,div_yield) > real :: spot,strike,volatility,time_to_exp,rate,div_yield > ! some code > end function call_price ... I prefer writing a smaller obvious name or abbreviation with a comment line when necessary. That way it serves twin purposes of brevity and explanation. === Subject: Re: Why (or not) use single letter variable names? >>Why do mathematicians use single letter variable names (even when >> their >>work is intended to be read by others) when programmers are >typically >>taught to use descriptive variable names? >.... >> If I am trying to program the Black Scholes formula (for example), it is much better for me to >> define a function (in Fortran, for illustration) >> real function >> call_price(spot,strike,volatility,time_to_exp,rate,div_yield) >> real :: spot,strike,volatility,time_to_exp,rate,div_yield >> ! some code >> end function call_price >.... >I prefer writing a smaller obvious name or abbreviation with a comment >line when necessary. That way it serves twin purposes of brevity and >explanation. Absolutely! Note that your method also reduces the probability of making a typo with undesired side effects to almost nothing. That's why having a variable name that completely describes its purpose is a bad idea. The reason comments in program sources was invented was to communicate with the human reader, not the compiler nor assembler. Side note: if you're using a beta version of newsreader software, it has a bug that puts a space after the comma in the newsgroup list. Another example of the difference between rules of puncuation of compilers and human readable material. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Why (or not) use single letter variable names? >>Why do mathematicians use single letter variable names (even when their >>work is intended to be read by others) when programmers are typically >>taught to use descriptive variable names? >Seven replies and nobody's mentioned one compelling reason: >In mathematical notation juxtaposition denotes multiplication: >ab is a times b. If 'ab' were also allowed as a variable name >then there would be no way to tell whether ab was a single variable >or the product of two variables. >In the typical programming language, otoh, ab must be a single >variable name because the product of a and b is a*b. Right. A parser has to be able to tell the difference. In programming, parsing is set off by characters chosen by the spec; I hate languages that use a non-printing character to mean something other than separation. >>Mathematicians will use single letter variable names even for complex >>sructures like M for a differential manifold with Riemannian metric or >>H for the function that maps said objects to their (co)homology groups. >>Given a shortage of single letter variables mathematicians would rather >>change font (to Greek or Gothic) or add decorations (such as a prime, a >>'hat') to single letters rather than use longer names. They'll even use >>single letter variable names in pseudocode when they describe >>algorithms! >>Meanwhile in programming land people are happy to use long variables >>names and insist that code is unreadable without. Just a quibble for the OP. Not all programmers are happy with that. But this is definitely a subject for another newsgroup. :-) /BAH Subtract a hundred and four for e-mail. === Subject: Re: Why (or not) use single letter variable names? <98vlr0lem8g57f1vbe5hfk1cqhg0pek26t@4ax.com> <87u0qswltv.fsf@phiwumbda.org> Although I must confess to being a mathematician, I also program frequently and I have to say I disagree with *always* using long variable names. Of course I use them for function, module and abstract data type names (it's essential to do so); but for variables inside a procedure I always use single letters. This works fine if you write short procedures like you're supposed to and is, to my mind, much easier to read. Maybe I should say that in particular I disagree to use long names for procedure parameters: it just discourages writing adequate documentation. (But, let me empasize, it is necessary for functions, procedures, modules, etc.) === Subject: Re: Why (or not) use single letter variable names? >Although I must confess to being a mathematician, I also program >frequently and I have to say I disagree with *always* using long >variable names. Of course I use them for function, module and abstract >data type names (it's essential to do so); but for variables inside a >procedure I always use single letters. This works fine if you write >short procedures like you're supposed to and is, to my mind, much >easier to read. >Maybe I should say that in particular I disagree to use long names for >procedure parameters: it just discourages writing adequate >documentation. (But, let me empasize, it is necessary for functions, >procedures, modules, etc.) It is not that necessary for those latter; it is used for convenience. Computer misuse of ASCII characters also adds to the problem; hypergeometric functions use the ; to separate fields of arguments. And while IBM apologized for using ** for exponentiation, C and its descendents require a called function, which Fortran often avoided. And the use of for so many irrelevant purposes, where it has numerous uses in mathematics, does not help. Since Fortran, mathematicians have been kept out of the process. It shows. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 > answer. Over on usernet there was someone who needed to answer a > question real bad and gave the correct wrong answer. > Carl Sagan on worm holes on his show Cosmos. After his description he > said I have been to parties like that. We should enjoy though. But we > all have moments. > Merry Christmas, Pascal. Note, however, that Lynn's answer is also correct. Pascal would have loved combinations, if he didn't know about them. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Differential Geometry Lie Group Question Let G be a Lie Group acting on a manifold M. Let L_g : M -> M be the diffeomorphism multiplication by g in G. Then we can define, given p in M and omega a differential form in $M$, phi : G -> wedge^q T_p^* M by g |-> L_g^*omega_p. (where the last string of symbols mean the q-exterior power of the cotangent space at p of M). I've been trying to show that phi is continuous. Any help ? nojb. === Subject: How big is the electron? Jack, since you know so many physicists let me ask you some things. In your theory and in the physics community in general, 1. How big is an electron, what's its radius? The classical ball size of an electron is e^2/mc^2 ~ 10^-13 cm The quantum size is h/mc ~ 10^-11 cm The classical ball is surrounded by a virtual electron-positron plasma that extends out from 10^-13 cm to 10^-11 cm This is the NAIVE picture. Problem is that Bohr's Copenhagen -> Von Neumann interpretation says THERE ARE NO ELECTRONS as Newton's hard massy balls - there are only quantum waves or qubits that collapse to make localized clicks on counters. This is Wheeler's Smoky Dragon mysticism. That is, there are NO HIDDEN VARIABLES. leads to problems of infinite renormalization parameters. There is then the scattering theory of form factors - complicated. Also high energy scattering of electrons on protons and neutrons shows the latter have three point like scattering centers - consistent with quarks. Quarks and leptons are unified in gauge theory. Well not quite. Bohm's theory does allow the HIDDEN VARIABLES i.e. Newton's hard massy balls that are guided by the Bohr quantum pilot waves. It's like a ball rolling on a landscape of hills valleys and saddle point mountain passes. These are the Bohmian trajectories of the hidden variables. Then there is the idea that electrons and quarks are really little microgeons i.e. localized regions of pure vacuum with CONFORMAL CURVATURE fields like rotating charged black holes. Burinski is working on that now in Moscow. However you need strong short range gravity G* ~ 10^40G on the scale of e^2/mc^2 ~ 1 fermi for this to work. Maybe the extra space dimensions kick it for nuclear physics. Also for the stable electron you need a special solution that does not have Hawking radiation - that is part of Kerr-Newman. The microgeons will shrink if probed with a big kick in a high-resolution scattering Heisenberg microscope experiment. They look - at least up to a critical value, where maybe they start looking bigger again in Susskind's IR/UV duality of string theory. One nice thing about rotating micro-geons in strong short range gravity is that they automatically obey the universal Regge slope law Spin of hadronic resonance ~ alpha'E^2 alpha' ~ (1 Gev)^-2 This is the basic data for string theory and of course black holes and strings are two sides of the same coin! Also no hierarchy problem in this picture where the quantum gravity Planck scale Lp is actually at ~ 1Gev. I published this in 1973 and Abdus Salam invited me to ICTP because of that. 2. What is the equivalent lineal mass density (kg/m) of the vacuum? Meaningless question. The naive zero point energy of the vacuum WITHOUT the COHERENCE that is the basis for my theory of the emergence of Einstein's General Relativity as a More is Different post-inflationary macro-quantum coherent phase modulation is Vacuum Energy Density ~ hc/Lp^4 This is MUCH TOO BIG. The dark energy density is ~ hc/(c/H)^4 Where H = today's FRW Hubble parameter R^-1dR/dt. If you use Lp ~ 10^-33 cm Hal's PV gives is off by 122 Powers of Ten. That is a colossal failure compared to GR's colossal success of 10^-14 in the 1916 + 13 pulsar data that won a Nobel Prize! Hal Puthoff has no solution to this problem in PV. He hand waves it away as he genuflects to ET. :-) http://www.guntheranderson.com/v/data/thevatic.htm http://www.hyperdictionary.com/dictionary/genuflect I'm doing a comparison of GR and PV and, I'd like to compare it to your model as well. However, I'm getting mixed input about the radius of an fm. Milonni on the other hand says it is point-like and 10^-2 fm but its charge is spread out by the interaction with the ZPF to about the Compton wavelength, ~10^3 fm. The classical radius is bogus. Do you agree? Sort of. See above. PV is worthless, a complete waste of time. 1. It violates general coordinate transformation covariance of the dynamical action. 2. It is incompatible with the LOCAL equivalence principle used in GR (e.g. Cartan tetrads). 3. It is seriously incomplete e.g. it cannot describe rotating bodies and the now observed gravimagnetic Lense-Thirring frame drag of the Cartan tetrads. 