I am cramming for an exam urgently... I am reading the text book: The textbook is: Fundamentals on Complex Analysis with Application to Ehgineering and Science (third eddition) by E. B. Saff and A. D. Snider It is known from the Professor that the exam is mainly about Series expansion, analytic continution, Residue theory and evaluation some strange integrals... and there are a few extra problems... I read the book but feel the examples are inadequate for me to grasp all the tricks, esp. in the Residue theory and strange integral evalution... Can anybody give me a hand by pointing me to some (online/offline) resources that can familarize me with these tricks and strenthen my techniques in solving these problems? Best Michael. ==== > Dear all, > > I am cramming for an exam urgently... I am reading the text book: > > The textbook is: Fundamentals on Complex Analysis with > Application to Ehgineering and Science > (third eddition) > by E. B. Saff and A. D. Snider > > It is known from the Professor that the exam is mainly about Series > expansion, analytic continution, Residue theory and evaluation some strange > integrals... and there are a few extra problems... I read the book but feel > the examples are inadequate for me to grasp all the tricks, esp. in the > Residue theory and strange integral evalution... > > Can anybody give me a hand by pointing me to some (online/offline) resources > that can familarize me with these tricks and strenthen my techniques in > solving these problems? > > > Best > > > Michael. > IMHO you should not treat your exam as learning bunch of tricks how to solve some 'strange' integrals but you should try instead to really understand principles behind them - understanding of these 'tricks' will follow as consequence. Goran ==== You are absolutely right... But I am just not feel enough just by working through the principles and examples in the textbook... I think I need to work on more problems and learn more techniques for the sake of the hard exam... By the way, the self-test problems in the text book have no solutions... Can you still give me a hand on that? -Walala > Ü«é.91îæ.b9ß > > Dear all, > > I am cramming for an exam urgently... I am reading the text book: > > The textbook is: Fundamentals on Complex Analysis with > Application to Ehgineering and Science > (third eddition) > by E. B. Saff and A. D. Snider > > It is known from the Professor that the exam is mainly about Series > expansion, analytic continution, Residue theory and evaluation some > strange > integrals... and there are a few extra problems... I read the book but > feel > the examples are inadequate for me to grasp all the tricks, esp. in the > Residue theory and strange integral evalution... > > Can anybody give me a hand by pointing me to some (online/offline) > resources > that can familarize me with these tricks and strenthen my techniques in > solving these problems? > > > Best > > > Michael. > > > IMHO you should not treat your exam as learning bunch of tricks how to > solve some 'strange' integrals but you should try instead to really > understand principles behind them - understanding of these 'tricks' will > follow as consequence. > > Goran > > ==== > > >>Dear all, >> >>I am cramming for an exam urgently... I am reading the text book: >> >>The textbook is: Fundamentals on Complex Analysis with >>Application to Ehgineering and Science >> (third eddition) >>by E. B. Saff and A. D. Snider >> >>It is known from the Professor that the exam is mainly about Series >>expansion, analytic continution, Residue theory and evaluation some strange >> >> >>integrals... and there are a few extra problems... I read the book but feel >> >> >>the examples are inadequate for me to grasp all the tricks, esp. in the >>Residue theory and strange integral evalution... >> >>Can anybody give me a hand by pointing me to some (online/offline) resources >> >> >>that can familarize me with these tricks and strenthen my techniques in >>solving these problems? >> >> >IMHO you should not treat your exam as learning bunch of tricks how to >solve some 'strange' integrals but you should try instead to really >understand principles behind them - understanding of these 'tricks' will >follow as consequence. > > Maybe. It depends what the time pressures are. Also, it may not even be true. The first time my wife took statistics (well, a course labeled statistics), it turned out to be a tricky integrals course. She claims not to have learned a thing. Series expansion and analytic continuation are IMO important but straightforward, so I have no idea what the tricks are. Strange integrals are only strange until you've done enough of them to become familiar with them. The key frequently is to pick a curve that you can calculate the integral over (or estimate) and then let R->oo or 0. I always separated the singularities and evaluated them individually (that way I know all the residues individually), but I have known people who did more complicated calculations. I always got tied up in the algebra when I tried that. Frankly, I don't know what to do if the singularities aren't isolated. This looks pretty good: http://www.ecs.fullerton.edu/~mathews/c2000/c08/c08.html Probably the best thing to do is go to the math library (OP's return address is @yahoo.com, so I have no idea where walala is from, but large US universities have separate math libraries) and start looking through the complex analysis books until you see one or more that you like. You'll have to use the card catalogue to find them (just look up your textbook and go to that section and start looking through the books there). I'd guess (from your description of your text) that you probably need at least two -- one for the basics and one for the tricky integrals. Jon Miller ==== Dear all, 1. Does the principal branch sqrt(z) have a Laurent sereis expansion in the domain C{0}? And why? 2. Two power series: A=sum(a_k*z^k, k from 0 to inf) and B=sum(b_k*z^k, k from 0 to inf): Prove that if A=B for all real x in some open interval containing the origin, then a_k=b_k for all k... This problem statement seems obvious to me... but how to prove it? -Walala ==== > Dear all, > > 1. Does the principal branch sqrt(z) have a Laurent sereis expansion in > the domain C{0}? Is it continuous on that domain? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen ==== Dear all, This problem took me some time as I am practicing for an exam urgently... Given that Integrate[exp(-x^2), x from 0 to inf]=sqrt(pi)/2, integrate exp(i*z^2) around the boundary of the circular sector {z=r*exp(i*theta): theta from 0 to pi/4, r from 0 to R} and let R-> +oo to prove that Integrate[exp(i*x^2), x from 0 to inf] = sqrt(2*pi)*(1+i)/4 Anybody please help me! -Walala Approved: news-answers-request@MIT.EDU Summary: The Lorentz transforms themselves are proof t' and x' cannot possibly be just coordinates. Examination of their derivation verifies their identity as ==== 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. Archive-name: physics-faq/criticism/lorentz-intervals (SR) Lorentz t', x' = Intervals (c) Eleaticus/Oren C. Webster Thnktank@concentric.net ------------------------------ of the use of 'just coordinates' in any scientific formula. Defenders of the Special Relativity faith are especially fond of telling opponents of their space-time fairy tales that they do not know the difference between coordinates and 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 acts with which the SR faithful bedazzle the public, and establish their 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' because it always represents particular elapsed times that can be used in the (t-t0) form to calculate perfectly good time intervals (elapsed times). Alternatively, I could (re)set my clock to zero at the start of 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 very spot where Zeno first presented his first paradox to Parmenides. That makes my x equal a couple of thousands of miles, plus, and is not useful for such purposes unless you establish the starting x value, perhaps x0, and use the formula (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 intervals that can always be used in the (x-x0) form for any and every scientific 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' axis 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.] ------------------------------ 1. Introduction with the obvious debunking of the use of 'just coordinates' in any scientific formula. 