mm-362 X-Last-Updated: 1999/08/06 === Subject: : Invariant Galilean Transformations (FAQ) On All LawsSummary: All laws/equations are Galilean invariant when expressed in the generalized cartesian coordinates demanded by basic analytic geometry, vector algebra, and measurement theory.Originator: faqserv@penguin-lust.MIT.EDUDisclaimer: approval for *.answers is based on form, not content. Opponents of the content should first actually find out what it is, then think, then request/submit-to arbitration by the appropriate neutral mathematics authorities. Flaming the hard- working, selfless, *.answers moderators evidences ignorance and despicable netiquette.Archive-Name: physics-faq/criticism/galilean-invarianceVersion: 0.04.03Posting-frequency: 15 days Invariant Galilean Transformations (FAQ) On All Laws (c) Eleaticus/Oren C. Webster Thnktank@concentric.netAn obvious typo or two corrected.The Brittanica section revised to less'pussy-footing' and to more directlyanticipate the elementary measurementtheory and basic analytic geometrythat is applied to the transformationconcept.------------------------------ === Subject: : 1. PurposeThe purpose of this document is to provide the student of Physics,especially Relativity and Electromagnetism, the most basic princ-iples and logic with which to evaluate the historic justificationof Relativity Theory as a necessary alternative to the classicalphysics of Newton and Galileo.We will prove that all laws are invariant under the Galileantransformation, rather than some being non-invariant, afterwe show you what that means.We shall also show that another primal requirement that SRexist is nonsense: Michelson-Morley and Kennedy-Thorndike doindeed fit Galilean (c+v) physics.------------------------------ === Subject: : 2. Table of Contents 1. Foreword and Intent 2. Table of Contents 3. The Principle of Relativity 4. The Encyclopedia Brittanica Incompetency. 5. Transformations on Generalized Coordinate Laws 6. The data scale degradation absurdity. 7. The Crackpots' Version of the Transforms. 8. What does sci.math have to say about x0'=x0-vt? 9. But Doesn't x.c'=x.c? 10. But Isn't (x'-x.c')=(x-x.c) Actually Two Transformations? 11. But Doesn't (x'-x.c+vt) Prove The Transformation Time Dependent? 12. But Isn't (x'-x.c')=(x-x.c) a Tautology? 13. But Isn't (x'-x.c')=(x-x.c) Almost the Definition of a Linear Transform? 14. But The Transform Won't Work On Time Dependent Equations? 15. But The Transform Won't Work On Wave Equations? 16. But Maxwell's Equations Aren't Galilean Invariant? 17. First and Second Derivative differential equations.------------------------------ === Subject: : 3. The Principle of Relativity and TransformationIf a law is different over there than it is here,it is not one law, but at least two, and leaves usin doubt about any third location. This is thePrinciple of Relativity: a natural law must be thesame relative to any location at which a given eventmay be perceived or measured, and whether or not theobserver is moving.The idea of location translates to a coordinatesystem, largely because any object in motion couldbe considered as having a coordinate system originmoving with it. If you perceive me moving relativeto you - who have your own coordinate system - willyour measurements of my position and velocity fitthe same laws my own, different measurements fit?If a law has the same form in both cases it iscalled covariant. If it is identical in form, var-ables, and output values, it is called invariant.What we're asking is that if the x-coordinate, x,on one coordinate axis works in an equation, doesthe coordinate, x', on some other, parallel axiswork? Speaking in terms of the axis on which x isthe coordinate, x' is the 'transformed' coordinate.The situation is complicated because we're talkingabout coordinates - locations - but in most mean-ingful laws/equations, it is lengths/distances (andtime intervals) the equations are about, and x coord-inates that represent good, ratio scale measures ofdistances are only interval scale measures on the x'axis. [See Table of Contents for discussion of scales.]So, if we have an x-coordinate in one system, thenwe can call the x' value that corresponds to the samepoint/location the transform of x.In particular, the Principle of Relativity is embodiedin the form of the Galilean transformation, whichrelates the original x, y, z, t to x', y', z', t' bythe transform equations x'=x-vt, y'=y, z'=z, t'=t inthe simplified case where attention is focused onlyon transforming the x-axis, and not y and z. In thecase of Special Relativity, the x' transform is thesame except that x' is then divided by sqrt(1-(v/c)^2),and t'=(t-xv/cc)/sqrt(1-(v/c)^2). In either case, vis the relative velocity of the coordinate systems;if there is already a v in the equations being trans-formed use u or some other variable name.------------------------------ === Subject: : 4. The Encyclopedia Brittanica Incompetency.One example of the traditional fallacious ideathat an equation is not invariant under the galileantransformation comes from the Encyclopedia Brittanica:Before Einstein's special theory of relativitywas published in 1905, it was usually assumedthat the time coordinates measured in all inertialframes were identical and equal to an 'absolutetime'. Thus, t = t'. (97)The position coordinates x and x' were thenassumed to be related by x' = x - vt. (98)The two formulas (97) and (98) are called aGalilean transformation. The laws of nonrelativ-istic mechanics take the same form in all framesrelated by Galilean transformations. This is therestricted, or Galilean, principle of relativity.The position of a light wave front speeding fromthe origin at time zero should satisfy x^2 - (ct)^2 = 0 (99)in the frame (t,x) and (x')^2 - (ct')^2 = 0 (100)in the frame (t',x'). Formula (100) does nottransform into formula (99) using the transform-ations (97) and (98), however.................................................. Besides the trivially correct statement of what theGalilean 'transform' equations are, there is exactlyone thing they got right.I. Eq-100 is indeed the correct basis for discussing the question of invariance, given that eq-99 is the correct 'stationary' (observer S) equation. [Let observer M be the 'moving'system observer.] In particular, eq-100 is of exactly the same form [the square of argument one minus the square of argument two equals zero (argument three).]II. It is nonsense to say eq-99 should be derivable from eq-100; for one thing, the transforms are TO x' and t' from x and t, not the other way around, and the idea that either observer's equation should contain within itself the terms to simplify or rearrange to get to the other is ridiculous. As the transform equations say, the relationship of t', x' to t, x is based on the relative velocity between the two systems, but neither the original (eq-99) equation nor the M observer equation is about a relationship between coordinate systems or observers. One might as well expect the two equations to contain banana export/import data; there is no relevancy. The 'transform' equations are the relationships between x' and x, t' and t and have nothing to do with what one equation or the other ought to 'say'. The equations' content is the rate at which light emitted along the x-axes moves.III. Most remarkable, the True Believer SR crackpots who most despise the consequences of measurement theory (demonstrable fact) contained in this document are those who want to argue against our saying the Britt- anica got eq-100 right; They insist that the correct equation is derived directly from x'=x-vt and t'=t. Solve for x=x'+vt and replace t with t', then substitute the result in eq-99: (x'+vt')^2 - (ct')^2 = 0. Besides the fact that this results in an equation with arguments exactly equal to eq-99, they will insist the transform is not invariant.IV. A major justification they have for their idea of the correct M system equation on which to base the the discussion of invariance, is that the variables are M system variables, never mind the fact that the arguments are S system values. That argument of theirs is arrant nonsense. The velocity v that S sees for the M system relative to herself is the negative of what the M system sees for the S system relative to himself. In other words, x'+vt' is a mixed frame expression and it is x'+(-v)t' that would be strictly M frame notation, and that equation is far off base. [Work it out for yourself, but make sure you try out an S frame negative v so as not to mislead yourself.]V. In I. we said: given that eq-99 is the correct 'stationary' equation. Let's look at it closely: x^2 - (ct)^2 = 0 (99) This whole matter is supposed to be about coordinate transforms. Is that what t is, just a coordinate? No. It isn't, in general. Suppose you and I are both modelling the same light event and you are using EST and I'm using PST. 'Just a time coordinate' is just a clock reading amd your t clock reading says the light has been moving three hours longer than my clock reading says. Well, that's what the idea that t is a coordinate means. Eq-99 works if and only if t is a time interval, and in particular the elapsed time since the light was emitted. Thus, that equation works only if we understand just what t is, an elapsed time, with emissioon at t=0. However, we don't have to 'understand' anything if we use a more intelligent and insightful form of the equation: (x)^2 - [ c(t-t.e) ]^2 = 0, where t.e is anyone's clock reading at the time of light emission, and t is any subsequent time on the same clock. Similarly, x is not just a coordinate, but a distance since emission. (x-x.e)^2 - [ c(t-t.e) ]^2 = 0 (99a)VI. In the spirit of 'there is exactly one thing they got right', the correct M system version of eq-99a is eq-100a: (x'-x.e')^2 - [ c(t'-t.e') ]^2 = 0 (100a) Every observer in the universe can derive their eq-100a from eq-99a and vice versa, not to mention to and from every other observer's eq-99a. Now, THAT's invariance. [You do realize that every eq-100a reduces to eq-99a, when you back substitute from the transforms, right? t.e'=t.e, x.e'=x.e-vt.]------------------------------ === Subject: : 5. Transformations on Generalized Coordinate LawsThe traditional Gallilean transform is correct: t' = t x' = x - vt.But remember this: a transform of x doesn't effectjust some values of x, but all of them, whether theyare in the formula or not. This is important if youwant to do things right. The crackpot position isstrongly against this sci.math verified position, andthe apparently standard coordinate pseudo-transformationthey suggest is perhaps the result. {See Table ofContents.]Let's use a simple equation: x^2 + y^2 = r^2, which isthe formula for a circle with radius r, centered at alocation where x=0.But what if the circle center isn't at x=0? Well, we'dwant to use the form analytic geometry, vector algebra,and elementary measurement theory tells us to use, a formwhere we make explicit just where the circle center is,even if it is at x=x0=0: (x-x0)^2 + (y-y0)^2 = r^2.The circle center coordinate, x0, is an x-axis coordinate,just like all the x-values of points on the circle.So, in proper generalized cartesian coordinate formsof laws/equations we want to transform every occurenceof x and x0 - by whatever name we call it: x.c, x_e,whatever.So, what is the transformed version of (x-x0)? Why,(x'-x0'); both x and x0 are x-coordinates, and everyx-coordinate has a new value on the new axis.So, what is the value of (x'-x0') in terms of the originalx data?is also true for x0'=x0-vt: (x'-x0')=[ (x-vt)-(x0-vt) ]=(x-x0).In other words, when we use the generalized coordinate formspecified by analytic geometry, we find that the value of(x'-x0') does not depend on either time or velocity in anyway, shape, form, or fashion.Similarly for (y-y0).We can treat time the same way if necessary: (t-t0).The above is a proof that any equation in x,y,z,t isinvariant under the galilean transforms. Just use thegeneralized coordinate form, with (x-x0)/etc, in thetransformation process, not the incompetently selectedprivileged form, with just x/etc.[The form is privileged because it assumes the circlecenter, point of emission, whatever, is at the origin ofthe axes instead at some less convenient point. Aftertransform the coordinate(s) of the circle center/originare also changed but the privileged form doesn't makethis explicit and screws up the calculations, whichshould be based on (x'-x0') but are calculated as (x'-0).]The value of (x'-x0') is the same as (x-x0). That makessense.Draw a circle on a piece of paper, maybe to the rightside of the paper. On a transparent sheet, draw x and ycoordinate axes, plus x to the right, plus y at the top.Place this axis sheet so the y-axis is at the left sideof the circle sheet.Now answer two questions after noting the x-coordinate ofthe circle center and then moving the axis sheet to the right:(a) did the circle change in any way because you movedthe axis sheet (ie because you transformed the coordin-nate axis)?(b) did the coordinate of the circle center change?The circle didn't change [although SR will say it did];that means that (x'-x0') does indeed equal (x-x0).The coordinate of the circle center did change, and itchanged at the same rate (-vt) as did every point onthe circle. That means that x0'<>x0, and the fact thecircle center didn't change wrt the circle, means thatthe relationship of x0' with x0 is the same as that ofany x' on the circle with the corresponding x: x'=x-vt;x0'=x0-vt.This is to prepare you for the True Believer crackpots thatsay 'constant' coordinates can't be transformed; some evensay they aren't coordinates. These crackpots include somethat brag about how they were childhood geniuses, btw.QED: The galilean transformation for any law ongeneralized Cartesian coordinates is invariant underthe Galilean transform.The use of the privileged form explains HOW the transformedequation can be messed up, the next === Subject: explains whatthe screwed up effect of the transform is, and how useof the generalized form corrects the screwup.------------------------------ === Subject: : 6. The data scale degradation absurdity.The SR transforms and the Galilean transforms bothconvert good, ratio scale data to inferior intervalscale data. The effect is corrected, allowed for,when the transforms are conducted on the generalizedcoordinate forms specified by analytic geometry andvector algebra.Both sets of transforms are 'translations' - lateralmovements of an axis, increasing over time in thesecases - but with the SR transform also involving arescaling. It is the translation term, -vt in the xtransform to x', and -xv/cc in the t transform to t',that degrades the ratio scale data to interval scaledata. In general, rescaling does not effect scalequality in the size-of-units sense we have here.SR likes to consider its transforms just rotations,however - in spite of the fact Einstein correctly saidthey were 'translations' (movements) - and in the caseof 'good' rotations, ratio scale data quality is indeedpreserved, but SR violates the conditions of good ro-tations; they are not rigid rotations and they don'tappropriately rescale all the axes that must be rescaledto preserve compatibility.The proof is in the pudding, and the pudding is thecombination of simple tests of the transformations.We can tell if the transformed data are ratio scaleor interval.Ratio scale data are like absolute Kelvin. A measure-ment of zero means there is zero quantity of thestuff being measured. Ratio scale data support add-ition, subtraction, multiplication, and division.The test of a ratio scale is that if one measurelooks like twice as much as another, the stuffbeing measured is actually twice as much. Withabsolute Kelvin, 100 degrees really is twice theheat as 50 degrees. 200 degrees really is twiceas much as 100.Interval scale data are like relative Celsius, whichis why your science teacher wouldn't let you use itin gas law problems. There is only one mathematicaloperation interval scales support, and that has tobe between two measures on the same scale: subtraction.100 degrees relative (household) Celsius is not twiceas much as 50; we have to convert the data to absoluteKelvin to tell us what the real ratio of temperaturesis.However, whether we use absolute Kelvin or relativeCelsius, the difference in the two temperature readingsis the same: 50 degrees.Thus, if we know the real quantities of the 'stuff'being measured, we can tell if two measures are ona ratio scale by seeing if the ratio of the twomeasures is the same as the ratio of the known quant-ities.If a scale passes the ratio test, the interval scale testis automatically a pass.If the scale fails the ratio test, the interval scaletest becomes the next in line.It isn't just the bare differences on an intervalscale that provides the test, however. Differencesin two interval scale measures are ratio scale, soit is ratios of two differences that tell the tale.Let's do some testing, and remember as we do that ourconcern is for whether or not the data are messed up,not with 'reasons', excuses, or avoidance.-------------------------------------------------- ----Are we going to take a transformed length (difference)and see whether that length fits ratio or interval scaledefinitions?Of course, not. Interval scale data are ratio afterone measure is subtracted from another. That is themajor reason the SR transforms can be used in science.Let there be three rods, A, B, C, of length 10, 20, 40,respectively. These lengths are on a known ratio scale,our original x-axis, with one end of each rod at theorigin, where x=0, and the other end at the coordinatethat tells us the correct lengths.Note that these x-values are ratio scale only becauseone end of each rod is at x=0. That may remind you ofthe correct way to use a ruler or yard/meter-stick:put the zero end at one end of the thing you aremeasuring. Put the 1.00 mark there instead of the zero,and you have interval scale measures.Let A,B,C, be 10, 20, 40.Let a,b,c be x' at v=.5, t=10.x'=x-vt.A B C a b c---------------- --------------------10 20 40 5 15 35---------------- --------------------B/A = 2 b/a = 3C/A = 4 c/a = 7C/B = 2 c/b = 2.333 Obviously, the transformed values are no longer ratio scale. The effect is less on the greater values.C-A = 10 b-a = 10C-A = 30 c-a = 30C-B = 20 c-b = 20 Obviously, the transformed values are now interval scale. This will hold true for any value of time or velocity.(C-A)/(B-A) = 3 (c-a)/(b-a) = 3(C-B)/(B-A) = 2 (c-b)/(b-a) = 2 Obviously, the ratios of the differences are ratio scale, being identical to the ratios of the corresponding original - ratio scale - differences.The main difference between these results and the SRresults is that the differences do not correspond soneatly to the original, ratio scale, differences.This is due only to the rescaling by 1/sqrt(1-(v/c)^2).The ratios of the differences on the transformed valuesdo correspond neatly and exactly to the ratio scaleresults.Using the generalized coordinate form, such as (x-x0),the transform produces an interval scale x' and aninterval scale x0'. That gives us a ratio scale (x'-x0'),just like - and equal to - (x-x0).------------------------------ === Subject: : 7. The Crackpots' Version of the Transforms.It has become apparent - whether misleading or not -that the crackpot responses to the obvious derive froma common source, whether it be bandwagoning or theirSR instructors.Below, in the sci.math subject, we see that all sci.mathrespondents agree with the basic controversial positionof this faq: every coordinate is transformed, whether asupposed constant or not.Think about it, the generalized coordinate of a circlecenter, x0, applies to infinities upon infinities ofcircle locations (given y and z, too); it is a constantonly for a given circle, and even then only on a givencoordinate axis.And even variables are often held 'constant' duringeither integration or differentiation.The utility of a variable is that you can discuss allpossible particular values without having to single outjust one. That utility does not make particular - singledout - values on the variable's axis not values of thevariable just because they have become named values.In any case, all that is preamble to the incompetent ideathey have proposed for a transform of coordinates. It isbased on the idea that the circle center, point of emission,whatever, has coordinates that cannot be transformed.Let there be an equation, say (x)^2 - (ict)^2 = 0.What is the transformed version of that equation?Answer: (x')^2 - (ict')^2 = 0. That's the one thing theBrittanica got right. Note that the leading crackpot justcriticized this faq for presuming to correct the Britt-anica, but it then and before poses the incompetent pseudo-transform we analyze here in this section.x to x' and t to t' are obviously coordinate transforms;the x and t coordinates have been replaced by the coord-inates in the primed system.A tranform of an equation from one coordinate system toanother is NOT a substitution of the/a definition of xfor itself; that is not a coordinate transformation.The most that can said for such a substitution is thatit is a change of variable.But the crackpots are calling this a coordinate trans-form of the original equation: (x'+vt)^2 - (ict')^2 = 0.It is not a coordinate transform, of course, exceptaccidentally. (x'+vt) is not the primed systemcoordinate, it is another form/expression of x. Theyget that substitution by solving x'=x-vt for x; x=x'+vt.So, by incompetent misnomer, they accomplish what theyhave been railing against all along.It has been the generalized coordinate form in question allthis time: (x-x0)^2 - (ict)^2 = 0.Here they substitute for x instead of transforming to theprimed frame: (x'+vt-x0)^2 - (ict')^2. ----- ^ | ^ |It is still x ^ but see what they have accomplishedby their mis/malfeasance: [x'+vt-x0]=[x'+(vt-x0)]=[x'-(x0-vt)]. =[x'-x0']The crackpots have been bragging about how you don'thave to transform the circle center's coordinate totransform the circle center's coordinate. Braggingthat what they were doing was not what they saidthey were doing.This does give us insight as to some of the crackpotvariations on their x0'<>x0-vt theme, which in all thevariations will be discussed in later sections..They are used to seeing the mixed coordinate form,(x'+vt-x0) without realizing what it respresented,so - accompanied with a lack of understanding ofthe term 'dependent' - they are used to seeing justthe one vt term, and not the one hidden in the defi-nition of x' and are used to imagining it makes thewhole expression time dependent and thus not invariant.About which, let x=10, let, x0=20, v=10, and tvariously 10 and 23:(x-x0)=-10. Using their (x'+vt-x0):For t=10, we have (x'+vt-x0) = [ (10-10*10) + (10*10) - (20) ] = -90 + 100 - 20 = -10 = (x-x0)For t=23, we have (x'+vt-x0) = [ (10-10*23) + (10*23) - (20) ] = -220 + 230 - 20 = -10 = (x-x0)The result depends in no way on the value of time;we showed the obvious for a couple of instances of tjust so you can see that the crackpots not only donot understand the obvious logic of the algebra{ (x'-x0')=[ (-vt)-(x0-vt) ]=(x-x0) } - which showsthat the transform has no possible time term effect -but they don't understand even a simple arithmeticdemonstration of the facts.Oh. Their (x'+vt-x0) or (x'+vt'-x0) reduces the sameway since t'=t: (x-vt+vt-x0)=(x-x0).Their process, which says (x'+vt') is the transformof x, says that (x'+vt') is the moving system locationof x, but it can't be because x is moving further inthe negative direction from the moving viewpoint.That formula will only work out with v<0 which is indeedthe velocity the primed system sees the other moving at.However, that formula cannot be derived from x'=x-vt,the formula for transformation of the coordinates fromthe unprimed to the primed,------------------------------ === Subject: : 8. What does sci.math have to say about x0'=x0-vt?The crackpots' positions/arguments were put to sci.mathin such a way that at least two or three who posted re-sponses thought it was your faq-er who was on the idiot'sside of the questions.Their responses:-------------------------------------------------- --------I. x0' = x0. In other words: x0' <> x0-vt, or constant values on the x-axis are not subject to the transform.AA: == == No. x0' = x0 - vt. Well, if you want, you could define constant values on the x-axis, butin the context of the question that is not relevant. The relevant fact isthat if the unprimed observer holds an object at point x0, then theprimed observer assigns to that object a coordinate x0' which isnumerically related to x0 by x0'= x0 -vt.AA: == ==EE: == ==What does this mean? The line x=x0 will give x'=x-v*t=x0-vt', so if x0'is to give the coordinate in the (x',t',)-system, it will be given byx0'=x0-v*t': ie., it is not given by a constant. Thus, being at rest(constant x-coordinate) is a coordinate-dependent concept.EE: == ==GG: == ==Sounds very false. We can say that the representation of the point X0 isthe number x0 in the unprimed system, and x0' in the primed system.Clearly x0 and x0' are different, if vt is not zero. However one may saythat (though it sounds/is stupid) the point X0 itself is the samethroughout the transformation. However that expression soundsmeaningless, since a transform (ok, maybe we should call it a change ofbasis) is only a function that takes the point's representation in onesystem into the same point's representation in another system. It ispreferrable to use three notations: X0 for the point itself and x0 andx0' for the points' representations in some coordinate systems.GG: == ==------------------------------ === Subject: : 9. But Doesn't x.c'=x.c?That idea is one of the most idiotic to come up, and it doesso frequently. And in a number of guises.The idea being that x.c' <> x.c-vt, with x.c being whatwe have called x0 above; the notation makes no difference.Some crackpots have managed to maintain that position evenafter graphs have illustrated that such an idea means thatafter a while a circle center represented by x.c' could beoutside the circle.The leading crackpot just make that explicit, as far asone can tell from his befuddled post in response to a lineabout active transforms, which are actually moving bodysituations, not coordinate transformations:-------------------------------------------- ------------------------e>An active transform is not a coordinate transform, ... Right, it is a transform of the center (in the opposite direction) done to effect the change of coordinates without a coordinate transform. ...E: Transform of the center? Center of a circle? He really is saying a circle center moves in the opposite direction of the circle! Right?------------------------------------------------------ --------------If r=10 and x.c was at x.c=0, then the points on the circle(10,0), (-10,0), (0,10) and (0,-10) could at some time become(-10,0), (-30,0), (-20,10), and (-20,-10), but with x.c'=x.c,the circle center would be at (0,0) still! The circle is herebut its center is way, way over there! Indeed, although a changeof coordinate systems is not movement of any object described inthe coordinates, the x.c'=x.c crackpottery is tantamount to thecircle staying put but the center moving away. Or vice versa.------------------------------ === Subject: : 10. But Isn't (x'-x.c')=(x-x.c) Actually Two Transformations?One crackpot puts the (x'-x.c')=(x-vt - x.c+vt) relationshiplike this: (x-vt+vt - x.c).See, he says, that is transforming x (with x-vt - x.c) and thenreversing the transform (x-vt+vt - x.c).That's just another crackpot form of the idiocy thatx.c' <> x.c-vt. You'll have noticed the implicationis that there is no transform vt term relating to x.c.------------------------------ === Subject: : 11. But Doesn't (x'-x.c+vt) Prove The Transformation Time Dependent?That particular crackpottery is perhaps more corrupt thanmoronic, since it includes deliberately hiding a vt term fromview, and pretending it isn't there. [However, we have seenabove that it is a familiar incompetency, and not likely anoriginal.]Look, the crackpots say, there is a time term in thetransformed (x' - x.c+vt). The transform isn't invariant!It's time dependent!Just put x' in its original axis form, also, which revealsthe other time term, the one they hide: (x'-x.c+vt) = (x-vt - x.c+vt) = (x-x.c).So, at any and all times, the transform reduces to theoriginal expression, with no time term on which to bedependent.Then there is the fact that if you leave the equationin any of the various notation forms - with or withoutreducing them algebraicly - the arithmetic always comesdown to the same as (x-x.c). That means nothing to crack-pots, but may mean something to you.------------------------------ === Subject: : 12. But Isn't (x'-x.c')=(x-x.c) a Tautology?My dictionary relates 'tautology' to needless repetition.That's another form of the x.c' <> x.c-vt idiocy.The repetition involved is the vt transformation term.Apply the -vt term to the x term, and it is needlessrepetition to apply it anywhere again? The 'again' isto the x.c term. The x.c' = x.c crackpot idiocy.The repetition of the vt terms is required by the presenceof two x values to be transformed.Be sure to note the next section.------------------------------ === Subject: : 13. But Isn't (x'-x.c')=(x-x.c) Almost the Definition of a Linear Transform?Now, how on earth can we relate a tautology to a basicdefinition in math?we get this definition:------------------------------------------------- -------------A linear transformation, A, on the space is a method of corr-esponding to each vector of the space another vector of thespace such that for any vectors U and V, and any scalarsa and b, A(aU+bV) = aAU + bAV.-------------------------------------------------------- -----Let points on the sphere satisfy the vector X={x,y,z,1},and the circle center satisfy C={x.c,y.c,z.c,1}. Let a=1,and b=-1.Let A= ( 1 0 0 -ut ) ( 0 1 0 -vt ) ( 0 0 1 -wt ) ( 0 0 0 1 )A(aX+bC) = aAX + bAC. aX+bC = (x-x.c, y-y.c, z-z.c, 0 ).The left hand side: A( x - x.c , y - y.c, z - z.c, 0 ) = ( x-x.c , y-y.c, z-z.c, 0 ).The right hand side: aAX= ( x-ut, y-vt, z-wt, 1 ). bAC= (-x.c+ut, -y.c+vt, -z.c+wt, -1 ).and aAX+bAC = ( x-x.c, y-y.c, z-z.c, 0 ).Need it be said?Sure: QED. On the galilean transform thedefinition of a linear transform, A(aU+bV)=aAU + bAV,is completely satisfied.The generalized form transforms exactly andnon-redundantly - with ONE TRANSFORM, not atransform and reverse transform - and non-tautologically, just as the very definitionof a linear transform says it should.And does so with absolute invariance, with thisgalilean transformation.------------------------------ === Subject: : 14. But The Transform Won't Work On Time Dependent Equations?The main crackpot that has asserted such a thing was referringto equations such as in === Subject: 4, above. The Light Sphereequation; for which we have shown repeatedly elsewhere that thenumerical calculations are identical for any primed values asfor the unprimed values.The presence - before transformation - of a velocity termseems to confuse the crackpots. It turns out there is ex-treme historical reason for this, as you will see in thesubject on Maxwell's equations.------------------------------ === Subject: : 15. But The Transform Won't Work On Wave Equations?See === Subject: 17, below, for a discussion of Second Derivativeforms and the galilean transforms.------------------------------ === Subject: : 16. But Maxwell's Equations Aren't Galilean Invariant?Oh? Just what is the magical term in them that prevents(x'-x.c')=(x-vt - x.c+vt)=(x-x.c) from holding true?It turns out not to be magic, but reality, that interfereswith the application of the galilean transforms to the gen-eralized coordinate form(s) of Maxwell: there are no coordi-nates to transform!When True Believer crackpots are shown the simpledemonstration that the galilean transform ongeneralized cartesian coordinates is invariant,their first defense is usually an incredibly stupidx0'=x0, because the coordinate of a circle center,or point of emission, etc, is a constant and can'tbe transformed.The last defense is but Maxwell's equations are notinvariant under that coordinate transform. Whenasked just what magic occurs in Maxwell that wouldprevent the simple algebra (x'-x0')=[ (x-vt)-(x0-vt) ]=(x-x0)from working, and when asked them for a demonstration,they will never do so, however many hundreds oftimes their defense is asserted.The reason may help you understand part of Einstein's1905 paper in which he gave us his absurd SpecialRelativity derivation:THERE ARE NO COORDINATES IN THE EQUATIONS TO BE TRANSFORMED.Einstein gave the electric force vector as E=(X,Y,Z)and the magnetic force vector as B=(L,M,N), where theforce components in the direction of the x axis areX and L, Y and M are in the y direction, Z and N inthe z direction.Those values are not, however, coordinates, but valuesvery much like acceleration values.BTW, the current fad is that E and B are 'fields', havingbeen 'force fields' for a while, after being 'forces'.So, when Einstein says he is applying his coordinatetransforms to the Maxwell form he presented, he iseither delusive or lying.