4. PV fails every experimental test beyond the 3 trivial classic tests in new observations like 1913+16 pulsar where GR works beautifully to a precision of 10^-14! The above opinion on Hal's PV is the consensus of several top physicists at GR 17 e.g. Cliff Will, Matt Visser, Bill Unruh ... Since at less than 1/2 the Compton wavelength we get the possibility of e-p pair creation, this would seem to fit well with your model of the vacuum. So I'm hoping you can give me some specifics to compare to. main_engineering === Subject: Re: How big is the electron? > Jack, since you know so many physicists let me ask you some things. > In your theory and in the physics community in general, > 1. How big is an electron, what's its radius? > The classical ball size of an electron is > e^2/mc^2 ~ 10^-13 cm > The quantum size is > h/mc ~ 10^-11 cm > The classical ball is surrounded by a virtual electron-positron plasma > that extends out from 10^-13 cm to 10^-11 cm > This is the NAIVE picture. > Problem is that Bohr's Copenhagen -> Von Neumann interpretation says > THERE ARE NO ELECTRONS as Newton's hard massy balls - there are only > quantum waves or qubits that collapse to make localized clicks > on counters. This is Wheeler's Smoky Dragon mysticism. That is, there > are NO HIDDEN VARIABLES. > leads to problems of infinite renormalization parameters. > There is then the scattering theory of form factors - complicated. > Also high energy scattering of electrons on protons and neutrons shows > the latter have three point like scattering centers - consistent with > quarks. Quarks and leptons are unified in gauge theory. Well not quite. > Bohm's theory does allow the HIDDEN VARIABLES i.e. Newton's hard massy > balls that are guided by the Bohr quantum pilot waves. It's like a > ball rolling on a landscape of hills valleys and saddle point mountain > passes. These are the Bohmian trajectories of the hidden variables. > Then there is the idea that electrons and quarks are really little > microgeons i.e. localized regions of pure vacuum with CONFORMAL > CURVATURE fields like rotating charged black holes. Burinski is working > on that now in Moscow. However you need strong short range gravity G* ~ > 10^40G on the scale of e^2/mc^2 ~ 1 fermi for this to work. Maybe the > extra space dimensions kick it for nuclear physics. Also for the stable > electron you need a special solution that does not have Hawking > radiation - that is part of Kerr-Newman. > The microgeons will shrink if probed with a big kick in a > high-resolution scattering Heisenberg microscope experiment. They look > - at least up to a critical value, where maybe they start looking bigger > again in Susskind's IR/UV duality of string theory. > One nice thing about rotating micro-geons in strong short range gravity > is that they automatically obey the universal Regge slope law > Spin of hadronic resonance ~ alpha'E^2 > alpha' ~ (1 Gev)^-2 > This is the basic data for string theory and of course black holes and > strings are two sides of the same coin! > Also no hierarchy problem in this picture where the quantum gravity > Planck scale Lp is actually at ~ 1Gev. I published this in 1973 and > Abdus Salam invited me to ICTP because of that. > 2. What is the equivalent lineal mass density (kg/m) of the vacuum? > Meaningless question. > The naive zero point energy of the vacuum WITHOUT the COHERENCE that is > the basis for my theory of the emergence of Einstein's General > Relativity as a More is Different post-inflationary macro-quantum > coherent phase modulation is > Vacuum Energy Density ~ hc/Lp^4 > This is MUCH TOO BIG. The dark energy density is > ~ hc/(c/H)^4 > Where H = today's FRW Hubble parameter R^-1dR/dt. > If you use Lp ~ 10^-33 cm Hal's PV gives is off by 122 Powers of Ten. > That is a colossal failure compared to GR's colossal success of 10^-14 > in the 1916 + 13 pulsar data that won a Nobel Prize! > Hal Puthoff has no solution to this problem in PV. He hand waves it away > as he genuflects to ET. :-) > http://www.guntheranderson.com/v/data/thevatic.htm > http://www.hyperdictionary.com/dictionary/genuflect > I'm doing a comparison of GR and PV and, I'd like to compare it to your > model as well. However, I'm getting mixed input about the radius of an > fm. Milonni on the other hand says it is point-like and 10^-2 fm but its > charge is spread out by the interaction with the ZPF to about the Compton > wavelength, ~10^3 fm. The classical radius is bogus. Do you agree? > Sort of. See above. PV is worthless, a complete waste of time. > 1. It violates general coordinate transformation covariance of the > dynamical action. > 2. It is incompatible with the LOCAL equivalence principle used in GR > (e.g. Cartan tetrads). > 3. It is seriously incomplete e.g. it cannot describe rotating bodies > and the now observed gravimagnetic Lense-Thirring frame drag of the > Cartan tetrads. > 4. PV fails every experimental test beyond the 3 trivial classic tests > in new observations like 1913+16 pulsar where GR works beautifully to a > precision of 10^-14! > The above opinion on Hal's PV is the consensus of several top physicists > at GR 17 > e.g. Cliff Will, Matt Visser, Bill Unruh ... > Since at less than 1/2 the Compton wavelength we get the possibility of e-p > pair creation, this would seem to fit well with your model of the > vacuum. So > I'm hoping you can give me some specifics to compare to. > main_engineering The electron is a spiral-shaped cloud extending outward from the nucleus *in exactly the same way* that a galaxy's arm is made. They are the same thing. John galaxy model for the atom http://www.petcom.com/~john/ === Subject: Deep Thoughts # 37: I just realized what's wrong with the Continuum Hypothesis The cardinality of a set is not well-defined. What is the cardinality of: 1. A = { x | x = N v (eY)YeA ^ x=P(Y) } 2. A = { x | x = N v x = P(A) } 3. A = { x | ~(x e A) } 4. A = { x | x e A } 5. A = { x | x e (A u {0}) } 6. A = { x | x e x } 7. A = { x | ~ (x e x) } Which have less cardinality than which? Also include P(A) , A u B , A n B etc. for A and B from this list. To tell if A 0$, then the two series both converge or diverge. -------------------------------------------------- My question would be, what happens if the limit doesn't exist. If the limit doesn't exist, then is it just inconclusive, or implies something about the relationship between the two sums? delta01211 === Subject: Re: Limit Comparison Test > Limit Comparison Test: > Given sum_a_{k} and sum_b_{k}, suppose > lim_{k to infty}frac{a_{k}}{b_{k}} = rho. > If $rho$ is finite and $rho > 0$, then the two series both > converge or diverge. > -------------------------------------------------- > My question would be, what happens if the limit doesn't exist. > If the limit doesn't exist, then is it just inconclusive, or > implies something about the relationship between the two sums? In part, the answer would be, you have to use common sense. Given below are some examples in which we are unable to apply your version of the Test. Yet we can make definitive conclusions. If a(n) and b(n) are positive for all large n, and sum b(n) diverges, then when lim[a(n)/b(n)] is infinite, a(n) clearly diverges too by ordinary comparison test. If a(n) and b(n) are positive for all large n, and sum b(n) converges, then when lim[a(n)/b(n)] = 0, a(n) clearly converges too by ordinary comparison test. === Subject: Re: Limit Comparison Test > Limit Comparison Test: > Given sum_a_{k} and sum_b_{k}, suppose > lim_{k to infty}frac{a_{k}}{b_{k}} = rho. > If $rho$ is finite and $rho > 0$, then the two series both > converge or diverge. > -------------------------------------------------- > My question would be, what happens if the limit doesn't exist. > If the limit doesn't exist, then is it just inconclusive, or > implies something about the relationship between the two sums? You are assuming, I take it, that both a_k and b_k are positive for all k. I guess the same conclusion can be made if the limsup and liminf are both finite and positive. -- Dae-jung Yoo (IGNJSA YOO) (Delete DELETE to reply by e-mail) === Subject: Re: Limit Comparison Test days. My association with the Department is that of an alumnus. >Limit Comparison Test: >Given sum_a_{k} and sum_b_{k}, suppose >lim_{k to infty}frac{a_{k}}{b_{k}} = rho. >If $rho$ is finite and $rho > 0$, then the two series both >converge or diverge. >-------------------------------------------------- You are assuming, I take it, that both a_k and b_k are positive for almost all k. >My question would be, what happens if the limit doesn't exist. >If the limit doesn't exist, then is it just inconclusive, or >implies something about the relationship between the two sums? It depends on how if fails to exist. If the limit fails to exist because the limit is infinite, then the divergence of sum(b_k) implies the divergence of sum(a_k) but that's the best you can say. Likewise, if the limit is 0 and sum(b_k) convergences, then sum(a_k) converges. Both the above are assuming the series are of positive terms. I believe that if the limit fails to exist but is not infinity, then the test is simply inconclusive. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: what are the correct set theory axioms? > This is to unambiguously determine the scope of the quantifier Ay. Yes, that's a good idea. [snip similar helpful words] > In the absence of the Axiom of Regularity, the set {a,{a,b}} is not > guaranteed to uniquely determine a and b. If a = {{a,b},c} for some c, > and a is not equal to {a,b}, then {a,{a,b}} determines two distinct > possibilities for the ordered pair corresponding to the set {a,{a,b}} > (i.e. (a,b) and ({a,b},c)). On the other hand, the Kuratowski definition > of the ordered pair (a,b) := {{a},{a,b}} uniquely determines a and b, even > without the Axiom of Regularity. I can just use Kuratowski, but is regularity normally insisted on? I though people just wanted to acoid sets getting too big, does non-regularity lead to some kind of contradiction? [snip more bad scoping on my part] >Separation: AF Ax Ey Az [(zey) <=> (zex & F(z)=0)] > Are you sure that you want to use a function, or a class-type function > (for want of a better way of describing a function which can be a proper > class, instead of just a set), here? It seems to me that using a > function may be begging the question somewhat, since it could be that > {zex : F(z) = 0} may be used in the definition of F. In ZFC, the axiom > scheme states that for all formulae P(z) (possibly with other free > variables, but not x or y), > Ax Ey Az [zey <=> (zex & P(z))]. I wanted SO axioms, so I was trying to write it in the leading existential part of SOL, since I thought that was the only part I needed. So the capital F is a SO function, and the only SO functions I need to state the axioms are the ones such that F(x) is a set y such that Q(x,y) is a true FO sentance for the sets x and y unless there are no such sets, in which case there is probably a trivial function that suffices, I'll have to look into it, I'm going to think about that separation from replacement thing you mentioned below. [snip yet more bad scopes on my part] >Choice: Ax (x=0) or (Ey yex & y=0 ) or [Ez (Aa Ab (aex & bex) => (Ec >Ed {a,{a,c}}ez & {b,{b,d}}ez & [a=b => c=d] & cea) & (Ar rez => [Et Eu >tex & r={t,{t,u}}])))] > This should be rewritten as: > Ax [x=0 or (Ey (yex & y=0)) or [Ez (Aa Ab [(aex & bex) => (Ec > Ed ({a,{a,c}}ez & {b,{b,d}}ez & [a=b => c=d] & cea))] & > (Ar (rez => [Et Eu (tex & r={t,{t,u}})])))]] > to unambiguously determine the scopes of the quantifiers. > This is not actually the Axiom of Choice, or an equivalent to the Axiom of > Choice, although I can see how the individual component formulae relate to > the Axiom of Choice. In line with my previous observations about > ordered pairs in the absence of the Axiom of Regularity, I will use > {{a},{a,b}} for the ordered pair (a,b). Try the following for the Axiom > of Choice: > Ax [x = 0 or Ey (yex & y = 0) or Ez (Aa (aex => Ec ({{a},{a,c}}ez > & cea)) & Aa Ac Ad (({{a},{a,c}}ez & {{a},{a,d}}ez) => c = d) & > Ar (rez => Et Eu (tex & r={{t},{t,u}})))] You've already been most kind, was the error with (a,b) the only problem or do you think I might have some conceptual misunderstanding of the axiom of choice? >Replacement: AF [(Aa Ax [aex => (Ey Ab bey <=> F(a,b)=0)]) => (Ez [ >(Aw [wex => Ec {w,{w,c}}ez]) & (Ar [rez <=> ([Et Eu tex & r={t,{t,u}}] >& [Ad deu <=> F(d,t)=0])])])] > This is incorrect. The scope of the quantifier Ax is aex => Ey Ab (bey > <=> F(a,b) = 0]), and x appears as a free variable outside this scope. > The quantifier for x and the appearance of x in the antecedent is > superfluous, since the antecedent, Aa Ax [aex => (Ey Ab (bey <= F(a,b)=0))], is equivalent to Aa Ey Ab (bey <=> F(a,b)=0), and it is > simply the statement that for all a, the class of all b such that > F(a,b) = 0 is a set. This means that for any F such that, for some a, > the class of all b such that F(a,b) = 0 is a proper class, the antecedent > is false, and so the implication is true. > The formula in the final square brackets, Ad (deu <=> F(d,t)=0), states > that the elements of u are exactly those d such that F(d,t) = 0. Note > that t and u are both free variables in the formula in the square > brackets, and furthermore, these occurrences of t and u are free in the > full formula as you have stated it. > In the second last set of square brackets, the formula Et Eu (tex & > r={t,{t,u}}) tells us that r is an ordered pair (taking note of any > modifications required to account for any absence of the Axiom of > Regularity) in which the first component is an element of x. > The scope of the quantifier Ar therefore states that r is an element of > z iff r is an ordered pair in which the first component is an element > of x and the elements of u are exactly those d such that F(d,t) = 0 > (although I am unsure about how this relates to r). > It follows that the elements of z are those ordered pairs such that the > first component is an element of x and such that the elements of u are > exactly those d such that F(d,t) = 0. Consequently, if the elements of > u are exactly those d such that F(d,t) = 0, then there is no set z > satisfying the condition, and if the elements of u are not exactly those > d such that F(d,t) = 0, then z must be the empty set. > The first part of the the scope for the quantifier Ez tells us that for > each element of x, there there is an ordered pair in z with that element > as its first component. This requires that z be nonempty if x is > nonempty. > It follows that the conclusion in your statement above can only be true if > x is empty and if the elements of u are not exactly those d such that > F(d,t) = 0. > So your full formula above is the statement that x is empty and that the > elements of u are not exactly those d such that F(d,t) = 0. This is > not the Axiom of Replacement. > My favoured version for the Axiom of Replacement is that for all formulae > Q(x,y) (possibly with other free variables, but not u or v), > [Ax Ay Az ((Q(x,y) & Q(x,z)) => y=z)] => Au Ev Ay (yev <=> Ex > (xeu & Q(x,y))) > The Axiom of Separation actually follows as a consequence of this > statement of the Axiom of Replacement (set Q(x,y) to be P(x) & x=y). Doesn't (Q(x,y) & Q(x,y))=>(y=z) just mean that for every x there is either a unique set y than makes Q(x,y) true, or that there is no such set y that makes Q(x,y), so that sounds functionally equivalent to that proper class of SO function F(x)=y that satisfy sentances. So would AF Aw Ev Ay (yev <=> Ex (xew & F(x)=y)) be the second order equivalent of the correct axiom of replacement scheme? The problem I see is for sentances like Q(x,y)=(xey & ~xey) which are never true, there is no function F(x), such that Q(x,y) <=> F(x)=y, so the functions don't seem to work do all the same work as the sentances, but I'd like a single SO axiom instead of a scheme of FO axioms. In this case, the set y is the empty set, so an additional axims Ey Az ~zey would cover it, and actually infinity covers that, so is that a sufficient correct single SO axiom given the other axioms? (And I can drop the SO separation axiom?) >Are these correct? > Your statements of the Axioms of Choice and Replacement were wrong. > Without the Axiom of Regularity (or any of its equivalents), it is > not guaranteed that your expression {a,{a,b}} properly represents an > ordered pair. The expression {{a},{a,b}} does properly represent an > ordered pair. If you do have the Axiom of Regularity, then {a,{a,b}} > does properly represent an ordered pair. >Are there more? > ZFC also has the Axiom of Regularity: Ax [(Ey yex) => Ez (zex & At (tez > => ~tex))]. David, are the second order versions actually stronger in proving more, or are they just a finite collection of axioms instead of an infinite scheme? === Subject: Prime numbers problem When I divide any prime number by 30, the remainder is always prime number. Can anybody help me to prove this? Is there any helpful theorem for this issue? Can anybody point me to some links on Web with theorems for prime number theory? Waldek === Subject: Re: Prime numbers problem > When I divide any prime number by 30, the remainder is always prime > number. > Can anybody help me to prove this? > Is there any helpful theorem for this issue? > Can anybody point me to some links on Web with theorems for prime number > theory? Primes>30 are coprime to 30. So are of the form 30n+{1,5,7,11,13,17,91,23,29} All of the above remainders, apart from 1, are prime. Primes<30 are prime. Therefore, apart from numbers of the form 30n+1, your statement is true, but that exception makes it false. If you used non-composite instead of prime, then you'd be OK. Phil -- The gun is good. The penis is evil... Go forth and kill. === Subject: Re: Prime numbers problem days. My association with the Department is that of an alumnus. >When I divide any prime number by 30, the remainder is always prime >number. >Can anybody help me to prove this? Doubtful, since 31 is a prime but 1 is not. Now, let us assume instead that you want to show that if p is a prime, then the remainder of dividing p by 30 is either 1 or a prime. Well, this is obviously true if p<30. For p>30, p must be odd, so it must be congruent to an odd number modulo 30. In addition, it cannot be congruent to 3 (because 3+ 30k is divisible by 3), 5 (5+30k is a multiple of 5), 9 (9 + 30k is divisible by 3), 15 (divisible by 5), 21 (divisible by 3), 25 (divisible by 5), or 27 (divisible by 3). That leaves as the only possibilities 1, 7, 11, 13, 17, 19, 23, and 29, which are all either prime or equal to 1. Note that in addition, there are infinitely many primes of the form 30k+1, so 1 occurs infinitely often. After 31, it occurs at 61, 151, 181, 211, etc. >Is there any helpful theorem for this issue? The observation that 30 is divisible by 3 and 5, is about it. >Can anybody point me to some links on Web with theorems for prime number >theory? You can start at the Prime Pages: http://www.utm.edu/research/primes/ -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Prime numbers problem >When I divide any prime number by 30, the remainder is always prime >number. >Can anybody help me to prove this? > Doubtful, since 31 is a prime but 1 is not. > Now, let us assume instead that you want to show that if p is a prime, > then the remainder of dividing p by 30 is either 1 or a prime. > Well, this is obviously true if p<30. For p>30, p must be odd, so it > must be congruent to an odd number modulo 30. In addition, it cannot > be congruent to 3 (because 3+ 30k is divisible by 3), 5 (5+30k is a > multiple of 5), 9 (9 + 30k is divisible by 3), 15 (divisible by 5), 21 > (divisible by 3), 25 (divisible by 5), or 27 (divisible by 3). > That leaves as the only possibilities 1, 7, 11, 13, 17, 19, 23, and > 29, which are all either prime or equal to 1. > Note that in addition, there are infinitely many primes of the form > 30k+1, so 1 occurs infinitely often. After 31, it occurs at 61, 151, > 181, 211, etc. >Is there any helpful theorem for this issue? > The observation that 30 is divisible by 3 and 5, is about it. >Can anybody point me to some links on Web with theorems for prime number >theory? > You can start at the Prime Pages: > http://www.utm.edu/research/primes/ Supplementary: Prove that 30 is the largest integer with this property. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Prime numbers problem > When I divide any prime number by 30, the remainder is always prime > number. False for primes smaller than 30. Tapio > Can anybody help me to prove this? > Is there any helpful theorem for this issue? > Can anybody point me to some links on Web with theorems for prime number > theory? > Waldek === Subject: Re: Prime numbers problem days. My association with the Department is that of an alumnus. >> When I divide any prime number by 30, the remainder is always prime >> number. >False for primes smaller than 30. Huh? If p is a prime and p<30, then the remainder is p, which is a prime. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Golf competition The undermentioned solution was sent by e-mail by Ross Moore. It satisfies the two most important criteria and I accept it as the best possible solution. {{2, 5, 7, 9}, {0, 1, 11, 6}, {3, 4, 8, 10}} {{3, 1, 10, 7}, {4, 5, 9, 6}, {0, 2, 11, 8}} {{5, 0, 6, 10}, {2, 3, 11, 9}, {4, 1, 7, 8}} {{0, 4, 11, 7}, {5, 3, 8, 6}, {1, 2, 10, 9}} {{0, 3, 9, 8}, {2, 4, 6, 7}, {1, 5, 11, 10}} Nobody partners the same player more than once and every player plays in the same group with every other player at least once - this was termed full socialisation by Ross and it was to achieve this that I posted the problem in the first place. Does this answer Dave Rusin's postscript? Bobby === Subject: Re: Golf competition >{{2, 5, 7, 9}, {0, 1, 11, 6}, {3, 4, 8, 10}} >{{3, 1, 10, 7}, {4, 5, 9, 6}, {0, 2, 11, 8}} >{{5, 0, 6, 10}, {2, 3, 11, 9}, {4, 1, 7, 8}} >{{0, 4, 11, 7}, {5, 3, 8, 6}, {1, 2, 10, 9}} >{{0, 3, 9, 8}, {2, 4, 6, 7}, {1, 5, 11, 10}} >every player plays >in the same group with every other player at least once When does player 0 play with players 4 or 7? === Subject: Re: Golf competition >>{{2, 5, 7, 9}, {0, 1, 11, 6}, {3, 4, 8, 10}} >>{{3, 1, 10, 7}, {4, 5, 9, 6}, {0, 2, 11, 8}} >>{{5, 0, 6, 10}, {2, 3, 11, 9}, {4, 1, 7, 8}} >>{{0, 4, 11, 7}, {5, 3, 8, 6}, {1, 2, 10, 9}} >>{{0, 3, 9, 8}, {2, 4, 6, 7}, {1, 5, 11, 10}} >>every player plays >>in the same group with every other player at least once >When does player 0 play with players 4 or 7? Third game, isn't it? -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: An interesting matrix by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBDH7TL25166; >Here is an interesting matrix at www.guffy.net/matrix.htm. >It is constructed from a very simple algorithm with the result being that >the matrix is upper triangular with the appearance of diagonals that have >the same pattern as the rows from which the matrix is constructed. >I have no conclusions about the matrix but simply wanted to share it with >others. David, I discovered (with mathematica) that the inverse of your curious matrix had the same structure of its mate, and could be computed recursively : Define your matrix : In[1]:=mat[n_]:=Table[Take[Flatten[PadLeft[{#},i]& /@ Table[1,{n}]],n],{i,1,n}]; First line of inverse : In[2]:=inv1[1]={1}; inv1[n_]:=Append[inv1[n-1],-inv1[n-1]. Drop[mat[n][[All,-1]],-1]] Inverse filled the same way as mat: In[4]:=inv[n_]:=Table[Take[Flatten[PadLeft[{#},i]& /@ inv1[n]],n],{i,1,n}] Example with a 10x10 matrix : In[5]:=inv[10] Out[5]= {{1,-1,-1,0,-1,1,-1,0,0,1}, {0,1,0,-1,0,-1,0,0,0,-1}, {0,0,1,0,0,-1,0,0,-1,0}, {0,0,0,1,0,0,0,-1,0,0}, {0,0,0,0,1,0,0,0,0,-1}, {0,0,0,0,0,1,0,0,0,0}, {0,0,0,0,0,0,1,0,0,0}, {0,0,0,0,0,0,0,1,0,0}, {0,0,0,0,0,0,0,0,1,0}, {0,0,0,0,0,0,0,0,0,1}} In[6]:=inv[10].mat[10] == mat[10].inv[10] == IdentityMatrix[10] Out[6]=True On the first line one gets 0 if position is a multiple of 4 and -1 if position is a prime. Valeri === Subject: Re: Extrapolation > If x is the number of days for which you are forecasting sales, then in > a simple model where there is no trend, just an average number of sales > per day and a standard deviation (for the daily sales distribution), > then extrapolating to multiple days (based on the normal distribution > behavior) would mean the error term grows like SQRT(x). So, for my forcast I want: y = mx + (sd)(sqrt(x)) ? > So, your intuition is right. (1) where the error term is a constant 2 > * sd is too low, and (2) where you let the error term grow like a > factor of x is too high. > Seems like I should be dragging Goldilocks into the story. === Subject: Re: physical significance by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBDGJVX20260; Often. Example, mechanics, energy is conserved iff the hamiltonian is unchanged as a result of time reversal, that is H(-t)=H(t). joccis >Hello Niel, >Thanx for