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 ------------------------------ Special Relativity's space-time circus is based on the 'transformation' equations by which it is believed one can relate a nominally 'stationary' system's space and time coordinates to those of an inertially (not accelerating) 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 just the spatial coordinate axis that lies parallel to the direction of motion, and with time, Einstein arrived at these formulas that relate the moving system measures or coordinates (x' and t') to the stationary 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 in that direction, and all axes parallel the other system's corresponding axis. We used vv to mean the square of v but might use v^2 for that purpose below. Similarly for c. Because it is believed that no physical object can reach or exceed c, the square-root term in both denominators is presumed always less than one, which means that the formulas say both x' and t' will tend to be greater than x and t, respectively. However, SRians call the x' result 'contraction' - which means shortening - and the t' result 'dilation' - which means increasing. ------------------------------ The 'just coordinates' argument is so patently ridiculous that even opponents have a hard time accepting just how simple and obvious its debunking can be, as shown in this section. 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 a duration, not a time interval; it's just an arbitrary clock reading. But what if the moving system observer comes speeding by while you make your annual 'spring forward' or 'fall back' change? The formula says that the 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 and both are watching the moving observer. You'll both be using the same v because you are at rest wrt (with respect to) each other. You're on Eastern Standard Time and your twin is on Pacific Standard Time maybe. 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 of your t-times: t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) Sure, the idea that you can change someone else's clock 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 elapsed time formula, or the basis of a t'/t ratio, then there is no implication that one clock's reading has 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. ------------------------------ Since we're going to be comparing measurements on two coordinate systems in the next section, let's go to our supply cabinet and get our yard-stick (which we use 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 is hanging on, right at the 3.5 inch mark of the yard- stick. Let's see if I can wave the stick around enough that she'll let go. Nope. However, before I gave up I waved the stick and the ant '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 the stick, however far and wide I have transported her. Neither of those 3.5 facts means very much. Of the two, the distance aspect meant almost nothing. So the distance was 3.5 from the end. So what? That length, distance, was not in use. And only maybe the ant might have been concerned with just what location, 'just coordinate', on the stick she was at. Just so with x and t. So, is the 3.5 reading just a coordinate? Or a distance/length? It's ambiguous in and of itself, and really makes no difference what you say until you try to make use of the number. Hey, my address is 5047 Newton Street. If you are looking for me and you're at 4120 Newton, it is helpful information, because it tells you which direction to go. Is that 'just coordinate'? Where it really becomes useful, perhaps, is in telling you how far away I am. That's not just a 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 distance or length; it is impossible to be otherwise with an 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 useful place. As in the introduction, we should either put it at the starting location/end, or use two readings from it: (x-x0). ------------------------------ Taking care to not damage our brave little ant, I place my yard-stick onto the table, zero end to the left, 36 end to the right. Now I place the 'just coordinate' meter-stick on the table in the same orientation, in a random location, and find that the ant's coordinate on the meter-stick is 51. The formula relating centimeters to inches is cm=i*2.54 but 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. 8.89 is 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 systems to have their zero points aligned, in which case saying the two measures are not intervals is pure idiocy. Noe that with both zero points at the same position both x' and x are great measures for scientific purposes, in any and every case where we were smart enough to put those zero points at a useful location. There is one extension of x'=x/g that will let us use the meter-stick in arbitrary position. When the cm reading was 51, the zero point of the yard-stick read (51-8.89=) 42.11 cm. If we call that point 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 distance from the x' coordinate of the location where x=0. An interval. Just as obviously, the fact that we now have the correct formula for relating an x interval to an arbitrary x' coordinate, 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') in the scientific formula. (x'-x.z') equals the useful, Ratio Scale value x/.3937. So, we have discovered a basic fact: a transformation formula like x'=x/g works only if the two zero points of the coordinate systems coincide. That makes it non- sense to say the two coodinates are only coordinates and not intervals. Both must be values that represent distances from their respective zero points unless you take the proper steps to adjust for the discrepancy. Make sure you understand that although the inclusion of 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 be that we're measuring a sheet of paper with one end at x=0 and the other at x=3.5; x'=51 is nonsense as a centimeter measure of the paper. But, you say, the Lorentz transform contain a -vt term. ------------------------------ We discovered x'=x.z' + x/g as the correct formula for 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 our right. 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 SRians wouldn't imagine it was. It is an interval, the distance over which the moving system has moved since t=0. And, of course, x/.3937 is the distance of our brave little ant from the point where x=0 and the centimeter reading is x.z'-vt/.3937. Yes, every minute the meter- stick moves to the right and the meter-stick coordinate of the spot where x=0 gets less and less - and eventually negative. 