(a) there are no coordinates in the transform equations he gives us for the Maxwell transforms, where B=beta=1/sqrt(1-(v/c)^2): X'=X. L'=L. Y'=B(Y-(v/c)N). M'=B(M+(v/c)Z). Z'=B(Z+(v/c)M). N'=B(N-(v/c)Y). X is in the same direction as x, but is not a coordinate. Ditto for L. They are not locations, coordinates on the x-axis, but force magnitudes in that direction. Similarly for Y and M and y, Z and N and z.(b) the v of the coordinate transforms is in Maxwell before any transform is imposed; Einstein's transform v is the velocity of a coordinate axis, not the velocity he touched it.(c) if they were honest Einsteinian transforms, they'd be x, which means it is X and L that are supposed to be transformed, not Y and M, and Z and N. And when SR does transform more than one axis, each axis has its own velocity term; using the v along the x-axis as the v for a y-axis and z-axis transform is thus trebly absurd: the axes perpendicular to the motion are not changed according to SR, the v used is not their v, and the v is not a transform velocity anyway.(d) as everyone knows, the effect of E and B are on the direction. Both the speed and direction are changed by E and B, but v - the speed - is a constant in SR.As absurd as are the previously demonstrated Einsteinianblunders, this one transcends error and is an incredibleexample of True Believer delusion propagating over decades.The components of E and B do differ from point to point,and in the variations that are not coordinate free,they are subject to the usual invariant galilean trans-formation when put in the generalized coordinate form.------------------------------------------------------- ------The SR crackpots don't know what coordinates are. Thevarious things they call coordinates include coordin-nates, but also include a variety of other quantities.------------------------------------------------- -----1. One may express coordinates in a one-axis-at-a-time manner [like x^2+y^2=r^2] but it is the use of vector notation that shows us what is going on. In vector notation the triplet x,y,z [or x1,x2,x3, whatever] represents the three spatial coordinates, but there are so-called basis vectors that underlie them. Those may be called i,j,k. Thus, what we normally treat as x,y,z is a set of three numbers TIMES a basis vector each.2. These e*i, f*j, g*k products can have a lot of meanings. If e, f, j are distances from the origin of i,j,k then e*i, f*j, g*k are coordinates: distances in the directions of i,j,k respectively, from their origin. That makes the triplet a coordinate vector that we describe as being an x,y,z triplet; perhaps X=(x,y,z). The e*i, f*j, g*k products could be directions; take any of the other vectors described above or below and divide the e,f,g numbers by the length of the vector [sqrt(e^2+f^2+g^2)]. That gives us a vector of length=1.0, the e,f,g values of which show us the direction of the original vector. That makes the triplet a direction vector that we describe as being an x,y,z triplet; perhaps D=(x,y,z). The e*i, f*j, g*k products could be velocities; take any of the unit direction vectors described above and multiply by a given speed, perhaps v. That gives a vector of length v in the direction specified. That makes the triplet a velocity vector that we describe as being an x,y,z triplet; perhaps V=(x,y,z). Each of the three values, e,f,g, is the velocity in the direction of i,j,k respectively. The e*i, f*j, g*k products could be accelerations; take any of the unit direction vectors described above and multiply by a given acceleration, perhaps a. That gives a vector of length a in the direction specified. That makes the triplet an acceleration vector that we describe as being an x,y,z triplet; perhaps A=(x,y,z). Each of the three values, e,f,g, is the acceleration in the direction of i,j,k respectively. The e*i, f*j, g*k products could be forces (much like accel- erations); take any of the unit direction vectors described above and multiply by a given force, perhaps E or B. That gives a vector of length E or B in the direction specified. That makes the triplet a force vector that we describe as being an x,y,z triplet; perhaps E=(x,y,z) or B=(x,y,z). Each of the three values, e,f,g, is the force in the direction of i,j,k respectively.Einstein's - and Maxwell's - E and B arenot coordinate vectors. There is another variety of intellectual befuddlement thatmisinforms the idea that Maxwell isn't invariant under thegalilean transform: confusions about velocities.Velocities With Respect to Coordinate Systems.----------------------------------------------- Aaron Bergman supplied the background in a post to a sci.physics.*newsgroup:=== Imagine two wires next to each other with a current I in each.Now, according to simple E&M, each current generates a magneticfield and this causes either a repulsion or attraction betweenthe wires due to the interaction of the magnetic field and thecurrent. Let's just use the case where the currents are parallel.Now, suppose you are running at the speed of the current betweenthe wires. If you simply use a galilean transform, each wire,having an equal number of protons and electrons is neutral. So,in this frame, there is no force between the wires. But this is acontradiction.=== =First of all, the invariance of the galilean transform (x'-x.c')=(x-x.c), insures that it is an error to imagine there is anydifference between the data and law in one frame and in another;the usual, convenient rest frame is the best frame and only framerequired for universal analysis. [Well, (x'<>x, x,c'<>x.c, but(x'-x.c')=(x-x.c).]Second, given that you decide unnecessarily to adapt a law toa moving frame, don't confuse coordinate systems with meaningfulphysical objects, like the velocity relative to a coordinatesystem instead of relative to a physical body or field.In other words, what does current velocity with respect to acoordinate system have to do with physics?Nothing. Certainly not anything in the example Bergman gave.What is relevant is not current velocity with respect to acoordinate system, but current velocity with respect to wiresand/or a medium. The velocity of an imaginary coordinate sys-tem has absolutely nothing to do with meaningful physical vel-ocity. You can - if you are insightful enough and don't violateitem (e) - identify a coordinate system and a relevant physicalobject, but where some v term in the pre-transformed law isin use, don't confuse it with the velocity of the coordinatetransform.Velocities With Respect to ... What?-----------------------------------------------Albert Einstein opened his 1905 paper on Special Relativitywith this ancient incompetency:=== The equations of the day had a velocity term that was takenas meaning that moving a magnet near a conductor would createa current in the conductor, but moving a conductor near awire would not. This was belied by fact, of course.The important velocity quantity is the velocity of themagnet and conductor with respect to each other, not tosome absolute coordinate frame (as far as we know) andnot to an arbitrary coordinate system.One possible cause was the idea: but the equation says the magnetmust be moving wrt the coordinate system or ... the absoluterest frame.There not being anything in the equation(s) to say either ofthose, it is amazing that folk will still insist the velocityterm has nothing to do with velocity of the two bodies wrteach other.------------------------------------------------------ ----------------------------------- === Subject: : 17. First and Second Derivative differential equations.One of the intellectually corrupt ways ofdenying the very simple demonstration ofgalilean invariance of all laws expressedin the generalized coordinate form demandedby analytic geometry, vector analysis, andmeasurement theory [ (x'-x.c')=[ (x-vt)-(x.c-vt) ]=(x-x.c) ]is the assertion that those equations 'over there'(usually Maxwell or wave) are somehow immune tothe elementary laws of algebra used to demon-strate the invariance. [Unfortunately, theassertions are never accompanied by referenceto the magical math that makes elementary al-gebra invalid. Wonder why that is?]Part of the time it is based on the old lorebased on the incompetent transformation ofthe privileged form of an equation insteadof the correct form. [Evidence of this isany reference to an effect due to the velocityof the transform; it falls out algebraicly- as you see above - and cancels out arith-metically - as you can see above.]But usually it is just whistling in the dark,waving the cross (zwastika, I'd say) atthe mean old vampire.The most general equation that could be conjuredup is a differential with either First or SecondDerivatives.Let's examine the plausibility of such magicalmagical, non-invariance assertions.(a) to get a Second Derivative you must have a First Derivative.(b) to get a First Derivative you must have a function to differentiate.(c) to get a Second Derivative you must have a function in the second degree.So, let us examine the question as to whetherany such common Maxwell/wave equation willdiffer for(a) the common, privileged form, represented as ax^2, with a being an unknown constant function.(b) the generalized cartesian form, represented as a(x-x.c)^2 = ax^2 -2ax(x.c) + ax.c^2, with a being an unknown constant function.(c) the transformed generalized cartesian form, represented as a(x-vt -x.c+vt)^2, same as for (b), = ax^2 -2ax(x.c) + ax.c^2, of course, with a being an unknown constant function.I. for (a), remembering that x.c is a constant, and that this version is only correct because x.c=0, otherwise (b) is the correct form: d/dx ax^2 = 2ax (d/dx)^2 ax^2 = 2aII. for (b), remembering that x.c is a constant. d/dx (ax^2 -2ax(x.c) + ax.c^2) = 2ax - 2ax.c (d/dx)^2 (ax^2 -2ax(x.c) + ax.c^2) = 2aIII. for (c); same as for (b).So, what we have seen so far is(1) differential equations in the second degree- the wave equations - must clearly be the same forall forms: the privileged form in x, the generalizedcartesian form in x and the centroid, x.c, or thetransformed generalized cartesian form.That is, anyone who imagines that correct usagegives different results for galilean transformedframes is at first showing his ignorance, and inthe end showing his intellectual corruption.(2) As far as the First Derivatives are concerned, theonly cases in which there really is a difference betweenthe two forms is where x.c <> 0, and in that case, theuse of the privileged form is obviously incompetent.So, how do you correctly use the differential equations?If you are using rest frame data with the centroidat x=0, etc, you can't go wrong without trying togo wrong.If you are using rest frame data with the centroidnot at x=0, you must use (x-x.c) anyplace x appearsin the equation.If you are using moving frame data, you must use themoving frame centroid as well as the light front(or whatever) moving frame data itself, perhaps firstcalculating (x'-x.c'), which equals (x-x.c) which isobviously correct, and which is obviously the plain oldcorrect x of the privileged form.Unless, of course, there really is some magical termor expression that invalidates the obvious and elemen-tary algebra of the invariance demonstration.Or maybe you just whistle when you don't want basicalgebra to hold true.Eleaticus!---?---!---?---!---?---!---?---!---?---!--- ?---!---?---!---?---!---?! Eleaticus Oren C. Webster ThnkTank@concentric.net ?! Anything and everything that requires or encourages systematic ?! examination of premises, logic, and conclusions ?!---?---!---?---!---?---!---?---!---?---!---?---!---?---!-- -?---!---?Supersedes: X-Last-Updated: 1999/10/17 === Subject: : (SR) Lorentz t', x' = IntervalsSummary: The Lorentz transforms themselves are proof t' and x' cannot possibly be just coordinates. Examination of their derivation verifies their identity as intervals.Originator: faqserv@penguin-lust.MIT.EDUDisclaimer: approval for *.answers is based on form, not content. Opponents should first actually find out what the content is, then think, then request/submit-to arbitration by the appropriate neutral mathematics authorities. Flaming the hard- working, selfless, *.answers moderators evidences ignorance and atrocious netiquette.Version: 0.02.1Archive-name: physics-faq/criticism/lorentz-intervalsPosting-frequency: 15 days (SR) Lorentz t', x' = Intervals (c) Eleaticus/Oren C. Webster Thnktank@concentric.net------------------------------ === Subject: : 1. Introduction with the obvious debunking of the use of 'just coordinates' in any scientific formula.Defenders of the Special Relativity faith are especiallyfond of telling opponents of their space-time fairy talesthat they do not know the difference between coordinatesand magnitudes. That may often be so, but the fault lies ultimately with SR dogma. The Lorentz-Einstein transformations cannot possibly be 'just coordinates', which is the interpre-tation required to support the many sideshow carnival actswith which the SR faithful bedazzle the public, and establishtheir moral and intellectual superiority.If I get in my car and drive steadily for a few hours at 50 kilometers per hour, is 50t the distance I travel? Of course not. The last time my hours-counting 'just coord-inates' clock was set to zero was when Zeno first reported one of his paradoxes to Parmenides. That was a long time ago, so my t is not useful for such purposes unless you also use my clock to established the starting time, perhaps t0, and use the formula 50(t-t0) to calculate the distance. In any case, my t is even then not 'just a coordinate' becauseit always represents particular elapsed times that can beused in the (t-t0) form to calculate perfectly good timeintervals (elapsed times).Alternatively, I could (re)set my clock to zero at the startof some meaningful time interval, in which case my t shows a scientifically perfect current and/or end time. In which case, the Lorentz-Einstein t'=(t-vx/cc)/g is a function of an elapsed time interval (not 'just a coordinate') and a time interval (-vx/cc; the interval amount the t' clock is being screwed up at time t) and thus cannot be 'just a coordinate' since neither of the independent variables is such a 'just' thing. {Their meaning is shown below, step-by-step.]If it takes me 50 minutes to cross the Interstate highway,was x/50 my velocity crossing it?Of course not. The origin of all my axes is at the veryspot where Zeno first presented his first paradox to Parmenides. That makes my x equal a couple of thousands ofmiles, plus, and is not useful for such purposes unless you establish the starting x value, perhaps x0, and use theformula (x-x0)/50 to calculate my velocity. In any case, even then my x is not 'just a coordinate' because it always repesents particular distance intervalsthat can always be used in the (x-x0) form for any and everyscientific purose.Alternatively, I could move my x-axis origin to the starting(zero) point of some meaningful distance, in which case my x shows a scientifically perfect current and/or end distance.In which case, the Lorentz-Einstein x'=(x-vt)/g is a function of a current/ending distance interval (not 'just a coordinate') and a distance interval (-vt; the interval amount the x' axisis being screwed up at time t) and thus cannot be 'just a coordinate' since neither of the independent variables is such a 'just' thing. {Their meaning is shown below, step-by-step.]------------------------------ === Subject: : 2. Table of Contents 1. Introduction with the obvious debunking of the use of 'just coordinates' in any scientific formula. 2. Table of Contents. 3. The Lorentz-Einstein transforms. 4. The 'just coordinates' argument. 5. Single-system, little-purpose ambiguity. 6. Relating two coordinate measures/systems. 7. Distances and moving coordinate axes. 8. Time intervals. 9. Einstein's (1905) derivations. 10. A word about intervals. 11. Intervals versus the Twins Paradox. 12. Summary------------------------------ === Subject: : 3. The Lorentz-Einstein transformsSpecial Relativity's space-time circus is based onthe 'transformation' equations by which it is believedone can relate a nominally 'stationary' system's spaceand time coordinates to those of an inertially (notaccelerating) moving other observer. That moving observer's own physical body and coordinate system might have been identical in size to those of the stationary observer before the traveller began moving, but are 'seen' as very different by the stationary observer when the relative velocity of the two is great enough, a high percentage of the velocity of light.Concerning ourselves - as is customary - with justthe spatial coordinate axis that lies parallel tothe direction of motion, and with time, Einsteinarrived at these formulas that relate the movingsystem measures or coordinates (x' and t') to thestationary system coordinates (x and t): x' = (x - vt)/sqrt(1-vv/cc) (Eq 1x) t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t)The v is for the two systems' relative velocity as seen by the stationary observer, and is positive if the dir-ection is toward higher values of x. By concensus,the moving system x'-axis higher values also lie inthat direction, and all axes parallel the other system'scorresponding axis.We used vv to mean the square of v but might use v^2for that purpose below. Similarly for c.Because it is believed that no physical object canreach or exceed c, the square-root term in bothdenominators is presumed always less than one, which means that the formulas say both x' and t' will tend tobe greater than x and t, respectively. However,SRians call the x' result 'contraction' - which meansshortening - and the t' result 'dilation' - whichmeans increasing. ------------------------------ === Subject: : 4. The 'just coordinates' argumentThe 'just coordinates' argument is so patently ridiculousthat even opponents have a hard time accepting just howsimple and obvious its debunking can be, as shown in thissection. However, further sections take a more arithmet-ical approach that you'll maybe find more professorial.The 'just coordinates' argument is that t is mot aduration, not a time interval; it's just an arbitraryclock reading. But what if the moving system observercomes speeding by while you make your annual 'springforward' or 'fall back' change? The formula says thatthe moving system clock's 'just coordinate' reading can be calculated from yours: t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t)Imagine the moving system oberver's confusion if his clock changes its reading while he's looking at it! If his clock doesn't change when yours does, the formula is wrong; if it is truly a 'just coordinates' formula. And then what happens if you realize you were a day early and put your clock back to what it had said previously?And suppose you are in NYC and your twin in LA andboth are watching the moving observer. You'll both beusing the same v because you are at rest wrt (withrespect to) each other. You're on Eastern StandardTime and your twin is on Pacific Standard Timemaybe. You have three hours more on your clock than does your twin. On which 'just coordinate' clock will the Lorentz transforms base the 'just coordinate' time the moving system clock says? The formula applies to both ofyour t-times: t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t)Sure, the idea that you can change someone else'sclock with no connection of any kind is really ridiculous, but Eqs 1x and 1t aren't MY equations. Are they yours? And we aren't the ones to say x, t, x', and t' are just coordinates.If the t' formula is actually either an elapsedtime formula, or the basis of a t'/t ratio, thenthere is no implication that one clock's readinghas anything to do with the other's. It can only be rates of clock ticking, or how one time INTERVAL compares to the other that the formula is about.------------------------------ === Subject: : 5. Single-system, little-purpose ambiguity.Since we're going to be comparing measurements on twocoordinate systems in the next section, let's go toour supply cabinet and get our yard-stick (which weuse to measure things in inches) and our meter-stick(which we use to measure things in centimeters).Here, I'm getting mine. Oh! Oh!There's an ant on mine, and he ... she ... sure ishanging on, right at the 3.5 inch mark of the yard-stick.Let's see if I can wave the stick around enough thatshe'll let go. Nope.However, before I gave up I waved the stick and theant 'all over the place.Always, however, the ant was at the 3.5 mark on the yard-stick, and always 3.5 away from the end of thestick, however far and wide I have transported her.Neither of those 3.5 facts means very much. Of thetwo, the distance aspect meant almost nothing. Sothe distance was 3.5 from the end. So what? Thatlength, distance, was not in use. And only maybethe ant might have been concerned with just whatlocation, 'just coordinate', on the stick she wasat.Just so with x and t.So, is the 3.5 reading just a coordinate? Or adistance/length? It's ambiguous in and of itself,and really makes no difference what you say untilyou try to make use of the number. Hey, my address is 5047 Newton Street. If youare looking for me and you're at 4120 Newton, itis helpful information, because it tells you whichdirection to go. Is that 'just coordinate'? Where it really becomes useful, perhaps, is intelling you how far away I am. That's not justa coordinate value, that's a distance, length,interval.However, it is subtracting 4120 from 5047 that tells you which direction and how far. It is only because both 5047 and 4120 are distances from the same point - ANY same point - that the result means anything.My x - my yardstick reading - is always a distanceor length; it is impossible to be otherwise withan honest, competently designed yardstick.Whether or not its reading is of good use in some particular scientific formula depends on whether I put the zero end of the yardstick at some usefulplace. As in the introduction, we should eitherput it at the starting location/end, or use tworeadings from it: (x-x0).------------------------------ === Subject: : 6. Relating two coordinate measures/systems.Taking care to not damage our brave little ant, I placemy yard-stick onto the table, zero end to the left, 36end to the right.Now I place the 'just coordinate' meter-stick on the tablein the same orientation, in a random location, and findthat the ant's coordinate on the meter-stick is 51.The formula relating centimeters to inches is cm=i*2.54but we want a formula similar to x'=(x-vt)/sqrt(1-vv/cc).That would be c=i/.03937 approximately, but let's use x'for the meter-stick reading, and x for the inch reading: x'=x/.3937. 3.5/.3937 = 8.89 Wait a minute. It's not just science but definition that says c=i/.3937=8.89, so something is wrong. 8.89is not 51.We already knew that 51 cm was just an arbitrary coordinate. Arbitrary not because that point isn't 51 cm from the zero end of the meter-stick, but because the zero point was in an arbitrary position.Let's put the meter-stick in a position where it's zero point is at the yard-stick zero point.What is the centimeter coordinate now? Hey. 8.89,just like the formula says.The only way for a 'transform' like x'=x/g to work, whatever g might be, is for both coordinate systemsto have their zero points aligned, in which casesaying the two measures are not intervals is pureidiocy.Noe that with both zero points at the same positionboth x' and x are great measures for scientificpurposes, in any and every case where we were smartenough to put those zero points at a useful location.There is one extension of x'=x/g that will let ususe the meter-stick in arbitrary position. When the cm reading was 51, the zero point of theyard-stick read (51-8.89=) 42.11 cm. If we call thatpoint x.z' we get x' = x.z' + x/.3937. = 42.11 + 3.5/.3937 = 42.11 + 8.89 = 51.Obviously, in this formula x/.3937 is the distancefrom the x' coordinate of the location where x=0. An interval.Just as obviously, the fact that we now have thecorrect formula for relating an x interval to anarbitrary x' coordinate, does not mean that x'is anything more than nonsense for use in anyscientific formula.Unless we were smart enough to put the x zeropoint in a useful location, and use (x'-x.z') inthe scientific formula. (x'-x.z') equals the useful,Ratio Scale value x/.3937.So, we have discovered a basic fact: a transformationformula like x'=x/g works only if the two zero pointsof the coordinate systems coincide. That makes it non-sense to say the two coodinates are only coordinatesand not intervals. Both must be values that representdistances from their respective zero points unless youtake the proper steps to adjust for the discrepancy.Make sure you understand that although the inclusionof x.z' made it possible to correctly calculate x',the result is nonsense when it comes to use of x'for general length/distance purposes; it is x'-x.z' that is a useful number in such cases. It could bethat we're measuring a sheet of paper with one endat x=0 and the other at x=3.5; x'=51 is nonsense asa centimeter measure of the paper.But, you say, the Lorentz transform contain a -vt term.------------------------------ === Subject: : 7. Distances and moving coordinate axes.We discovered x'=x.z' + x/g as the correct formulafor relating one coordinate to another system's.But the Lorentz transform contains another term, -vt/sqrt(1-vv/cc). What is it?Let's start with our x'=51 cm, x=3.5, x.z'=42.11 example.Every minute, let's move the meter-stick one inch to ourright.At minute 0, the cm reading was 51 cm.At minute 1, the cm reading is now 50 cm.At minute 2, the cm reading is now 49 cm.In this instance, v=1 inch/minute. And t was 0, 1, 2.What has happened is that we have made our x.z' a lie,and increasingly so. -vt/.3937 is the change in x.z'. x' = (x.z - vt/.3937) + x/.3937.Obviously, vt/.3937 is not a coordinate; even most SRianswouldn't imagine it was. It is an interval, the distanceover which the moving system has moved since t=0.And, of course, x/.3937 is the distance of our bravelittle ant from the point where x=0 and the centimeterreading is x.z'-vt/.3937. Yes, every minute the meter-stick moves to the right and the meter-stick coordinateof the spot where x=0 gets less and less - and eventuallynegative.Make sure you understand that every minute the x' coordinate, because of -vt/g, becomes a better measure of, say, the 3.5 paper we might be measuring with the yard-stick, given that 51 was too big a number and-vt is negative. That is, until the two origins coincide at x'=x=0, and then it gets worse and worse.With -vt positive (because v<0) the situation is different.With 51 and -vt positive, x' just gets worse and worseover time.Quite obviously, the fact that we now have thecorrect formula for relating an x interval to anarbitrary x' coordinate even when the x' axis ismoving, does not mean that x'is anything more than nonsense for use in any scientific formula.Unless we were smart enough to put the x zero point in a useful location, and use (x'-x.z'+vt/.3937) inthe scientific formula. (x'-x.z'+vt/.3937) equals the useful, Ratio Scale value x/.3937.------------------------------ === Subject: : 8. Time intervals.Instead of using our sticks, let's get out two clocks.Mind you, we're not going to deal with different clockrates here, just establish the same basics as for distance.Your clock says 9:00 Eastern Standard Time (EST) and we note that t=540 minutes when we put down the clock.Blindly, let's turn the setting knob of your twin's Pacific Standard Time clock and put it down before us.According to what we see, EST's 540 minutes (9:00) corre-sponds to PST's 14:30; t'=870.We know the formula relating PST to EST is t' (pacific)= t (eastern) - 180 (minutes). Thus, it is not correct that the second clock can have an arbitrary setting, because 870 <> 540-180. We know that the two clocks are related by t' = t/1 since both are using the same second, hour, etc units. But 870 (14:30 in minutes) is not 540/1-180, so once again we know something is wrong. However, t'=t.z' + t/1 works. EST midnight equals PST 0.0 (midnite) - 180, so t.z' = -180, and t' = -180 + 540/1 = 360.Since EST-180=PST, 9:00 EST is 6:00 PST = 360 minutes.We see thus that like distance measures/coordinates, timeaxis origins (zero points) must either be 'lined up' or adjusted for. So, the Lorentz/Einstein t'=t/sqrt(1-vv/cc) must be the moving system elapsed time interval since the time axes were both at a common zero. There is no t.z' adjustment: t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t)Make sure you understand that in the clock case, if theEST is showing a good number for elapsed time since thetravelling observer passed NYC, then the PST clock issilliness. t.z' must be zero or must be taken out oftime lapse calculations for the PST clock to be usedintelligently, just as was true for x.z'.What is lacking as yet for Lorentz t' is the -vx/cc term thatcorresponds to the x' formula -vt term.Break it up into two parts: v/c and x/c. v/c is a scaling factor that changes velocity from whatever kind of unit you are using over to fractions of c.x/c is distance divided by velocity, which is time. x/cis thus the time interval since the two time axeshad a common zero point - which they have to have in theLorentz transforms which do not have the t.z' term welearned to use above.Thus, (-vx/cc)/sqrt(1-vv/cc) is the interval amount the moving system clock has been changed - since the common/adjusted time - over and beyond the elapsed time intervalrepresented by x/sqrt(1-vv/cc).We have discovered that the only way for t' to be t/gis for t' and t to have a common zero point, just asfor x' and x. It would be otherwise if the t' formulacontained an adjustment t.z' under some name or other,but the necessity to include such a term correlates100% with t' numbers that aren't directly usable.As for x and x', our knowledge of how to setup a properformula relating t and t' is of no use unless we usethe knowledge in scientific formulas; (t'-t.z'+xv/gcc)gives us the only directly useful value: t/g.------------------------------ === Subject: : 9. Einstein's (1905) derivations.When we return to Einstein's derivations of the transformformulas with a well-focused eye, we find he was a wee bitconfused - or at least self-contradictory.When he set up his (at first unknown) tau=moving systemtime formulas, he created three particular instances of tau.Tau.0 is the time at which light is emitted at the movingorigin toward a mirror to the right that is moving at rest wrt that moving origin and at a constant distance from that origin. He lets the stationary time slot have the value t,a constant, the stationary system starting time.Tau.1 is the time at which the light is reflected. Helets the stationary time be t+x'/(c-v); t is still aconstant and x'/(c-v) is the time interval since t.Tau.2 is the time at which the light gets (back) to themoving origin. The stationary time value is put as t +x'/(c-v) + x'/(c+v); t is still a constant and x'/(c-v)+ x'/(c+v) is the time interval since t.On the thesis that the moving observer sees the time tothe mirror as the same as the time back to the origin,he sets .5[ tau.0 + tau.2 ] = tau.1.Tau.0 completely drops out of the analysis and leavesno trace, and has no effect.Further, the t you see in tau.0, tau.1, and tau.2 also completely drops out with no trace and no effect, leaving us with exactly what you'd get if you had explicilty said t' is an interval and so is t.What doesn't drop out in the stationary time values isx'/(c-v) and x'/(c+v), the time interval it takes forlight to get to the fleeing mirror, and the time intervalit takes for light to get back to the approaching origin.Thus, his resultant t' formula is strictly based on time intervals in the stationary system. Time intervals since some starting time, yes, but time intervals.There is absolutely nothing in the derived formulas that depends on arbitrary coordinates like the constant t in the stationary time arguments.Let's look at the x dimension; it is x'=x-vt [as x increasesby vt, the effect over time is x'=(x+vt)-vt)], which Einstein explicitly sets up as a constant stationary distance.He uses that x' not just in the time interval parts of the stationary time arguments, but also in the x (distance) stationary system argument for the tau at the time light is reflected. x' can't be the stationary system coordinate of the mirror at that time. That value is x'+vt.x' is explicitly an interval, distance. Thus, the whole tau derivation of the t' formula is fully andexplicitly based on x' - a spatial length/distance/interval -and the two time interals x'/(c-v) and x'/(c+v).While we're at it, if the starting t is not zero, his x'=x-vt formula is complete nonsense also. Given thatthere was some L that was the mirror x-location and lengthwhen the light is emitted, if t was already, say, 500, thenx'=L-vt could have been a very negative length.