the post. >I am seeking whether any abstract theory of mathematics or physics use the >concept of complex time? If so how? >Shashi === Subject: Re: terminology question >> If a^n is the nth POWER of a, what is n called in a^(1/n), where n is > an >> integer? I need the one word, if there is one. >> To explain why I'm asking, I'm trying to find a good name (just n > won't >> do) for such a variable in a computer program I'm writing. >> Jack > if a^n is the n-th power of a then n is called the exponent. If a^(1/n) > is the n-th root of a then n is called the index > http://mathworld.wolfram.com/Index.html Jack === Subject: Re: terminology question > radicand ? Jack === Subject: Re: terminology question >> radicand ? > Jack Oops. I was a bit hasty. Index is the answer. Jack === Subject: Re: arithmetic function with sepcial properties > given 2 natural numbers x, y, I am searching for an arithmetic function f > that using the input x, y has the following result: > 1.) x = y => f(x,y) = 1 > 2.) x != y => f(x,y) = 0 > I was searching a while for an arithmetic function that has these 2 > properties, but sadly my math-skills are not that superior. I was thinking > about using the following: > f(x,y) = (x-y)^2 + 1 > which satisfy the property 1.) but sadly not the second property. Maybe > there is any more complex arithmetic-function which satisfy the 2 propertys? > Anika If we evaluate (x=y) logically using conventional truth values (True=1 and False=0), we can simplify your function to: f(x,y) = (x=y) BTW, most programming languages permit this type of logical evaluation. === Subject: On Selfcontained Systems Consider the following: 1. Set Theory in which you only have sets. That is, every element of every set is a set. 2. Recursion Theory in which the input and output of programs are programs. 3. Logic in which we replace free variables in a formula with the symbol for the Godel number of a formula and consider whether the formula for a given number is provable. Examples: 1: {} and {{}} and {{},{{}}} are sets. 2: A program can input a program and output a program that produces the same result as running the first program and then using its output as input and running it again. f(x) => f(f(x)) Call them Selfcontained Systems. Note that each of these is a subset of a non-Selfcontained system! 1. What other Selfcontained Systems do we know of? 2. What systems have no Selfcontained subset? 3. How do we formally define each of these? 4. Abstract: What do they have in common and what does each have besides what they have in common? 5. Formalize their commonality and differences. Develop a higher level system in which theorems translate into theorems in every Selfcontained System. C-B === Subject: Re: Starting Calculus in January... bring with me to study and on lunch breaks at work, highlight, etc... >> I'll just call you man too cheap to buy a printer. >What's with the hostility here? Printing a website is hardly a >substitute for a book. Maybe I don't want to print some website but >would rather have a decent book recommended? Sorry, didn't mean to sound hostile, I should have added a smiley to that statement. Sometimes the best info available comes in the form of a website. When I was trying to learn Hungarian and couldn't find a good reference, I printed a 20-page tutorial I found on the web. It was much much better than nothing. On the original topic, I wouldn't recommend any of the calc text's I've used for uninstructed study as they were all intolerably dry. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Starting Calculus in January... >...at university. Any recommendations on material I can read until then >to get a leg up on the course? It sounds like you are looking for a book to read in December before you buy your textbook at university. If that's the case, I can recommend the following. Don't let the name bother you. I was looking around the bookstore this weekend and happened to see The Idiot's Guide to Calculus. It appeared to be very good as a quick introduction to Calculus. It will probably be made fun of here but would be a good book to have around for a couple of weeks just to get a quick overview of the major concepts. It's also less than $20. Then study very hard from your textbook when class starts and make sure you work problems every day. In math, working problems is the key to learning. The more problems you work, the more you learn. Buying an expensive textbook now when you are going to be buying one in a month might be a waste of money if your money is tight like many students. You surely won't have time for two when classes start. Good luck at school. If you do buy the book, it would be nice if you could post a message letting others know if it was helpful for your purpose. Bill === Subject: Re: Starting Calculus in January... > ...at university. Any recommendations on material I can read until then > to get a leg up on the course? There are hosts of introductory calculus texts, but if you're really interested in mathematics you may benefit from reading something entirely different. The little book by Otto Toeplitz, The Calculus: a Genetic Approach works through the history of elementary calculus, showing what originally motivated various ideas. Some bits are harder than others, but you could find it a more rewarding book than the usual texts. Don't neglect the exercises which are collected together near the end of the book. After that, you're likely to have much better insights into the calculus you learn formally next year. Good luck! Ken Pledger. === Subject: Re: Starting Calculus in January... >> ...at university. Any recommendations on material I can read until then >> to get a leg up on the course? > There are hosts of introductory calculus texts, but if you're >really interested in mathematics you may benefit from reading something >entirely different. The little book by Otto Toeplitz, The Calculus: a >Genetic Approach works through the history of elementary calculus, >showing what originally motivated various ideas. Some bits are harder >than others, but you could find it a more rewarding book than the usual >texts. Don't neglect the exercises which are collected together near >the end of the book. > After that, you're likely to have much better insights into the >calculus you learn formally next year. Mr. Pledger, please don't be insulted when I ask my question because I don't know you and don't have any idea if you're one those pests ;-). Is this recommendation a good one? It sounds like a book that would be worthwhile buying at full price. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Starting Calculus in January... Calculus writing style has changed a lot in the past fifty years. Back then, textbooks were slimmer -- amd smaller format -- because you were being taught the essentials, rather than Everything you ever wanted to know about calculus and were afraid to ask. You might proceed more speedily if you could lay your hands on such a book. I might also mention that the first edition of Leithold, although a thick volume, gives a lot of explanation in plain English. David Ames === Subject: Re: Starting Calculus in January... > Calculus writing style has changed a lot in the past fifty years. Back > then, textbooks were slimmer -- amd smaller format -- because you were > being taught the essentials, rather than Everything you ever wanted to > know about calculus and were afraid to ask. That's NOT what I remember. They all weighed 10 pounds. They had depth. But maybe my university liked big books with heavy content. -- BTW: In Australia, Calculus was taught in high school (I was learning 1D calculus at 15, including the background analysis, theorems, limits, etc). At Uni, we had overseas students from different parts of the US and [parts of] the UK who were way out of depth. European and Asian overseas students usually had no major problems with calculus. Tomasso. === Subject: Re: Starting Calculus in January... >> Calculus writing style has changed a lot in the past fifty years. Back >> then, textbooks were slimmer -- amd smaller format -- because you were >> being taught the essentials, rather than Everything you ever wanted to >> know about calculus and were afraid to ask. >That's NOT what I remember. They all weighed 10 pounds. They had depth. >But maybe my university liked big books with heavy content. I never met a calc intro book that wasn't heavy with both--content and mass. The ones I've been picking up at the dump are getting published in two volumes and each are the size of the Thomas I used in 1968. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Starting Calculus in January... >>Be sure that you know how to use trig identities like sin(a+b) = >>sin(a)cos(b) + sin(b)cos(a), and can expand expressions like (2x + >>3)*(2x^2+7), remember how to use exponents and logs, generally that your >>basic algebra is in place - it will make it a lot easier. >Cool. It's been a semester since my precalc/trig class, so I'll >definitely go over the trig identities again. Any theorems that will >definitely come up often in a first year calc. course? >> If you want to see theorems, find a Thomas text. >Okay. What's the author's first name? >into an Applied Math degree. I took HIS121 since it's required and was >convenient for me this term. The CIS115 class my adviser said I had to >take, but it turns out I didn't need it and so spent an entire semester >bored to tears on Monday nights... :-/ >> Yea, well. Think of it as one of those hoops you have to prove >> before you get handed the real fun stuff. >> Since you have this kind of work experience, another text that >> gives lots and lots and lots of application examples using >> calc is the author Munen. He doesn't get into the formal math >> as much as Thomas. >Excellent! I'll look up both authors at the library straight away. >Munem only has one book in our county libraries and it's unrequestable. It was the text the last time I took calc (I did the same thing you are doing and went back but had to retake all my math.) I bought the teacher's edition of Munen; it had more answer to the problems and an extra four chapters of material. [emoticon goes off to its library to retrieve texts] Well, it also would help if I could spell correctly. Author: Munem and Foulis; ISBN: 0-87901-236-6. This is the second edition and 1984 by Worth Publishers. George B. Thomas, Jr. The one I have is the fourth edition but I saw a 7th (I think) at our dump. >Thomas didn't show up in the list. Good grief! What is the world coming to? > .. But, I did find _Introduction To >Calculus_ by Joan Dykes. I appreciate the response. You're welcome. If your town has a recycle place for books, look for discarded texts. The local public library book sales usually have tons of textbooks for $1 or less. Pick up as many texts as you can since every author has a different style of presentation. /BAH Subtract a hundred and four for e-mail. === Subject: Need your help to solve a PDE. Hi All, I need to solve an equation. It would be a great help if you solve this. The problem is mentioned at http://www.geocities.com/satyaranjandas77/PDE.pdf. I had to go this way because I was not able to send an attachment. I had made many attempts myself, but since it is more than 5 years I am out of college I need to relearn how to solve PDE. :( Is it possible to solve this equation using mathmatica? Even the power series solution will work just fine for me. satya --- Below are some more related info ---- I did try Phi(r, theta) = F(r)G(theta), but this did not work. I had tried the followings too: Phi(r, theta) = (1/r) Psi(r, theta) -> to reduce the equation to look more like solvable. Phi(r, theta) = Psi(r sin(theta)/r_0, theta) -> to make the equation dimensionless, and from the equation it seems like r sin(theta) is more