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 worse over time. Quite obviously, the fact that we now have the correct formula for relating an x interval to an arbitrary x' coordinate even when the x' axis is moving, 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) in the scientific formula. (x'-x.z'+vt/.3937) equals the useful, Ratio Scale value x/.3937. ------------------------------ Instead of using our sticks, let's get out two clocks. Mind you, we're not going to deal with different clock rates 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, time axis 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 the EST is showing a good number for elapsed time since the travelling observer passed NYC, then the PST clock is silliness. t.z' must be zero or must be taken out of time lapse calculations for the PST clock to be used intelligently, just as was true for x.z'. What is lacking as yet for Lorentz t' is the -vx/cc term that corresponds 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/c is thus the time interval since the two time axes had a common zero point - which they have to have in the Lorentz transforms which do not have the t.z' term we learned 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 interval represented by x/sqrt(1-vv/cc). We have discovered that the only way for t' to be t/g is for t' and t to have a common zero point, just as for x' and x. It would be otherwise if the t' formula contained an adjustment t.z' under some name or other, but the necessity to include such a term correlates 100% with t' numbers that aren't directly usable. As for x and x', our knowledge of how to setup a proper formula relating t and t' is of no use unless we use the knowledge in scientific formulas; (t'-t.z'+xv/gcc) gives us the only directly useful value: t/g. ------------------------------ When we return to Einstein's derivations of the transform formulas with a well-focused eye, we find he was a wee bit confused - or at least self-contradictory. When he set up his (at first unknown) tau=moving system time formulas, he created three particular instances of tau. Tau.0 is the time at which light is emitted at the moving origin 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. He lets the stationary time be t+x'/(c-v); t is still a constant and x'/(c-v) is the time interval since t. Tau.2 is the time at which the light gets (back) to the moving 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 to the 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 leaves no 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 is x'/(c-v) and x'/(c+v), the time interval it takes for light to get to the fleeing mirror, and the time interval it 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 increases by 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 and explicitly 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 that there was some L that was the mirror x-location and length when the light is emitted, if t was already, say, 500, then x'=L-vt could have been a very negative length. ------------------------------ There are intervals, and there are intervals. If we put our yard stick zero point at one end of a piece of paper and read off the coordinate at the other end of the paper, we have a good measure of the paper's length, a Ratio Scale measure. [Absolute temperature scales are ratio scale.] If instead we put the one end of the paper at the one inch mark (or the zero end of the stick one inch 'into' the length of the paper) we get measures that are one inch off the true, ratio scale length. The two messed up measures are still intervals, but they are Interval Scale measures. [Household temperature scales are interval scale, which is why your physics and chemistry professors won't let you use them without first converting to the ratio 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 t and 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 x and a good ratio scale t. Look for the (SR) Lorentz t', x' = degraded measures document soon at a newsgroup near you. ------------------------------ t'=(t-vx/cc)/g shows t' being greater than t. The reason Special Relativity will not allow the use of its basic time equation in determining what SR has to say about the twins' ages, is that t' and x' 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, not just 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 their transformation formulas. Look for the (SR) Lorentz t', x' = degraded measures document at a newsgroup near you. ------------------------------ A. 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 ? !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---?---!---? ==== I understand this has been discussed at great, staggering, exhausting lengths here before, but no satisfactory conclusion seems to ever be reached. The general concensus seems to be that it is a non-issue and not worth arguing about ad infinitum, or that it is arbitrary and can be whatever we want it to be, or both or some combination. However, I was pondering it and realized that if we do define it (regardless of what we define it as, so long as we make it some real or even complex number) then it allows us to write certain piecewise-defined equations without the piecewise-defined clause. For, if we say 0^0 = 0 then we have the following: for any real x, x^0 = 1 iff x != 0, else x = 0 But, if we say 0^0 = 1, then we have: for any real x, 0^|x| = 0 iff x != 0, else x = 0 And if we say 0^0 = r, r neither 1 nor 0, then we have: for any real x, (0^|x|)/r = 0 iff x != 0, else x = 0 Consider, for example, the function H defined as H(x) = 1 if x = 5 H(x) = 3 if x = 10 H(x) = 9 for all other real x Then, if we say 0^0 = 0, we have (after a minimal amount of trivial algebra): H(x) = 1 + {2 + [6 * 0^(x-10)]}*0^(x-5) A bit ugly, indeed. However, the important thing is, H, a function originally written by a piecewise equation and very difficult/impossible to transcribe using just one non piecewise equation, is, in fact, written as a single non piecewise equation, in fact one which uses nothing but the 4 very elementary arithmetic operations (addition, subtraction, multiplication, exponent). If one adds various other allowable elementary operations (such as the floor or ceiling function or the ability to do finite summations with a variable upper limit for the index), it is very possible to write out some surprising things with just those tools: a formula for the nth prime, a formula for the nth rational in Cantor's famous proof, a formula for H(x)=1 if x is rational, 0 otherwise. But the key and the cornerstone is that in order to do this, we must define 0^0 as some real number- it doesn't matter what, although 0^0=0 obviously leads to (by far) the most simple equations and is unique in that any other definition would require us to add the absolute value function to our equations. In fact, after setting some notation and getting some extremely trivial theorems whose proofs are insultively easy, the act of deriving an equation for the nth prime that uses only addition, subtraction, multiplication, exponents, the floor function, and finite summations with variable upper limits.. requires no ingenuity whatsoever and is comparable to doing a complicated but trivial differentiation problem in Calc 1. (Of course, this equation for pi(n) is exceedingly long, ugly, and unwieldy.) But only if we define 0^0. (And, if we define 0^0 such that 0^0 is not 0, we must also include the absolute value function in our equations.) ==== How about defining 0/0 instead? Won't that work just as well? ==== > > However, I was pondering it and realized that if we do define it > (regardless of what we define it as, so long as we make it some real > or even complex number) then it allows us to [etc.] > Actually, many mathematicians