------------------------------ === Subject: : 10. A word about intervals.There are intervals, and there are intervals.If we put our yard stick zero point at one endof a piece of paper and read off the coordinateat the other end of the paper, we have a goodmeasure of the paper's length, a Ratio Scalemeasure. [Absolute temperature scales are ratioscale.]If instead we put the one end of the paper at theone inch mark (or the zero end of the stick oneinch 'into' the length of the paper) we get measuresthat are one inch off the true, ratio scale length.The two messed up measures are still intervals,but they are Interval Scale measures. [Householdtemperature scales are interval scale, which iswhy your physics and chemistry professors won'tlet you use them without first converting to theratio scale absolute temperatures.)t'=t/g and x'=x/g represent ratio scale measures,given that t and x were ratio scalae to start with.t'=t.z'+t/g and t'=t/g-vx/gcc are both interval scale measures, even given a good ratio scale tand a good ratio scale x.x'=x.z'+x/g and x'=x/g-vt/g are both interval scale measures, even given a good ratio scale xand a good ratio scale t.Look for the (SR) Lorentz t', x' = degraded measuresdocument soon at a newsgroup near you.------------------------------ === Subject: : 11. Intervals versus the Twins Paradox.t'=(t-vx/cc)/g shows t' being greater than t.The reason Special Relativity will not allow theuse of its basic time equation in determining whatSR has to say about the twins' ages, is that t' andx' are supposedly just coordinates, and they say you have to take the coordinate pairs (t',x') and (x,t)into consideration in both the time and place the twins' separation started and the time and place the twins reunited.Since t' and x' are actually both intervals, notjust coordinates, the 'excuse' is spurious, and is so even without use of the obvious (x_b-x_a) and(t_b-t_a) usages.However, SR is right to be embarrassed by theirtransformation formulas.Look for the (SR) Lorentz t', x' = degraded measuresdocument at a newsgroup near you. ------------------------------ === Subject: : 12. SummaryA. t'=t/g and x'=x/g can be almost 'just coordinates' in the sense that the values obtained may not be of much use except in the most primal and useless way: how long and how far since/from the time/ place they were zero. Even here, however, the zero points within each of the two scale pairs (t',t) and (x'.x) must have been lined up. If the zero points have been intelligently selected (such as at the starting point and time of a trip) they can be rationally used 'as is' in any valid sci- entific equation.B. Even the interval scale t'=t.z' - xv/gcc + t/g and x'=x.z' - vt/g + x/g are not 'just coordinates'. They can be used to good effect by establishing the relevant starting times/points and using (t'-t.z'+xv/gcc) and (x'-x.z'+vt/g), as the situation may require.C. When you see vx/gcc or vt/g in use in any guise with non-zero values, you know the resultant t' or x' is a degraded, interval scale value.E-X: Anytime you do not see what amounts to t.z' and xv/gcc in the time case, or x.z' and vt/g in the distance case, you know that the t' and/or x' in use are intervals. Period.Y: Either set your clock to zero at the start of the relevant time interval, or use (t-t0), with both being readings on the same clock. Either move your x-axis origin to the starting end or point, or use (x-x0), with both being readings on the same axis.Z: In _(SR) Lorentz t', x' = Degraded (Interval) Scales_ we see that t' and x' satisfy the mathematical tests for/of interval scales when -vt and -vx/cc are not zero; thus, they must be intervals. When -vt and -vx/cc are zero, t' and x' satisfy the much better mathematical definition of ratio scales, and are thus not just mere intervals, but (rescaled) good ones.Eleaticus!---?---!---?---!---?---!---?---!---?---!--- ?---!---?---!---?---!---?! Eleaticus Oren C. Webster ThnkTank@concentric.net ?! Anything and everything that requires or encourages systematic ?! examination of premises, logic, and conclusions ?!---?---!---?---!---?---!---?---!---?---!---?---!---?---!-- -?---!---? === Subject: : details of joke wantedCan somebody help me recall the details of an old maths joke? It is one ofthose situations where a computer scientist, a physicist and a mathematician(or some such combination) have to solve a certain problem, and themathematician starts by doing something crazy in order to reduce to aproblem that is already solved.Derek Holt. === Subject: : Re: details of joke wanted>Can somebody help me recall the details of an old maths joke? It is one of>those situations where a computer scientist, a physicist and a mathematician>(or some such combination) have to solve a certain problem, and the>mathematician starts by doing something crazy in order to reduce to a>problem that is already solved.>>Derek Holt.There are lots of variations and the mathematician is not always the one whodoes something crazy. Take your pick:+++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++=6. THE MATHEMATICIAN, THE PHYSICIST AND THE ENGINEER (AND OTHERS)MPE__________________________________________________ ______________________jwest@jwest.ecen.okstate.edu:A mathematician, a physicist, and an engineer were all given a red rubberball and told to find the volume. The mathematician carefully measuredthe diameter and evaluated a triple integral. The physicist filled abeaker with water, put the ball in the water, and measured the totaldisplacement. The engineer looked up the model and serial numbers inhis red-rubber-ball table.If it was my company: The engineer tried to look up the model and serialnumbers, couldn't find them, so told his manager that it's just not goingto work.MPE____________________________________________________ ____________________So a mathematician, an engineer, and a physicist are out huntingtogether. They spy a deer(*) in the woods.The physicist calculates the velocity of the deer and the effect ofgravity on the bullet, aims his rifle and fires. Alas, he misses; thebullet passes three feet behind the deer. The deer bolts some yards,but comes to a halt, still within sight of the trio.Shame you missed, comments the engineer, but of course with anordinary gun, one would expect that. He then levels his specialdeer-hunting gun, which he rigged together from an ordinary rifle, asextant, a compass, a barometer, and a bunch of flashing lights whichdon't do anything but impress onlookers, and fires. Alas, his bulletpasses three feet in front of the deer, who by this time wises up andvanishes for good.Well, says the physicist, your contraption didn't get it either.What do you mean? pipes up the mathematician. Between the two ofyou, that was a perfect shot!(*) How they knew it was a deer:The physicist observed that it behaved in a deer-like manner, so itmust be a deer.The mathematician asked the physicist what it was, thereby reducing itto a previously solved problem.The engineer was in the woods to hunt deer, therefore it was a deer.MP_____________________________________________________ ____________________A mathematician and a physicist agree to a psychological experiment.The mathematician is put in a chair in a large empty room and abeautiful naked woman is placed on a bed at the other end of the room.The psychologist explains, You are to remain in your chair. Everyfive minutes, I will move your chair to a position halfway between itscurrent location and the woman on the bed. The mathematician looksat the psychologist in disgust. What? I'm not going to go throughthis. You know I'll never reach the bed! And he gets up and stormsout. The psychologist makes a note on his clipboard and ushers thephysicist in. He explains the situation, and the physicist's eyeslight up and he starts drooling. The psychologist is a bit confused.Don't you realize that you'll never reach her? The physicist smilesand replied, Of course! But I'll get close enough for all practicalpurposes!ME____________________________________ _____________________________________A businessman needed to employ a quantitative type person.He wasn't sure if he should get a mathematician, an engineer,or an applied mathematician. As it happened, all theapplicants were male. The businessman devised a test.The mathematician came first. Miss How, the administrativeassistant took him into the hall. At the end of the hall,lounging on a couch, was a beautiful woman. Miss How said,You may only go half the distance at a time. When youreach the end, you may kiss our model.The mathematician explained how he would never get there ina finite number of iterations and politely excused himself.Then came the engineer. He quickly bounded halfway down thehall, then halfway again, and so on. Soon he declared he waswell within accepted error tolerance and grabbed the beautifulwoman and kissed her.Finally it was the applied mathematician's turn. Miss Howexplained the rules. The applied mathematician listenedpolitely, then grabbed Miss How and gave her a big smooch.What was that about? she cried.Well, you see I'm an applied mathematician. If I can'tsolve the problem, I change it!PEA____________________________________________________ ____________________Three men, a physican, a engineer and a computer scientist, aretravelling in a car. Suddenly, the car starts to smoke and stops.The three atonished men try to solve the problem:- Physican says: This is obviously a classic problem of torque. It has been overloaded the elasticity limit of the main axis.- Engineer says : Let's be serious! The matter is that it has been burned the spark of the connecting rod to the dynamo of the radiator. I can easily repair it by hammering.- Computer scientist says : What if we get off the car, wait a minute, and then get in and try again?EA____________________________________________________ _____________________There are comp sci student, an engineering student and a meterology studentgoing through the desert in a jeep.Suddenly the jeep stops and they're left sitting there wondering whathappened..The Eng student pipes up, must be the fan belt thats broken..the engine has overheated...so we'lljusthave to wait till it cools down, bodge the fan belt and we'll be fine.The meterology replies,naw, it's not that...its just the ambient heat in this place. It's notallowing the engine to breath correctly...we just have to wait till nighttime..The comp sci student thinks about this for a minute then says,yeah, you might be right, but I've got an idea....What say we all getout..then get back in again?MEA_________________________________________________ _______________________An engineer, a mathematician, and a computer programmer are drivingdown the road when the car they are in gets a flat tire. The engineersays that they should buy a new car. The mathematician says theyshould sell the old tire and buy a new one. The computer programmersays they should drive the car around the block and see if the tirefixes itself.MBA__________________________________________________ ______________________A biologist, a statistician, a mathematician and a computer scientist areon a photo-safari in Africa. They drive out into the savannah in theirjeep, stop and scour the horizon with their binoculars.The biologist: Look! There's a herd of zebras! And there, in the middle:a white zebra! It's fantastic! There are white zebras! We'll be famous!The statistician:It's not significant. We only know there's one white zebraThe mathematician:Actually, we know there exists a zebra which is white on one sideThe computer scientist:Oh no! A special case!MPA__________________________________________________ ______________________A philosopher, a physicist, a mathematician and a computer scientist weretravelling through Scotland when they saw a black sheep through thewindow of the train.Aha, says the philosopher, I see that Scottish sheep are black.Hmm, says the physicist, You mean that some Scottish sheep areblack.No, says the mathematician, All we know is that there is at leastone sheep in Scotland, and that at least one side of that one sheep isblack!Oh, no! shouts the computer scientist, A special case! Sherlock Holmes and Dr. Watson were travelling on the same trainwhen they passed the same field full of sheep. Look at that solitary black sheep among all those white ones saidWatson to Holmes. Yes Watson, the ratio of black sheep to white in that field isone black to three hundred and seventeen white replied Holmes. But how can you be so precise said Watson, flabbergasted. Elementary, my dear Watson replied Holmes, I counted all of thelegs and divided by four!MPE__________________________________________________ ______________________A mathematician, an engineer, and a physicist are being interviewed for ajob. In each case, the interview goes along famously until the lastquestion is asked: How much is one plus one?Each of them suspects a trap, and is hesitant to answer.The mathematician thinks for a moment, and says I'm not sure, butI think it converges.The physicist says I'm not sure, but I think it's on the order of oneThe engineer gets up, closes the door to the office, and says How muchdo you want it to be?.M_____________________________________________________ _____________________A doctor, a lawyer and a mathematician were discussing the relativemerits of having a wife or a mistress.The lawyer says: For sure a mistress is better. If you have a wifeand want a divorce, it causes all sorts of legal problems.The doctor says: It's better to have a wife because the sense ofsecurity lowers your stress and is good for your health.The mathematician says: You're both wrong. It's best to have both sothat when the wife thinks you're with the mistress and the mistressthinks you're with your wife --- you can do some mathematics.MPB_____________________________________________ ___________________________A Mathematician, a Biologist and a Physicist are sitting in a street cafewatching people going in and coming out of the house on the other sideof the street.First they see two people going into the house. Time passes.After a while they notice three persons coming out of the house.The Physicist: The measurement wasn't accurate..The Biologists conclusion: They have reproduced.The Mathematician: If now exactly 1 person enters the house then it will beempty again.ME__________________________________________________ _______________________There were two men trying to decide what to do for a living. Theywent to see a counselor, and he decided that they had good problemsolving skills.He tried a test to narrow the area of specialty. He put each man in aroom with a stove, a table, and a pot of water on the table. He saidBoil the water. Both men moved the pot from the table to the stoveand turned on the burner to boil the water. Next, he put them into aroom with a stove, a table, and a pot of water on the floor. Again,he said Boil the water. The first man put the pot on the stove andturned on the burner. The counselor told him to be an Engineer,because he could solve each problem individually. The second manmoved the pot from the floor to the table, and then moved the pot fromthe table to the stove and turned on the burner. The counselor toldhim to be a mathematician because he reduced the problem to apreviously solved problem.E___________________________________________________ _______________________Three engineering students were gathered together discussing the possibledesigners of the human body.One said, ``It was a mechanical engineer. Just look at all the joints.''Another said, ``No, it was an electrical engineer. The nervous system hasmany thousands of electrical connections.''The last said, ``Actually it was a civil engineer. Who else would run atoxic waste pipeline through a recreational area?''MPE__________________________________________________ ______________________An engineer, a physicist, and a mathematician are shown a pasturewith a herd of sheep, and told to put them inside the smallestpossible amount of fence. The engineer is first. He herds the sheepinto a circle and then puts the fence around them, declaring, Acircle will use the least fence for a given area, so this is thebest solution. The physicist is next. She creates a circular fence ofinfinite radius around the sheep, and then draws the fence tight aroundthe herd, declaring, This will give the smallest circular fence aroundthe herd. The mathematician is last. After giving the problem a littlethought, he puts a small fence around himself and then declares, Idefine myself to be on the outside!MPE_______________________________________________ _________________________One day a farmer called up an engineer, a physicist, and a mathematicianand asked them to fence of the largest possible area with the leastamount of fence. The engineer made the fence in a circle andproclaimed that he had the most efficient design. The physicist madea long, straight line and proclaimed 'We can assume the length isinfinite...' and pointed out that fencing off half of the Earth wascertainly a more efficient way to do it. The Mathematician justlaughed at them. He built a tiny fence around himself and said 'Ideclare myself to be on the outside.'EC_________________________________________________ ________________________Four men were sitting one day discussing how smart their dog's were.The first man was an Engineer, who said his dog could do math. His dogwas named T-Square, and he told him to get some paper and draw a square,a circle, and a triangle, which the dog did with no sweat.The Accountant said that his dog was better. His dog, Slide Rule, wastold to fetch a dozen cookies, bring them back, and divide them intopiles of 3, which Slide Rule did with no problem.The Chemist said his dog was smarter, his dog named Measure, was told toget a quart of milk, and pour 7 ounces into a 10 ounce glass. The dogdid this with no trouble at all, and all three men agreed that theirdog's were equally smart.Then they turned to the Union Member and asked, what can your dog do?The Union Member called his dog, who was named Coffee Break, and said,Show the fellows what you can do.Coffee Break went over and ate the cookies, drank the milk, on thepaper, ed the other dogs, and claimed he injured his back whiledoing so, filed a grievence report for unsafe working conditions, put infor Workmens Compensation, and left for home on sick leave.MP____________________________________________________ _____________________A mathematician and a physicist are given the task of describing a room.They both go in, and spend hours meticulously writing down every detail,each turning in nearly a ream of paper. The next day, the room is changed,and they are again given the task. The physicist spends the better partof the day, but the mathematician, amazingly enough, leaves within aminute. he hands in a single sheet of paper with the followingdescription: Put picture back on wall to return to previously solved state.ME____________________________________________________ _____________________To tell a difference between a mathematician and an engineer, performthis experiment. Put an empty kettle in the middle of the kitchenfloor and tell your subjects to boil some water.The engineer will fill the kettle with water, put it on the stove, andturn the flame on. The mathematician will do the same thing.Next, put the kettle already filled with water on the stove, and askthe subjects to boil the water. The engineer will turn the flame on.The mathematician will empty the kettle and put it in the middle ofthe kitchen floor... thereby reducing the problem to one that hasalready been solved!MPE__________________________________________________ ______________________A Mathematician (M) and an Engineer (E) attend a lecture by aPhysicist. The topic concerns Kulza-Klein theories involving physicalprocesses that occur in spaces with dimensions of 9, 12 and evenhigher. The M is sitting, clearly enjoying the lecture, while the Eis frowning and looking generally confused and puzzled. By the endthe E has a terrible headache. At the end, the M comments about thewonderful lecture. The E says How do you understand this stuff?M: I just visualize the process.E: How can you POSSIBLY visualize something that occurs in9-dimensional space?M: Easy, first visualize it in N-dimensional space, then let N go to 9.MPE_____________________________________________________ ___________________When considering the behaviour of a howitzer:A mathematician will be able to calculate where the shell will land.A physicist will be able to explain how the shell gets there.An engineer will stand there and try to catch it.MPE______________________________________________________ __________________(Blame translation from German on Joachim)A physicist, an engineer and a mathematician make their first parachutejump. Before the jump the instructor explains exactly what they must do:Jump out of the plane, count until three and pull the line.The physicist jumps. For him counting till three is too unexact and tooprimitive. Instead, he calculates out of his height, angle and velocitythe exact moment he should pull the line for a soft landing and arrivesoptimally.The engineer is a practical man and thinks calling to three is toounreliable and therefore dangerous... He jumps and pulls the lineimmediately. He takes a bit longer than the physicist but he landssafely.Both see jump the mathematician jump out of the plane. He falls ... andfalls ... and falls ...No parachute opens and finally he falls on the ground. Fortunately, helands in a haystack. The physicist and engineer walk alarmed to thefrom complete induction: 3MPCB_____________________________________________________ __________________The USDA once wanted to make cows produce milk faster, to improve thedairy industry.So, they decided to consult the foremost biologists and recombinantDNA technicians to build them a better cow. They assembled this teamof great scientists, and gave them unlimited funding. They requestedrare chemicals, weird bacteria, tons of quarantine equipment, therewas a horrible typhus epidemic they started by accident, and, 2 yearslater, they came back with the new, improved cow. It had a milkproduction improvement of 2% over the original.They then tried with the greatest Nobel Prize winning chemists around.They worked for six months, and, after requisitioning tons of chemicalequipment, and poisoning half the small town in Colorado where theywere working with a toxic cloud from one of their experiments, theygot a 5% improvement in milk output.The physicists tried for a year, and, after ten thousand cows weresubjected to radiation therapy, they got a 1% improvement in output.Finally, in desperation, they turned to the mathematicians. Theforemost mathematician of his time offered to help them with theproblem. Upon hearing the problem, he told the delegation that theycould come back in the morning and he would have solved the problem.In the morning, they came back, and he handed them a piece of paperwith the computations for the new, 300% improved milk cow.The plans began:A Proof of the Attainability of Increased Milk Output from Bovines:Consider a spherical cow......MPCBE____________________________________________ _________________________An assemblage of the most gifted minds in the world were all posed thefollowing question:What is 2 * 2 ?The chemist says immediately circa 10 to the power 1.The engineer whips out his slide rule (so it's old) and shuffles itback and forth, and finally announces 3.99.The physicist consults his technical references, sets up the problemon his computer, and announces it lies between 3.98 and 4.02.The mathematician cogitates for a while, oblivious to the rest of theworld, then announces: I don't what the answer is, but I can tellyou, an answer exists!.Philosopher: But what do you _mean_ by 2 * 2 ?Logician: Please define 2 * 2 more precisely.Accountant: Closes all the doors and windows, looks around carefully, then asks What do you _want_ the answer to be?Computer Hacker: Breaks into the NSA super-computer and gives the answer.Stress engineer: Well I know it's 4, but let's call it 50 anyway.......(blame JV for translation)The psychologist: Why do you wish to know that?The sociologist: I don't know, but is was nice talking about it.Behavioral Ecologist: A polygamous mating system.Medical Student : 4All others looking astonished : How did you know ??Medical Student : I memorized it.ME_______________________________________________________ _________________An Engineer, Statistician and Economist were asked what does 2 + 2 equal?They answered as follows:Engineer: With a safety factor of 2x, 2 + 2 = 8Statistician: With a degree of freedom of 1, 2 + 2 = anywhere from 1 to 7,but I can't be sure.Economist: What would you like it to equal?MP____________________________________________________ _____________________A mathematician, a physicist and a doctor were posed the questin 2*2. The physicist takes a notebook and starts scribbling. After 3 days of themost complex calculations he finds with use of the Earth radius, thegravitation constant : Somewhere between pi and 2 times the square rootof 3. The mathematican comes back after a week with dark rings under his eyesand proclaims: Colleges, their is a solution. The doctor says simple :4The others answer: Oh well you memorized it.MPA____________________________________________________ ____________________(Blame Joachim for translation from German)And yet another variation:A Physicist, a computer scientist and a mathematician must calculate whatis 2 + 2.The physicist constructs out of slopes and balls etcetera a complicatedmeasuring system and finds 3.99998 as solution.Measuring errors are possible, of course4.000001 as solutions.Going from a binary to a decimal system and back can cause inaccuracies.expressions on thousands pieces of papers. Then he proofs that there isonly one solution, and it is calculable.MP_______________________________________________ __________________________Three people answered an add for a an open job - an engineer, aphysicist and a statistician. When the engineer went in, he was asked:Q: What is two plus two?A: Four.When the physisict went in, he was asked the same question:Q: What is two plus two?A: Four.The statistician went in next. When the question was posed to him, helooked around furtively, shut the door and drew the blinds closed. Hisresponse:What do you want it to be?MPE____________________________________________________ ____________________The Board of Trustees, not convinced by the performance in a previous joke,decides to test the Profs. again. First they take a Math Prof. and put himin a room. Now, the room contains a table and three metal spheres about thesize of softballs. They tell him to do whatever he want with the balls andthe table in one hour. After an hour, he comes out and the Trustees look inand the balls are arranges in a triangle at the center of the table.Next, they give the same test to a Physics Prof. After an hour, they lookin, and the balls are stacked one on top of the other in the center of thetable.Finally, the give the test to an Engineering Prof. After an hour, they lookin and one of the balls is broken, one is missing, and he's carrying thethird out in his lunchbox.PE_________________________________________________ ________________________An economist, an engineer, and a physicist are marooned on a desertedisland. One day they find a can of food washed up on the beach andcontrive to open it. The engineer said: let's hammer the can openbetween these rocks. The physicist said: that's pretty crude. We canjust use the force of gravity by dropping a rock on the can from thattall tree over there. The economist is somewhat disgusted at thesedeliberations, and says: I've got a much more elegant solution. All wehave to do is assume a can-opener.E______________________________________________ ____________________________In some foreign country a priest, a lawyer and an engineer areabout to be guillotined. The priest puts his head on the block,they pull the rope and nothing happens -- he declares that he'sbeen saved by divine intervention -- so he's let go. The lawyeris put on the block, and again the rope doesn't release theblade, he claims he can't be executed twice for the same crimeand he is set free too. Theygrab the engineer and shove his head into theguillotine, he looks up at the release mechanism and says, Waita minute, I see your problem......MPE__________________________________________ ______________________________An engineer, a mathematician, and a physicist went to the races oneSaturday and laid their money down. Commiserating in the bar afterthe race, the engineer says, I don't understand why I lost all mymoney. I measured all the horses and calculated their strength andmechanical advantage and figured out how fast they could run...The physicist interrupted him: ...but you didn't take individualvariations into account. I did a statistical analysis of theirprevious performances and bet on the horses with the highestprobability of winning......so if you're so hot why are you broke? asked the engineer. Butbefore the argument can grow, the mathematician takes out his pipe andthey get a glimpse of his well-fattened wallet. Obviously here was aman who knows something about horses. They both demanded to know hissecret.Well, he says, between puffs on the pipe, first I assumed all thehorses were identical and spherical...MPB___________________________________________ _____________________________A group of wealthy investors wanted to be able to predict the outcome of ahorse race. So they hired a group of biologists, a group of statisticians,and a group of physicists. Each group was given a year to research theissue. After one year, the groups all reported to the investors. Thebiologists said that they could genetically engineer an unbeatableracehorse, but it would take 200 years and $100 billion. The statisticiansreported next. They said that they could predict the outcome of any race,at a cost of $100 million per race, and they would only be right 10% of thetime. Finally, the physicists reported that they could also predict theoutcome of any race, and that their process was cheap and simple. Theinvestors listened eagerly to this proposal. The head physicist reported,We have made several simplifying assumptions... first, let each horse be aperfect rolling sphere...MPE______________________________________________ __________________________A group of scientists were doing an investigation into problem-solvingtechniques, and constructed an experiment involving a physicist, anengineer, and a mathematician.The experimental apparatus consisted of a water spigot and two identicalpails, one of which was fastened to the ground ten feet from the spigot.Each of the subjects was given the second pail, empty, and told to fill thepail on the ground.The physicist was the first subject: he carried his pail to the spigot,filled it there, carried it full of water to the pail on the ground, andpoured the water into it. Standing back, he declared, There: I havesolved the problem.The engineer and the mathematician each approached the problem similarly.Upon finishing, the engineer noted that the solution was exact, since thevolumes of the pails were equal. The mathematician merely