natural. Phi(r, theta) = Psi(r_0 sin(theta)/r, theta) -> here r and r_0 have changed their position. I did not try power series because the equation is in two variables. Boundary condition is that Phi vanishes as r->oo. Other condition on Phi is that it is symmetric about theta = pi/2 => Phi(r, theta) = Phi(r, pi - theta). I even tried a trial function: Phi(r, theta) = (r /(r_0^2 sin^2 theta)) (1 - exp(-r^2/(r_0^2 sin^2 theta))), but this does not satisfy the equation. You may ignore my attempts because I am not too sure if I was doing the right thing. I realized that my mathematical capabilities have been deteriorated during last five years. satya === Subject: Re: Another Galois group question by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBDELrZ06844; >Hi everyone, >I'm trying to find the Galois group of f(x) = x^4 + x^2 + x + 1. I have >found that there are no real roots. Thus, the Galois group contains a >product of two disjoint transpositions. Also, I have found that f is sure. >irreducible. Thus, the Galois group contains a 4-cycle. Why ? Are you claiming that A_4 is not a possible Gaolis group ? >But I couldn't find anything else. How can I finish this problem? The standard idea is to start by analyzing the splitting field of Ferrari's cubic - or any other cubic whose splitting field, with generic coefficients, is the field corresponding to the group S_3. write f=x^4+ax^3+bx^2+cx+d=(x-x_1)(x-x_2)(x-x_3)(x-x_4), and express the coefficients of g:=(x-x_1x_2-x_3-x_4)(x-x_1x_3-x_2x_4)(x-x_1x_4-x_2x_3) in terms of a,b,c,d. Now. Let K:=split(f), L:=split(g), note that Gal(L/Q)=Gal(K/Q)/{Id,(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)} 1) check if g is reducible 2) calculate Disc(g) 3) from 1,2 calculate L, Gal(L/Q) 4) express f as a product of two quadratics q_1(x)q_2(x) in L 5) complete the square for q_1(x),q_2(x) getting q'_1(x),q'_2(x) 6) check if the free coefficients of q'_i(x) are squares in L thus getting K and Gal(K/Q). Alternative solution: maple galois(x^4+x^2+x+1,x); HTH D.L. === Subject: convergence in L^2 Let f_n be a sequence of L^2(AxA) points. We now that if Sum(n=1,inf) ||f_n|| is convergent than Sum(n=1,inf) f_n is also convergent in L^2(AxA). Here the question: [I denote with f(x,.) the one variable function we get fixing x] In the same hypothesis, is it true that, for a.e. x in A, also the Sum(n=1,inf) f_n(x,.) is convergent in L^2(A)? How to prove this result? Can you give me some reference? === Subject: Ho! Ho! Ho! Merry Christmas! Ho! Ho! Ho! Merry Christmas! http://www.operationlettertosanta.com/new%20years/newyear1.gif Garry W. Denke Geologist.Geophysicist Denoco Inc. of Texas Wildcat Station, P.O. Box 866488 Plano, Texas 75086.6488 Tel #972.422.8268 Fax #972.423.6337 Cell #940.521.1667 GarryDenke@hotmail.com DenocoInc@hotmail.com === Subject: Re: inverse image >[...] >> Let >> X subset R^n and >> Y subset R^m be >> semi--algebraic sets. >> A map h: X -> Y is said to be >> ''semi-algebraic'' if its graph >> is a semi-algebraic subset of R^{n+m}. >> May you help me in proving the following, please? >> For any given semi-algebraic set >> A subset Y >> and h: X -> Y semi-algebraic as above, >> then h^{-1}(A) subset X is semialgebraic. > Consider the two projections > p: R^{n+m} = R^n x R^m ---> R^n > q: R^{n+m} = R^n x R^m ---> R^m > then h^{-1}(A) = p( A cap q^{-1}(A) ) _____________________^^^ sorry for the typo. I meant to write: h^{-1}(A) = p( Graph(f) cap q^{-1}(A) ) So you only need to consider (1) the inverse images of a semi-algebraic sets _along_the_projection_ (2) the intersection of two semialgebraic sets (3) the projetion of a semialgebraic set. Only (3) is difficult, but it has certainly been done before in your text or the lecture. Marc === Subject: physical significance Hello All, I am a beginner in mathematics and i am stuck in the following concept i found in a book of complex numbers: Consider a child throwing a ball into the air. For example, assume that the ball is thrown straight up, with an initial velocity of 9.8 meters per second. One second after it leaves the child's hand, the ball has reached a height of 4.9 meters, and the acceleration of gravity (9.8 meters per second2) has reduced its velocity to zero. The ball then accelerates toward the ground, being caught by the child two seconds any instant of time is given by: h = (-g*t^2)/2 + v*t where h is the height above the ground (in meters), g is the acceleration of gravity (9.8 meters per second2), v is the initial velocity (9.8 meters per second), and t is the time (in seconds). t ' 1± 1&h/4.9 Now, suppose we want to know when the ball passes a certain height. Plugging in the known values and solving for t: For instance, the ball is at a height of 3 meters twice: t =0.38 (going up) and t = 1.62 seconds (going down). As long as we ask reasonable questions, these equations give reasonable answers. But what happens when we ask unreasonable questions? For example: At what time does the ball reach a height of 10 meters? This question has no answer in reality because the ball never reaches this height. Nevertheless, plugging the value of h = 10 into the above equation gives two answers: t = 1+ sqrt(-1.041) and t = 1- sqrt(-1.041). My question is, in the above example what is the Physical significance of the complex time? I believe this quantity of complex time would be used in some physical concept or theory....How do we analyse this complex time in the real world? Thanx in advance. Shashi === Subject: Re: physical significance > Hello All, > I am a beginner in mathematics and i am stuck in the following concept i > found in a book of complex numbers: > Consider a child throwing a ball into the air. > For example, assume that the ball is thrown straight up, with an initial > velocity of 9.8 meters per second. One second after it leaves the child's > hand, the ball has reached a height of 4.9 meters, and the acceleration of > gravity (9.8 meters per second2) has reduced its velocity to zero. The ball > then accelerates toward the ground, being caught by the child two seconds at > any instant of time is given by: > h = (-g*t^2)/2 + v*t > where h is the height above the ground (in meters), g is the acceleration of > gravity (9.8 meters per second2), v is the initial velocity (9.8 meters per > second), and t is the time (in seconds). > t ' 1± 1&h/4.9 > Now, suppose we want to know when the ball passes a certain height. > Plugging in the known values and solving for t: > For instance, the ball is at a height of 3 meters twice: t =0.38 (going up) > and t = 1.62 seconds (going down). > As long as we ask reasonable questions, these equations give reasonable > answers. But what happens when we ask unreasonable questions? For > example: At what time does the ball reach a height of 10 meters? This > question has no answer in reality because the ball never reaches this > height. > Nevertheless, plugging the value of h = 10 into the above equation gives two > answers: t = 1+ sqrt(-1.041) and t = 1- sqrt(-1.041). > My question is, in the above example what is the Physical significance of > the complex time? > I believe this quantity of complex time would be used in some physical > concept or theory....How do we analyse this complex time in the real world? Note that the three basic equations* for motion with uniform acceleration mathematically model physical system only if the system is internally consistent. One can expect a real solution with physical significance only if one is talking about system that is internally consistent (meaning one can at the least observe such system experimentally). On earth, ball sent up with initial velocity (u) of 9.8 m/s cannot travel upwards longer than 1 s as it is decelerating to a final volocity v =0. One cannot observe upward motion of ball for 10 s. Solution to the final quadratic equation in terms of t does not yield real roots with physical significance due to inconsistent setup of experiment. As far as concept of complex t is concerned - whenever equation of the type t^2 + 1 = 0 is found one cannot find t on the real number line where the numbers are _really number entities_ that are either positive or negative or zero. With t^2 + 1 = 0, number t can be neither be positive nor negative nor zero and therefore, t has no place on the real number line. Mathematics concludes that t must be imaginary or complex. > Thanx in advance. > Shashi * v=u + at v^2-u^2 = 2as s = ut + (1/2) at^2 -- Respectfully, Mohan Pawar MIO Instruments LLC (920) 277-6037 === Subject: Re: physical significance (example snipped) >My question is, in the above example what is the Physical significance of >complex time? >I believe this quantity of complex time would be used in some physical >concept or theory....How do we analyse this complex time in the real world? According to Hawking and Hartle's theory, the Big Bang singularity - like all singularities - is only a singularity in ordinary time. In imaginary time, it is simply a point analogous to the north pole on the Earth. Although ordinary time begins with the Big Bang, it cannot be truly said that the north polebegins the Earth's surface, so imaginary time has no beginning or end. Therefore, when rephrased in imaginary time, the beginning and end of ordinary time are no more special, unique, or problematic than any other point in time. The dual-time theory clears up the singularity of the Big Bang by restating it in higher dimensions, now of time instead of space. quoted from: http://library.thinkquest.org/27930/time.htm?tqskip1=1 John Bailey http://home.rochester.rr.com/jbxroads/mailto.html === Subject: Re: physical significance Hello John, Thanx for the post. I looked up the links and the matter is very intriguing. Is there a practical engineering application of this imaginary time? I am looking at it from the perspective of complex frequency(1/time) from signal processing where the complex parts of the frequency determines the phase. I know that my queries are very vague and divergent. But I hope u could help me out. Shashi === Subject: Re: physical significance > ... As long as we ask reasonable questions, these equations give reasonable > answers. But what happens when we ask unreasonable questions? For > example: At what time does the ball reach a height of 10 meters? This > question has no answer in reality because the ball never reaches this > height. > Nevertheless, plugging the value of h = 10 into the above equation gives two > answers: t = 1+ sqrt(-1.041) and t = 1- sqrt(-1.041). > My question is, in the above example what is the Physical significance of > the complex time? > I believe this quantity of complex time would be used in some physical > concept or theory....How do we analyse this complex time in the real world? Apparently, even when the text says the question is unreasonable, still it is asked in this newsgroup. === Subject: Re: physical significance Hello Niel, Thanx for the post. I am seeking whether any abstract theory of mathematics or physics use the concept of complex time? If so how? Shashi === Subject: Re: physical significance > I am seeking whether any abstract theory of mathematics or physics use the > concept of complex time? If so how? Hawking's _A Brief History of Time_ mentions the concept of imaginary time, but it's well known that the book contains no equations other than E = mc^2. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: physical significance Hi Dave, Thanx for the post, incidentally my experiences with Abrief history of time reveales the same. I am looking it in the perspective of complex time as the inverse of complex frequency from signal processing. Am I touching the Nomans Land ? please comment. Shashi > I am seeking whether any abstract theory of mathematics or physics use the > concept of complex time? If so how? > Hawking's _A Brief History of Time_ mentions the concept of imaginary time, > but it's well known that the book contains no equations other than E = mc^2. > -- > Dave Seaman > Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. > === Subject: Re: physical significance Have you heard of imaginary energy? A quantum wave moves at a frequency related to it's energy, but if it's energy were imaginary, then the wave amplitude decreases with time instead of being constant. === Subject: Re: Deep Thoughts # 17: Liar Paradox is a Formal Metamathematical Theorem > You _really_ need to work on the irony detector. Truth > and provability are _different_ things - when you define > one to be the other you exhibit amazing ignorance of the > most basic issues. > ************************ > David C. Ullrich The definitions of Turing Machines and Lambda Calculus are very different, too, aren't they? C-B === Subject: Moments over a Simplex The midpoint of a line segment (x1,x2) is (x1+x2)/2 The midpoint of a triangle (r1,r2,r3) is (r1+r2+r3)/3 The midpoint of a tetrahedron (r0,r1,r2,r3) is (r0+r1+r2+r3)/4 The midpoint of a simplex in N dimensions is sum(k<=N) r_k/N ? The variance in x over a line segment in 1-D is: (x2-x1)^2/12 The variance in x over a triangle in 2-D is, if I made no errors: ((x2-x1)^2 + (x3-x1)^2 + (x3-x2)^2)/36 Question: What is the variance in x over a simplex in N dimensions? Is there a sensible way to make these calculations (multiple integrals) more amenable to treatment? Han de Bruijn === Subject: Re: Moments over a Simplex > The midpoint of a line segment (x1,x2) is (x1+x2)/2 > The midpoint of a triangle (r1,r2,r3) is (r1+r2+r3)/3 > The midpoint of a tetrahedron (r0,r1,r2,r3) is (r0+r1+r2+r3)/4 > The midpoint of a simplex in N dimensions is sum(k<=N) r_k/N ? > The variance in x over a line segment in 1-D is: (x2-x1)^2/12 > The variance in x over a triangle in 2-D is, if I made no errors: > ((x2-x1)^2 + (x3-x1)^2 + (x3-x2)^2)/36 What do you mean by the variance over a multi-dimensional simplex? For example, suppose (X1, X2, X3) is uniformly distributed over {x in R^3: min x_i >= 0, x1 + x2 + x3 =1} You can come up with a (singular) covariance matrix, but what do you mean by one scalar variance? > Question: What is the variance in x over a simplex in N dimensions? > Is there a sensible way to make these calculations (multiple integrals) > more amenable to treatment? -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Moments over a Simplex >The midpoint of a simplex in N dimensions is sum(k<=N) r_k/N ? The centroid of an N-simplex with vertices r_0, r_1, ..., r_N is sum_{k=0}^N r_k/(N+1). >The variance in x over a line segment in 1-D is: (x2-x1)^2/12 >The variance in x over a triangle in 2-D is, if I made no errors: >((x2-x1)^2 + (x3-x1)^2 + (x3-x2)^2)/36 >Question: What is the variance in x over a simplex in N dimensions? For the N-simplex with vertices r_k, k=0...N, we can use coordinates x_1,...,x_N such that all x_i >= 0 and sum_{i=1}^N x_i <= 1, so that (x_1,...,x_N) corresponds to the point R(x) = sum_{i=1}^N x_i r_i + (1 - sum_{i=1}^N x_i) r_0 = sum_{i=0}^N x_i r_i where x_0 = 1 - sum_{i=1}^N x_i. Integration over the simplex corresponds to integrating with the volume element dx_1 ... dx_N over S = {(x_1,...,x_N): all x_i >= 0, sum_{i=1}^N x_i <= 1}. Since int_S dx_1 ... dx_N = 1/N!, the average value of a function F on the simplex is E[F] = N! int_S F(R(x)) dx_1 ... dx_N. In particular, with mu = N! int_S R(x) dx_1 ... dx_N = (r_0 + ... + r_N)/(N+1), the variance is V = E[R(x)^2] - mu^2. Now I get E[x_i x_j] = 1/((N+1)(N+2)) for i <> j, while E[x_i^2] = 2/((N+1)(N+2)) and, if I haven't made a mistake, the variance is sum_{0<=i Hello All, > A new online quiz is on Wx Plotter's web site. > Find out how nerdy you really are, and get a free prize (an image) at > the end to show all your friends and family. > Take the Nerd Quotient quiz at: > http://www.wxplotter.com/ft_nq.php?ng > Other quizzes are available, take them all. > P.S. If you feel like sharing, place your score in a follow up message > to this posting. 1% scored higher, 0% scored the same, and 99% scored lower. What does this mean? Your nerdiness is: All hail the monsterous nerd. You are by far the KING NERD GOD!!! === Subject: Re: Nerd Quotient Quiz 35 for me, my sig other got an 11 (and she's a teacher) === Subject: Re: Nerd Quotient Quiz <%IOud.109494$8G4.39211@tornado.tampabay.rr.com> > 35 for me, my sig other got an 11 (and she's a teacher) Then I would be like your child- 20, nerd wanna-be. === Subject: Re: Disappointed (i) g_sigma >= 0. (ii) int g_sigma = 1. > (iii) If a > 0 is fixed then > int_{|t-T| > a} g_sigma(t-T) dt -> 0 as sigma -> 0. OK. Got that. But I don't see (immediately) how to prove the diaphragm property (Dutch: diafragma eigenschap) with the above knowledge i.e.: int(-oo,+oo) f(t).g_sigma(t-T) dt = f(T) Han de Bruijn === Subject: Re: Disappointed >(i) g_sigma >= 0. >(ii) int g_sigma = 1. >> (iii) If a > 0 is fixed then >> >> int_{|t-T| > a} g_sigma(t-T) dt -> 0 as sigma -> 0. >OK. Got that. But I don't see (immediately) how to prove the diaphragm >property (Dutch: diafragma eigenschap) with the above knowledge i.e.: >int(-oo,+oo) f(t).g_sigma(t-T) dt = f(T) Well of course you can't prove that because it's not true - by now I know that you mean the limit of that as sigma -> 0, but the fact that you mean that doesn't make the notation correct. How to prove that int(-oo,+oo) f(t).g_sigma(t-T) dt -> f(T) as sigma -> 0 is a standard thing in harmonic analysis - look for approximate identity arguments. Let's see. Of course we need some technical hypotheses of f; let's assume that f is bounded and continuous. Using the fact that the gaussian has integral 1 we need to show that int(-oo,+oo) (f(t) - f(T)).g_sigma(t-T) dt -> 0. Let's assume T = 0 just to save typing - we need to show that int(-oo,+oo) (f(t) - f(0)).g_sigma(t) dt -> 0 as sigma -> 0. So it's more than enough to show that int(-oo,+oo) |f(t) - f(0)|.g_sigma(t) dt -> 0 Let epsilon > 0. Choose delta > 0 so that |f(t) - f(0)| < epsilon whenever |t| < 0. Write the last integral as the sum of two integrals, one where |t| < delta and one where |t| > delta. The integral over |t| < delta is less than epsilon for all sigma, by our choice of delta and the fact that the gaussian has integral 1. On the other hand the fact that f is bounded and property (iii) above shows that the integral over |t| > delta is < epsilon if sigma is small enough. QED. (more details if you want.) >Han de Bruijn ************************ David C. Ullrich === Subject: Re: Disappointed Differential > Let epsilon > 0. Choose delta > 0 so that |f(t) - f(0)| > < epsilon whenever |t| < 0. Write the last integral |t| >= 0, I think you mean whenever |t| > 0. > as the sum of two integrals, one where |t| < delta and > one where |t| > delta. That's just the only thing I could understand if it was wrong. O || ^ _ t _ Dave, if you wanted to express all known non-linear analytic methods on one page, how would you go about doing that? For example you mention the technical hypotheses, there are boundary conditions, stuff like that. Assuming I'm quite ignorant, which I am, or another is, how do you outline non-linear differential methods in a nutshell, all of them. I'd appreciate that, I'd like to learn about this chaos theory someday. Ross F. === Subject: Re: Disappointed > Let's see. Of course we need some technical hypotheses > of f; let's assume that f is bounded and continuous. As is demonstrated further on, this assumption is essential. > Using the fact that the gaussian has integral 1 we need > to show that > int(-oo,+oo) (f(t) - f(T)).g_sigma(t-T) dt -> 0. > Let's assume T = 0 just to save typing - we need to show > that > int(-oo,+oo) (f(t) - f(0)).g_sigma(t) dt -> 0 > as sigma -> 0. So it's more than enough to show that > int(-oo,+oo) |f(t) - f(0)|.g_sigma(t) dt -> 0 > Let epsilon > 0. Choose delta > 0 so that |f(t) - f(0)| > < epsilon whenever |t| < 0. Write the last integral > as the sum of two integrals, one where |t| < delta and > one where |t| > delta. I suppose you mean |t| < delta instead of |t| < 0 . Mind your typo's! This is SCI.MATH! And I would rather say _three_ integrals instead of two, but, anyway, I understand what you mean. > The integral over |t| < delta is less than epsilon > for all sigma, by our choice of delta and the fact > that the gaussian has integral 1. On the other > hand the fact that f is bounded and property (iii) > above shows that the integral over |t| > delta is > < epsilon if sigma is small enough. QED. (more details > if you want.) This is correct, as far as I can see. No more details needed. Thanx! Han de Bruijn === Subject: Re: Disappointed > Publish the text here. It's a discussion group. No. Enough is enough. Han de Bruijn === Subject: Re: Poll: Are PCs Turing Machines? A Turing machine is NOT a finite state machine,although the converse is true. This is because the TM combines with finite states an unbounded tape which can be extended as needed (NOT an infinite tape) and the ability to change squares. Finite state machines do not have unbounded tapes and cannot change the contents of their squares. They cannot even back up. === Subject: Re: Poll: Are PCs Turing Machines? <93b23afabc3c546801c32d93f4bb0d55.48257@mygate.mailgate.org> This is a good example of how Platonism is terroristically enforced. The poster is told, point blank when in fact this Dolan individual, long a troll, hasn't confronted him at all. Instead, he's left a note. Ignorance poses as knowledge. Turing machines DO NOT support infinite tapes. Instead, they extend the tape which ALWAYS remains FINITE. The System cannot say these things. It speaks in pear-shaped tones. But on the ground, human sacrifice is demanded. Through the mechanism of flaming, the Internet has become the real terrorism. === Subject: Re: Poll: Are PCs Turing Machines? Trivially, of course, PCs can square a large number x, simply by displaying or printing out x**2. This objection is not frivolous. In many cases of numerical computation alone, the answer is more illuminating as a formula and NOT a sequence of numbers. The claim being made is that there is a class of problems P such that Turing machines can solve members of P, and that PCs cannot. The claim is FALSE. And in fact if one believes it, one doesn't understand Thing One about computer science and should stay in physics, always assuming you can, today, get a job with a major in physics...that does NOT involve, at least until your school loans are paid, mega programming. Let us consider large number calculation, and the silly introduction of Silly Physics to prove that there are problems in large number calculation that belong to P. universe to tell its user, whoever that might be, the answer to the age-old question, what is the factorial of 10e1000? All known PCs are unable to solve this problem but strangely, in any meaningful sense, so is the goddamn TM. This is because computing is a social relation. Any TM which uses all solution to some silly question hasn't solved a goddamn thing. It's destroyed the universe, because any technology which used, perhaps probability destroy mankind. Again, I am being serious. To claim that there is or even could be such an abomination reveals the problems of Platonism, its anti-human character because it rests on a deep assumption that the bigger and more ideal, the closer to an inhuman abstraction, a thing or concept is, it is more real and true. And I use the German word Einsatz advisedly, full well aware of its relation to the holocaust. This is because Platonism and modernism equals Fascism, where the inhuman (the out of control TM) is compared with the human, and naturally respondents to this idiot poll think that the out of control TM must be more real. In Derrida's terms, these respondents are Pavlovian