define 0^0 to be some value (in certain contexts). Very often 0^0 is set to 1. R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics, p.162: Some textbooks leave the quantity 0^0 undefined, because the functions x^0 and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x^0=1 for all x, _if_ the binomial theorem is to be valid when x = 0, y = 0, and/or x = -y . F. ==== Donald E. Knuth, Two notes on notation, Amer. Math. Monthly 99 (1992), 403-422. ==== > How about defining 0/0 instead? Won't that work just as well? If we said 0/0 = 1, then it could work, but why 0/0 instead of 0^0? There are some extremely good reasons NOT to define 0/0, which noone credible will refute, whereas there are no serious reasons not to define 0^0, in fact there are many books where 0^0 *IS* defined. In fact, it has been shown on this newsgroup in the past, that different math textbooks, all of which were completely credible and legit, had different definitions for 0^0 (mostly either 1 or 0). ==== >> How about defining 0/0 instead? Won't that work just as well? > If we said 0/0 = 1, then it could work, but why 0/0 instead of 0^0? > There are some extremely good reasons NOT to define 0/0, which noone > credible will refute, whereas there are no serious reasons not to > define 0^0, in fact there are many books where 0^0 *IS* defined. In > fact, it has been shown on this newsgroup in the past, that different > math textbooks, all of which were completely credible and legit, had > different definitions for 0^0 (mostly either 1 or 0). If 0^0 has a value, say 0^0 = x, then x * x = 0^0 * 0^0 = 0 ^ (0+0) = 0^0 = x and therefore x must be 0 or 1. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== In sci.math, Trishia Rose : > I understand this has been discussed at great, staggering, exhausting > lengths here before, but no satisfactory conclusion seems to ever be > reached. The general concensus seems to be that it is a non-issue and > not worth arguing about ad infinitum, or that it is arbitrary and can > be whatever we want it to be, or both or some combination. > However, I was pondering it and realized that if we do define it > (regardless of what we define it as, so long as we make it some real > or even complex number) then it allows us to write certain > piecewise-defined equations without the piecewise-defined clause. Well, part of the problem is that lim (x -> 0+) x^0 = 1 but lim (y -> 0+) 0^y = 0 so there is a very fundamental discontinuity that precludes us saying lim ((x,y) -> (0+,0+)) x^y = L for any L. However, Rudin, in his work _Real and Complex Analysis_ (a textbook I happened to learn Lebesgue integral theory from in college), does assume 0 * oo = 0, presumably for his convenience but it does make things a lot simpler. > > For, if we say 0^0 = 0 then we have the following: > for any real x, x^0 = 1 iff x != 0, else x = 0 > > But, if we say 0^0 = 1, then we have: > for any real x, 0^|x| = 0 iff x != 0, else x = 0 > > And if we say 0^0 = r, r neither 1 nor 0, then we have: > for any real x, (0^|x|)/r = 0 iff x != 0, else x = 0 > > Consider, for example, the function H defined as > H(x) = 1 if x = 5 > H(x) = 3 if x = 10 > H(x) = 9 for all other real x > > Then, if we say 0^0 = 0, we have (after a minimal amount of trivial > algebra): > > H(x) = 1 + {2 + [6 * 0^(x-10)]}*0^(x-5) Erm, H(5) = 1 + {2 + [6 * 0^(5-10)]}*0^(5-5) = infinity (0^(-n)) H(10) = 1 + {2 + [6 * 0^(10-10)]}*0^(10-5) = 1 H(x), x < 10, fails miserably. If you want to go this route use 0^abs(x-n). You can then write things like H(x) = 6*0^abs(x-5) + 8*0^abs(x-10) - 5 At least that way it more or less works. :-) There are, however, other ways of handling the problem, which may be simpler. The sgn(x) function is defined as a real -> (-1,0,1) mapping, where sgn(x) = -1 if x < 0, +1 if x > 0, and 0 for x = 0. Using this function and abs, it's trivial to set up various flavors of H(x). Noting that 1 - sgn(abs(x-n)) is 1 if x == n and 0 everywhere else, we can write: H(x) = 9 - 8 * (1 - sgn(abs(x-5))) - 6 * (1 - sgn(abs(x-10))) = 8 * sgn(abs(x-5)) + 6 * sgn(abs(x-10)) - 5 and voila. Of course H(x) = 9 almost everywhere anyway, and lim (x -> a) H(x) = 9 for any a on the real line (or, for that matter, on the complex plane) -- *including* the two points of discontinuity -- so as far as analysis goes (or my understanding thereof) H(x) is a relatively uninteresting function. > > A bit ugly, indeed. However, the important thing is, H, a function > originally written by a piecewise equation and very > difficult/impossible to transcribe using just one non piecewise > equation, is, in fact, written as a single non piecewise equation, > in fact one which uses nothing but the 4 very elementary arithmetic > operations (addition, subtraction, multiplication, exponent). Maybe six: +, -, *, /, to the power of, and exponent. :-) > > If one adds various other allowable elementary operations (such as the > floor or ceiling function or the ability to do finite summations with > a variable upper limit for the index), it is very possible to write > out some surprising things with just those tools: a formula for the > nth prime, a formula for the nth rational in Cantor's famous proof, I'm assuming you're referring to the countability of the rationals, not the uncountability of the reals. > a > formula for H(x)=1 if x is rational, 0 otherwise. That would be a challenge, even given 0^0 = 0, mostly because it's not clear to me how to eliminate the non-normalized fractions. If one were to, for instance, naively sum H'(x) = sum(i=1,oo) sum(j=0,i) (1 - abs(x - j/i)^0) it wouldn't work, although you'd get the interesting result that H'(x) = oo if x rational, 0 otherwise, on the interval [0,1]. How one might normalize that, I'm not entirely certain. > But the key and > the cornerstone is that in order to do this, we must define 0^0 as > some real number- it doesn't matter what, although 0^0=0 obviously > leads to (by far) the most simple equations and is unique in that any > other definition would require us to add the absolute value function > to our equations. > > In fact, after setting some notation and getting some extremely > trivial theorems whose proofs are insultively easy, the act of > deriving an equation for the nth prime that uses only addition, > subtraction, multiplication, exponents, the floor function, and finite > summations with variable upper limits.. requires no ingenuity > whatsoever and is comparable to doing a complicated but trivial > differentiation problem in Calc 1. (Of course, this equation for > pi(n) is exceedingly long, ugly, and unwieldy.) > > But only if we define 0^0. (And, if we define 0^0 such that 0^0 is > not 0, we must also include the absolute value function in our > equations.) You'd have to anyway. 