noted that hehad proven that a solution exists.Now, the experimenters altered the parameters of the task a bit: the pailon the ground was still empty, but the subjects were presented with a pailthat was already half-filled with water.The physicist immediately carried his pail over to the one on the ground,emptied the water into it, went back to the spigot, *filled* the pail, andfinally emptied the entire contents into the pail on the ground,overflowing it and spilling some of the water. Upon finishing, hecommented that the problem should have been better stated.The engineer, in turn, thought for some time before going into action. Hethen took his half-filled pail to the spigot, filled it to the brim, andfilled the pail on the ground from it. Again he noted that the problem hadan exact solution, which of course he had found.The mathematician thought for a long time before stirring. At last hestood up, emptied his pail onto the ground, and declared, The problem hasbeen reduced to one already solved.A__________________________________________________ ________________________ A doctor, an architect, and a computer scientist were arguingabout whose profession was the oldest. In the course of theirarguments, they got all the way back to the Garden of Eden, whereuponthe doctor said, The medical profession is clearly the oldest, becauseEve was made from Adam's rib, as the story goes, and that was a simplyincredible surgical feat. The architect did not agree. He said, But if you look at theGarden itself, in the beginning there was chaos and void, and out ofthat, the Garden and the world were created. So God must have been anarchitect. The computer scientist, who had listened to all of this said,Yes, but where do you think the chaos came from?MPBE_________________________________________________ ______________________The biologist says I study the principles of life.The psychologist says You are controlled by the principles of life.The businessman says My business can use its force to control the economy.The economist says The forces of the economy will control your business.The engineer says: My equations are a model of the universe.The physicist says: The universe is a model of my equations.The mathematician says: I don't care.PCE__________________________________________________ ______________________A chemist, a physicist and an Engineer went on a camping trip, accompaniedby a guide. The were brought to a cabin in the deep Canadian wilderness.Inside the cabin was a wood-burning stove, but it was set up on bricksabout 60 cm above the floor of the cabin. The three scientists speculatedabout the function of the high placement of the stove. The chemist said,Obviously, the guide has anticipated the convection currents of the heatan placed the stove in a raised position to maximize the heat flow in thesemi-adiabatic system. The Physicist believed, No, it's far simplerthan that, the guide placed the stove higher so movement from thecountertops to the stove would be minimized and energy conserved. Theengineer believed he had the true answer, Obviously, you fellows don't domuch camping. The stove is place higher so we can bring in wood and putit under the stove to dry. The guide soon returned and all threescientists were eager to find out who was right. The guide replied,Well, we was bringin' the dang thing up the river and part of the chimneypipe fell off the boat, so we had to put it up for the pipe to reach theceiling.PS: If you know all the words in this essay, your English is better than99% of native Americans.MPE_______________________________________________ _________________________An Engineering Student, a Physics Student, and a Mathematics student wereeach given $150 dollars and were told to use that money to find out exactlyhow tall a particular hotel was. All three ran off, extremely keen on how to do this. The Physicsstudent went out, purchased some stopwatches, a number of ball bearings,a calculator, and some friends. He had them all time the drop of ballbearings from the roof, and he then figured out the height from the timeit took for the bearings to accelerate from rest until they impacted withthe sidewalk. The Math student waited until the sun was going down, then shetook out her protractor, plumb line, measuring tape,and scratch pad,measured the length of the shadow, found the angle the buildings roofmade from the ground, and used trignometry to figure out the height ofthe building. These two students bumped into the Engineering student the nextday, who was nursing a really bad hangover. When asked what he did tofind the height of the building he replied: Well, I walked up to the bell hop, gave him 10 bucks, asked himhow tall the hotel was, and hit the bar inside for happy hour!MP___________________________________________________ ______________________A math student and a physics student are camping. The physics students takeshis turn to do the cooking first. He makes a tasty stew, but in so doing,uses up all the water.The next day, it is the math student's turn to do the cooking. The physicsstudent watches him go to the creek to fetch the water. He puts the waterinto the pot and then stops and goes off to do something else.Puzzled, the physics student asks the math student when he is going tofinish making dinner. The math student tells him that there is nothingleft to do as now it has been reduced to a problem which has already beensolved.MPE____________________________________________ ____________________________A mathematician, a physicist and an engineer were all umpiring a softballgame. The batter hit a fly ball to the outfield that was not caught. Allthe runners who were on base scored easily and the batter tried to turn itinto an inside the park home run. It became clear that there would be aclose play at the plate and all three umpires rushed into position to makethe call. They all called the batter out. The captain of the batting teamwent out to argue and demanded Why is he out?The engineer said He looked out to me, so he's out.The physicist said I watched very carefully, and I saw that, at the momentthat the batter was tagged, he had not touched home plate; so he's out.The mathematician said He's out because I called him out.ME____________________________________________________ _____________________Ask a surveyor, a statistician, and an engineer to measure a 4 cm piece ofstring:Surveyor gets out his tripod, gets an assistant to hold the rod, thencompensates for temperature and declares that the string is 4.000 cm long.Statstician takes a ruler marked in metres and makes (n^-1)/(1-1/n)!measurements before declaring that the string is between 1 cm and 10 cm 90percent of the timeEngineer takes out a pair of scissors and asks How long do you want it tobe?MPE________________________________________________ ________________________A Mathematician, Physicist and an Engineer all have to nip to theloo. The M has a leak, and then sprinkles a few drops of water onhis hands, turns to the attendant and says 'Mathematicians learnto be concise'. The P has a turn, spends 5 minutes scrubbing hishands, then turns to the attendant and says 'Physicists learn tobe thorough'. The engineer has a wee, doesn't bother washing hishands, turns to the attendant and say 'Engineers learn not to peeall over their hands'.MPE__________________________________________________ ______________________A mathematician, a physician and an engineer are on vacation in Paris attheir friend's Jean-Pierre.- How high exactly is that Eiffel Tower? asks the mathematician- I've got an idea, replied Jean-Pierre. How about guessing it, and thewinner wins a good dinner in a good restaurant?, what do you think?- All right, says the physician,...but let's leave us some time andmeet tomorrow at 10 a.m., Ok?- Ok.As the mathematician and the physician stay to think on the problem, theengineer leaves: Sorry, I've got a date, see you tomorrow .The next morning, the friends meet at the bottom of the Eiffel Tower.- So, what's your estimation ? asked Jean-Pierre.- Well, says the mathematician, I measured the length of the shadow ofa simple trigonometric calculation gave me 320,68 metres.- Not a bad idea, replied Jean-Pierre, but not quite the right answer.What about you?- Well, says the physician, I climbed the stairs up to the top of thetower, then I started a chronograph and dropped it immediately. As ithit the ground, it broke, indicating the duration of the fall.Considering the Newton equations and the viscosity of the air, mycalculations gave me 321,9 metres.- That's a bit better, but not the right answer, says Jean-Pierre. But,where is our engineer?The engineer arrives:- Sorry, I'm late, but, woahoo, what a night I had! .- So, what about our little bet ? asked the physician.- Our bet? What bet? Oh yes, the Eiffel Tower! I forgot...err...justwait here a moment.He turns back and comes again 2 minutes later:- The Eiffel Tower is 321,50 metres high.- That's absolutely right, says Jean-Pierre, you won the bet!The mathematician and the physician are puzzled:- How did you do it?And the engineer replies:- Oh...well...quite simple, in fact... I just went to that caf.8e overthere...and asked the waiter... .EA_________________________________________________________ ________________Once upon a time, in a kingdom not far from here, a king summoned twoof his advisors for a test. He showed them both a shiny metal box withtwo slots in the top, a control knob, and a lever. What do you thinkthis is?One advisor, an engineer, answered first. It is a toaster, he said.The king asked, How would you design an embedded computer for it?The engineer replied, Using a four-bit microcontroller, I would writea simple program that reads the darkness knob and quantizes itsposition to one of 16 shades of darkness, from snow white to coalblack. The program would use that darkness level as the index to a16-element table of initial timer values. Then it would turn on theheating elements and start the timer with the initial value selectedfrom the table. At the end of the time delay, it would turn off theheat and pop up the toast. Come back next week, and I'll show you aworking prototype.The second advisor, a computer scientist, immediately recognized thedanger of such short-sighted thinking. He said, Toasters don't justturn bread into toast, they are also used to warm frozen waffles. Whatyou see before you is really a breakfast food cooker. As the subjectsof your kingdom become more sophisticated, they will demand morecapabilities. They will need a breakfast food cooker that can alsocook sausage, fry bacon, and make scrambled eggs. A toaster that onlymakes toast will soon be obsolete. If we don't look to the future, wewill have to completely redesign the toaster in just a few years.With this in mind, we can formulate a more intelligent solution tothe problem. First, create a class of breakfast foods. Specialize thisclass into subclasses: grains, pork, and poultry. The specializationprocess should be repeated with grains divided into toast, muffins,pancakes, and waffles; pork divided into sausage, links, and bacon;and poultry divided into scrambled eggs, hard- boiled eggs, poachedeggs, fried eggs, and various omelet classes.The ham and cheese omelet class is worth special attention because itmust inherit characteristics from the pork, dairy, and poultryclasses. Thus, we see that the problem cannot be properly solvedwithout multiple inheritance. At run time, the program must create theproper object and send a message to the object that says, 'Cookyourself.' The semantics of this message depend, of course, on thekind of object, so they have a different meaning to a piece of toastthan to scrambled eggs.Reviewing the process so far, we see that the analysis phase hasrevealed that the primary requirement is to cook any kind of breakfastfood. In the design phase, we have discovered some derivedrequirements. Specifically, we need an object-oriented language withmultiple inheritance. Of course, users don't want the eggs to get coldwhile the bacon is frying, so concurrent processing is required, too.We must not forget the user interface. The lever that lowers the foodlacks versatility, and the darkness knob is confusing. Users won't buythe product unless it has a user-friendly, graphical interface. Whenthe breakfast cooker is plugged in, users should see a cowboy boot onthe screen. Users click on it, and the message 'Booting UNIX v.8.3'appears on the screen. (UNIX 8.3 should be out by the time the productgets to the market.) Users can pull down a menu and click on the foodsthey want to cook.Having made the wise decision of specifying the software first in thedesign phase, all that remains is to pick an adequate hardwareplatform for the implementation phase. An Intel 80386 with 8MB ofmemory, a 30MB hard disk, and a VGA monitor should be sufficient. Ifyou select a multitasking, object oriented language that supportsmultiple inheritance and has a built-in GUI, writing the program willbe a snap. (Imagine the difficulty we would have had if we hadfoolishly allowed a hardware-first design strategy to lock us into afour-bit microcontroller!).The king wisely had the computer scientist beheaded, and they alllived happily ever after.MPE___________________________________________________ _____________________A mathematician and a physicist are trying to measure the height of aflag pole using a long tape measure. The mathematician takes the tapemeasure, walks up to the flag pole, and begins to shinny up the pole. Ashort way up, he slips and falls down.The physicist notices a ladder lying nearby in the bushes. He leans theladder against the pole, but it reaches only half way up. He climbs theladder and tries to shinny up from there, but he also slips and falls.While they sit near the pole scratching their heads, an engineerwalks by, so the mathematician and the physicist tell him their problem.The engineer notices a crank at the base of the flag pole. He turns thecrank, and the flag pole tilts over until it lies on the ground. Theengineer stretches out the tape measure, cranks the pole back up, andtells the mathematician and the physicist: 'It is 15 meters.'As the engineer walks off into the distance, the mathematician looks atthe physicist and says: 'Isn't that just like an engineer? You ask himfor the height, and he gives you the length.'BUT SOME PEOPLE BELIEVE THE STORY GOES LIKE THIS:A team of engineers were required to measure the height of a flagpole. They only had a measuring tape, and were getting quitefrustrated trying to keep the tape along the pole. It kept fallingdown, etc.A mathematician comes along, finds out their problem, and proceeds toremove the pole from the ground and measure it easily.When he leaves, one engineer says to the other: Just like amathematician! We need to know the height, and he gives us thelength!MPE___________________________________________ _____________________________A mathematician, scientist, and engineer are each asked:Suppose we define a horse's tail to be a leg. How many legs does ahorse have?The mathematician answers 5; the scientist 1; and the engineersays But you can't do that!MPE__________________________________________________ ______________________There are three umpires at a baseball game. One is an engineer, oneis a physicist, and one is a mathematician. There is a close play athome plate and all three umpires call the man out. The manager runsout of the dugout and asks each umpire why the man was called out.The physicist says He's out because I calls 'em as I sees 'em.The engineer says He's out because I calls 'em as they are.And the mathematician says He's out because I called him out.PE____________________________________________________ _____________________In an effort to determine the department which produces the mostintelligent graduates, a university president threw down a challenge to thedeans of the schools of science, engineering, and business. He asked eachto send him their brightest student from the current graduating class tocompete in solving a simple problem.The next day, three students showed up at the university president'soffice. He explained the problem as follows:I want you to determine the height of the university's newest residencetower. I am giving each of you only three tools to work with: a stopwatch, a ruler and a ball of string. You are each to devise your ownsolution to the problem and report back here by the end of the day.Whoever has the most accurate answer wins.The three students set off to the new residence tower. The science manorwent immediately to the roof of the building and dropped the ruler over theside, carefully timing its descent with the stop watch. Factoring in theaerodynamic properties of the ruler, the science major calculated theheight of the building within six inches.Next the engineering major, still panting from running up all the stairs tothe roof, took his turn. He tied the stop watch onto the end of the ballof string and gently lowered it until it just touched the ground. Reelingthe string back up, he measured it carefully with the ruler, makingadjustments for its elasticity under the weight of the stop watch, andcalculated the height of the building within two inches.At that point, the science major turns to the engineering major and asks,What happened to the kid from the business school? I thought he was rightbehind us.They head back down to the building lobby and there, sitting comfortably inan upholstered chair, is the business major.So, what are you going to do? asks the science major.Oh, I'm done, says the business major, unfolding a piece of paper onwhich is written the height of the building expressed to the lastone-eighth inch.How did you do that? asks the engineering major.Simple, replies the student from the business school. While you guyswere screwing around up on the roof, I went down to the basement and foundthe building superintendant. I told him I'd give him a nice stop watch ifhe'd let me look through the architectural plans for the building.There were a number of these kind of stories (which are somewhat similarin stucture to the many There was a priest, a minister and a rabbi...anecdotes).MPE__________________________________ ______________________________________An engineer, a physicist, a mathematician, and a mystic were asked to namethe greatest invention of all time. The engineer chose fire, which gavehumanity power over matter. The physicist chose the wheel, which gavehumanity the power over space. The mathematician chose the alphabet, whichgave humanity power over symbols. The mystic chose the thermos bottle.Why a thermos bottle? the others asked.Because the thermos keeps hot liquids hot in winter and cold liquids coldin summer.Yes -- so what?Think about it. said the mystic reverently. That little bottle -- howdoes it *know*?MPE________________________________________________ ________________________An engineer, a physicist and a mathematician find themselves in ananecdote, indeed an anecdote quite similar to many that you have nodoubt already heard. After some observations and rough calculationsthe engineer realizes the situation and starts laughing. A fewminutes later the physicist understands too and chuckles to himselfhappily as he now has enough experimental evidence to publish a paper.This leaves the mathematician somewhat perplexed, as he had observedright away that he was the subject of an anecdote, and deduced quiterapidly the presence of humour from similar anecdotes, but considersthis anecdote to be too trivial a corollary to be significant, letalone funny.++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++=6.1 THE LOCKED ROOM AND THE TIN CANMPB______________________________________________________ __________________Three men with degrees in mathematics, physics and biology are lockedup in dark rooms for research reasons.A week later the researchers open the a door, the biologist steps outand reports: `Well, I sat around until I started to get bored, thenI searched the room and found a tin which I smashed on the floor.There was food in it which I ate when I got hungry. That's it.'Then they free the man with the degree in physics and he says:`I walked along the walls to get an image of the room's geometry, thenI searched it. There was a metal cylinder at five feet into the roomand two feet left of the door. It felt like a tin and I threw it atthe left wall at the right angle and velocity for it to crack open.'Finally, the researchers open the third door and hear a faint voiceout of the darkness: `Let C be an open can.'MP_____________________________________________________ ____________________(Blame JV for translation from german)Take a mathematician, a physicist and a theologian and locks each of themin his own room with a can of spinach, but withouth a can opener. After aweek one opens the doors. What has happened:With the theologian: Bumps in the can, scratches on the wall, everywhere spinach.=> He threw the can so long against the wall till it broke - He survived.With the physicist: On the wall are calculations and in a corner a small scratch with spinach.=> He calculated the force and the angel needed to open the can. - Hesurvived.With the mathematician: Dead! On the wall a text: I define: The can is open.Which proves you cannot eat from open cans.Which proves before all, that both theologians and physicists can live formone can of spinach and need no water. The mathematician, as a moredemanding lifeform understood immediately, that an open can spinach wouldnot reach for a week and made a last joke.MP_____________________________________________________ ____________________The mathematician's version of the story:A mathematician and a physicist are in a room an try to open a can ofbeans. The physicist calculates the exact angle to throw the can againstthe wall, that is necessary to open it. He throws, the can hits the wall,it bumps back to the physicist and throws him against the floor.When he is standing again, he sees the mathematician eating the beans.He asks: How did you do that?The mathematician: O, I just defined the can open.M____________________________________________________ ______________________And yet another variation:The room with the topologist mathematician, who was locked in the roomwith the closed can is found empty, with the can still closed. Suddenlysomething knocks in the can. The can is opened, the mathematician comes outwith the word: , sign error.MPE_________________________________________________ _______________________There was a mad scientist ( a mad ...social... scientist ) whokidnapped three colleagues, an engineer, a physicist, and amathematician, and locked each of them in seperate cells with plentyof canned food and water but no can opener.A month later, returning, the mad scientist went to the engineer'scell and found it long empty. The engineer had constructed a canopener from pocket trash, used aluminum shavings and dried sugar tomake an explosive, and escaped.The physicist had worked out the angle necessary to knock the lids offthe tin cans by throwing them against the wall. She was developing agood pitching arm and a new quantum theory.The mathematician had stacked the unopened cans into a surprisingsolution to the kissing problem; his desiccated corpse was proppedcalmly against a wall, and this was inscribed on the floor in blood: Theorem: If I can't open these cans, I'll die. Proof: assume the opposite...MP_______________________________________________ __________________________A philosopher, a physicist and a mathematician are scheduled to prove theirsurviving abilities. Therefore they are to be incarcerated in different cellswith each having a closed can of corned beef and no opener. First goes thephilosopher. After a time of two weeks, the cell is opened and thephilosopher found dead, the can still closed. He must have died thinkingabout a way to open it. Second goes the physicist. Two weeks later he isfound with the whole wall filled with formulas, munching happily on hiscorned beef. Last goes the Mathematician. Two weeks later the door isopened. The can is still there, untouched but the mathematician hasdisappeared.Suddenly the Jailor hears a knocking from inside the can. After he has gotan opener, he discovers the mathematician sitting inside, scratching hisbeard and mumbling, There must have been something wrong with the prefix.++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++=6.2 FIREMPE_____________________________________________________ ___________________An engineer, physicist, and mathematician are all challenged with aproblem: to fry an egg when there is a fire in the house. Theengineer just grabs a huge bucket of water, runs over to the fire, andputs it out. The physicist thinks for a long while, and then measuresa precise amount of water into a container. He takes it over to thefire, pours it on, and with the last drop the fire goes out. Themathematician pores over pencil and paper. After a few minutes hegoes Aha! A solution exists! and goes back to frying the egg.Sequel: This time they are asked simply to fry an egg (no fire). Theengineer just does it, kludging along; the physicist calculatescarefully and produces a carefully cooked egg; and the mathematicianlights a fire in the corner, and says I have reduced it to theprevious problem.MPE_______________________________________________ _________________________An engineer, a mathematician, and a physicist are staying in threeadjoining cabins at a decrepit old motel.First the engineer's coffee maker catches fire on the bathroom vanity.He smells the smoke, wakes up, unplugs it, throws it out the window,and goes back to sleep.Later that night the physicist smells smoke too. He wakes up and seesthat a cigarette butt has set the trash can on fire. He says tohimself, Hmm. How does one put out a fire? One can reduce thetemperature of the fuel below the flash point, isolate the burningmaterial from oxygen, or both. This could be accomplished by applyingwater. So he picks up the trash can, puts it in the shower stall,turns on the water, and, when the fire is out, goes back to sleep.The mathematician, of course, has been watching all this out thewindow. So later, when he finds that his pipe ashes have set thebedsheet on fire, he is not in the least taken aback. He immediatelysees that the problem reduces to one that has already been solved andgoes back to sleep.MPE___________________________________________________ _____________________An Engineer, a Physicist, and a Mathematician all go the sameConference. University budgets being what they are, they all stay inthe same cheap hotel. Each room has the same floor plan, has the samecheap TV, the same cheap bed, and a small bathroom. Instead ofa sprinkler system, the hotel has opted for Fire Buckets.The Engineer, Physicist, and Mathematician are all asleep in bed. Atthe corner of the room and sees that the TV set is on fire! He dashesinto the bathroom, fills the Fire Bucket to overflowing with water, anddrenches the TV set. The fire goes out, and the Engineer goes back tosleep.A little while later, the Physicist wakes because he smells smoke. Helooks in the corner and sees that the TV set is on fire. He grabs ahandy envelope, estimates the BTU output of the fire, scribbles a quickcalculation, then dashes into the bathroom and fills the Fire Bucketwith just enough water to douse the flames. He puts the fire out andgoes back to sleep.In a little while, the Mathematician wakes up to the smell of smoke.He looks in the corner and sees the TV on fire. He looks into thebathroom and sees the Fire Bucket. Having determined that a solutionexists, he goes back to sleep.MPE___________________________________________________ _____________________A physicist, an engineer and a mathematician were all in a hotelsleeping when a fire broke out in their respective rooms.The physicist woke up, saw the fire, ran over to his desk, pulledout his CRC, and began working out all sorts of fluid dynamicsequations. After a couple minutes, he threw down his pencil, gota graduated cylinder out of his suitcase, and measured out aprecise amount of water. He threw it on the fire, extinguishingit, with not a drop wasted, and went back to sleep.The engineer woke up, saw the fire, ran into the bathroom, turnedon the faucets full-blast, flooding out the entire apartment,which put out the fire, and went back to sleep.The mathematician woke up, saw the fire, ran over to his desk,began working through theorems, lemmas, hypotheses , you -name-it,and after a few minutes, put down his pencil triumphantly andexclaimed, I have *proven* that I *can* put the fire out!He then went back to sleep.ME____________________________________________________ _____________________ The difference between an Engineer and a Mathematician : The Engineer walks in her office and finds her trash can on fire. Shegets the fire extinguisher and puts out the fire. The Mathematician walks in his office and finds his trash can on fire.He gets the fire extinguisher and puts out the fire. The following day : The Engineer walks in her office and finds the trash can on fire ontop of her desk. She gets the fire extinguisher and put out the fire. The Mathematician walks in his office and finds the trash can on fireon top of his desk. He takes the trash can and puts it on the floor.He has reduced the problem to a previously solved state. Too solve itagain would be redundant.MP________________________________________________ _________________________A physicist and a mathematician setting in a faculty lounge.Suddenly, the coffee machine catches on fire. The physicist grabs abucket and leaps towards the sink, fills the bucket with water andputs out the fire. The second day, the same two sit in the samelounge. Again, the coffee machine catches on fire. This time, themathematician stands up, gets a bucket, hands the bucket to thephysicist, thus reducing the problem to a previously solved one.MPE_____________________________________________________ ___________________An engineer, physicist, and mathematician were playing cards in a parlor.A fire breaks out. The engineer start to calculate how much water it takesto put out the fire. The physicist figures out the best theory on how toput out the fire. The mathematician tries to prove the fire doesn't exist.MPE___________________________________________________ _____________________An engineer, a physicist, a mathematician, and a statistician are taken,one at a time, into a room to undergo a psychological test. In the room isa table (upon which is a pad and pencil), a chair, a bucket of water, and awaste basket rigged so that it can be set ablaze from an adjacent room inwhich the psychologists watch.The engineer is first, and the basket is set ablaze. The engineerimmediately jumps up, grabs the bucket of water and dashes the entire thingonto the fire, flooding the entire room and extinguishing the fire.The physicist is next. The basket ignites, the physicist quickly calculatesexactly how much water is required to extinguish the flames and poursexactly that amount, neatly extinguishing the flames.The mathematician next. The basket blazes up, the mathematician calculatesexactly how much water is required to put out the fire, and then walks outof the room.The statistician is last. The basket is ignited. He grabs the bucket, pourshalf on one side, half on the other, and announces, It's out.MPCE__________________________________________________ _____________________Four professors (An engineer, a physicist, a chemist, and a statistician)are called in to see their dean. Just as they arrive the dean is called outof his office, leaving the three professors there. The professors see withalarm that there is a fire in the wastebasket.Brute force is the answer says the engineer. If we hit it enough we canput it out.The physicist says, I know what to do! We must cool down the materialsuntil their temperature is lower than the ignition temperature and then thefire will go out.The chemist says, No! No! I know what to do! We must cut off the supply ofoxygen so that the fire will go out due to lack of one of the reactants.While they debate what course to take, they are alarmed to see thestatistician running around the room starting other fires. They bothscream, What are you doing?To which the statistician replies, Trying to get an adequate sample size.ME__________________________________________________ _______________________Institute of technology is burning. Engineers realize they should pour some wateron the fire to stop it. As usual they make arather rough calculation on the amount ofwater and pour too much of it. They destroythe whole departement but manage to save their lives. Applied Mathematicians, using an brand-new UFWT(Ultra fast wavelet transform) technique calculatewith a high degree of accuracy the ammountof water required and so, they save their livesAND the whole departement of applied mathematics. Pure Mathemticians are all dead! Why? Well, in 2 minutes they found a very simple prooffor the existence of the solution... they lost then3 hours trying to prove unicity.MP__________________________________________________ _______________________A mathematician and a physicist were asked the following question: Suppose you walked by a burning house and saw a hydrant and a hose not connected to the hydrant. What would you do?P: I would attach the hose to the hydrant, turn on the water, and put out the fire.M: I would attach the hose to the hydrant, turn on the water, and put out the fire.Then they were asked this question: Suppose you walked by a house and saw a hose connected to a hydrant. What would you do?P: I would keep walking, as there is no problem to solve.M: I would disconnect the hose from the hydrant and set the house on fire, reducing the problem to a previously solved form.MPE____________________________________________________ ____________________A company was going to hire someone to do a job. Given the circumstances,and the nature of the job, they were not sure if they needed an engineer,a scientist, or a mathematician. So the manager devised a testto decided which one to hire (I know- a manager with imaginationis kind of hard to swallow, but bear with me).First, the manager brought in he engineer. Without warning, themanager tossed a lit match into the waste paper basket, causing an immediateconflagration. Oh my ghod! shouted the engineer, who the promptly ranout to the hall, returning with a fire extinguisher he had noticed there(having broken the glass with his slide rule- who says those things areworseless?). Several seconds later, the fire was out, with bits of burnedpaper are fire extinguisher foam everywhere.After the engineer had left, the manager brought in the scientist. Onceagain, the basket of waste organocarbon films were induced to exothermicallyreact with the di-oxygen molcules in the atmosphere (whoosh). Oh my thescientist caused to vibrate in the atmospheric substrate. The scientistthen proceeded to make several measurements of the rapidly occurringexcthermic reactions, related mainly to the rate of said reactions, andthen caused said reactions to cease with a controlled flow ofdihydrooxygen obtained from the faucet in the mens room. This causedanother data point to be graphed in the experiment the manager wasperforming.After the scientists had left, the mathematician was brought in. Onceagain, the manager caused a graduate thesis to be throughly reviewed(whoosh). Upon viewing the nature of the problem, the mathematiciansaid The problem is obviously trivial, and will be left as anexercise to the student.The first round of tests being completed, the manager was still leftundecided as to which to hire (that should make the manager morebelievable). So another round was decreed, but this time there wouldbe a twist. Before being lit, the recycling container would first beput underneath a horizontal workarea (desk), preferably of woodenconstruction, so as to add a greater sense of urgency (the possibiltyof said horizontal work-area itself catching fire) and greaterdifficulty (the horizontal work area restricting access to therecycling bin).Once again, the engineer was brought in, and the waste paper basketlit. Once again, the engineer retrieved the fire extinguisher fromthe hall, removed the danger of the desk catching fire and theinterference of his aim by desk by kicking said desk over, and put thefire out. The engineer then left to post several flames to usenetanout people who post long, stupid jokes to alt.folore.computers.Once again, the scientist was brought in, and the exothermic reactionresumed. Once again, said reaction was dampened by the measured applicationof dihydro-oxygen, once said reaction was removed from underneath thedesk. The scientist left to write a paper entitle On the effects ofthe application of dihydro-oxgen to the continuing exothermic reactionsof dioxygen and organocarbon polymers.Once again, the mathematician was brought in, and the thesis reviewed.Sensing a punchline in the offing, the mathemitican removed the reviewingthesis from underneath the desk, whereupon he stated I have reducedthe problem to a trivial problem, and left.