privileging machines who are systematically deluded into sorting things into a favored category, favored no matter how inhuman, favored BECAUSE inhuman, and a marginal category. You see, Turing wasn't thinking about monster machines. He was thinking in an Intuitionist style in which finished infinities cannot be assumed. And, the most amusing paradox in these silly objections that a TM could calculate some incomprehensible answer is that in actual phenomenological fact, PCs are MORE and not less powerful than any possible, actually existing Turing machine! You see, once you fail to grok the meaning of is unlike Clinton, once you are unclear on the concept of simulation, you think to answer the question posed by The Silly Poll by setting a physical object, a PC from Dell or something, in an absurd race with an all-powerful TM that...does not goddamn well exist. But once you level the playing field, you realize that in order to be Fair and Balanced, in order to avoid comparing apples and oranges, you must pit the Dell PC against AN ACTUALLY CONSTRUCTED Turing Machine. [Socialism sounded like a fabulous idea once: free beer on steroids. Then it was ACTUALLY CONSTRUCTED by thugs. Then its victims compared the working prototype with Prada and Reebok and the rest of capitalismus, and they were like, I'm outa here. The moral seems to be don't compare apples and oranges and DON'T compare PCs with TMs in a simple minded fashion.] The Turing machine AS ACTUALLY CONSTRUCTED fails EVERY TIME to do what the Dell downmarket PC can do! While the Dell PC is cheerfully calculating, if not 10 e1000 factorial, then 100 factorial, using a simple program written in Visual Basic, the TM is stripping its gears...which Babbage's experience proved would be incapable of supporting the wear and tear of the simplest computation if based exclusively on mechanical motion. Even if you implement the PC in special purpose electronics or biocomputing, it STILL would be necessarily slower than the Dell, simply because the Turing machine is bound to do everything with the RISC of move right, move left, stamp a symbol, change state!! D'oh, you might say, it's faster. This manages to NOT UNDERSTAND computing complexity where you have to factor out specific time constants and compare...complexity. In the useful and phenomenological sense, a PC running Intel machine language is faster than the TM because instead of quintuples, the PC uses CISC instructions! Viewed through Platonic glasses, the Turing machine is more powerful than the Dell. Viewed through almost any other philosophical lens (mathematical intuitionism, phenomenology, and Marxism, for starters) the TM is less powerful. This is without apology a deconstruction, a reading against the grain, which exposes deep fault lines in the standard line. What's interesting about similar skylarking in the work of the late Derrida is the precise way this intellectual mechanism works. inculcated first as ideology in the hope that students will mature and grow. Instead the students never embark on any philosophical examination-of-concepts because part of the Platonic world-view is a strong closure condition: this is all ye need to know and subsequent enquiry is useless, even dangerous. Stop thinking and go watch The Apprentice. The world-view itself never overtly promulgates recognizable nonsense. Instead, the nonsense emerges under pressure and at its margins and is enforced demotically...in Foucault's sense. No responsible compsci prof would, I think, make the absurd claim that a PC doesn't simulate a TM because there are, in a physical sense, problems that a TM can solve by destroying the universe and a Dell cannot. Instead, the real work of nonsense occurs lower down in which double majors conduct absurd polls. The system, as Derrida showed in the case of deSaussure in De La Grammatology, always reverts to barbarism at this level. Saussure evinced a curious rage, in his own book Cours de la Langue General, against the very possibility that speech might depend, in any way on writing. Here, the interesting and profoundly illuminating fact that a PC is a TM when suitably explained becomes in the system's terms, terrorism of the same sort that exercised de Saussure. Viewed as computer science, the TM is equivalent to the PC in an interesting way. In fact, I am writing Yet Another simulator for a TM, this time with a small C-like language which I propose to compile to TM quintuples, mostly for s and giggles but also because I learned recently that in biocomputing this might have practical application. I suggest this foolish discussion stop with this conclusion. It really sounds like a bunch of physics majors. In college I took Serious Physics rather than Physics for Boneheads even though I only needed Physics for Boneheads as a philosophy major. For swots, physics becomes an all-purpose dodge and explanatory tool. not phenomenological. Is takes on one meaning as it did for Ken Starr. But poetry teaches one to REJECT the claim that is is not inherently multivocal. The first use of is wasn't reductionistic, it was metaphor in which the unlike was, like, wow, the same. Again, I am serious. Teaching kids how to dodge wild animals and find berries was taught as a songline in which everything was everything else and nothing was a Likewise, the Turing machine is a poetic songline which shows us how to recognize the turds of noncomputability. We are told that there are problems which cannot be solved, and they have nothing to do with big numbers. I am reminded in this connection of how lousy, how really bad, some physics majors are in programming which is best taught as a mythos and not at all reductionistically. Noncomputability of the interesting sort takes place in self-reflection when the TM examines itself or another TM for halting conditions. This poll is typical of the way in which the Internet destroys understanding. === Subject: Re: Poll: Are PCs Turing Machines? > universe to tell its user, whoever that might be, the answer to the > age-old question, what is the factorial of 10e1000? > All known PCs are unable to solve this problem but strangely, in any > meaningful sense, so is the goddamn TM. > This is because computing is a social relation. Any TM which uses all > solution to some silly question hasn't solved a goddamn thing. It's > destroyed the universe, because any technology which used, perhaps > probability destroy mankind. artifact, not a physical one (unlike a PC, which is both). A TM doesn't give ANY answers to any problems in the real world, because it doesn't exist in the real world. One can create a conceptual TM to solve arbitrary problems which cannot be solved by real PCs, but such a conceptual TM can't really do any calculations or produce any answers, unless it is (partially) simulated by a real PC or human. So what? Saying that there are TMs which can't be simulated by PCs doesn't mean TMs are better than PCs. Saying that (10^1000)! can never be calculated in the real world doesn't mean there is no such number. There does exist a (conceptual) TM that can perform that calculation, but it will never be built or simulated in this universe. --Mark === Subject: Re: Poll: Are PCs Turing Machines? Interesting, because you assume based on present knowledge that there exists at least one calculation that always and forever will require an unbounded number of digits. This may be true OR it may be an artifact of our knowledge. In the case of pi the dependency, I understand, is on something called the Riemann hypothesis, the solution of which may crack the problem of digit patterns recurring in pi, in which case, a TM could calculate it to arbitrary precision WITHOUT running out of storage. But at heart there is a major confusion when you make the answer to the question can a PC simulate a Turing machine? dependent on physics. Whether a PC simulates a Turing machine is a question unrelated to physical and empirical data. You see, the assertion that the universe's possible worlds because an infinite universe is physically possible, if not verifiable. act like a Turing machine and these silly ass physical arguments only serve to confuse the issue. For you don't understand computer science until you understand not only that PCs simulate TMs but also that there is no possible PC that goes beyond the limits described by Turing's formalism. I realize that it is popular to naturalize and fetishize technical development and to claim that one goes beyond TM formalisms using usually some clap trap from AI which turns out on investigation to be spaghetti code, no more and no less. Recall that we said simulate. When an ordinary computer simulates another, it may not be as powerful without the validity of a simulation being placed in question. So it is for the Turing machine. There is an unpleasant, even nasty theological undertone here in which the Turing machine, based as it was on the daily struggles of working people (for Turing was probably inspired by a Bletchley or City clerk at work) is taken from the people and made into something which at this juncture we simply do not need. This is another goddamn Platonic myth in which one is supposed to stand in awe of a completed infinity, pi in the sky, and the social effect of these myths is of course to reinforce Guardian power. Ultimately, the reaction towards Turing's discovery is emotional, and it varies. To some, it is a finished infinity and as such the concern only of a professoriat which is permitted to concern itself with such marginalities. To others it is an illumination, a story, a legend which teaches a usable Truth. >> I didn't say that a PC couldn't handle an input of X=10^1000. I >> said that it couldn't perform a calculation that requires 10^1000 >> _storage cells_. (For example, multiplying two numbers, each of >> which has 10^1000 digits.) There isn't enough matter in the universe >> to even begin to build a computer that could hold a 10^1000 digit >> number. > Doesn't have to hold complete numbers: think events. Is sent digits > just in time. Emits significant digits of product and revises them > when carry forward occurs. > You're trying to enhance the PC by adding some kind of input / output device > to hold unprocessed input and temporary results. I would consider that > equivalent to just making the PC's memory bigger. But where in the universe > are you going to hold the input before you give it to the TM, and where will > you store the intermediate results? The problem is NOT that a PC is not big > enough to hold a 10^1000 digit number -- the problem is that the universe > isn't big enough. Moving the storage outside the PC doesn't solve the > problem. > But even if you want to have and to hold a 10e4 number, think > biocomputation and the coming ability to inscribe data on molecules, > on cells thereby oops creating designer diseases. > I deliberately chose the example to invalidate this kind of solution. Even there's a > lot more matter in the universe than we think -- even if the observable > universe was packed solid with electron-sized memory cells, there still > wouldn't be nearly enough memory to hold a 10^1000 digit number (by which I > mean an integer containing 10^1000 digits -- I'm not sure what you mean by > a 10e4 number). > --Mark === Subject: Re: Poll: Are PCs Turing Machines? > Interesting, because you assume based on present knowledge that there > exists at least one calculation that always and forever will require > an unbounded number of digits. Yes, I do make such an assumption. This assumption seems fairly trivial. In my example, I don't even care about the calculation itself, only that the input is a 10^1000 digit number. There's no conceivable way to even store such a number without a corresponding amount of memory. > This may be true OR it may be an