0^x is infinity for x < 0, as it results in the value 1/0. -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== JSH recently made the following post on his site: ==================== Realizing that it might help for people to know more about me, I've opened up the photo section of the group, and setup a spot where group members can put in their own photos. Yes, this means you can put a photo of yourself on the group, and yes, if you put up something obnoxious it's a quick way to get dumped out of the group, and your photo will go with you. The group is rated general, which means it is open to audiences of all ages. James Harris ==================== A photo of Mr Harris can be found here: http://groups.msn.com/AmateurMath/memberspictures.msnw?action=ShowPhoto&Phot oID=2 ==== >A photo of Mr Harris can be found here: > >http://groups.msn.com/AmateurMath/memberspictures.msnw?action=ShowPhoto&Pho toID=2 And we should care because .... ? -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com/ You find yourself amusing, Blackadder. I try not to fly in the face of public opinion. ==== >Message-id: > >>A photo of Mr Harris can be found here: >> > >>http://groups.msn.com/AmateurMath/memberspictures.msnw?action=ShowPhoto& PhotoID=2 > >And we should care because .... ? http://members.aol.com/owagiveaway/ace.htm > >-- >Stan Brown, Oak Road Systems, Cortland County, New York, USA > http://OakRoadSystems.com/ >You find yourself amusing, Blackadder. >I try not to fly in the face of public opinion. -- Mensanator 2 of Clubs http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm ==== >>A photo of Mr Harris can be found here: >> >>http://groups.msn.com/AmateurMath/memberspictures.msnw?action=ShowPhoto&Ph otoID=2 > >And we should care because .... ? Why we should care about the photo is not clear to me either. Otoh the fact that he said Realizing that it might help for people to know more about me, I've opened up the photo section of the group is kind of amusing - like knowing what he looks like is going to help us understand The Proof. ************************ David C. Ullrich ==== Do please trim your quotes. >A photo of Mr Harris can be found here: >http://groups.msn.com/AmateurMath/memberspictures.msnw?action=ShowPhoto& >PhotoID=2 >>And we should care because .... ? >http://members.aol.com/owagiveaway/ace.htm And we should care because .... ? -- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.com/ You find yourself amusing, Blackadder. I try not to fly in the face of public opinion. ==== >Message-id: > > >Do please trim your quotes. > > >>A photo of Mr Harris can be found here: >>http://groups.msn.com/AmateurMath/memberspictures.msnw?action=ShowPhoto& >>PhotoID=2 > >And we should care because .... ? > > >>http://members.aol.com/owagiveaway/ace.htm > >And we should care because .... ? When you say we, are you presuming to speak for everyone? My reply to the first question demonstrates that you do _not_, in fact, speak for everyone. There is at least one person who takes delight in being informed of the posting of a photograph of JSH. > >-- >Stan Brown, Oak Road Systems, Cortland County, New York, USA > http://OakRoadSystems.com/ >You find yourself amusing, Blackadder. >I try not to fly in the face of public opinion. -- Mensanator 2 of Clubs http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm ==== Let rho be a function [0,1]->[0,1], regular enough so that T(rho)(x) := int_0^{sqrt{1-x^2}} rho(sqrt{x^2+y^2}) dy = int_x^1 rrho(r)/sqrt{r^2-x^2} dr is defined for all x in [0,1]. This yields another function [0,1]->[0,1]: then can T be inverted explicitly in terms of expressions of the kind of those appearing in the equation above? (under suitable assumptions on the functions T and T^{-1} are applied, of course) [I don't know if the problem is phrased correctly - think of it, roughly speaking, in terms of translating a radial density into a linear one] Michele -- > Comments should say _why_ something is being done. Oh? My comments always say what _really_ should have happened. :) - Tore Aursand on comp.lang.perl.misc ==== >Let rho be a function [0,1]->[0,1], regular enough so that >T(rho)(x) := int_0^{sqrt{1-x^2}} rho(sqrt{x^2+y^2}) dy > = int_x^1 rrho(r)/sqrt{r^2-x^2} dr >is defined for all x in [0,1]. This yields another function >[0,1]->[0,1]: then can T be inverted explicitly in terms of >expressions of the kind of those appearing in the equation above? >(under suitable assumptions on the functions T and T^{-1} are applied, >of course) Expand rho in a Bessel series rho(r) = sum_{n=1}^infinity c_n J_0(a_n r) where a_n are the zeros of J_0. I believe you should be able to find the coefficients by int_0^1 T(rho)(x) cos(a_m x) dx = sum_{n=1}^infinity c_n int_0^1 r J_0(a_n r) int_0^r cos(a_m x)/sqrt(r^2-x^2) dx dr = pi/2 sum_{n=1}^infinity c_n int_0^1 r J_0(a_n r) J_0(a_m r) dr = pi/2 c_m int_0^1 x J_0(a_m r)^2 dr = pi/4 c_m J_1(a_m)^2 Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== One way to approach the prime numbers is to look at a number and see if it has any factors other than itself. Another way to approach the problem is to look at all of the numbers that are NOT prime and conclude that anything left over IS prime. This latter approach is the basis of my conjecture. See http://tln.net/~reriker/prime.html for more details. ==== > One way to approach the prime numbers is to look at a number and see > if it has any factors other than itself. Another way to approach the > problem is to look at all of the numbers that are NOT prime and > conclude that anything left over IS prime. This latter approach is the > basis of my conjecture. > > See http://tln.net/~reriker/prime.html for more details. This need not be a conjecture. [n(n + (2m+1))/2] either n is odd (o), or it is even (e) case 1, n is odd n(n + (2m+1))/2 o(o + (2m+1))/2 (2p+1)((2p+1) + (2m+1))/2 (2p+1)(p + 1/2 + m + 1/2) (2p+1)(p + m + 1) case 2, n is even n(n + (2m+1))/2 e(e + (2m+1))/2 2p(2p + (2m+1))/2 2p(p + m + 1/2) p(2p + 2m + 1) This is pretty elementary no? ==== > > This is pretty elementary no? > I think it was a hoax, but it seems you took it seriously ;) ==== problem in the subject line, quoting a message in The Math Forum citing this reference: >TWO MORE REPRESENTATION PROBLEMS >by Andrew Bremner and Richard Guy >published in Proc. Edinburgh Math. Soc. vol 40 1997 pp 1-17. For anyone interested in this problem: that's a fine paper to read; it includes quite a references to the literature, tables of numerical solutions, and a discussion of the elliptic-curve techniques needed to carry out the computations. But as they say, a week in the laboratory can frequently save an hour's trip to the library!. I worked out most of the details myself and had access to a bundle of computers, so I have quite a complete set of data now. I'm not going to post it all to the newsgroup, but I will make it available at http://www.math.niu.edu/~rusin/research-math/abcn/ Here are some excerpts: The following is the complete set of the 111 values of n under 200 for which the equation x^3+y^3+z^3=nxyz can be solved in nonzero integers: 3, 5, 6, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 30, 31, 35, 36, 38, 40, 41, 44, 47, 51, 53, 54, 57, 62, 63, 64, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 83, 84, 86, 87, 92, 94, 96, 98, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 112, 113, 116, 117, 119, 120, 122, 123, 124, 126, 127, 128, 129, 130, 132, 133, 136, 142, 143, 145, 147, 148, 149, 151, 154, 155, 156, 158, 159, 160, 161, 162, 164, 166, 167, 172, 174, 175, 177, 178, 181, 185, 186, 187, 189, 190, 191, 192, 195, 196, 197 Except for the case n=5, there are infinitely many solutions for each n. Except for n=142 and n=177, we have explicit solutions. The following is a complete list of the values of n under 200 for which the equation x^3+y^3+z^3=nxyz can be solved in POSITIVE integers: 3, 5, 6, 9, 10, 13, 14, 17, 18, 19, 21, 26, 29, 30, 38, 41, 51, 53, 54, 57, 66, 67, 69, 73, 74, 77, 83, 86, 94, 101, 102, 105, 106, 110, 113, 117, 122, 126, 129, 130, 133, 145, 147, 149, 154, 158, 161, 162, 166, 174, 178, 181, 186, 195, 197 except for the possibility that the values n=147 and n=177 must be added. (We can know that as soon as we have explicit solutions for these two n. That, in turn, can be accomplished by sufficient computing power, probably to be measured in CPU-weeks. N.B. -- We have already expended over one hundred CPU-days for this project!) Pfoertner goes on to say: >N = 62 and N = 64 both have solutions but they are quite large. What, a dozen or