-- This post is free post; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. Joachim Verhagen (jcdverha@xs4all.nl)WWW http://www.xs4all.nl/~jcdverha/ (Science Jokes) === Subject: : Re: details of joke wanted> Can somebody help me recall the details of an old maths joke? It is> one of those situations where a computer scientist, a physicist and a> mathematician (or some such combination) have to solve a certain> problem, and the mathematician starts by doing something crazy in> order to reduce to a problem that is already solved.Then there's this one:A physicist, an engineer, and a mathematician are on a stakeout outside ahouse. Through previous observation, they know that there are two peopleinside, and they have to wait until nobody is in the house before they cango in and search it. After some time, one person comes out and drivesoff. A little later, two people come out and drive off.The physicist says There must have been another person in the house thatwe didn't know about.The engineer says There must be another entrance to the house that wedon't know about.The mathematician says We just have to wait until someone enters thehouse. Then there will be zero people in the house, and we can go in.----I've also seen a variant on this one with a biologist instead of anengineer. The biologist says The original two people must havereproduced.-Mike === Subject: : Re: details of joke wanted> Can somebody help me recall the details of an old maths joke? It is one of> those situations where a computer scientist, a physicist and a mathematician> (or some such combination) have to solve a certain problem, and the> mathematician starts by doing something crazy in order to reduce to a> problem that is already solved.The ones with the compsci, physicist and mathmatician are as follows:} A computer scientist, a physicist and a mathmatician are woken up in the} middle of the night by fires in their rooms.}} The computer scientist throws water on the fire until it goes out, and} goes back to bed.}} The physicist works out exactly how much water he needs to put out the} fire, puts it out and goes back to bed.}} The mathmatican considers the above problem, shows that a solution} exists, and then goes back to bed.} A computer scientist, a physicist and a mathmatician take a day trip} to Wales. The compsci spots a black sheep in a field.}} 'Look!', he exclaims, 'All sheep in Wales are black!'}} 'Not quite', says the physicist, 'there is at least one black sheep in} Wales'.}} 'Not quite', says the mathmatician, 'there is at least one sheep in} Wales which is at least one-half black'.} Q: What is pi?}} Compsci: about 3} Physicist: 3.14159 +/- 0.000005} Mathmatician: the ratio of the circumference of a circle to its} diameter.There exist many variants of these.The 'reduction to previous problem' joke is:} 1. You are given a hose, a fire hydrant, and a house on fire. What do} you do?}} PHYSICIST: connect the hose to the hydrant, turn it on, put out the} fire.} MATHEMATICIAN: connect the hose to the hydrant, turn it on, put out the} fire.}} 2. You are given a hose attached to a fire hydrant and a house on fire.} What do you do?}} PHYSICIST: turn on the hydrant.} MATHMATICIAN: disconnect the hose, reducing the problem to the previous} case.}} 3. You are now given a hose, a fire hydrant, and a house not on fire.} What do you do?}} PHYSICIST: Nothing.} MATHMATICIAN: Set fire to the house.There are many variants of this.-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will find these people and we will bring them to justice. === Subject: : Re: details of joke wantedmareg@mimosa.csv.warwick.ac.uk scribbled the following:> Can somebody help me recall the details of an old maths joke? It is one of> those situations where a computer scientist, a physicist and a mathematician> (or some such combination) have to solve a certain problem, and the> mathematician starts by doing something crazy in order to reduce to a> problem that is already solved.I heard the one about the house on fire. The mathematician strikes amatch and sets a piece of paper on fire. Then he gets a glass of water,and empties its contents on the burning paper. The paper stops burning,turning into a black mess of coal. Ah, thinks the mathematician, asolution exists. Then he goes back to bed.-- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland ---------- http://www.helsinki.fi/~palaste --------------------- rules! --------/That's no raisin - it's an ALIEN! - Tourist in MTV's Oddities === Subject: : Re: Calculating the expected range of results> I have a normal distribution of known mean and standard deviation.> For a certain case, a finite number of results will be drawn from this> distribution. Is there a mathematical formula for calculating the> expected range of these results? What exactly do you mean by expected range?> I could interpret the question literally: What is the> expectation value of the range?> Suppose you are drawing n independent samples, X_1, X_2,...> ,X_n.> Here's the cdf of the maximum of the X's:> P[max(X_1, X_2, ..., X_n)<=x] = > = P(X_1 <=x & X_2 <=x & ... & X_n <=x]> = P(X_1 <= x)^n> So the pdf is p(x)= dP/dx> = n*P_x(x)^(n-1)*p_x(x)> where P_x(x) and p_x(x) refer to the normal distribution> of individual samples.> The expectation value is therefore> integral(-inf,inf) n*P_x(x)^(n-1)*p_x(x)*x dx> So you can in principle calculate E[max(x_i)].> Similarly, you can work out a formula for E[min(x_i)]> in terms of P(x_i >= x) = 1 - P_x(x).> Thus, E[max - min] = E[max] - E[min].> - RandyThanks very much everyone. === Subject: : Re: Calculating the expected range of results> I have a normal distribution of known mean and standard deviation.> For a certain case, a finite number of results will be drawn from this> distribution. Is there a mathematical formula for calculating the> expected range of these results? What exactly do you mean by expected range?> I could interpret the question literally: What is the> expectation value of the range?> Suppose you are drawing n independent samples, X_1, X_2,...> ,X_n.> Here's the cdf of the maximum of the X's:> P[max(X_1, X_2, ..., X_n)<=x] = > = P(X_1 <=x & X_2 <=x & ... & X_n <=x]> = P(X_1 <= x)^n> So the pdf is p(x)= dP/dx> = n*P_x(x)^(n-1)*p_x(x)> where P_x(x) and p_x(x) refer to the normal distribution> of individual samples.> The expectation value is therefore> integral(-inf,inf) n*P_x(x)^(n-1)*p_x(x)*x dx> So you can in principle calculate E[max(x_i)].> Similarly, you can work out a formula for E[min(x_i)]> in terms of P(x_i >= x) = 1 - P_x(x).> Thus, E[max - min] = E[max] - E[min].> - RandyYes, I think that is what the OP wanted. Note that in this case, itsimplifies a little. Since we are talking about the distribution ofthe range (=max - min), we may assume WLOG that the mean of thedistribution is 0. Since the normal is symmetric, it is easy to showthat the min has the same distribution as the negative of the max,whence the expected range is twice the expectation of the max. Sincethe max(a x1, a x2,..., a x_n) = a max(x1, x2,..., x_n) when a > 0, we may scale the result for the standard normal. Putting it alltogether, the expected range is2n s int(x=-infty..infty, x [F(x)]^(n-1) f(x))where n is the sample size, s is the standard deviation, and f and F are respectively the pdf and cdf of the standard normal.I do not think this can be expressed in closed form (as HR has noted).By the way, another formula for the expectation of the max isint(x=0..infty, 1 - [F(x)]^n - [F(-x)]^n)(Multiply by 2s for the expected range.) === Subject: : Banach limit and almost convergenceHi!It was noted several times in Google Groups (by Ronald Bruck), thenuniqueness of Banach limit of a sequence is equivalent to almostconvergence of this sequence. Can you give me a hint where to find aproof of this result. (Web resources are prefered, but also book orpaper in a journal would help.)Thanks in advance!Martin Sleziak === Subject: : Re: Banach limit and almost convergence>Hi!>It was noted several times in Google Groups (by Ronald Bruck), then>uniqueness of Banach limit of a sequence is equivalent to almost>convergence of this sequence. Can you give me a hint where to find a>proof of this result. (Web resources are prefered, but also book or>paper in a journal would help.)Have you tried searching the _web_ using Google? If you typeBanach limit almost convergenceinto the search box you get a few hits, one of which says thisresult was proved by Lorentz in A contribution to the theoryof divergent sequences, Acta Math 80(1948).>Thanks in advance!>Martin SleziakDavid C. Ullrich === Subject: : Re: Probability of a Run> I am trying to calculate the probability that a gambler with capital C> and who uses a Martingale betting strategy will be wiped out in m> turns at the game. This would happen with a run of n consecutive> In addition to the recurrence approach, you can also use a markov> process approach to this problem; this has come up before on sci.math> in the context of coin flipping.> > Using this approach, I get, with p = 0.5, n = 30, and m = 1488522243, that you have a 50/50 chance of never getting 30 consecutive events in> m trials.> Cheers - Chas> I like the simplicity of the idea behind this method of calculation.> There are no cancellation problems since we are always adding. It> actually executes reasonably quickly but I appear to be getting a lot> of rounding error problems.> My first reaction was to check my results. I added an extra row and> column to get a double check on my results.> I assume you mean here an extra r/c of 0's (equivalent to aforbidden state of s_n)? I don't see how that would change theresult...> I got > .499999999139885 from adding up the values for 0 to 29> .500000002118337 from the extra row/column> and> .500000000053831 via ProbnmpApprox.> I'm pretty sure that .500000000053831 is accurate, but the rounding> errors via the matrix multiplication method seems a bit drastic.> Can you supply the values you got, if possible it would be nice if you> could add the extra row to confirm whether the problem is genuine or> in my code.> I get 0.499999997774158 regardless of whether or not I add the extrarow/column; but I'm not surprised that they are the same - adding anextra row or column of zeros _shouldn't_ make any difference (whichdoesn't mean that it _wouldn't_).The closest I get to 0.5 is actually at 1488522233 trials, with aresult of 0.500000000102465 (I had previously used the same number oftrials as the other poster had).As another check, I get that the matrix A refered to above has(A^(2^30))[0,0] = 0.389400394801501.I can't really attest to the accuracy of the implementation, as Icoded this up in (believe it or not) Lingo, which is the scriptinglanguage of macromedia's director/shockwave. It's great as a quickprototyping, interactive language; although slow as mollasses formathematical use (it took about 10 seconds to calculate the above). Atsome base level though, it surely uses native floating point; so itshould be as accurate as a one would expect.I'll code it up in C when I get a chance and post the results.Cheers - Chas> Ian Smith === Subject: : Re: Probability of a Run> I am trying to calculate the probability that a gambler with capital C and who uses a Martingale betting strategy will be wiped out in m> turns at the game. This would happen with a run of n consecutive> > In addition to the recurrence approach, you can also use a markov> process approach to this problem; this has come up before on sci.math> in the context of coin flipping.> > Using this approach, I get, with p = 0.5, n = 30, and m = 1488522243,> that you have a 50/50 chance of never getting 30 consecutive events in> m trials. Cheers - Chas> I like the simplicity of the idea behind this method of calculation.> There are no cancellation problems since we are always adding. It> actually executes reasonably quickly but I appear to be getting a lot> of rounding error problems.> My first reaction was to check my results. I added an extra row and> column to get a double check on my results. I assume you mean here an extra r/c of 0's (equivalent to a> forbidden state of s_n)? I don't see how that would change the> result...> I got > .499999999139885 from adding up the values for 0 to 29> .500000002118337 from the extra row/column> and> .500000000053831 via ProbnmpApprox.> I'm pretty sure that .500000000053831 is accurate, but the rounding> errors via the matrix multiplication method seems a bit drastic.> Can you supply the values you got, if possible it would be nice if you> could add the extra row to confirm whether the problem is genuine or> in my code. I get 0.499999997774158 regardless of whether or not I add the extra> row/column; but I'm not surprised that they are the same - adding an> extra row or column of zeros _shouldn't_ make any difference (which> doesn't mean that it _wouldn't_).> The closest I get to 0.5 is actually at 1488522233 trials, with a> result of 0.500000000102465 (I had previously used the same number of> trials as the other poster had).> As another check, I get that the matrix A refered to above has> (A^(2^30))[0,0] = 0.389400394801501.> I can't really attest to the accuracy of the implementation, as I> coded this up in (believe it or not) Lingo, which is the scripting> language of macromedia's director/shockwave. It's great as a quick> prototyping, interactive language; although slow as mollasses for> mathematical use (it took about 10 seconds to calculate the above). At> some base level though, it surely uses native floating point; so it> should be as accurate as a one would expect.> I'll code it up in C when I get a chance and post the results.> Cheers - Chas Ian SmithI wasn't thinking too brightly when I asked about the accuracy of thematrix multiplication approach. If I had been, I would have realisedthat rounding error problems are inherent when raising a matrix to alarge power. The errors you are getting are actually smaller than onemight expect due to the fact that the probabilities in the originalmatrix can be held exactly.Basically the method is reasonably quick and the error propagationproperties are known. As long as you don't use it for huge numbers oftrials you'll get accurate answers.Ian Smith === Subject: : Re: I NEED HELP BADLY (sorry, maths not psych)>>>> If fields acted instantaneously, we would be>> able to use them for instantaneous communication.>>Please explain how you can use the fact that a force is>a static electric field, for instantaneous communication.>>Paul, puzzled> I'm sure some bright QMian would find a way.Backing out again, Henry?You cannot defend your assertion,but you will repeat it, won't you?And you will back out once more whenasked to defend your repeated assertion, won't you?That's Henry Wilson's eternal circle of fleeingrestatements of fled assertions.Paul, not surprised === Subject: : Re: What is Advanced Calculus?> I am a bit confused.> It seems as though Multivariable Mathematics is just another name> for Advanced Calculus, and the same applies to it as well.> The Implicit Function Theorem, the Invese Function Theorem, the>> Taylor's Theorem in n-dimensions, derivatives as linear maps,>> 2nd derivatives as bilinear maps, differentiability as being>> distinct from having derivatives, Frobenius' Theorem, maybe>> some Distribution Theory. On the integral side, Measure>> Theory, different definitions for integrals, etc.>>> Thanks for that list.>> Can somebody recommend a mostly formal textbook on these topics? > Hopefully something about two inches thick, second or third edition, > with everything proven, lots of examples, tons of problems and a > complete solutions manual. I loved Ellis & Gulick, Calculus and > Analytic Geometry, 2nd Ed.> Buck, Advanced Calculus.> Spivak, Calculus on Manifolds.> You will not find any with solution manuals.> Others will recommend introductory analysis texts, probably> Rudin, Principles of Mathematical Analysis.> Personally, I find that rough reading for self-teaching, but I have no > alternative to offer. (Goldberg is readable but does no multivariable > analysis.) Besides, it sounds to me as if you are more interested in > the calculus direction than the analysis direction (though this > distinction is somewhat vague). I really think the two books I > recommend are what you seek.In 1965-6, we used Goldberg, which was new at the time. I understoodthat the professor used two books the previous year, one of which wasApostol, Mathematical Analysis.David Ames === Subject: : Fastest factorial algorithmWhat is the fastest algorithm for computing factorial, for very large numbers (e.g. 10000!)?Normally this would take n-2 multiplications, by multiplying out each term n(n-1)(n-2)(n-3)...3.2. Is a better way known? === Subject: : Re: Fastest factorial algorithm> What is the fastest algorithm for computing factorial, for very large > numbers (e.g. 10000!)?> Normally this would take n-2 multiplications, by multiplying out each > term n(n-1)(n-2)(n-3)...3.2. Is a better way known?This may be a useless reply, but is there a procedure which COMBINESsomehow, say, a good Stirling approximation to n! with the fact that agiven prime p divides n! a given (known) number of times?(If p^k is highest power of p dividing n!, thenk = sum{j>=1} floor(n/p^j) =(n - (sum of base-p digits of n))/(p-1).)I am wondering if we can narrow the possible values of n! down byusing the Stirling approximation, and given that we know some otherthings about n! such as what some of its divisors are. So, can wereally use this two-step algorithm to get n! (for n = large positiveinteger)??Leroy Quet === Subject: : Re: Fastest factorial algorithm> What is the fastest algorithm for computing factorial, for very large > numbers (e.g. 10000!)?> Normally this would take n-2 multiplications, by multiplying out each > term n(n-1)(n-2)(n-3)...3.2. Is a better way known?Since you get lots of 2's in the factorization of n! there is a bit of divide and conquer you can do.n! = 1*2*3*4*...(n-1)*n = 1* 3* 5* *...*(n-1) 2* 4* 6* ... *n = 1* 3* 5* *...*(n-1) 1* 2* 3* ... *n/2 * 2^(n/2) = 1* 3* 5* *...*(n-1) 1* 3* ... *n/2 * 2^(n/2) 1* 2*... * 2^(n/4)etc. So remember all your partial products in computing 1*3*5*...so very roughly n/2 + 3 lg n mults.n/2 to multiply the odds, 3 lg n to compute the powers of 2 and multiply them with the partial products and then multiply all together.I think this is called binary splitting.Theoretically, extend this idea (for all primes, not just 2) and you get the algorithm in (I think):Borwein, Peter B.On the complexity of calculating factorials.J. Algorithms 6 (1985), no. 3, 376--380.Mitch === Subject: : Re: Fastest factorial algorithm> What is the fastest algorithm for computing factorial, for very large> numbers (e.g. 10000!)?> Normally this would take n-2 multiplications, by multiplying out each> term n(n-1)(n-2)(n-3)...3.2. Is a better way known?Don't know, however if you wait until the contest athttp://www.ni.com/devzone/lvzone/codingchallenge.htmis finished and the results presented, you'll likelyhave some good ones. === Subject: : Combinatorial proofShow with combinatorial reasoning thatC(3n,3) = 3*C(n,3)+6n*C(n,2)+n^3, where C(n,k) is n choose k.Am I supposed to show that both sides of the equation counts the thing? Does it qualify as combinatorial reasoning if I use combinatorial identities or prove it by induction?Any good hints, by the way?-- Be sure to include the word zebra in the subject line of any mail sentto my email adress fergusprint@casino.com since it will be automaticallyremoved otherwise! === Subject: : Re: Combinatorial proof> Show with combinatorial reasoning that> C(3n,3) = 3*C(n,3)+6n*C(n,2)+n^3, where C(n,k) is n choose k.> Am I supposed to show that both sides of the equation counts the thing?> Does it qualify as combinatorial reasoning if I use combinatorial> identities or prove it by induction?> Any good hints, by the way?Big hint (maybe even a spoiler):You have 3 jars with n numbered marbles each.You must pick 3 marbles.There are 3 ways to do it: 1) take 3 marbles from one jar. 2) take 2 marbles from one jar and 1 from another one 3) take 1 marble from each jar.These 3 ways are independent, so you can add the numbers...Dirk Vdm === Subject: : Programmer ax+by=c en CSalut tout le monde,Ce n'est peut-.90tre pas tr.8fs compliqu.8eMais je me suis pris la t.90te .88 programmer en C ax+by=co.9d a,b,c,x et y sont tous des nombres entiers.On conna.94t a,b et c et l'objectif c'est d'obtenir x et y.D'abord je commence par le fait que le pgcd(a,b) est un diviseur aussi de c.Quand je le fais .88 la main c'est .8evident mais pour le programme je suis bloqu.8e ici.Si quelqu'un sait comment faire et peut m'aider? === Subject: : Re: Programmer ax+by=c en C>Salut tout le monde,>Ce n'est peut-.90tre pas tr.8fs compliqu.8e>Mais je me suis pris la t.90te .88 programmer en C ax+by=c>o.9d a,b,c,x et y sont tous des nombres entiers.>On conna.94t a,b et c et l'objectif c'est d'obtenir x et y.>D'abord je commence par le fait que le pgcd(a,b) est un diviseur aussi de c.>Quand je le fais .88 la main c'est .8evident mais pour le programme je suis bloqu.8e ici.>Si quelqu'un sait comment faire et peut m'aider?Si j'ai bien compris c'est le th.8eor.8fme de B.8ezout dont vous parlez, pasl'.8equation d'une ligne droite :) Cherchez sur google 'algorithmeEuclide .8etendu:En C:// R.8esoudre ax+by=pgcd(a,b), a et b connus:void Euclide_Etendu (unsigned int a,unsigned int b, int *pRet_d, /* pgcd(a,b) */ int *pRet_x, /* coefficient x */ int *pRet_y){ /* coefficient y */// Ces variables doivent etre signees, car on peut avoir des negatifs int x, x1, x2, y, y1, y2, q, r; if (b==0) { *pRet_d=a; *pRet_x=1; *pRet_y=0; return; } x2=y1=1; x1=y2=0; while (b>0) { q=a/b; r=a-q*b; x=x2-q*x1; y=y2-q*y1; a=b; b=r; x2=x1; x1=x; y2=y1; y1=y; } *pRet_d=a; *pRet_x=x2; *pRet_y=y2;} === Subject: : Re: Programmer ax+by=c en CVoulez vous typez en anglais s.v.p.?|| >Salut tout le monde,| >Ce n'est peut-.90tre pas tr.8fs compliqu.8e| >Mais je me suis pris la t.90te .88 programmer en C ax+by=c| >o.9d a,b,c,x et y sont tous des nombres entiers.| >On conna.94t a,b et c et l'objectif c'est d'obtenir x et y.| >D'abord je commence par le fait que le pgcd(a,b) est un diviseur aussi de c.| >Quand je le fais .88 la main c'est .8evident mais pour le programme je suisbloqu.8e ici.| >Si quelqu'un sait comment faire et peut m'aider?|| Si j'ai bien compris c'est le th.8eor.8fme de B.8ezout dont vous parlez, pas| l'.8equation d'une ligne droite :) Cherchez sur google 'algorithme| Euclide .8etendu:|| En C:| // R.8esoudre ax+by=pgcd(a,b), a et b connus:| void Euclide_Etendu (unsigned int a,unsigned int b,| int *pRet_d, /* pgcd(a,b) */| int *pRet_x, /* coefficient x */| int *pRet_y){ /* coefficient y */| // Ces variables doivent etre signees, car on peut avoir des negatifs| int x, x1, x2, y, y1, y2, q, r;| if (b==0) {| *pRet_d=a;| *pRet_x=1;| *pRet_y=0;| return;| }|| x2=y1=1;| x1=y2=0;|| while (b>0) {| q=a/b;| r=a-q*b;| x=x2-q*x1;| y=y2-q*y1;| a=b;| b=r;| x2=x1;| x1=x;| y2=y1;| y1=y;| }|| *pRet_d=a;| *pRet_x=x2;| *pRet_y=y2;|| }| === Subject: : Re: Programmer ax+by=c en CSalut branko tu t'es tromp.8e de newgroups, il faut aller sur fr.sci.maths oucarr.8ement sur un newsgroups de programmation.-GS-> Salut tout le monde,> Ce n'est peut-.90tre pas tr.8fs compliqu.8e> Mais je me suis pris la t.90te .88 programmer en C ax+by=c> o.9d a,b,c,x et y sont tous des nombres entiers.> On conna.94t a,b et c et l'objectif c'est d'obtenir x et y.> D'abord je commence par le fait que le pgcd(a,b) est un diviseur aussi dec.> Quand je le fais .88 la main c'est .8evident mais pour le programme je suisbloqu.8e ici.> Si quelqu'un sait comment faire et peut m'aider? === Subject: : A question in projective geomtryFirst of all, I'm not sure who this newsgroup is intended for, so I hope I'mnot going to ask a question that will be too trivial... but I need help withthis subject, so I'll go ahead anyway :-)How to prove that in finite projective geomtry, there is an equal number ofpoints on all the lines?Thank you very much in advance. === Subject: : Re: A question in projective geomtryX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 03:51 PM, Arie Levit said:>How to prove that in finite projective geomtry, there is an equal>number of points on all the lines?Is this a homework question? Do you know what a collineation is? That may already be more of a hintthan I should have given, so I'll let it go at that.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to spamtrap@library.lspace.org === Subject: : Re: A question in projective geomtry> First of all, I'm not sure who this newsgroup is intended for, so I hope I'm> not going to ask a question that will be too trivial... but I need help with> this subject, so I'll go ahead anyway :-)> How to prove that in finite projective geomtry, there is an equal number of> points on all the lines?> Thank you very much in advance.What are the axioms for (finite) projective geometry?Two different lines intersect in exactly one point.Two different points lie on exactly one line.These are what you have to use in your proof. === Subject: : Re: A question in projective geomtryI know that my proof should be based on the axioms (they're are quitesimple, as well). Thats why its bothering I can't prove it. Perhaps you cangive some tip about the body of the proof... === Subject: : Re: A question in projective geomtry> First of all, I'm not sure who this newsgroup is intended for, so I hope I'm> not going to ask a question that will be too trivial... but I need help with> this subject, so I'll go ahead anyway :-) How to prove that in finite projective geomtry, there is an equal number of> points on all the lines?> Thank you very much in advance.>>What are the axioms for (finite) projective geometry?>>Two different lines intersect in exactly one point.>>Two different points lie on exactly one line.>>These are what you have to use in your proof.>I know that my proof should be based on the axioms (they're are quite>simple, as well). Thats why its bothering I can't prove it. Perhaps you can>give some tip about the body of the proof...You can't prove it based on those axioms alone, as a simple exampleshows: draw three collinear points in the plane, and a fourth pointthat is not collinear with the first three. Now draw lines through allpairs of points. There are four lines in all. One line contains fourpoints, the remaining lines each contain two points. The axioms aboveare clearly satisfied.You need another axiom: each line contains at least three points. Nowconsider two (distinct) lines, L and M. Choose distinct points P on Land Q on M (prove that this is possible). Draw a line through P and Q(prove that this line is distinct from L and M), and choose a thirdpoint R on this line, distinct from P and Q. Now use R to map pointson L to points on M (using a simple natural construction). Show thatthis mapping is 1-1. The same method gives a 1-1 map from M to L .Voila!John Mitchell === Subject: : Re: Newsgroup survey: Math and personality assessment <3c65f87.0311120412.33636da4@posting.google.com> Discussion, linux)> Notice above that I referenced Merriam Webster Online, a great> resource for quick access to definitions on the go--access at the> speed of the Internet!> Merriam-Webster Online> http://www.m-w.com/I think you should put away that marketing book. It's getting to you.