artifact of our knowledge. In the > case of pi the dependency, I understand, is on something called the > Riemann hypothesis, the solution of which may crack the problem of > digit patterns recurring in pi, in which case, a TM could calculate > it to arbitrary precision WITHOUT running out of storage. I don't think the randomness of digits in pi has anything to do with the RH, but I could be wrong. But calculating pi wasn't my example. > Whether a PC simulates a Turing machine is a question unrelated to > physical and empirical data. You see, the assertion that the > true in all possible worlds because an infinite universe is > physically possible, if not verifiable. This is true, but the question was, can a PC simulate a TM. I assumed PC meant a real PC, existing in our universe. If you want to hypothesize a universe in which there is an infinite amount of matter, which can be formed into an infinite amount of memory units in a finite time, and in which there is no speed-of-light limit on information transfer, then, yeah, a PC can probably simulate a TM. But I don't think that was the question that was asked. --Mark === Subject: Re: tetrahedral-cartesian transform > There are four unit vectors S0, S1, S2, S3. > They are rays from the center of a tetrahedron to its corners. > This exists in 3D cartesian space( X0, X1, X2 ). [ ... snip ... ] Define : (X0,X1,X2) = S0 + (S1-S0).xi + (S2-S0).eta + (S3-S0).zeta . And solve for (xi,eta,zeta), being three equations with three unknowns. Then (xi,eta,zeta) are your Cartesian coordinates, as the tetrahedron is transformed into a rectangular coordinate system with the origin at S0 -> (0,0,0) in (xi,eta,zeta)-space and likewise S1 -> (1,0,0) ; S2 -> (0,1,0) ; S3 -> (0,0,1) . Han de Bruijn === Subject: Acceptance Criteria of A Simulated Annealing Algorithm In the literature, most of the works use the following acceptance function for SA: IF ( The modifiedsolution is better than the currentsolution ) OR ( random[0..1] < exp(-deltachanges/T) THEN currentsolution=modifiedsolution Note: deltachanges=cost(currentsolution)-cost(modifiedsolution) However, a few works propose the following acceptance creteria: IF ( random[0..1] < 1.0/(1.0+exp(deltachanges/T)) ) THEN currentsolution=modifiedsolution What is the difference between them? Which one is preferable? === Subject: Re: Acceptance Criteria of A Simulated Annealing Algorithm > In the literature, most of the works > use the following acceptance function for SA: > IF > ( The modifiedsolution is better than the currentsolution ) > OR > ( random[0..1] < exp(-deltachanges/T) > THEN currentsolution=modifiedsolution right! > Note: deltachanges=cost(currentsolution)-cost(modifiedsolution) > However, a few works propose the following acceptance creteria: > IF ( random[0..1] < 1.0/(1.0+exp(deltachanges/T)) ) > THEN currentsolution=modifiedsolution > What is the difference between them? Which one is preferable? thats just the *second part* of the 1st solution with a positive exponent, you have to accept all improvements otherwise its a wasted search so it can't just be random. they just added 1 so it doesn't divide by 0. the trick with annealing is to make small jumps most of the time and periodic larger jumps. i think this is what deltachange is scaling the acceptance by. say you have a sequence through a network. 5 7 3' 1 2' 9 6 4 8 pick two nodes to swap, 3 and 2 5 7 2' 1 3' 9 6 4 8 you can also reverse the sequence between them so its a viable change in the path, this is TSP. the trick is not to make rash big changes all the time, I did something like city2 = city1 + rnd*rnd*5. that way its very smooth anneal, but will test sporadically if it can make large adjustments, I got a big improvement on the local minima with that trick. Herc === Subject: Re: Don't Feed the Troll! (was Re: Impossible sampling theory!) |He was actually in court once? Keen. If you like to see amateur mathematics in court, maybe you would enjoy the case of T. J. Rout: http://www.austlii.edu.au/au/other/hca/transcripts/2002/C4/1.html Here's his own take on it: http://home.pacific.net.au/~t_rout/h.c_justice_gleeson_appeal.htm You see, we all have the right to multiply and divide by zero.... Keith Ramsay === Subject: Re: Don't Feed the Troll! (was Re: Impossible sampling theory!) >|He was actually in court once? Keen. >If you like to see amateur mathematics in court, maybe you would >enjoy the case of T. J. Rout: >http://www.austlii.edu.au/au/other/hca/transcripts/2002/C4/1.html >Here's his own take on it: >http://home.pacific.net.au/~t_rout/h.c_justice_gleeson_appeal.htm Rout guy is obviously a crackpot. >You see, we all have the right to multiply and divide by zero.... >Keith Ramsay ************************ David C. Ullrich === Subject: Re: Don't Feed the Troll! (was Re: Impossible sampling theory!) >>|He was actually in court once? Keen. >>If you like to see amateur mathematics in court, maybe you would >>enjoy the case of T. J. Rout: >>http://www.austlii.edu.au/au/other/hca/transcripts/2002/C4/1.html >>Here's his own take on it: >>http://home.pacific.net.au/~t_rout/h.c_justice_gleeson_appeal.htm > Rout guy is obviously a crackpot. No, it's because the Rout guy is obviously a minority of one. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: What is a proof, exactly? |since any expression x=y can be replaced with Az (zey <=> zex) |for some free variable z. In extensional set theories, yes, you usually can proceed this way. Note however that to keep it genuinely equivalent, one needs then to include the case (x=y & x in z) -> y in z of the logical axiom for equality now as nonlogical axiom: (Ax)(Ay)(Aw) ((Az) (z in y <-> z in x) & x in w) -> y in w. Keith Ramsay === Subject: Re: What is a proof, exactly? I've heard that before, but is it really required. For instance, if someone says that Ex ((Ay ~y in x) & x in W) & ( x in W => Ez ((x in z) & (z in W)) do we care if someone has a set X where W in X and another set Y with some set similar to W, say V, such that V in Y. How does it matter? By foundation, each set grounds and it is the strucute of grounding that eventually it gets down to a set x such that Ay ~ y in x, and if they are the same empty set or different, it hardly matters to anything we do. The whole idea of whether the set {{}} contains a particular empty set or all of them is immaterial since with equality as a defined principle instead of a logical principle, one has x in {{}} & y in {{}}=> x=y it just means that they are both empty in the set sense rather than that they are equal in the logical sense. But that's enough. I think the idea of whether there is one empty set or many is silly and unecissary, since a FO language with no logical equality and just a binary relation can't distinguish between any sets x that make the sentance Az (z in x <=> S[z]) true, so to ask of a set y that makes the sentance Ax ((Az(z in x <=> S[z])) => x in y) true whether it has one such x or many is a non issue. And I've been personally flamed a bunch for asking questions that can't be stated in the FO language of set theory, so distinguishing between a y that contained one x and a y that contained a different x is a non question. And distinguishing between a y that contains one x and a y that contains both is likewise something that can't even be stated. Yes one can't dinstinguish between a naive bijection from {{}} to {{}}, because one empty set could be sent to another empty x, but the formal level can't even discuss these naive concerns. === Subject: Re: What is a proof, exactly? >|since any expression x=y can be replaced with Az (zey <=> zex) >|for some free variable z. >In extensional set theories, yes, you usually can proceed this way. >Note however that to keep it genuinely equivalent, one needs then >to include the case (x=y & x in z) -> y in z of the logical axiom for >equality now as nonlogical axiom: > (Ax)(Ay)(Aw) ((Az) (z in y <-> z in x) & x in w) -> y in w. Strictly speaking, that should be (Ax)(Ay)(Aw) (((Az) (z in y <-> z in x) & x in w) -> y in w). David ----- === Subject: Re: logic is innate? > Maybe people > do learn how to reason. Or maybe there's an innate capacity for > reason (a logic module, analogous to a face recognition area of the > brain) which, through exposure, we develop well. Is there a brain > lesion/freezing study that shows where modus ponens lies?. Answering my own post, for the record, of course there's been work done here. Well, maybe not specifically logic, but about math (they are the same, right? ;) ). For just one example, there's Dehaene's group: http://www.unicog.org/main/pages.php?page=Stanislas_Dehaene and his popularization The Number Sense. -- Mitch Harris (remove q to reply) === Subject: OT: Software for plotting graphs? I need a program that can break the axis into segments, preferably more than two segments. What software can I use? Jaco === Subject: What is maximal rank of this matrix? I'm not sure what you problem is. The rank of a matrix is first of all defined separately for the rows and the columns. In any good textbook on matrices, you will find the proof that the row rank of a matrix is equal to its column rank. Since you generate your matrix using only two vectors v and w, it is clear that the column rank of the resulting matrix cannot exceed 2. Zigoteau. === Subject: Re: Inplace Matrix Transpose? > I am writing code and need to be able to transpose non-square > matricies. This is easy to do if you create a second matrix and copy > the rows from the input matrix to the columns of the output matrix. > However, my matricies are large and take up a lot of memory. I would > prefer not to create a second matrix. I am sure there is a way to do an > inplace transpose, but deriving an algorithm is more difficult than I > thought. [ ... snip ... ] > But how can it be done inplace without creating a second matrix? Maybe this is a stupid remark, but I cannot imagine that someone will ever be in need of a transposed matrix. Why not simply use different indexing? I mean, M[j,i] instead of M[i,j]. Or maybe something like: function Transpose(i,j : integer) : double; begin Transpose := M[j,i]; end; Isn't it better to fake the system than re-organize your whole memory!? But I suppose the real problem is that you are using a standard package and have no choice? Han de Bruijn === Subject: Re: Inplace Matrix Transpose? A B C A D G J D E F => B E H K G H I C F I L J K L NewHeight = CurrentWidth = OldWidth NewWidth = CurrentHeight = OldHeight for i = 0, to NewHeight for j = i to NewWidth if ( i >= CurrentHeight ) { ExpandHeight of M from CurrentHeight to i and update CurrentHeight } if (j >= CurrentWidth) { Expand Width of M from CurrentWidth to j } Swap([i,j], [j, i]) done done Maybe if you alter this strategy by moving from bottom to top starting from the right instead of the left you can shrink the other dimension of M as you go -J. Hess === Subject: Re: Inplace Matrix Transpose? ... > But how can it be done inplace without creating a second matrix? Actually that can not be done in a program. The best you can come with is to create an additional 1 bit matrix that notes whether an element has been visited or not, and walk through all cycles. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Inplace Matrix Transpose? > ... > > But how can it be done inplace without creating a second matrix? > Actually that can not be done in a program. The best you can come > with is to create an additional 1 bit matrix that notes whether an > element has been visited or not, and walk through all cycles. Square matrixes can be transposed in place if you can interchange two elements: for i := 1 to n do for j := i + 1 to n do interchange(a[i,j],a[j,i]); For non-square matrices, it can only be done if the given matrix is embedded in a square matrix, at least during the transposition process.