two digits? Piffle! The techniques we discuss show that the equation a/b + b/c + c/a = 112 can be solved with explicit integers a,b,c; what appears to be the simplest solution requires (90+)-digit numbers: a = 44488222032517984080347242004206223609176772084 4845203037340381653808676781078204185344064777425 b = 180001063934056147663194703762128694791524068 4971323481294582383858472523311320365128373281158 c = -1331809157685411330016283859165784168699351 9959959149070559988026538909081959649861205201860 dave ==== I've read that the CH is equivalent to the assertion that the plane is the union of two sets, one being countable along any line parallel to the ordinate-axis, the other being countable along any line parallel to the abscissa-axis. Is it possible to give constructive definitions of two particular such sets? ==== > >I've read that the CH is equivalent to the assertion that the >plane is the union of two sets, one being countable along any >line parallel to the ordinate-axis, the other being countable >along any line parallel to the abscissa-axis. Is it possible >to give constructive definitions of two particular such sets? Sure thing. If the reals have cardinality aleph_1, that means that it is possible to set up a one-to-one correspondence between the reals and the countable ordinals (ordinals less than aleph_1). So let ord(x) be the countable ordinal corresponding to real x. Then we let A = { | ord(x) < ord(y) }, and let B = { | ord(x) >= ord(y) }. (where is the point on the plane with coordinates x,y) Every horizontal line intersects A in only countably many points (because if you fix y, there are only countably many x such that ord(x) < ord(y)). Every vertical line intersects B in only countably many points (because if you fix x, there are only countably many y such that ord(x) >= ord(y)). -- Daryl McCullough Ithaca, NY ==== >I've read that the CH is equivalent to the assertion that the >plane is the union of two sets, one being countable along any >line parallel to the ordinate-axis, the other being countable >along any line parallel to the abscissa-axis. Is it possible >to give constructive definitions of two particular such sets? I'm not sure whether you mean to be asking for a constructive definition, _assuming_ CH. If so: CH says that c = aleph_1; it follows that there's a well-ordering < on R such that every element of R has only countably many predecessors (note that here < has nothing whatever to do with the usual < relation.) Let A be the set of all (x,y) such that x < y, and let B be the set of all (x,y) such that y <= x. ************************ David C. Ullrich ==== > I've read that the CH is equivalent to the assertion that the > plane is the union of two sets, one being countable along any > line parallel to the ordinate-axis, the other being countable > along any line parallel to the abscissa-axis. Is it possible > to give constructive definitions of two particular such sets? I have no idea about your question. I didn't even read it. I was too distracted by the format of your post. How long did it take to get a perfect rectangle from your post? -- Yup, as far as I'm concerned, if you live out your lives smiling the entire time full of pride in your *believed* accomplishments, when you never had any, well that's ok with me. --James Harris, a man of remarkable accomplishments. ==== >> I've read that the CH is equivalent to the assertion that the >> plane is the union of two sets, one being countable along any >> line parallel to the ordinate-axis, the other being countable >> along any line parallel to the abscissa-axis. Is it possible >> to give constructive definitions of two particular such sets? > > I have no idea about your question. I didn't even read it. I was too > distracted by the format of your post. > > How long did it take to get a perfect rectangle from your post? It actually looks like an accident to me. There aren't any extra spaces anywhere. I reckon it could happen by accident, in a five-line post, once every few thousand tries. ==== 115/70 mi. East, 1/14 mi. South. > A supermarket chain has 3 outlets in the suburbs of a large town. The > locations of these outlets relative to the town centre (in miles) are as > follows: > > Outlet North South East West > 1 5 5 > 2 6 3 > 3 4 3 > > The supermarket chain wish to site a warehouse so that it is centrally > located relative to the positions of the outlets. Advise the supermarket > chain on the optimum location of the warehouse relative to the town centre. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== > A supermarket chain has 3 outlets in the suburbs of a large town. The > locations of these outlets relative to the town centre (in miles) are as > follows: > > Outlet North South East West > 1 5 5 > 2 6 3 > 3 4 3 > > The supermarket chain wish to site a warehouse so that it is centrally > located relative to the positions of the outlets. Advise the supermarket > chain on the optimum location of the warehouse relative to the town centre. > > Since your columns do not line up on my screen, Do your data mean: Outlet 1 is North 5mi and East 5mi from town center, Outlet 2 is South 6mi and East 3mi from town center, Outlet 3 is South 4mi and West 3mi from town center ? And is distance to be measured in straight lines fom the warehouse or using a taxi metric (sum of North_South and East-West distances)? ==== I'm assuming this is not a homework problem. In sci.math, Frank Dewhurst : > A supermarket chain has 3 outlets in the suburbs of a large town. The > locations of these outlets relative to the town centre (in miles) are as > follows: > > Outlet North South East West > 1 5 5 > 2 6 3 > 3 4 3 Your formatting is a little messy; I'm assuming you're using a proportionally-fonted newsreader variant. Store 1: North 5 mi, East 5 mi Store 2: North 6 mi, East 3 mi Store 3: North 4 mi, East 3 mi > > The supermarket chain wish to site a warehouse so that it is centrally > located relative to the positions of the outlets. Advise the supermarket > chain on the optimum location of the warehouse relative to the town centre. > Assuming the large town is gridlike and has all two-way streets, the distance of the warehouse from each store is simply the Manhattan distance. (I'm assuming the problem is an economic one; the cheapest site, all other factors being equal [you didn't mention where the warehouse is going to be supplied from, for instance] is the site where the truck drivers from the warehouse travel the minimum distance. Since trucks can't fly (old DHL adverts notwithstanding :-) ) one is restricted to the grid.) This can be represented as D(S,W) = abs(x_S - x_W) + abs(y_S - y_W) The idea would be to minimize F(W) = D(S1,W) + D(S2,W) + D(S3,W) assuming all stores are the same size. (If they're not, one can place a size multiplier on each term, complicating the analysis.) Since that's a little unwieldly to manipulate (although it could be done) I'm assuming the other posters are using Euclidean distance, which means one has to minimize f(W) = d(S1,W) + d(S2,W) + d(S3,W) where d(A,B) = sqrt( (x_A - x_B)^2 + (y_A - y_B)^2). However, one obvious result already is that x_W has to be 5 [*], if I'm reading your input correctly, again assuming same-sized stores. This simplifies life considerably since now I only have to minimize a single variate equation, and is regardless of distance model used, mostly because the perpendicular bisector of the line between the two western stores just happens to fall on a thoroughfare. (Had you specified different coordinates for your stores I may not have been so lucky.) Therefore the optimum location is x_W = 5, y_W = 3, right between the