-- Jesse HughesYou see 300 of something, anything, and you go `[Man], that's a lot ofstuff.' -- Jim Bigler, quoted in the Pittsburgh Post-Gazette. === Subject: : Re: Newsgroup survey: Math and personality assessment Discussion, linux)>> It seems to me that there have been debates over math concepts I>> thought basic, so here's a quick survey:> James is the person who's always whining about social issues in> mathematics and accusing mathematicians of caring more about each other's> opinions than about the Truth. So why is he also the only person who> conducts these ridiculous opinion surveys? Does he *really* think that> *this time* he's going to find that the masses are on his side? And why> does he care *what* everyone else thinks, if he's so sure he's right?Duh. He's doing marketing research. -- Well *supposedly* a correct and profound math paper can get publishedin a 'reputable journal' which means that the journals I've faced sofar may lose a lot of their luster once the full story comes out.s--- , on the quality of math journals rejecting his paper === Subject: : Fundamental groups and liftings help please!I am somewhat confused now...maybe someone can pleeeeease help.My homework problem says let p : E --> B be a covering map and fix b_0 in B.Let p^{-1}(b_0) x pi_1 (B ; b_0) --> p^{-1}(b_0) be given by x * [f] = f '(1), where f ' : I ---> E is the unique lift of f to a path based at x inp^{-1}(b_0).I have to show that this is a well defined group action of pi_1 (B ; b_0) onthe fiber p^{-1}(b_0).(Note : pi_1 (B ; b_0) denotes the fundamental group of B based at b_0 andp^{-1}(b_0) is merely the inverse image of b_0 under p)Ok, I am really confused. It is obvious that if you take two loops f and gthat are not path homotopic, then you get x * [f] = f ' (1) and x * [g] = g' (1), and f ' (1) does not equal g ' (1) since f and g are not pathhomotopic.Is this all that is needed to show that this group action is well-defined?My problem is that f ' (1) DEPENDS on x, doesn't it? f ' is a lift of fthat starts at x, but there are a bunch of different x's sitting indifferent slices in E. So, f ' (1) is not the same for any x inp^{-1}(b_0). Right? I mean, if you pick a lift that starts in one certainslice you get a certain f ' (1), but if you pick a lift that sits in adifferent (but homeomorphic) slice that is disjoint, you get literally adifferent element of E! Or maybe I am REALLY confused!Help! pleaseMike === Subject: : Re: Fundamental groups and liftings help please!|I am somewhat confused now...maybe someone can pleeeeease help.||My homework problem says let p : E --> B be a covering map and fix b_0 in B.|Let p^{-1}(b_0) x pi_1 (B ; b_0) --> p^{-1}(b_0) be given by x * [f] = f '|(1), where f ' : I ---> E is the unique lift of f to a path based at x in|p^{-1}(b_0).||I have to show that this is a well defined group action of pi_1 (B ; b_0) on|the fiber p^{-1}(b_0).||(Note : pi_1 (B ; b_0) denotes the fundamental group of B based at b_0 and|p^{-1}(b_0) is merely the inverse image of b_0 under p)||Ok, I am really confused. It is obvious that if you take two loops f and g|that are not path homotopic, then you get x * [f] = f ' (1) and x * [g] = g|' (1), and f ' (1) does not equal g ' (1) since f and g are not path|homotopic.yep, you're a bit confused, but it's probably not fatal.(the secret way to really understand this stuff is a bit different inspirit from the approach that your teacher seems to be using, but iwon't worry about that for now, since you can probably understand yourteacher's approach if you just keep working at it a bit longer.)actually it's _not_ obvious that if loops f and g aren'tpath-homotopic then f'(1) and g'(1) can't be equal, and in fact it'sfalse; for example consider the special case where the covering map pis the identity map of the base space; in this case _all_ points inp^[-1](b_0) are equal to each other. (maybe you were thinking of theopposite extreme special case of the _universal_ covering space, wherewhat you said is true more or less by definition?)|Is this all that is needed to show that this group action is well-defined?again, that's _not_ what you have to show, and it's a good thing youdon't have to show it because it's not always true. rather, what youhave do to show that the action is a well-defined map is to showthat if loops f and g _are_ path-homotopic (meaning with fixedend-points), then f'(1) and g'(1) _are_ equal.i'll leave it at that for now, in case that helps. by the way,depending on how the question is interpreted there's probably a bitmore that you have to do after showing that the action is awell-defined map; you probably have to show that it satisfies thealgebraic definition of action as well.-- [e-mail address jdolan@math.ucr.edu] === Subject: : Re: Fundamental groups and liftings help please!>I am somewhat confused now...maybe someone can pleeeeease help.>My homework problem says let p : E --> B be a covering map and fix b_0 in B.>Let p^{-1}(b_0) x pi_1 (B ; b_0) --> p^{-1}(b_0) be given by x * [f] = f '>(1), where f ' : I ---> E is the unique lift of f to a path based at x in>p^{-1}(b_0).>I have to show that this is a well defined group action of pi_1 (B ; b_0) on>the fiber p^{-1}(b_0).>(Note : pi_1 (B ; b_0) denotes the fundamental group of B based at b_0 and>p^{-1}(b_0) is merely the inverse image of b_0 under p)>Ok, I am really confused. It is obvious that if you take two loops f and g>that are not path homotopic, then you get x * [f] = f ' (1) and x * [g] = g>' (1), and f ' (1) does not equal g ' (1) since f and g are not path>homotopic.That's not true, nor is it what you have to prove. Consider, forexample, a double-cover of a circle (E = S^1, B = S^1). The loop thatgoes twice around the B and the loop that just sits at the base pointof B both induce the identity map on the fiber p^{-1}(b_0), but theyare not homotopic loops.Well-defined means that if f and g are homotopic, then f'(1) =g'(1). You can use the homotopy lifting theorem to show this. You thenhave to show that the action is a group action, which isn't hard.John Mitchell>Is this all that is needed to show that this group action is well-defined?>My problem is that f ' (1) DEPENDS on x, doesn't it? f ' is a lift of f>that starts at x, but there are a bunch of different x's sitting in>different slices in E. So, f ' (1) is not the same for any x in>p^{-1}(b_0). Right? I mean, if you pick a lift that starts in one certain>slice you get a certain f ' (1), but if you pick a lift that sits in a>different (but homeomorphic) slice that is disjoint, you get literally a>different element of E! Or maybe I am REALLY confused!>Help! please>Mike === Subject: : About big numbers...I got to thinking about that big numbers thread. Here's one quite bignumber.Let f: N^2->N be f(i, 0)=i, f(i, j)=f(i, j-1)! if j>0.My number is:f(9, f(9, f(9, f(9, f(9, f(9, f(9, f(9, f(9, f(9, f(9, 9))))))))))).There are 11 nested iterations of f.How big is this number? Is it smaller or bigger than Graham's number?It ends in a rather long strip of zeroes but I haven't the foggiestwhat it starts with.-- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland ---------- http://www.helsinki.fi/~palaste --------------------- rules! --------/Show me a good mouser and I'll show you a cat with bad breath. - Garfield === Subject: : Re: What constitutes an implicit use of a function? (Was: a confusing group t||>if one has a given property, ||What do you mean by has a given property? Your usage seems to|transcend the values of the propositional functions, in which case two|propositional functions with the same values need not have the same|properties.A term often used in this context is extensionality. Two properties areconsidered extensionally equivalent if the same objects satisfy them. Soanother way to put it is that you (Sniz Pilbor) are consideringnon-extensional aspects of your property, and shouldn't suppose that twoextensionally equivalent properties will make implicit use of the samefunctions (and so on).Don't get too hung up on trying to make the notion of implicit usageprecise. Using a little mathematical logic one could no doubt cook up adescription of certain ways in which a function could be said to appearimplicitly in a proposition or term. You could characterize some restrictedway in which the existence of a function could be deduced from thestatement.That would be too rigid, however. The way the phrase implicitly uses isordinarily used is informal, and whether some bit of mathematics would beconsidered implicitly to use a function would depend in a fuzzy way on themanner in which it's expressed. It's a matter of using your intuition tojudge what the essential ingredients are.Keith Ramsay === Subject: : measurable functionX-Filename: sci.math/A0VR4L90 Let f(x,y) be a function defined on the unit square 0<=x<=1, 0<=y<=1 which is continuous in each variable separately. Show that f is a measurable function of (x, y). Please give me a hint. Thank you!-- === Subject: : Re: measurable function> Let f(x,y) be a function defined on the unit square 0<=x<=1, 0<=y<=1> which is continuous in each variable separately. Show that f is a> measurable function of (x, y).For each n, divide the square into n equally spaced horizontal strips. On each strip, make f_n independent of y by setting it equal to f's values on the lower edge of the strip. Now you have a sequence f_n of measurable functions, and fn -> f a.e. (in fact everywhere). === Subject: : Re: Factorial/Exponential Identity, InfinityYou see, fish or cut bait doesn't mean stop or go. It means go.Either fish, ie troll, or cut bait, disassembling the logicalprogression of our discussion of sequences with equal zero and onedensity vis-a-vis sequences normal to base two, cutting the bait thusthat I may use it to troll sci.math.Either way the point is to answer the questions about whethersequences normal to a base other than two, or a power or multiple oftwo, have equal zero and one densities, or to ask them.For you to not even address this besides claiming ignorance ishilarious.Sets of numbers contain only points. http://www.tiki-lounge.com/~raf/math/CLICAL.tar.gzSolve with CLICAL, the Clifford Calculator.Ross === Subject: : Re: Cardinality of 2^n numbers?> You're misusing cantor's theorum.Could it be that rather than everyone in this threadmisusing Cantor's theorem, that you alone are misunderstanding it?> I never said that K wasn't greater> than K. I said N was equal in size to the power set of K. Cantor's> theorum does not involve this kidn of thing.Cantor's theorem does not involve the relativecardinalities of K and its power set? Huh?Cantor's Theorem says |P(K)| > |K|. Since |K| = |N|,Cantor's Theorem tells us immediately that N issmaller in cardinality than the power set of K. Itis very much involved with this kind of thing. Whatmakes you think it isn't involved with either cardinality of power sets or the cardinality of N?> The power set of K is related to the natural numbers if you do this:> For each element in the power set of K, add all the numbers together.> You will always get a unique natural number.K is an infinite set. Most of its subsets (i.e. most ofthe elements of P(K)) are infinite. There is no correspondingnatural number.> For instance, take K = (1, 2, 4). Take the power set of K = (Null, 1,> 2, 1&2, 4, 1&4, 2&4, 1&2&4). Now this relates to the natural numbers> Num = (0, 1, 2, 3, 4, 5, 6, 7) easily. Now K is smaller than the power> set of K. However, the power set of K, as the size of the power set> approaches infinity, also approaches the set of the natural numbersNope, you're taking an invalid limit argument again. Allyou can tell by this argument is that as K grows withoutbound, so does |P(K)|. You can't tell anything aboutthe cardinality of the infinite set. Limit argumentsonly concern themselves with what happens at finitevalues. They say nothing about at infinity.Here's another mapping for you. Consider the infiniteset K = {1, 2, 4, ... }. For any given subset of K, say{1, 2^4, 2^5} construct a binary fraction with a 1bit if K contains 2^(n-1). Thus, for {2^0, 2^4, 2^5}I get the number 0.100011.There is clearly a bijection between such fractionsand the subsets of K. How many such fractions arethere? Well, there is one for every value in [0,1],rational or irrational. And each one of thosecorresponds to a subset of K, a member of P(K).How many numbers do you think there are in [0,1]? - Randy === Subject: : Re: Cardinality of 2^n numbers?> You are very unhelpful. Are you sure you're not just here to be as> completely useless as possible?You are very helpless. Are you sure you're not just here to be ascompletely useless as possible?-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: : Re: Cardinality of 2^n numbers?> I never said 2^K WAS the set of all natural numbers. I said 2^K was> BIJECTIVE with the set of all natural numbers.And you were wrong in saying so.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: : Re: Cardinality of 2^n numbers?> The only way I see there being a contradiction is if you require that> K and N both be of cardinality N_0.> However, I don't see why there can't be a non-N_0 cardinality of K> such that when its power set is taken, its power set is of cardinality> N_0. That will also have to be explained to me.There isn't.Is it worth explaining this to you? In the sense that you mightpossiibly grasp the explanation?-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: : Re: Cardinality of 2^n numbers? permission for an emailed response.> The only way I see there being a contradiction is if you require that> K and N both be of cardinality N_0.I *proved* that K and N have the same cardinality. === Subject: : Re: Cardinality of 2^n numbers?> I never said 2^K WAS the set of all natural numbers. I said 2^K was> BIJECTIVE with the set of all natural numbers.> (...Starblade Riven Darksquall...)For K any infinite subset of the naturals,if 2^K represents {2^k : k in K} you get one cardinality, butif 2^K represents the set of all functions from {0,1} to K, you get an entirely different cardinality.In the first case, you can do such a bijection, but in the second case not. === Subject: : Re: Cardinality of 2^n numbers?> However, I don't see why there can't be a non-N_0 cardinality of K> such that when its power set is taken, its power set is of cardinality> N_0. That will also have to be explained to me.> (...Starblade Riven Darksquall...)The power set of any finite set has finite cardinality, and the power set of any infinite set has cardinality greater than the smallest possible infinite cardinal.If N_0 represents smallest non-finite cardinality, what is K? === Subject: : Re: Factorization dispute, againIn sci.math, G Frege:>>> There are plenty of examples of the converse as well,>> where somebody proves that something in general need not>> be true, but James comes up with an example where it is>> true and claims that proves the general result.>> Well, actually this is a valid proof method (in modern Harrisanism).> Let x be an arbitrary number. Now consider 0 = x. Obviously we have > 0 = 0. With other words, x = 0. (Note that 0 is a constant! Hence x is a> constant too!) Now since x has been arbitrary, this means x = 0 for any> x! This actually proves FLT. qed.> ...> Hmm...beats the usual divide by zero and square root sign proofs... :-)But not by much. :-)As it is, JSH's latest attempts appear to be Let . Then . Then. Now [pick one: 1, 2, 4, 23]of in are obviously divisible by[pick one: 2, 3, 7, 11, 97] but there's only[pick one: 1, 2, 3, 22] factors of [pick one: 2, 3, 7, 11, 97]in the equation, therefore the algebraic numbers are equivalentto C and . QED(Queerly Elicited and Debatable):-)-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: : Re: Factorization dispute, againX-ID: buP2q0ZHgerWLJECOBhJSW1ULHOnTZgiw9w7il0dU554I-v26uJrcr> As it is, JSH's latest attempts appear to be > Let . Then . Then> . Now [pick one: 1, 2, 4, 23]> of in are obviously divisible by> [pick one: 2, 3, 7, 11, 97] but there's only> [pick one: 1, 2, 3, 22] factors of [pick one: 2, 3, 7, 11, 97]> in the equation, therefore the algebraic numbers are equivalent> to C and . QED> Well let's hear the master himself: What you have just seen is a major advance in mathematical thinking where I've used a rather simple abstraction and a special polynomial to analyze the *roots* of another *different* polynomial. It is a powerful tool that is new to mathematical analysis, as I discovered it only a few years ago. Unfortunately the concepts seem advanced enough to attract posters who react with fury when they can't quite get it, who refuse to acknowledge the basic principles, who also post a LOT!!! They've created a false picture that mathematicians as a group have refuted the information provided here, when it's impossible to refute mathematical logic. Basically, a few people with an agenda, posting a lot, have created an atmosphere of confusion and distrust which has fed upon itself. ()I like best his rambling about the false picture that mathematicians asa group have refuted the information provided here. Right! Actually,only a few people with an agenda, posting a lot, have created anatmosphere of confusion and distrust which has fed upon itself. Indeed!Though looking at the facts from this _point of view_ certainly won'texplain the existence of http://www.crank.net/harris.html.After all It's not every braying jackass that gets a whole page atcrank.net (Uncle Al). === Subject: : Re: Factorization dispute, againI guess my point is that this is part of his problem: he has a bad tendencynot to check calculations and to see if what he is deriving is actuallycorrect (and for a physicist this is odd).Sure, the technique he uses is correct IF USED BY SOMEONE WHO IS COMPETENTAND KNOWS HOW TO USE IT. But he isn't, and he doesn't. If I wanted todisabuse someone of something in a manner which was clear and indisputable,I would provide a simple direct counter-example. IF JSH tried to do this, hewould often realize that he is wrong (since he wouldn't be able to come upwith one, and his examples would show him that he ain't gettin what hethinks he should be gettin), and prevent these back and forth posts adnauseum that end up with him realizing after weeks that he's made somestupid assumption/error.MB> Here's a nugget for the did you ever notice bin:> Did you ever notice, how Mr. Harris gives refutations based on general> relations, whereas those who dispute his claims give exact, concrete> counter-examples?> MB> Notice,>> (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) =>> 49(300125 x^3 - 18375 x^2 - 360 x + 22)>> where you see that the constant terms match as now you have 7(7)(22) => 1078, which is the constant term of the polynomial>> 49(300125 x^3 - 18375 x^2 - 360 x + 22).>> Various people have debated me about what happens when you divide off> 49, where for some odd reason, some of them seem to believe that you> can have>> w_1(x), w_2(x), and w_3(x) such that w_1(x) w_2(x) w_3(x) = 49, and>> (5 a_1(x) + 7)/w_1(x) (5 a_2(x) + 7)/w_2(x) (5 b_3(x) + 22)/w_3(x) =>> 300125 x^3 - 18375 x^2 - 360 x + 22> where the w's vary as x varies, which is a rather naive notion.>> That's because you can multiply *everything* out, and simplify to get>> (7/w_1(x)) (7/w_2(x)) (22/w_3(x)) = 22>> which should be simple enough for all of you.>> Now those of you who usually work in the field of complex numbers may> think that it's not a big deal, as you may think it doesn't matter if> w_3(x) has some factor factor of 7, despite *seeing* (22/w_3(x)) but> you see, as 22 is coprime to 7 in the ring of algebraic integers, if> w_3(x) isn't coprime to 7, (22/w_3(x)) does not exist in the ring.>> You know, it's like how in integers 1/2 doesn't exist. It's not an> integer, so it's not in the ring.>> So you see, my argument is correct and simple, and mathematicians are> indeed running from a little gut check in their field. They're> pussies too scared to handle the truth.>> But you should also understand, some people will be able to see that,> which is part of my plan. I can let mathematicians destroy themselves> proving they can't be trusted based on what they *see*, while they> forget what they can't see: the wearing down of the mathematician> mystique.>> http://mathforprofit.blogspot.com/ === Subject: : Solving a 2nd order transfer functionI have a second order transfer function that looks like this:P(s) = k/((tau1*s+1)*(tau2*s+1)) * exp(-td*s)In order to reverse laplace transform this I will have to multiply P(s) with1/sThe resulting Laplace transform will be:p(t)=k*(1+tau1*exp(-(t-td)/tau1)/(tau2-tau1)-tau2*exp(-( t-td)/tau2)/(tau2-tau1))When tau1, tau2 and td are known it is possible to construct the curve p(t).This is not difficult.However, imagine that you have this curve, but do not know the values of tau1,tau2 and td that constructed the curve.In my field this is the problem that I have. I have a second order system thatgenerates a curve p(t) as output when a step function is input into the system.The problem is that I do not know the time constants (tau1 and tau2) nor thedead time (td). I want to analyse the curve and based on this find the valuesof tau1, tau2 and td. But, I have no idea how to do this. The general solutionfor p(t), presented above, doesn't allow me to solve for t.For a first order system it is all much easier as it will be possible tolinearize the function for t and then it is just a question of solving a systemof linear equations. This does not seem to be possible for a second ordersystem.How do I find tau1, tau2 and td given a curve with values for t and p(t)?Thanks in advance,-- Someone === Subject: : Factorization, basic to advanced conceptsIf you saw(c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = 49(x^3 + 5x^2 + 3x + 1)with the c's algebraic integers, I think few of you would have aproblem realizing that only two of the c's have 7 as a factor.But, of course, you're looking at *functions* of x, as you have f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, so I could also write it as(f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1).Notice that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1as long as you're in a ring where 7 is not a factor of 1.That's an important point and represents an issue over which I'vefound a lot of people willing to argue, and it may seem vague to moveto functions, even though in the previous example they were actuallyfunctions but they weren't being *called* functions.So some rules need to be outlined in generalizing from the basicpolynomial factors with their simple functions like c_1 x, where c_1is constant, to more complicated ones that we might not even haveimagined yet.One thing that's clearly important is that the functions *must* equal0 when x=0, as then you have factors of the constant term oppositethem.For instance(f_1(x) + 7) versus (f_3(x) + 1), where at x=0, both functions equal0, and 1 and 7 are both factors of the constant term.Next, the factorization must multiply out correctly, which just meansthat(f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1)multiplies out to give49(x^3 + 5 x^2 + 3x + 1).so f_1(x) f_2(x) f_3(x) = 49, for instance.I'm abstracting and generalizing to functions because I've facedarguments with a much more complicated example, where the basicprinciples are the same:(5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)where the a's are roots ofa^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)so they are functions of x, and since one of the roots equals 3 atx=0, I haveb_3(x) = a_3(x) - 3, so that I can see all the constant term factors.What you have just seen is a major advance in mathematical thinkingwhere I've used a rather simple abstraction and a special polynomialto analyze the *roots* of another *different* polynomial.It is a powerful tool that is new to mathematical analysis, as Idiscovered it only a few years ago.Unfortunately the concepts seem advanced enough to attract posters whoreact with fury when they can't quite get it, who refuse toacknowledge the basic principles, who also post a LOT!!!They've created a false picture that mathematicians as a group haverefuted the information provided here, when it's impossible to refutemathematical logic.Basically, a few people with an agenda, posting a lot, have created anatmosphere of confusion and distrust which has fed upon itself.But now I hope to reach other, more sophisticated and intelligentpeople who can consider the actual facts.My math discoveries, found for profithttp://mathforprofit.blogspot.com/ === Subject: : Re: Factorization, basic to advanced concepts > If you saw > > (c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = > 49(x^3 + 5x^2 + 3x + 1) > > with the c's algebraic integers, I think few of you would have a > problem realizing that only two of the c's have 7 as a factor.If they realise that by the two 7's in the factorisation they are toomuch relying on intuition. Change the constants 1 to 2 and it isno longer true in the algebraic integers. In fact with (c1 x + 7)(c2 x + 7)(c3 x + 2) = 49(x^3 + 5x^2 + 3x + 2)there are *no* c's possible that are also algebraic integers.However, in this case the roots of the cubic (r1, r2 and r3) are units, meaning that 1/r1, 1/r2 and 1/r3 are also algebraic integers. And wehave: c1 = 7/r1, c2 = 7/r2 and c3 = 1/r3. > But, of course, you're looking at *functions* of x, as you have > > f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, This is an irrelevant red herring. > so I could also write it as > > (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1). > > Notice that dividing both sides by 49 gives > > (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1 > > as long as you're in a ring where 7 is not a factor of 1.And this is another red herring. It is true in *every* ring thatcontains the roots of x^3 + 5x^2 + 3x + 1. > That's an important point and represents an issue over which I've > found a lot of people willing to argue, and it may seem vague to move > to functions, even though in the previous example they were actually > functions but they weren't being *called* functions.My problem with your position is that you claim that is the *only* wayyou can distribute 1/49 over the three factors while the factors remainfunctions from the algebraic integers to the algebraic integers. Thereare indeed other ways. Moreover, this example is a complete red herringas the roots of the cubic are units. > So some rules need to be outlined in generalizing from the basic > polynomial factors with their simple functions like c_1 x, where c_1 > is constant, to more complicated ones that we might not even have > imagined yet.No, first you need to outline it in a generalised form where the rootsof the cubic are *not* units. Next you can get to the complication ofarbitrary functions from the algebraic integers to the algebraic integers.But I will give an alternative factorisation:Define: w3(x) = gcd(c3 x + 1, 7) (and pray note that this is not always 1) w2(x) = gcd(c2 x + 7, 7)/w3(x); P(x)/7 = [(c1 x + 7)/7] [(c2 x + 7)/w2(x)] [(c3 x + 1)/w3(x)]is *also* a valid factorisation. > One thing that's clearly important is that the functions *must* equal > 0 when x=0, as then you have factors of the constant term opposite > them. > For instance > (f_1(x) + 7) versus (f_3(x) + 1), where at x=0, both functions equal > 0, and 1 and 7 are both factors of the constant term. > Next, the factorization must multiply out correctly, which just means > that > (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) > multiplies out to give > 49(x^3 + 5 x^2 + 3x + 1). > so f_1(x) f_2(x) f_3(x) = 49, for instance.Eh? Is the product f1(x) f2(x) f3(x) *independent* of x? In your caseit si *not* 49, it is 49 x^3. > I'm abstracting and generalizing to functions because I've faced > arguments with a much more complicated example, where the basic > principles are the same: > > (5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = > > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > > where the a's are roots of > > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > > so they are functions of x, and since one of the roots equals 3 at > x=0, I have > > b_3(x) = a_3(x) - 3, > > so that I can see all the constant term factors.I never complained about your discussion upto this point. Nor have I seenother people object to this. So I do not see what you mean with this. Itis your step 6 that is broken, and that is beyond this stage.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: : Re: Factorization, basic to advanced concepts> Crank Information http://www.crank.net/harris.html http://www.crank.net/usenet.html http://www.google.com/search?q=harris+site%3Awww.crank.net http://www.google.com/search?q=%22james+harris%22+site% 3Ausers.pandora.be === Subject: : Re: Factorization, basic to advanced concepts> If you saw> (c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = > 49(x^3 + 5x^2 + 3x + 1)> with the c's algebraic integers, I think few of you would have a> problem realizing that only two of the c's have 7 as a factor.That's dumb.If r_1, r_2, and r_3 are roots of x^3 + 5x^2 + 3x + 1then lettingc_1 = 7(-1/r_1), c_2 = 7(-1/r_2), c_3 = -1/r_3will result in your factorization. Obviously c_1 and c_2 willhave factors of 7 and c_1, c_2, c_3 will all be algebraicintegers because you've chosen a polynomial FOR WHICHTHE ROOTS ARE ALL ALGEBRAIC INTEGER UNITS.More importantly, examples are a waste of time when yourproof is in front of us, and the proof itself is sosimply wrong.What I hate about your going off on a tangent is yourneed to waste time. Obviously you have an agenda which isto cast your work in a favorable light, but rather thanface your errors directly, you start chattering aboutsomething else, claiming it's related.But over time it's clear that there's some key difference,or worse, even when your own examples fail to supportyour case you simply fudge.So your purpose is simply to obfuscate, which is why youhide from the clear and direct counterexamples in front ofyou to send people on wild goose chases.Rather than show some common decency, you simply lookfor another route to continue hiding the truth.If you had any, you'd be posting retractions,rather than working still to hide the truth.Rick === Subject: : Re: Factorization, basic to advanced concepts> If you saw> (c_1 x + 7)(c_2 x + 7)( c_3 x + 1) => 49(x^3 + 5x^2 + 3x + 1)> you'd next seehttp://www.crank.net/harris.html It's not every braying jackass that gets a whole page at crank.net-- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The Net! === Subject: : Re: If anyone cares about interesting questions on today's GRE subject Math testHi Mike. I need to that this same test. Just wondering - do you haveany additional questions from this test. If so, how can I get a copy. I need all the help that I can get.> Some people may think that this test is irrelevant for admissions committees> for pure math, but there were some interesting questions that were asked on> the exam (it was held today and I took it)...> For instance,> 1. Hom(Z/2Z x Z/2Z, S_3) is isomorphic to what?> 2. Let f(x) = x^2 + Bx + C, let B and C be independent random variables> uniformly distributed on [0,1]. What is the probability that the roots of> f(x) are distinct and real?> There were other interesting ones as well, but I can't remember them...These> are pretty easy if you're not under pressure to do them in less than average> 3 minutes each (or maybe you think 30 seconds under pressure is> sufficient...but give undergraduates some slack :-) )> It's good to see that the test requires more than memorization and some> thinking...needless to say I got the 1st one wrong and the 2nd one right...> In any case, I don't think I did as well as I would have liked, and hope> that doesn't impact my chances at some of the top schools in representation> theory and algebra in general (princeton, harvard, berkeley), but I have> heard that this test isn't looked at very much, it's more recommendations> and extra stuff/research...only I have heard that if you do poorly it raises> a red flag (but I'm confident I don't have to worry about that)> Just a post if anyone's interested,> Mike === Subject: : A neat computer programint main( void ){ int n; printf( Let B(n) be the nth busy beaver number.nr ); printf( The busy beaver numbers are:nr ); for ( n = 1; ; n = n+1 ) printf( B(%d)nr, n );}This program lists the busy beaver numbers :-) (just not in decimal) === Subject: : C++ Simulator of a Nondeterministic Turing MachineC++ Simulator of a Nondeterministic Turing Machine has been added at :* http://alexvn.freeservers.com/s1/turing.html* http://sourceforge.net/projects/turing-machine/Currently those sites contain C++ Simulators for both Deterministic and Nondeterministic Turing Machines.The Simulators contain examples of Turing Machines as well.1. A (Deterministic) Turing Machine example : Recognition of Palindromes from 'The Design and Analysis of Computer Algorithms [1976]' by A.V.Aho, J.E.Hopcroft, J.D.Ullman (See examples 1.8, 1.9)2. A Nondeterministic Turing Machine example : Partition Problem from 'The Design and Analysis of Computer Algorithms [1976]' by A.V.Aho, J.E.Hopcroft, J.D.Ullman (See example 10.1)-- = Alex Vinokur mailto:alexvn@connect.to http://mathforum.org/library/view/10978.html = === Subject: : Re: Usenet Posting Guide?> With all respect, sir, I can guess your politics right across the> board. Just for openers, you're a Sierra Club, save-the-whales type.> Gotcha, huh?> You're about as far off the mark as it is possible to get. Not that I see> the relevance, but I'm a very conservative, Religious Right type -- not> far removed from what most people would call a fundamentalist (though> I don't use that label myself). I'm irrevocably opposed to both abortion> and gun control, voted for Bush (both of them) and supported the Iraq war> (both of them), and certainly do not support the Sierra Club, Greenpeace,> PETA, or any other environmentalist or animal-rights group. And no,> I don't care to discuss or defend any of those positions in sci.math.> You're right as right can be. Taking what you say at face value, which> I do, I was indeed off the mark in that assessment. My apologies. You> also sound like my kind of guy, for that describes me absent the> religious overtones. I despise political liberals, and I'll let it go> at that as well in that most writers are political liberals,> enviro-whackos and PETA types.Not to get into a flame war, but coming from sci.mathI have to object to what seems to me to be some falsetheorems in your reasoning, namely that politically liberal => PETA Sierra Club => environmental terrorist environmentalist => every wacko animal rights sentimentI would describe myself as a political liberal and anenvironmentalist (to the point of bicyling acrossPhiladelphia to work on occasion). But I don'tidentify with anything on the right hand side.I refuse to get into an argument about my positions,or yours. I just want to point out that you havedismissed the other side of the political spectrumas not being a spectrum. - Randy === Subject: : Re: Usenet Posting Guide?> You're right as right can be. Taking what you say at face value, which> I do, I was indeed off the mark in that assessment. My apologies. You> also sound like my kind of guy, for that describes me absent the> religious overtones. I despise political liberals, and I'll let it go> at that as well in that most writers are political liberals,> enviro-whackos and PETA types. UrsulaNo apologies needed, though I thank you for them anyway. By off themark I simply meant that you were mistaken, not that you'd offended me.Wayne-- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers, === Subject: : Sex and Math recently came up with the excellent idea of using modernadvertising techniques to get his theories through to the public. Ihave decided that this is an excellent way for me to promote myanti-Cantorian doctrine to the public. Now, it is well known that sexsells. Therefor, without further ado (please maximize your window): ,ad8888888888888888ba, ad88888888888888888888888a, a888888888888888888888888888, CANTOR IS WRONG! ,8888 P88888888888888888888b, d88 `P88888888888888888, ,8888b 88888888888888, CANTOR IS EVIL! d8P''' ,aa, 888888888b 888bbdd888888ba, ,I 88888888, 8888888888888888ba8 ,88888888b NEVER BELIEVE ,888888888888888888b, ,8888888888 CANTOR'S LIES! (88888888888888888888, ,88888888888, d888888888888888888888, ,8 8888888b 88888888888888888888888 .;8' (888888 8888888888888I8888888P ,8 ,aaa, 888888 MEN WHO FOLLOW 888888888888I:8888888 ,8 'b8d' (88888 CANTOR WILL (8888888888I'888888P' ,8) 88888 NEVER SLEEP 88888888I 8888P' ,8) 88888 WITH THIS 8888888I' 888 ,8 ( ) 88888 YOUNG LADY! (8888I 88, ,8 ,8888P 888I' P8 ,8 ____ ,88888) (88I' ,8 MM ,888888' ,8I ,8( aaaa ,8888888 ,8I' ,888a ,8888888) CANTOR IS FULL OF ,8I' ,888888, ,888888888 INCONSISTENCIES! ,8I' ,8888888'`-===-'888888888' ,8I' ,8888888 88888888 8I' ,8 88 888888P YOU SAY THERE IS NO SUCH8I ,8' 88 `P888 THING AS THE NEXT REAL8I ,8I 88 8ba,. NUMBER. HA! I LAUGH(8, ,8P' 88 888bma,.AT YOU. ARRANGE A 8I ,8P' 88, 8b P8ma, COUNTABLY (8, ,8d `88, 8b `8a INFINITE 8I ,8dP ,8X8, 8b. :8b NUMBER OF (8 ,8dP' ,I ,8XXX8, `88, 8) REALS IN 8, 8dP' ,I ,8XxxxX8, I, 8X8, ,8 BINARY AND 8I 8P' ,I ,8XxxxxxX8, I, `8X88,I8 PRECISELY ONE I8, ,I ,8XxxxxxxxX8b, I, 8XXX88I, DIAGONAL `8I I' ,8XxxxxxxxxxxxXX8 I 8XXxxXX8, NUMBER WILL 8I (8 ,8XxxxxxxxxxxxxxxX8 I 8XxxxxxXX8, BE ,8I I[ ,8XxxxxxxxxxxxxxxxxX8 8 8XxxxxxxxX8,DETERMINED. d8I, I[ 8XxxxxxxxxxxxxxxxxxX8b 8 (8XxxxxxxxxX8, THIS IS 888I `8,8XxxxxxxxxxxxxxxxxxxX8 8, ,8XxxxxxxxxxxX8 YOUR 8888, 88XxxxxxxxxxxxxxxxxxxX8)8I .8XxxxxxxxxxxxX8 NEXT ,8888I 88XxxxxxxxxxxxxxxxxxxX8 `8, ,8XxxxxxxxxxxxX8 REAL! d88888 `8XXxxxxxxxxxxxxxxxxX8' `8,,8XxxxxxxxxxxxX8 888888I `8XXxxxxxxxxxxxxxxX8' 88XxxxxxxxxxxxX8 88888888bbaaaa88XXxxxxxxxxxxXX8) )8XXxxxxxxxxXX8WOMEN LOVE MEN 8888888I, ``8888888888888888aaaaa8888XxxxxXX8WHO TURN AGAINST (8888888I, . ```88888P CANTOR 88888888I, ,8I 8, I8 88888I, ,8I' I8, ;8 `8I, ,8I' `I8, 8)CANTOR IS A PLAGIARIST `8I, ,8I' I8 :8' AND A CHILD MOLESTOR `8I, ,8I' I8 :8 `8I ,8I' `8 (8 8I ,8I' 8 (8; 8I ,8 I 88, CANTOR'S THEORY IS .8I ,8' 88, A DEAD END (PI '8 ,8,`8, .88' ,@ .a8X8,`8, (88 @@ ,a8XX888,`8, (888 @' ,d8XX8 b `8, .8888, a8XXX8 a `8,CANTOR'S WORKS .888X88 ,d8XX8I 9, `8,ARE SUPPORTED .88:8XX8, a8XxX8I' `8 `8,BY AL-QAEDA .88' 8XxX8a ,ad8XxX8I' ,8 `8, d8' 8XxxxX8ba, ,ad8XxxX8I 8 , `8, (8I 8XxxxxxX888888888XxxxX8I 8 II `8 8I' 8XxxxxxxxxxxxxxxxxxX8I' (8 8) 8; (8I 8XxxxxxxxxxxxxxxxxX8 (8 8) 8I JESUS 8P' (8XxxxxxxxxxxxxxX8I' 8, (8 :8 HATES (8' 8XxxxxxxxxxxxxxX8' `8, 8 8 CANTOR 8I `8XxxxxxxxxxxxX8' `8,8 ;8 8' `8XxxxxxxxxxX8' `8I ,8' 8 `8XxxxxxxxX8' 8' ,8' 8 `8XxxxxxX8' 8 ,8' 8 `8XxxxX8' d' 8' 8 `8XxxX8 8 8' 8 8X8' 8 8, `88 8 8I ,8' d) `8, d8 ,8 (b 8' ,8' 8, dP ,8' HITLER AND (b 8' ,8' STALIN DERIVED 8, d8 ,8' THEIR IDEOLOGIES (b 8' ,8' DIRECTLY FROM CANTOR 8, a8 ,8' (b 8' ,8' 8, ,8 ,8' (b 8' ,8' TO FOLLOW CANTOR IS TO 8, ,8 ,8' FOLLOW THE DEVIL HIMSELF (b 8' ,8' 8, d8 ,8' (b ,8' ,8' 8,,I8 ,8' `I8I ,8' BELIEVE IN CANTOR. MAYBE YOU I8' ,8' SHOULD ASK YOURSELF, WHY? 8 ,8' (8 ,8' 8I ,8' (b, 8, ,8) STUDIES SHOW THAT MEN WHO LAUGH AT `8I 88 ,8i8, CANTOR RATHER THAN TAKE HIM (b, ,88) SERIOUSLY SLEEP WITH, ON AVERAGE, `8I ,8 8) 8 8 25% MORE SUPERMODELS 8I 8I 8 8 (b 8I 8 8 `8 (8, b 8, 8 8) b8, 8 8( b8 8 I b8, 8 `8) HOW CAN WE INVEST BILLIONS OF 8 I8 DOLLARS INTO CANTOR'S BOGUS 8 (8 THEORY WHEN THERE ARE CHILDREN Ib 8) (8 I8 8 I8 8 I8 8, I8 EVARISTE GALOIS AND SRINIVASA (8 (8' OUTRIGHT BUT THEIR PROTESTS 8 I8 WERE SILENCED BY THE 8, 8I ILLUMINATI Ib (8' (8 I8 `8 8I 8 (8' 8, I8 CANTOR WAS ACKNOWLEDGED IN HIS OWN Ib 8I TIME AS A HERETIC AND A LAUGHING- (8 8' STOCK. WHY SHOULD HE BE GIVEN ANY 8, (8 MORE RESPECT TODAY? Ib I8 (8 8I (8 8I 8, 8' (b (8 DO YOU WANT THE BLEEDING HEART 8, I8 LIBERALS TO PREACH CANTOR IN I8 I8 SCHOOL TO YOUR INNOCENT CHILDREN? (8 I8 8 I8, 8 8 8, 8, 8 8' CANTOR'S TRANSFINITES AND EUCLID'S ,I8 8 PRIME NUMBERS ARE BOTH MUTUALLY ,88, b EXCLUSIVE OF EACHOTHER ,8' `8 8 ,8' 8 8, ,8' (a b ,8' `8 (b PEOPLE WHO LOVE CANTOR I8/ 8 8, HATE DEMOCRACY I8-/ 8 `8, (8/-/ 8 `8, 8I/-/ ,8 `8 CANTOR WOULD NEVER HAVE WON `8I/--,I8 -8) IF IT WERENT FOR THE `8I,,d8I -8) DISCREPANCIES OF DIMPLED bdI8, -I8 CHADS `8, -I8' `8,,--I8' `Ib,,I8' `I8I'I hope that you enjoyed this complimentary ASCII babe, brought to youfree as a gift from your friends at the mathematical advertisingsociety.Your dear friend,Nathaniel DeethAge 11 === Subject: : Re: Sex and Math> recently came up with the excellent idea of using modern> advertising techniques to get his theories through to the public. I> have decided that this is an excellent way for me to promote my> anti-Cantorian doctrine to the public. Now, it is well known that sex> sells. Therefor, without further ado (please maximize your window):> But wait!! Here's this, from Salon magazine, written by a woman:The greatest thrill I remember from my girlhood -- better than my first kiss, first airplane flight, first taste of mango, first circuit around the ice rink without clinging to a grown-up's sleeve -- was the heart-lifting moment when I first understood Georg Cantor's Diagonal Proof of the nondenumerability of the real numbers. === Subject: : Re: Sex and Math> Heck if it will put me in good with the babes, then I'm against Cantor too. === Subject: : Software NeededHello everyone,I am looking for a software that can find written in Chinese).Chinese).in English.Thank you in advance.Mike === Subject: : rearranging equationsHiCan someone help me solve the following equation for x. Could youplease put all the steps down so I can follow it (idiot's guide!)ThanksJoy = A1*exp(-x/t1) + A2*exp(-x/t2) + y0 === Subject: : Re: rearranging equations Adjunct Assistant Professor at the University of Montana.>Hi>Can someone help me solve the following equation for x. Could you>please put all the steps down so I can follow it (idiot's guide!)>Thanks>y = A1*exp(-x/t1) + A2*exp(-x/t2) + y0This looks like a function, not an equation, unless y is supposed tobe a constant. I assume you want to solve y = 0?Let z = exp(-x/(t1*t2)). Then you can rewrite y=0 asA1 z^{t2} + A2 z^{t1} + y0 = 0.If t1, t2 are integers, then this is a polynomial in z. Solve it byany of the usual methods, and once you have a value z=z_0, then setexp(-x/(t1*t2)) = z_0and solve for x by taking logarithms; note that only positivesolutions for z will yield solutions for x.== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ==Arturo Magidinmagidin@math.berkeley.edu === Subject: : Re: rearranging equations>Can someone help me solve the following equation for x. Could you>please put all the steps down so I can follow it (idiot's guide!)>y = A1*exp(-x/t1) + A2*exp(-x/t2) + y0> This looks like a function, not an equation, unless y is supposed to> be a constant. I assume you want to solve y = 0?Huh? An equation is something with an equals sign in the middle, so I'd say this qualifies as an equation. I assume when the poster asks for help solving for x, the poster wants the equation solved for x. Of course, the steps you gave are useful for my interpretation as well (except that you won't want to set y to zero). But when you write, > A1 z^{t2} + A2 z^{t1} + y0 = 0.> If t1, t2 are integers, then this is a polynomial in z. Solve it by> any of the usual methods, the usual methods are going to be tricky unless t1 & t2 have been chosen (by whoever set the problem) with some foresight.-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: : Re: Looking for HINTS on proving a limitIgnacio Larrosa Ca.96estro escribi.97> By the way ... It is possible to get directly that limit without use(e^x)'> = e^x?> I.e., it is possible to prove directly, using only the definition of> derivative, that (e^x)' = e^x?Well, Jos.8e H. Nieto give me a satisfactory answer in es.ciencia matematicas.Lim((1 + 1/x)^x, x, inf) = elet x = 1/t,Lim((1 + t)^(1/t), t, 0) = e ==>Log(Lim((1 + t)^(1/t), t, 0)) = Lim(Log((1 + t)^(1/t)), t, 0) = 1Lim(log(1 + t)/t, t, 0) = 1Then, let y = Log(1 + t),Lim(y/(e^y - 1), y, 0) = 1 ===> Lim((e^y - 1)/y, y, 0) = 1With that is possible study the derivative of exponential function beforethan the derivative of the logarithmic function and the composite or inversefunction.-- Ignacio Larrosa Ca.96estroA Coru.96a (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: : Re: How do you prove that the circumference of a circle is proportional to it's radius?> How do you prove that the circumference of a circle is proportional to > its radius? .... and how did the Greeks prove it? You've asked about history. You shall have history!on the 8th June 2001, so here's a copy of that.> Is there a straightforward way to prove that the ratio of> circumference to the diameter of any arbitrary circle is always a> constant? It feels like this should be easy but I have not seen such a> proof in any book so far.> That's good mathematical thinking! You've seen that rough ideas> about similarity aren't enough. It was the ancient Greeks who realized> that it's not straightforward at all, and invented the first technique of> limits to handle just such problems. You've already had limit arguments> outlined by DWIII and Peter M. Jack, so I'll just mention some of the> history.> The area was tackled before the circumference, perhaps because it's> easier. Euclid XII.2 (probably due to Eudoxus) proved that Circles are> to one another as the squares on the diameters. A modern statement of> that might say> (area of first circle)/(area of second circle)> = (diameter of first circle)^2/(diameter of second circle)^2.> This shows that our modern area formula (pi)(r^2) or (pi)(d^2)/4 > really does have the same constant pi for all circles. Euclid's proof> used inscribed and circumscribed polygons of more and more sides, and then> a rigorous limit argument entirely in Greek geometrical terms. It's not> easy!> Later, Archimedes used the same limit technique in his Measurement> of the Circle. He first proved that The area of any circle is equal to> a right-angled triangle in which one of the sides about the right angle is> equal to the radius, and the other to the circumference of the circle. A> modern version is area = (radius)(circumference)/2. Since each> circle's area = (pi)(r^2) with constant pi, it follows that its> circumference = 2(pi)r with the same constant pi. Then Archimedes went> on to calculate bounds for that constant, and proved that 3 1/7 < pi < 3> 10/71. Those bounds aren't very tight, but nothing like them had ever> before been rigorously proved.> So your unease about the constancy of pi was well justified. The> question needed some very great mathematical minds to handle.> Ken Pledger. === Subject: : Re: How do you prove that the circumference of a circle is proportional to it's radius? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hABKvka05954;>> How do you prove that the circumference of a circle is proportional to >> it's radius?Please refer to:http://mathforum.org/discuss/sci.math/a/m/107260/107288 andhttp://mathforum.org/discuss/sci.math/a/m/107260/107289 Panagiotis Stefanideshttp://www.stefanides.gr === Subject: : Compactness helpHi all,In Maxwell Rosenlicht's book Intro. to mathematical analysis, he claimsthat the union of the intervals (1/n, 1) covers (0,1), and since there isnot finite subcover, then (0,1) is not compact. How does he get that (1/n,1) covers (0,1) ? Wouldn't there have to be some limiting argument going onthere? No matter how big the union is, it will never actually cover 0,right? What am I missing?TIA,Lurch === Subject: : Re: Compactness helpDooohhh, do I feel stupid! I was looking at that for so long, I wasthinking that 0 was in (0,1). Must have been confusing it with [0, 1] whichis compact, and was discussed previously. Thanks for the help, and I amsorry to waste your time.Lurch> Hi all,> In Maxwell Rosenlicht's book Intro. to mathematical analysis, he claims> that the union of the intervals (1/n, 1) covers (0,1), and since there is> not finite subcover, then (0,1) is not compact. How does he get that(1/n,> 1) covers (0,1) ? Wouldn't there have to be some limiting argument goingon> there? No matter how big the union is, it will never actually cover 0,> right? What am I missing?> TIA,> Lurch === Subject: : Re: Compactness help> No matter how big the union is, it will never actually cover 0,> right?Fact: 0 is not in (0,1). === Subject: : Re: Compactness help Adjunct Assistant Professor at the University of Montana.>Hi all,>In Maxwell Rosenlicht's book Intro. to mathematical analysis, he claims>that the union of the intervals (1/n, 1) covers (0,1), and since there is>not finite subcover, then (0,1) is not compact. How does he get that (1/n,>1) covers (0,1) ? Wouldn't there have to be some limiting argument going on>there? No matter how big the union is, it will never actually cover 0,>right? But (0,1) does not include 0. It contains exactly all real numbers xsuch that 00,there exists a natural number n such that 1/n is smaller than x.(There are many ways to state the Archimedean Property; that's one ofthem. Another is to state that given any positive real number x, thereis a positive integer n such that n>x. Yet a third is to state thatgiven any two positive real numbers, x and y, there exists a positiveinteger n such that ny > x.)How do you know it covers (0,1)? Let x be any real number in (0,1); so00 suchthat 1/n < x. Then 1/n < x <1, so x is in the interval (1/n,1).Therefore, every element of (0,1) is in at least one of the intervals(1/n,1). So every element of (0,1) is in the union of the intervals(1/n,1) (by definition of union), so the intervals (1/n,1) cover (0,1)(by definition of cover).= =It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes)=== ===Arturo Magidinmagidin@math.berkeley.edu === Subject: : Re: Compactness help> Hi all,> In Maxwell Rosenlicht's book Intro. to mathematical analysis, he claims> that the union of the intervals (1/n, 1) covers (0,1), and since there is> not finite subcover, then (0,1) is not compact. How does he get that (1/n,> 1) covers (0,1) ? Wouldn't there have to be some limiting argument going on> there? No matter how big the union is, it will never actually cover 0,> right? What am I missing?> Pick any number in (0,1), call it x. For some sufficiently large n, 0 < 1/n < x < 1so that x is in (1/n, 1).In other words we have shown that ANY number x in (0,1) is in SOME interval (1/n, 1). So the union of all the (1/n, 1) covers (0,1). === Subject: : What is the explicit expression for this series?[series for n from 0 to N of] n*A^nfirstly with N as a finite number and then with N as infiniteA is a constant: 0 [series for n from 0 to N of] n*A^n> firstly with N as a finite number and then with N as infinite> A is a constant: 0 Is there an explicit expression?> Sure is. Here's a hint:1. Start with the series [sum n = 0, N] A^n. There is a closed formfor this (one for N finite and another for N infinite) whichyou either know or can look up.both sides with respect to A.3. Multiply both sides by A and you'll have[sum n = 0, N] n*A^n = (something else).Hope this helps.Rick === Subject: : Re: What is the explicit expression for this series?Rick Decker ha scritto nel messaggio> [series for n from 0 to N of] n*A^n>> firstly with N as a finite number and then with N as infinite> A is a constant: 0> Is there an explicit expression?> Sure is. Here's a hint:> 1. Start with the series [sum n = 0, N] A^n. There is a closed form> for this (one for N finite and another for N infinite) which> you either know or can look up.> both sides with respect to A.> 3. Multiply both sides by A and you'll have> [sum n = 0, N] n*A^n = (something else).> Hope this helps.Thank you very much. === Subject: : Re: What is the explicit expression for this series?> [series for n from 0 to N of] n*A^n> firstly with N as a finite number and then with N as infinite> A is a constant: 0 Is there an explicit expression?Yes. Multiply by 1 - A. Again. See what happens. Or, consider what happens when you differentiate sum(A^n) with respect to A.-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: : Re: What is the explicit expression for this series?Gerry Myerson ha scritto nel messaggio Is there an explicit expression?> Yes. Multiply by 1 - A. Again. See what happens.Uhmm......> Or, consider what happens when you differentiate sum(A^n) with respect> to A.This way seem better.Thank you. === Subject: : Homeomorphisms and compactnessI have a small question which I have been unable to solve on my own. Iknow that if f: M->N is a homeomorphism, thenK compact in M <=> f(K) compact in Nand I'd like to know if the opposite way holds, that is, if f isbijective andK compact in M <=> f(K) compact in Nis f then a homeomorphism? I have proven this to be true for locallycompact spaces, but I would like to know whether the result holdsgenerally.Any help?-Alexander === Subject: : Re: Homeomorphisms and compactness Adjunct Assistant Professor at the University of Montana.>I have a small question which I have been unable to solve on my own. I>know that if f: M->N is a homeomorphism, then>K compact in M <=> f(K) compact in N>and I'd like to know if the opposite way holds, that is, if f is>bijective and>K compact in M <=> f(K) compact in N>is f then a homeomorphism? I have proven this to be true for locally>compact spaces, but I would like to know whether the result holds>generally.>Any help?I know at least two different definitions of 'compact'. One is simplythat any open cover has a finite subcover; the second one alsorequires some separation properties. The fomer is sometimes calledquasi-compact, though.In the first case, it is easy to construct counterexamples: any finiteset will satisfy that any subset is compact, since there are onlyfinitely many open sets. So pick any set, and two different topologieson the set, and let f be the identity. For example, the indiscrete anddiscrete topologies on the two element set satisfy the condition Kcompact in the discrete topology if and only if K compact in theindiscrete topology, but they are certainly not homeomorphic.However, since you claim to have proven it for locally compact spaces,that suggests to me that you are actually using the definition ofcompact which includes separation axioms, namely that the space mustbe Hausdorff. Naturally, all the examples above collapse in thatinstance, since a finite topological space is Hausdorff iff thetopology is discrete... Is that the case?=== ===It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ==Arturo Magidinmagidin@math.berkeley.edu === Subject: : Re: Homeomorphisms and compactness>The fomer is sometimes called> quasi-compact, though.I guess the former is the one introduced by Bourbaki (the one we're usingin France) but in English literature the space isn't required to beHaussdorf, is it? === Subject: : Re: Homeomorphisms and compactness Adjunct Assistant Professor at the University of Montana.>>The fomer is sometimes called>> quasi-compact, though.>I guess the former is the one introduced by Bourbaki (the one we're using>in France) but in English literature the space isn't required to be>Haussdorf, is it?Depends on the author, and depends on the area. For instance, inAlgebraic Geometry, where a lot of the basic literature is byGroethendick, Serre, or people who studied in France or their students(Zariski, Hartshorne), they use quasicompact and they do assume thatcompact means Haussdorf; but then it is merely a slight point forthem, since the Zariski topology is not Haussdorf, and compactnessis quickly replaced by other similar notions with differentnames for schemes. I have seen both definitions from time to time inEnglish textbooks, though the ones I learned from always usedcompactness to mean exclusively the finite subcover property.=== ===It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes)=== ===Arturo Magidinmagidin@math.berkeley.edu === Subject: : Re: Structure of Frobenius Complement>I have read that Burnside was able to prove that the Frobenius>complement of a Frobenius group had the property that any Sylow>p-subgroup of the group had a unique subgroup of order p. Therefore,>any Sylow p-subgroup of such a group is cyclic unless p=2 and theSylow 2-subgroup is a generalized quaternion group. Can anyone give>me a reference (preferably in English) on how Burnside did this?>> I don't know how it compares with Burnside's proof, but there is a proof> on page 86 of the book John Dixon & Brian Mortimer, Permutation Groups> (Springer, 1996)> Derek Holt.This is not a proof. This is a statement of the theorem. I know thestatement of the theorem pretty well. When did you last look at thebook?---- David === Subject: : Re: Linear independence> Let aj = a_j = sqr j-th prime.> Show A = { aj | j in N } is linear independent subset of the reals as a> vector space over the rationals.> Related problem is to show A with 1 included is linear independent.I posted the same problem to this newsgroup in 1992, and got a good answer or two. Search Google Groups for the subject header, linear independence/square roots/primesand you should find the thread.-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: : Re: Linear independenceSubject: Re: Linear independence >{sqrt{a}: a in Z, a is squarefree} is linearly independent over Q.when considering C as a vector space over Q?n square free when for all integers m > 1, not m^2 | n.-1,1 square free; 0 not square free.B = { sqr n | n in N, n square free } is linearly independent over Qwhen considering R as a vector space over Q?If sum(j=0..n) (q_j sqr n_j) = 0 for some q_j's in Q, n_j's in B:1 = sum(j=1..n) [-q_j/q_0 * (n_0,n_j) sqr (n_0 n_j)/(n_0, n_j)]1 = sum(j=0..m) (r_j sqr m_j) for some r_j's in Q, m_j's in BWhat next? Assume m is the smallest such m?Clearly 1 /= r_0 sqr m_0.If 1 = r0 sqr m0 + r1 sqr m1, m0 /= m11 = r0^2 m0 + m1^2 m1 + 2r0.r1.sqr m0.m1but as both m0,m1 are square free, m0 = m1Beyond that, I'm stimied.---- === Subject: : determing the inside of a set of pointsI've got a list of points (in x,y form) that are the corners of a polygon,in order. It's not necessarily a convex polygon. Is there a standardalgorithm for determining whether a given point is inside or outsidethe polygon? All this takes place in the plane... For practical purposes, an algorithm is ok if it can handle up to 7 points decently.Many thanks -Adrian === Subject: : Re: determing the inside of a set of points> I've got a list of points (in x,y form) that are the corners of a> polygon, in order. It's not necessarily a convex polygon. Is there a> standard algorithm for determining whether a given point is inside> or outside the polygon? All this takes place in the plane... For> practical purposes, an algorithm is ok if it can handle up to 7 points> decently. > Many thanks -Adrian> This is probably in the faq at comp.programming.games.algorithms, or something like that. Google will probably also help. === Subject: : Re: determing the inside of a set of points> I've got a list of points (in x,y form) that are the corners of a polygon,> in order. It's not necessarily a convex polygon. Is there a standard> algorithm for determining whether a given point is inside or outside> the polygon? All this takes place in the plane... For practical > purposes, an algorithm is ok if it can handle up to 7 points decently.Google is your friend: http://www.google.com/search?q=inside+outside+polygon-- Andreas K.8ah.8ari === Subject: : Re: Largest number ever written down?I don't know exactly what JT has in mind here, but it could be that he considers a number to be written down only when all of its digits are written down. E.g., 10^10 doesn't count as being written down. The question becomes, what's the largest integer anyone has ever written out, writing down all the digits. Note that whatever the answer is, you can't beat it by just saying, 1 + whatever, you would actually have to write out the whole thing over again, changing the appropriate digit(s). It's still a pretty silly question, but maybe not quite as silly as we're making it out to be. Once upon a time - back in the 1950s, I think - someone published all the digits of n-factorial for various large values of n. I think the values of n may have gone up to 1000 or so. Those would have been numbers of around 7000 digits. That would be my nominee for the largest number ever written out in a mathematical journal.-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: : Re: Largest number ever written down?> If you allow finite numbers constructed from a series of mathematical> formulae, I think Graham's number takes the cake, by far.FYI,Harvey Friedman's lower bound for n(4) in his block subsequence theorem ismuch larger than Graham's number- it involves the Ackermann numbers A(n,n). let A(n) equal A(n,n) A(A(A........(A(1)..) where there are A(187,196) A's---cs === Subject: : Re: Largest number ever written down?> http://www.cs.berkeley.edu/~aaronson/bignumbers.htmlOoops. (regarding my previous post which hasn't returned to me yet.)He wasn't wearing his moose-horns, so I didn't know he was still hypothesising the imposible. The very next paragraph begins:<< http://www.cs.berkeley.edu/~aaronson/bignumbers.html<< Joona I Palaste scribbled the following:>> Quaternion scribbled the following:>> it? Thanks.> One of those people that is very fond of exclamation marks probably once> 9!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! at the end of a sentence,> in this thread, I think. I'm not sure.>> You have 30 ! signs there. This means that your number is smaller than>> 9^(9^(9^... containing 60 9's. I haven't counted the 9's in Jeroen>> Boschma's posting but I'm fairly sure he has more of them than you do.> Erm, no. I was wrong. n! is smaller than n^n, so if we replace n with> m!, we get that m!! is smaller than (m^m)^(m^m), and if we replace m> with l!, we get that l!!! is smaller than ((l^l)^(l^l))^((l^l)^(l^l)),> and so on. The number the variable occurs in the bigger number is 2 to> the power of the number of ! signs in the smaller number.> If those operations were all grouped like l^(l^(l^... then there would> be 1073741824 nested exponentations, which would make Quaternion's> number bigger than Jeroen's, but as they are not, I don't know which is> larger. Could some of you math gurus shed more light on this?I don't know it either. I suppose it's easier to work with 10 as base numberand then count the minimal amount of zeroes, but they are two entirelydifferent growths if you indeed write the exponential as n^n^n^n^.. Hard tocompare them.Note that the amount of zeroes grows with n(n-1)/2, for the recursivefactorial function, for each next n (since the amount of zeroes is sum(1 ton) with each new '!', if we take 10 as a base number (or (n-9)(n-10)/2 tobe precise))eventually become larger than n^^k, for rather small values of k.And, my post wasted less bandwidth and required no copy-pasting. It was alsomeant as a joke. Now, I want that cake that was mentioned in an other'spost.-- Quaternion === Subject: : Re: Algebraic number theory> Let q1,q2 algebraic integers such that q1^k=n(integer),q2^k=m(integer) let > K1=Q(q1),K2=Q(q2),Q:rational field.> Also suppose that m,n free from k powers and K1=K2. > Is there any relation of m,n?Do you have any examples where m and n are not equal?-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: : Re: Algebraic number theory Adjunct Assistant Professor at the University of Montana.>> Let q1,q2 algebraic integers such that q1^k=n(integer),q2^k=m(integer) let >> K1=Q(q1),K2=Q(q2),Q:rational field.>> Also suppose that m,n free from k powers and K1=K2. >> Is there any relation of m,n?>Do you have any examples where m and n are not equal?k=3, m=2, n=4. Then Q(q2) is clearly contained in Q(q1), and 2^{1/3} = (1/2)(4^{1/3})^2 so you get the reverse inclusion.If k=2, then m must equal n. If k=3, I believe that you must have that all primes that divide mdivide n as well, and vice-versa, and that will be both necessary andsufficient.For k=4 things seem to get a bit more complicated, since I think thatQ( (18)^{1/4}) is not the same as Q( (12)^{1/4}).If I had to guess, I would guess that the following are necessary(though I may be proven wrong in short order): every prime eitherdivides both n and m or does not divide either; and if p^r is thehighest power of p dividing n, and p^s is the highest power of p thatdivides m, 0>> Let q1,q2 algebraic integers such that q1^k=n(integer),q2^k=m(integer) let >> K1=Q(q1),K2=Q(q2),Q:rational field.>> Also suppose that m,n free from k powers and K1=K2. >> Is there any relation of m,n?>>Do you have any examples where m and n are not equal?> k=3, m=2, n=4. Then Q(q2) is clearly contained in Q(q1), and > 2^{1/3} = (1/2)(4^{1/3})^2 so you get the reverse inclusion.> If k=2, then m must equal n. > If k=3, I believe that you must have that all primes that divide m> divide n as well, and vice-versa, and that will be both necessary and> sufficient.Is Q(cube root 6) the same field as Q(cube root 12)?