two western stores. The slightly less logical Euclidean distance still results in x_W = 5 as before, and: f(y_W) = sqrt((5 - 5)^2 + (y_W - 5)^2) + sqrt((5 - 6)^2 + (y_W - 3)^2) + sqrt((5 - 4)^2 + (y_W - 3)^2) = abs(y_W - 5) + sqrt(1 + y_W^2 - 6*y_W + 9) + sqrt(1 + y_W^2 - 6*y_W + 9) = abs(y_W - 5) + 2 * sqrt(y_W^2 - 6*y_W + 10) Obviously y_W can't be greater than 5 so we can drop the abs(): F(y_W) = 5 - y_W + 2 * sqrt(y_W^2 - 6*y_W + 10) We now differentiate: F'(y_W) = -1 + 2 * (1 / (2 * sqrt(y_W ^ 2 - 6*y_W + 10))) * (2 * y_W - 6) To find a y_W such that F'(y_W) = 0 requires: 0 = -1 + (1 / sqrt(y_W ^ 2 - 6*y_W + 10)) * (2 * y_W - 6) 1 = (1 / sqrt(y_W ^ 2 - 6*y_W + 10)) * (2 * y_W - 6) 2 * y_W - 6 = sqrt(y_W ^ 2 - 6*y_W + 10) 4 * y_W^2 - 24 * y_W + 36 = y_W ^ 2 - 6*y_W + 10 3 * y_W^2 - 18 * y_W + 26 = 0 which results in y_W = (18 + sqrt(324-4*3*26)) / 6 = 3 + sqrt(12)/6 = 3 + sqrt(3) / 3 = 3.557 A quick test shows that this is indeed the optimum result: f(3) = 4 f(5) = 4.472 f(3.557) = 3.732 I'm not sure if my numbers are correct, nor do they match the other posters on this thread. However, I can at least state that they're based on fairly sound algebra, and I show my work above. :-) Of course the problem is incompletely specified (suppose there was a political friend of the mayor's at North 5 mi, East 4 mi? Or maybe a Superfund toxic cleanup site?) but wotthehell; math is a model, and the size of the grain of salt is dependent on the goodness of the model. :-) [*] x is the distance north-south, y the distance east-west. This is flipped from the standard representation but shouldn't affect the solution to any great extent. -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== In sci.math, Virgil > >> A supermarket chain has 3 outlets in the suburbs of a large town. The >> locations of these outlets relative to the town centre (in miles) are as >> follows: >> >> Outlet North South East West >> 1 5 5 >> 2 6 3 >> 3 4 3 >> >> The supermarket chain wish to site a warehouse so that it is centrally >> located relative to the positions of the outlets. Advise the supermarket >> chain on the optimum location of the warehouse relative to the town centre. >> >> > > Since your columns do not line up on my screen, Do your data mean: > > Outlet 1 is North 5mi and East 5mi from town center, > Outlet 2 is South 6mi and East 3mi from town center, > Outlet 3 is South 4mi and West 3mi from town center ? > > And is distance to be measured in straight lines fom the warehouse > or using a taxi metric (sum of North_South and East-West distances)? See my prior post in this thread for an answer to your second question. :-) Then again, you do bring up the interesting point that I've probably misread the problem myself, in light of this issue. Aargh. (Not that my analysis needs to be changed too much.) -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== > Simpson's dog: my recollection of that is that it came from a > Far Side cartoon. In one panel the dog's owner is saying > something like No, Ginger! You must not bark in the house, Ginger! > Do you hear me, Ginger?, and the dog hears Blah, Ginger! Blah > blah blah, Ginger! Blah blah blah, Ginger! I wonder which > came first - Far Side, or the Simpson's version? In any case, yes, > JSH has reinvented this also. > > Nora B. My guess is that Larson did it first. He covered most of the bases. I really like Jerry van Amerongen too. Gib ==== I'm sorry if this is the wrong place for this question, but it's the most appropriate spot I could find.... The problem is this: I know the Cartesian coordinates of the centre of a circle. There are two points on the circumferance of the circle, joined by a straight line. I know the coordinates of the midpoint of that line, and the length of the line. Is it possible to calculate the coordinates of the two points on the circumferance of the circle? ==== > I'm sorry if this is the wrong place for this question, but it's the > most appropriate spot I could find.... > The problem is this: > I know the Cartesian coordinates of the centre of a circle. There are > two points on the circumferance of the circle, joined by a straight > line. I know the coordinates of the midpoint of that line, and the > length of the line. Is it possible to calculate the coordinates of the > two points on the circumferance of the circle? that problem sounds really easy; you should be able to find the solution graphically with ruler and compass. As first step draw some arbitrary circle and a line segment. What can you say about that line and the line connecting the center of the circle with the midpoint of your line? So given only the center, the midpoint and the length of the line, how can you find out the points on the circle? Alois ==== > I'm sorry if this is the wrong place for this question, but it's the > most appropriate spot I could find.... > The problem is this: > I know the Cartesian coordinates of the centre of a circle. There are > two points on the circumferance of the circle, joined by a straight > line. I know the coordinates of the midpoint of that line, and the > length of the line. Is it possible to calculate the coordinates of the > two points on the circumferance of the circle? Yes, except in the case where the line is a diameter of the circle (when the midpoint is the centre). It's convenient to use vectors. Let v be the vector from the centre O of the circle to the midpoint M of the chord. The chord is perpendicular to v so let u be a unit vector perpendicular to v (if v = (a, b) you can take u = (-b/sqrt(a^2 +b^2), a/sqrt(a^2 + b^2)) ). Now if the chord has length l, the vectors from M to its endpoints are (l/2)u and (-l/2)u. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen ==== > I know the Cartesian coordinates of the centre of a circle. My mind has been corrupted by morons, trolls, and cranks: I first read that as Cantorian coordinates. Lee Rudolph ==== Messy but doable. General idea: First we find the equation of this line. Then we find out where this line intersects the cirle--Done. For simplicity I will asume that the circle is centered at the origin. Here goes: Let the midpoint of the given line be (a,b). The slope the line joining the two points is perpendicular to the line from the center of the circle to the midpoint (a,b). Now the slope of the line from the midpoint to the center is m= b/a. Hence the slope of the line joining the two points is m= -a/b. Suppose the length of the chord is c and the equationof the line is x^2 + y^2 = r^2. We suppose (x_1,y_1) is one point on the circumference and on the chord. Oh, so the distance from (x_1,y_1) to (a,b) [which equals c/2!!)] is sqrt{(x_1-a)^2 + (y_1-b)^2} = c/2 or (x_1-a)^2 + (y_1-b)^2 = c^2/4(= (c/2)^2). This last equation happens to an equation of a cicle that contains (a,b) and (x_1,y_1). Even more importantly this circle ONLY crosses the original circle at (x_1,y_1)-prove this. So know the problem reduces to find the point of intersection of the two circles. After finding this intersection point you should think about how you might find the other point on the circle. > I'm sorry if this is the wrong place for this question, but it's the > most appropriate spot I could find.... > The problem is this: > I know the Cartesian coordinates of the centre of a circle. There are > two points on the circumferance of the circle, joined by a straight > line. I know the coordinates of the midpoint of that line, and the > length of the line. Is it possible to calculate the coordinates of the > two points on the circumferance of the circle? > I'm sorry if this is the wrong place for this question, but it's the > most appropriate