-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: : JSH: My use of my initialsSome of you may have realized that I have remarkable power to drawattention on several newsgroups to the extent that I even have my owndedicated repliers, like Uncle Al or David Ullrich, who like toobsessively insult me! Anyone have a good term for people who justfollow around a popular poster insulting him? Well I call them critictrolls.In any event, so like I said I have this drawing power, and lots ofpeople besides Uncle Al and David Ullrich get upset with me overstrange things, and there's little doubt there are people out therebothered by my use of my initials JSH in subject lines, as if it'ssome kind of arrogant thing, so here's the story as to how I happenedto start using my initials in subject lines.Let me take you back a bit.I started looking for simple yet profound math discoveries back in1995. My thinking was that if I could look in places that othersthought were well-worked and find some spectacular, but basic result,I could make money.Ok, so yeah, I'm in it for the money.In any event, at first I talked directly to mathematicians, primarilyat math journals, and found that when I made mistakes with mymathematics, it was embarrassing and time consuming to find someoneelse to talk ro about new ideas, and then in 1996, I discoveredUsenet.It seemed like the perfect place! I could talk about mathematics onthe newsgroup sci.math where other people were talking about math. Maybe I'd even get some people who would be sympathetic to my idea.What I found was a lot of hostility.Apparently certain posters see the sci.math newsgroup as theirpersonal territory, and have a territorial reaction to people postingthere. They feel they can control the newsgroup content, bycontrolling posters by, you guessed it, insulting them.Well I ended up in lots of arguments, but meanwhile kept posting myideas, as I realized that maybe I'd signed on to a really big task,looking for a spectacular but simple proof, and I shifted more to ideageneration mode, you know--brainstorming.Well that *infuriated* the territorial posters who became moreenergetic in their insults, as like I said, they'd try to insultposters like me into shutting up.And I didn't exactly like all of those attacks, but hey, I kept atcoming up with ideas and posting them, but found myself first angry,then intrigued when one poster--angry at *accidentally* reading myposts--suggested that I give some identifier so that people could knowwhen I was the one who started a thread.So at first I replied haughtily about my freedom of speech, and rightto post whatever subject line I wanted. But then I thought, hey,that's not a bad idea!!!So I added my initials to the subject line to show that it was meposting, as requested.Now when I see a need, like now, to identify who started a thread, Iadd JSH to the front, but of course, Usenet sparks imitators, sosometimes other posters will stick *my* initials on a thread THEYstarted, which is just one of those things.In any event, in case you were wondering, no, I didn't think oftossing my initals on subject lines--I was *ordered* to do it by aposter angry at not always being able to tell threads that Istarted!!!Isn't Usenet a wacky, wacky world?My math discoveries, found for profithttp://mathforprofit.blogspot.com/ === Subject: : Re: JSH: My use of my initials> Some of you may have realized that I have remarkable power to draw> attention on several newsgroups attracts flies, boy. That doesn't give it value.http://www.crank.net/harris.html It's not every braying jackass that gets a whole page at crank.net-- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The Net! === Subject: : Re: JSH: My use of my initialsX-ID: ZKzY9kZboeJVO9UhBabwGARywLsPwNLKUS7kXJT+l646Mq7joufOrR Uncle Al> attracts flies, boy. Indeed... :-)F. === Subject: : Re: My use of my initials> Some of you may have realized that I have remarkable power to draw> attention on several newsgroups to the extent that I even have my own> dedicated repliers, like Uncle Al or David Ullrich, who like to> obsessively insult me! Anyone have a good term for people who just> follow around a popular poster insulting him?It is generally the unpopular posters who attract insults.Martin Hogbin === Subject: : Re: JSH: My use of my initials> Some of you may have realized that I have remarkable power to draw> attention on several newsgroupsYou are the uber-troll, for sure. to the extent that I even have my own> dedicated repliers, like Uncle Al or David Ullrich, who like to> obsessively insult me! Anyone have a good term for people who just> follow around a popular poster insulting him? JSH addicts. I'm one, I admit it. It's like watching a train wreck in progress. It's too horrible to watch, but you can't force yourself to look away.> In any event, so like I said I have this drawing power, and lots of> people besides Uncle Al and David Ullrich get upset with me over> strange things, and there's little doubt there are people out there> bothered by my use of my initials JSH in subject lines, as if it's> some kind of arrogant thing, so here's the story as to how I happened> to start using my initials in subject lines.> Hell no, most of us appreciate it. It lets people identify JSH-related posts so that they can either killfile them or read them, depending on their preference.> Let me take you back a bit.> I started looking for simple yet profound math discoveries back in> 1995. My thinking was that if I could look in places that others> thought were well-worked and find some spectacular, but basic result,> I could make money.> Ok, so yeah, I'm in it for the money.> Of all your delusions, that is the strangest and saddest. Do you understand that EVEN IF you were to somehow prove that the definition of the algebraic integers causes a contradiction in math; even if you were to convince people that you alone had the world's fastest prime counting algorithm; even if you showed up one day with a genuine elementary proof of FLT -- you would not make any money out of it. Math is many things, but it's certainly not a get-rich-quick scheme. If money is your motivation, you'd do better selling penis enlargement pills on the Internet.> In any event, at first I talked directly to mathematicians, primarily> at math journals, and found that when I made mistakes with my> mathematics, it was embarrassing and time consuming to find someone> else to talk ro about new ideas, and then in 1996, I discovered> Usenet.> Where you can embarrass yourself to your heart's content.> It seemed like the perfect place! I could talk about mathematics on> the newsgroup sci.math where other people were talking about math. > Maybe I'd even get some people who would be sympathetic to my idea.> What I found was a lot of hostility.> This is not true. You have over the years had many many learned and smart people patiently sorting out and clarifying your muddled expositions, and identifying the key errors in each. These people enrage you and you lash out with great hostility of your own. Which invites others to take cheap hostile shots at you. You reap exactly what you have sown.> Apparently certain posters see the sci.math newsgroup as their> personal territory, and have a territorial reaction to people posting> there. They feel they can control the newsgroup content, by> controlling posters by, you guessed it, insulting them.> Look directly into the mirror when you say that, and you will see that it's true.> Isn't Usenet a wacky, wacky world?The concept of free speech encompasses the right of people to say stupid and hateful things. The Supreme Court said you can burn the American flag or claim that Jerry Falwell had sex with his mother in an outhouse, and that those things are protected speech. I for one defend your right to say whatever the hell you want.But I think you would do yourself a world of good if you would simply pick up a copy of Herstein's Topics in Algebra, spend six months of your life working through it page by page, and then come back and talk to us about your great discoveries. === Subject: : Re: Uncle Al is Sadistic .>> You jokers are nuts! But then, this is the Usenet, a 24/7 cyber>> party, where prodigies that failed have to come to die.........>> Therefore, let'em sing.....all of them....it's a beautiful choir!>> Sig file!!!>bob, are you a wanna-be or closet Nazi? ....Only on Wednesdays. On Thursdays I'm a civil rights crusader, FridaysI'm antisemitic, Saturday's I'm a Jew in the morning and a supporterof radical Islam after dinner, Sundays I start out the day readingliberation theology and end it speaking in tongues while playing withrattlers. Mondays I'm a eco freak and Tuesdays I dump my garbage inTampa Bay on my way to work.It works for me.>....anyway, Bob, it would be Sieg Heil.I sit corrected.>> Usenet - where failed prodigies come to die.>> Mr. Shambat, how do you say that in Russian?>I don't know about any Mr. Shambat,If you were female he'd already have emailed you.>but to play with or to shave her Schambart, THAT is a lovely thing.>ahahaha.....ahahahansonHocken M.8anner auf Stellagenim milchigen Nebel aus Dampfund sitzen gespreizt, zum Zugriff bereit:jeder ein barbarinischer Faun.Etwa die Uhr noch, ein Amulett,der engsitzende Ehering blinkt auf.Haut weich, behaart,Gelenke und Muskeln,Waden und Schl.9fsselbein,Schenkel .9fberflossen vom Schwei.Langen sich M.8anner auf Stellagenentlang der Elle .9fbers Schulterblatt an, ber.9fhren den R.9fcken, die H.9ffte, den Bauch.Mit fiebrigen Augen, brennender Gierstarren sie auf das begehrte Geschmeide,das wachsende Goldhorn der Lust.Streicheln sich M.8anner auf Stellagenz.8artlich .9fber Schl.8afen und Brustkreuzund umfangen die Kugeln mit z.9fgigem Druck.Da ist der ge.9affnete Mund,marmorglatt und doch r.9ater als sonst:jeder ein barbarinischer Faun.Finger knorplig, verkrampft,Schambart und Kopfhaar,Sehnen und Knochen aus Stein,ein abgeschlagenes Glied. === Subject: : Re: Uncle Al is Sadistic . charset=iso-8859-1 dude,........ahahahahahaha........**you are alright, bob!**ahahahahaahahaha.....ahahahahanson>> You jokers are nuts! But then, this is the Usenet, a 24/7 cyber>> party, where prodigies that failed have to come to die.........>> Therefore, let'em sing.....all of them....it's a beautiful choir!>> Sig file!!!>bob, are you a wanna-be or closet Nazi? ....> Only on Wednesdays. On Thursdays I'm a civil rights crusader, Fridays> I'm antisemitic, Saturday's I'm a Jew in the morning and a supporter> of radical Islam after dinner, Sundays I start out the day reading> liberation theology and end it speaking in tongues while playing with> rattlers. Mondays I'm a eco freak and Tuesdays I dump my garbage in> Tampa Bay on my way to work.> It works for me.....anyway, Bob, it would be Sieg Heil.> I sit corrected.>> Usenet - where failed prodigies come to die.>> Mr. Shambat, how do you say that in Russian?>I don't know about any Mr. Shambat,> If you were female he'd already have emailed you.>but to play with or to shave her Schambart, THAT is a lovely thing.ahahaha.....ahahahanson> Hocken M.8anner auf Stellagen> im milchigen Nebel aus Dampf> und sitzen gespreizt, zum Zugriff bereit:> jeder ein barbarinischer Faun.> Etwa die Uhr noch, ein Amulett,> der engsitzende Ehering blinkt auf.> Haut weich, behaart,> Gelenke und Muskeln,> Waden und Schl.9fsselbein,> Schenkel .9fberflossen vom Schwei.> Langen sich M.8anner auf Stellagen> entlang der Elle .9fbers Schulterblatt an,> ber.9fhren den R.9fcken, die H.9ffte, den Bauch.> Mit fiebrigen Augen, brennender Gier> starren sie auf das begehrte Geschmeide,> das wachsende Goldhorn der Lust.> Streicheln sich M.8anner auf Stellagen> z.8artlich .9fber Schl.8afen und Brustkreuz> und umfangen die Kugeln mit z.9fgigem Druck.> Da ist der ge.9affnete Mund,> marmorglatt und doch r.9ater als sonst:> jeder ein barbarinischer Faun.> Finger knorplig, verkrampft,> Schambart und Kopfhaar,> Sehnen und Knochen aus Stein,> ein abgeschlagenes Glied. === Subject: : Re: Uncle Al is Sadistic .>>> >>>>> By the time I saw your response to my other post, I have already>>>replied to this post of yours. I would not have otherwsie.>>>>>Put it in your think head. You cannot demand; you can only request.>>>>> HUH?!>>>>Did you have to use the word 'f***ing'? >>>>You could have asked What countries were they from?>>> And you still haven't answered the damned^W question.>>I am not obligated to answer, am I?> Gawd, I hate cute. Nope, you aren't obliged to answer,> especially when you used the datum as the basis> of your argument. I said Africans. You said that Etheopia used to send their brighterst; I replied, theywere not Etheopians.I waited to say the countries name because I was forgetting where 2were from (I do remmeber that it wasn't Etheopia) and I decided tothink about it a bit before saying anything since I knew that I wouldcome to remember, at least one of that two.> When asked for a clarification of that> datum, you were rude > you deliberately ignore the question,You were rude and so I ignored to answer.> add all kinds> of red lines to divert the lack of an answer. What lack of answer? > Ergo, you> don't feel very strongly about the subject but do want> to get a free ride based on perceived discrimination.Note: No amount of your accusation will make me tell you the names ofthose countries. Live with it. /BAH> Subtract a hundred and four for e-mail. === Subject: : Re: Uncle Al is Sadistic .Apparently you don't have anything at all to say that'srelevant to any of the three newsgroups to which youcontinue to post. How *do* you justify your existence?Can you come up with even one post that on-topicsomewhere and avoid the otherwise inevitable trip toeveryone's killfile? === Subject: : Re: Uncle Al is Sadistic .> Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence? Feeling hurt..still? > Can you come up with even one post that on-topic> somewhere and avoid the otherwise inevitable trip to> everyone's killfile?Do it INSTAED OF whinning about it. I didn't expect that there would be people like you to take things outof context and attack me. Moreover, I realized that it was a mistakefor me to even bother to reply to people like.Now ... put me in your killfile instead of talkign about it. You willbe helping me greatly. === Subject: : Re: Uncle Al is Sadistic .>> Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence?> Feeling hurt..still?>> Can you come up with even one post that on-topic> somewhere and avoid the otherwise inevitable trip to> everyone's killfile?> Do it INSTAED OF whinning about it.> I didn't expect that there would be people like you to take things out> of context and attack me. Moreover, I realized that it was a mistake> for me to even bother to reply to people like.> Now ... put me in your killfile instead of talkign about it. You will> be helping me greatly.Naw... We prefer tracking and monitoring the enemy so please keepposting so a better tracking of your ISP usage and your politicalactivities can be followed closely... And please refrain from wearinga facial covering when leaving you house as its hard to satellite trackand use our facial recognition systems ( not imposable but a tad harder)We have decided to use the old concept of war... Our tacticswill be based on the enemies tactics and we will use theenemies rules of engagement as our rules of engagement...So next time you walk out your door we may decide its timeto have the mother of all stoning.... 100 B-52'a and a fewdozen B1-B's dropping a few tons of rocks should be a hootto watch when we stone Mecca.... === Subject: : Re: Uncle Al is Sadistic .>>> Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence?> Feeling hurt..still?You must be joking, Little One.>> Can you come up with even one post that on-topic> somewhere and avoid the otherwise inevitable trip to> everyone's killfile?> Do it INSTAED OF whinning about it.You would order me to do something? What amusingarrogance you display.> I didn't expect that there would be people like you to take things out> of context and attack me. Moreover, I realized that it was a mistake> for me to even bother to reply to people like.Like?Apparently you have little clue about a lot of things.And by the way, it's not necessary to take your words outof context in order to thoroughly ridicule your patheticstatements. They are the ramblings of an ernest but naiveadolescent.> Now ... put me in your killfile instead of talkign about it. You will> be helping me greatly.Sorry sweets, but I don't take orders from The Clueless.May I suggest that you take your whinging and moaningto one of the many pop psychology newsgroups? === Subject: : Re: Uncle Al is Sadistic .> Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence?She seems to be looking for a Muslim scientist to hiton her so she's advertising in sci newsgroups and forbait is trying to appear intelligent, passionate, anddesirable. === Subject: : Re: Uncle Al is Sadistic . Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence?> She seems to be looking for a Muslim scientist to hit> on her so she's advertising in sci newsgroups and for> bait is trying to appear intelligent, passionate, and> desirable.Is she cute? I mean, I'm not Muslim but ... === Subject: : Re: Uncle Al is Sadistic .>Apparently you don't have anything at all to say that's>relevant to any of the three newsgroups to which you>continue to post. How *do* you justify your existence?>>She seems to be looking for a Muslim scientist to hit>>on her so she's advertising in sci newsgroups and for>>bait is trying to appear intelligent, passionate, and>>desirable.> Is she cute? I mean, I'm not Muslim but ...There are precious few in this world who haveabsolutely nothing going for them. Scratchand see if it festers of blooms. Please reportback. === Subject: : Re: Uncle Al is Sadistic .> Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence?> She seems to be looking for a Muslim scientist to hit> on her so she's advertising in sci newsgroups and for> bait is trying to appear intelligent, passionate, and> desirable.So ..you are anti-Muslim. I understand why you must now attack me insuch personal way but still I am sure that that's very low of you.Most idiots like you have no clue that there are lots of Muslims incountries like Burma, Thailand, who don't deserve the animosity peoplelike you have for them for being Muslims. === Subject: : Re: Uncle Al is Sadistic .>>> Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence?>> She seems to be looking for a Muslim scientist to hit> on her so she's advertising in sci newsgroups and for> bait is trying to appear intelligent, passionate, and> desirable.> So ..you are anti-Muslim. I understand why you must now attack me in> such personal way but still I am sure that that's very low of you.Dear Clueless, Where in the above quote did Bill say anything at allthat was anti-Muslim? Unless of course you think thatany Muslim who would hit on you would be degradinghimself.> Most idiots like you have no clue that there are lots of Muslims in> countries like Burma, Thailand, who don't deserve the animosity people> like you have for them for being Muslims.Unwarranted assertion; you have not provided any evidenceshowing what people in general know or don't know aboutthe distribution of Muslims in the world. You do not haveany basis for claiming that most of any particular groupknow or do not know anything in particular. Provideyour evidence (references please) or recant. === Subject: : Re: Uncle Al is Sadistic .>Apparently you don't have anything at all to say that's>relevant to any of the three newsgroups to which you>continue to post. How *do* you justify your existence?>>She seems to be looking for a Muslim scientist to hit>>on her so she's advertising in sci newsgroups and for>>bait is trying to appear intelligent, passionate, and>>desirable.> So ..you are anti-Muslim. I understand why you must now attack me in> such personal way but still I am sure that that's very low of you.My comment was limited to your misuse of these newsgroups.> Most idiots like you have no clue that there are lots of Muslims in> countries like Burma, Thailand, who don't deserve the animosity people> like you have for them for being Muslims.So Hansen was right after all? Are you really justa rabid bitch attacking anything you don't likewith any stone that happens to be in your hand atthe moment? So why are you walking around withrocks in your hand in the first place then?Frankly, Amanda, behavior of the sort you just engagedin gives all of Islam a even more of a bad reputation.Calm down already, or is not having found a man to fillyour life the base cause of your outbursts?Here's a little instantaneous help for you:http://www.virtual-vibrator.com/index-ns.shtml# === Subject: : Re: Uncle Al is Sadistic .Apparently you don't have anything at all to say that'srelevant to any of the three newsgroups to which youcontinue to post. How *do* you justify your existence?>She seems to be looking for a Muslim scientist to hit>on her so she's advertising in sci newsgroups and for>bait is trying to appear intelligent, passionate, and>desirable.> So ..you are anti-Muslim. I understand why you must now attack me in> such personal way but still I am sure that that's very low of you.> My comment was limited to your misuse of these newsgroups.> Most idiots like you have no clue that there are lots of Muslims in> countries like Burma, Thailand, who don't deserve the animosity people> like you have for them for being Muslims.> So Hansen was right after all? Are you really just> a rabid bitch attacking anything you don't like> with any stone that happens to be in your hand at> the moment? So why are you walking around with> rocks in your hand in the first place then?> Frankly, Amanda, behavior of the sort you just engaged> in gives all of Islam a even more of a bad reputation.How so? You think I am a Muslim? ha...ha.. > Calm down already, or is not having found a man to fill> your life the base cause of your outbursts?Typical of you. It only shows that you got nothing else to attack mewith. Just as Rich assume that I am a Muslim after beleiving junks he saw,you assume it too.Since you are anti-Muslim, you can't help but reveal how cheap youare. Oh...I am sure that I got it right that you are anti-Muslim. And I am sure that you know how low you are for attacking me likethat. Men like you just can't help it but to be cheap, can you?> Here's a little instantaneous help for you:It only shows how depesrate you are to attack me in such a cheap waythat you failed to realize how you are revealing how low you are.> http://www.virtual-vibrator.com/index-ns.shtml# === Subject: : Re: Uncle Al is Sadistic . charset=iso-8859-1>How *do* you justify your existence?>>She is trying to appear intelligent, passionate, and>>desirable.> I understand why you must now attack me in such personal > way but still I am sure that that's very low of you.> Most idiots like you have no clue.....> Bill Vajk:> So Hansen was right after all? Are you really just> a rabid bitch attacking anything you don't like> with any stone that happens to be in your hand at> the moment? So why are you walking around with> rocks in your hand in the first place then?> Here's a little instantaneous help for you:> http://www.virtual-vibrator.com/index-ns.shtml#> AHAHahhaha.......ahahahaha.....AHAHAHHAHAHA....See, you are a good Samaritan after all, Bill. However,I doubt though that neither vibrations nor dildonisationswill help her condition, because as said, recommendationsfor dicktation didn't work for her, rectification was of no help neither and oralizations just made her even louder. She suffers from a very extreme case of rock retention.........which is the cause of all her tensions. Poor thing just can't get her rocks off ..........bad scene, Bill.She probably dreams of having some rock on her hands or it running down her chin.....which would be a good scene.ahahahaha........ahahahahansonPS: OTOH, in all reality the broad is getting the tinglingin her loins from all that attentions she's getting here.In that she's been successful: Broad 1 -- guys 0.Did you notice that she used the oldest trick in thebook?: It's the wheel that squeaks that gets the oilahahahahaha......AHAHAHHAHA.......ahahahahaha === Subject: : Re: Uncle Al is Sadistic .> Apparently you don't have anything at all to say that's> relevant to any of the three newsgroups to which you> continue to post. How *do* you justify your existence?> She seems to be looking for a Muslim scientist to hit> on her so she's advertising in sci newsgroups and for> bait is trying to appear intelligent, passionate, and> desirable.She has failed in at least two out of three of the aboveendeavors. === Subject: : Re: Uncle Al is Sadistic .>Apparently you don't have anything at all to say that's>relevant to any of the three newsgroups to which you>continue to post. How *do* you justify your existence?>>She seems to be looking for a Muslim scientist to hit>>on her so she's advertising in sci newsgroups and for>>bait is trying to appear intelligent, passionate, and>>desirable.> She has failed in at least two out of three of the above> endeavors.Often having the right equipment compensates. Perhaps theobvious feelings of inadequacy have something to do withthat..... === Subject: : Re: Grid Of Mystery Puzzle>I double double-checked the numbers, but *still* did not realize I had>exchanged the two digits.>>And there is a relatively simple rule for the 3-by-3 case, which by a>certain generalization can extend to all sequences of n positive>integers.> I'm still at a loss for what the rule is, but I believe I have determined> the placement of the 4x4 grid. Will you confirm whether this is correct?> 10 11 14 15> 9 12 13 16> 8 5 4 1> 7 6 3 2> -- ErickYou are correct regarding the placement of integers.(I should have had a rule that did not result in so much correlationbetween the output integers' relative magnitudes and the relativemagnitudes of the integers in the grid.):/I will reveal the rule I intended tomorrow or the next day. (As well,I will reveal the answer then at that time for my Array RecursionPuzzle.)leroy Quet === Subject: : Recommendations on Complex Analysis books?The subject says it -- however, I'd like to clarifyone detail: I'm looking for a book on *analysis*, asopposed to Calculus (i.e., that covers rigorouslythe concepts and proofs on Complex numbers andComplex variables functions).However, I'm just a hobbyist, so I'm not looking forthe ultimate, advanced reference book (i.e., I'm nota mathematician or even a student in Mathematics; I'man engineer, who already knows (at least *knew* verywell :-)) about complex numbers, but I'm beginningto appreciate and enjoy the rigorous side of maths,and I find that complex numbers do have a greatappeal.Thanks for any advice!Carlos-- === Subject: : Re: Recommendations on Complex Analysis books?> The subject says it -- however, I'd like to clarify> one detail: I'm looking for a book on *analysis*, as> opposed to Calculus (i.e., that covers rigorously> the concepts and proofs on Complex numbers and> Complex variables functions).> However, I'm just a hobbyist, so I'm not looking for> the ultimate, advanced reference book (i.e., I'm not> a mathematician or even a student in Mathematics; I'm> an engineer, who already knows (at least *knew* very> well :-)) about complex numbers, but I'm beginning> to appreciate and enjoy the rigorous side of maths,> and I find that complex numbers do have a great> appeal.I like Complex Variables and applications by Churchill, Brown, and Verhey.It is a pretty standard text for undergraduate courses, as far as I know.Lurch> Thanks for any advice!> Carlos> -- === Subject: : Turing machine questionA Turing machine question:Can you build a Turing machine if - instead of a pen and aneraser to mark on the tape you have a *finite* number of paperclips, which can be clipped on and off the tape?You could not use binary representations - you would haveto use large numbers to store things - stored in the form: 1: .X 2: ..X ...50: ..................................................X...etc. If the answer is no, then why not?Thanks for any assistance.-- __________ |im |yler http://timtyler.org/ tim@tt1lock.org Remove lock to reply. === Subject: : Sphinx Tiling Explanation?AllI'm looking at tilings right now, and at the sitehttp://astronomy.swin.edu.au/~pbourke/texture/ nonperiodic/about 2/5 of the way down the page there is something called a SphinxTiling, which is, according to the page, an obvious nonperiodictiling. Okay, call me dense, but it's not obvious to me. Can anyoneexplain why it's aperiodic?Justin === Subject: : Re: Sphinx Tiling Explanation?Hmmm, if you create another copy of the sphinx and rotate that copy 180degrees, the two sphinx shapes can be connected into a 3 by 2 trapezoid,which in turn clearly tiles the plane. So I guess I'm confused, too. Thesphinx seems to me to be periodic.Best,Kerry Soileau> All> I'm looking at tilings right now, and at the site> http://astronomy.swin.edu.au/~pbourke/texture/nonperiodic/> about 2/5 of the way down the page there is something called a Sphinx> Tiling, which is, according to the page, an obvious nonperiodic> tiling. Okay, call me dense, but it's not obvious to me. Can anyone> explain why it's aperiodic?> Justin === Subject: : Re: Other Gamma-Function Asymptotics> Most of us know that> x! = Gamma(x+1) ~ x^(x+1/2) exp(-x) sqrt(2 pi)> as x -> oo, by Stirling's asymptotical formula.> But what about the asymptotic behavior of Gamma(x+1) as x appoaches> other limits other than real positive infinity?> For example:> In the vicinity of x = 0,> Gamma(x+1) = exp(-c*x +x^2 pi^2/12 - x^3 zeta(3) +O(x^4)),> where c is Euler's constant. Whoops! I forgot to divide zeta(3) by 3. So... Gamma(x+1) = exp(-c*x +x^2 pi^2/12 - x^3 zeta(3)/3 +O(x^4))> Leroy> -> And, as x -> -oo,> Gamma(x+1) ~ -(-x)^(x+1/2) exp(-x) csc(pi x) sqrt(pi/2).> But I am not absolutely certain about this asymptotic formula, and> would not use it to investigate the behavior of Gamma(x+1) at x very> near negative integers.> -> So, what, for example, about the asymptotics if x = 1/2 + i y, where y> approaches infinity?> Or how about if x approaches any other interesting finite/infinite> complex/real constants?> Leroy QuetOf course, if x is a real > 1, and {x} is the fractional part of x({x} = x - floor(x)), then Gamma(x+1) = x*(x-1)*(x-2)*...(x-floor(x)+1)*exp(-c*{x} +{x}^2 pi^2/12 - {x}^3 zeta(3)/3 +O({x}^4)),which gives a better approximation (for finite number of terms takenin the sum) for smaller {x}, obviously.(And we can modify this for negative x to help investigate Gammafunction near negative integers, if we want.)LeroyQuet === Subject: : A dimension problem in vector spacesHiI'd be very grateful if someone could prove the following.Let f:V->V be an antilinear map (i.e. f(cv)=c*f(v) - c* is complex conjugateof c) of a complex vector space V into itself, and satisfies f^2=id (i.e.fof =id on V).Let W={v in V | f(v)=v} so that W is a *real* vector space.Prove that dim V (over C) = dim W (over R).Thanks so much, in anticipation.Ron Jones === Subject: : Re: some coin tossing:> Thanks a lot, that was really interesting though it took me some time to:> figure out what exactly he (Bryce Carlson) was saying. But I still don't:> understand how he computes those p1, p2, etc ... It seems that he is:> continuing some discussion here. Can you please send me the link where he:> does the analysis for 2 consecutive heads?: No.: Seriously, that one turned up in a google search, but: doesn't seem linked to the main structure of www.bjmath.com,: talking about.: However, this problem does get discussed here and there: around the web. Here's an entry at Ask Dr. Math.: http://mathforum.org/library/drmath/view/52217.html: Somewhere I've also seen it developed as a recursive: expression, but when I've tried to work that out myself: I get lost in the counting. One recursion goes something: like:: I can get m (or more) successive heads in N tosses: by either:: (a) getting m or more in the first (N-1) or: (b) Having the first (N-1) *not* contain a run: of m, but ending with (m-1) heads in a row, followed: by heads in the last toss.: I *think* that means: P(N,m) = P(N,m-1) + (1 - P(N, m-1))*(1/2)^m: With P(N,m) = probability of getting m or more heads: in a row in N tosses, and: P(N,m) = 0 if N < m.That should beP(N,m) = P(N-1,m) + (1-P(N-m-1, m))*(1/2)^(m+1)The P(N-1,m) describes case (a).For case (b), we have something like ..................0111111111.....111 N-m-1 flips m 1's where m heads do no occur in a rowThat 0 in the middle guarantees we do not count somethingin both case (a) and (b).The probability of the first N-m-1 flips not containing mheads in a row is 1-P(N-m-1,m). The probability that the first N-m-1 flips are followed by a 0 and m 1's is (1/2)^(m+1).Stephen