spot I could find.... > The problem is this: > I know the Cartesian coordinates of the centre of a circle. There are > two points on the circumferance of the circle, joined by a straight > line. I know the coordinates of the midpoint of that line, and the > length of the line. Is it possible to calculate the coordinates of the > two points on the circumferance of the circle? ==== > > I know the Cartesian coordinates of the centre of a circle. There are > two points on the circumferance of the circle, joined by a straight > line. I know the coordinates of the midpoint of that line, and the > length of the line. Is it possible to calculate the coordinates of the > two points on the circumferance of the circle? Denote the coordinates of the centre of the circle be (p, q) and of the line midpoint (r, s). Then shift coordinates via x', y' = x - p, y - q, so that the circle is centred on the origin, and in this new Ox'y' frame the coordinates of the line midpoint become (r - p, s - q). Let r be the radius of the circle, and let the line segment intersect the circle at points P, Q, and let the normal through the origin O to the line segment meet it at R, and finally denote angle POR by v and lengths OR, PR by D, L resp, so r.cos(t) = D and r.sin(t) = L, and consequently r^2 = D^2 + L^2. As D^2 = (r - p)^2 + (s - q)^2 and you say you know L, this gives you r. (Remember L is _half_ of the length of the line segment.) Letting tan(u) = (s - q)/(r - p) the coordinates of the line segment's intersections with the circle are (in Ox'y', and with signs conformable) (r.cos(u +/- v), r.sin(u +/- v). To return to the original coordinates, just let x, y = x' + p, y' + 1. --------------------------------------------------------------------------- John R Ramsden (jr@adslate.com) --------------------------------------------------------------------------- Eternity is a long time, especially towards the end. Woody Allen ==== I hope you don't mind this post but I could really do with some help! I am a University student in the UK trying to implement an relatively efficient (..oh...dear...) GMP version of AKS (off paper ver_3; change on bound and size of r etc..). However, I am not 100% sure how to do the: STEP V: reduce (X + a)^r by the ideal (n, X^r +1) I understand how to reduce an expanded polynomial, but since I appreciate that this will be exponential in the size of its input (N) this is the crappiest way ever as I will have to expand N coeffecients... I wondered if anyone could give me an idea of how to create a already monically reduced poly; I have read in the forums and some people are talking about it just being convolution, but I am unsure to what they mean and I am having no luck looking in either Knuth or Herstein. Any suggests or actual immplementations in C/java etc would be really appriecated - but really I would just love someone to fully explain it so that I may implement it my self. Phil ==== > I hope you don't mind this post but I could really do with some help! > I am a University student in the UK trying to > implement an relatively efficient (..oh...dear...) GMP version > of AKS (off paper ver_3; change on bound and size of r etc..). > > However, I am not 100% sure how to do the: > > STEP V: reduce (X + a)^r by the ideal (n, X^r +1) I hope you actually mean: compute (X + a)^n modulo . Use the binary trick, same as one does when computing a^{n-1} modulo n in the classical Fermat test. At each stage you have (X+a)^k modulo computed, then you go up to (X+a)^{2k} or (X+a)^{2k+1} by squaring, and then in the second case multiplying by (X+a). Heep doing this 'til you get to k = n. Of course the storage needed is about 2r numbers of the size of n^2. Quite big .... (but polynomial in log n). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen ==== > > Aha. But whereas the integers merely go *on* forever, the > irrationals also go *in* forever! Think of writing down the > decimal representations of all the irrational numbers: once > you start writing the digits of one irrational number, > you'll never finish even *that* number. > > (Rational numbers, on the other hand, don't go *in* forever. > Eventually, every rational number starts repeating, and we can > just make up a notation that says, this number repeats here, > and go on to the next one. That's why they're countable. > Sort of. ;) > > This is a pretty nice explanation! Well, not really. At least, *I* don't think it's very good. That's why the Sort of. and the smiley. It is a nice intuitive explanation, but in math you have to be *very* careful about nice intuitive results. For example, consider the set of numbers A = { pi } that is to say, the set whose only member is the number pi, 3.14159... Now, obviously this set has cardinality (or size) 1. It's only got one element! But if you apply the *exact* *same* *logic* that I applied to the set of irrationals above, you'll find that once you start writing the digits of pi, *they* never stop either! Should we conclude that A has an infinite size? Of course not! But even though the members of A go in forever, you reply, they don't go on forever anymore. So then take the set B = pi * Q that is to say, the set that you get by multiplying every rational number by 3.14159... That's the set B = { 1*pi, 2*pi, 0.5*pi, 3*pi, 1/3*pi, 2/3*pi, 4*pi, ... } Now each of these numbers is irrational (goes in forever), *and* the set itself is infinite (goes on forever), but the cardinality of the set is *still* only aleph_0! So even though my analogy was clever and intuitive (if I do say so myself) and I'm glad it helped you to understand cardinality a little better, please please PLEASE remember that clever analogies are no substitute for clever proofs! :) > I also read in another > aleph-1 guests would have to arrive. (Or, following the notation of some of the respondants to my post, say rather that more than aleph_0 guests would have to arrive. The exact choice of words is all mixed up with a statement called the Continuum Hypothesis, which is actually an axiom of certain mathematical *systems*, not an unalienable axiom of math itself -- kind of like Euclid's parallel postulate.) > Now, following the > logic that irrationals go *in* forever I guess I can > hotel has infinitely many rooms, it seems that each room is > of finite space, that's why aleph-1 guests would overflow > the hotel. Rooms in the hotel don't provide infinite space > (which would be required to store guests as large as an > irrational number). Nope! See, you are relying too heavily on the analogy. Suppose we had aleph_0 rooms, each room having aleph_0 beds. Let's number the rooms with numbers 0, 1, 2,... and so on, and let's number the beds in each room the same way. Then we can talk about bed 0 in room 0, bed 5 in room 42, and so on. In fact, we can abbreviate them by putting the room number first, then a dot, and then the bed number *backwards* (never mind why). Then Bed 0, room 0: 0.0 Bed 1, room 42: 42.1 Bed 5, room 50: 50.5 Bed 50, room 5: 5.05 -- the digits of 50 got flipped. Now, how many beds do we have in total? Well, notice how each bed is identified by a unique number in this system. (For example, bed 4.56 is the 65th bed in room 4.) And each of these bed numbers looks just like a rational number - 4.56 is 456/100, for example. And we *know* that the size of the set of rational numbers is just aleph_0, again! (In this case, it doesn't matter that some rational numbers -- for example, 10/3 -- don't correspond to beds in our hotel. All that matters is that each bed number corresponds to a unique rational number.) So even with aleph_0 beds in each of aleph_0 rooms, we *still* have no room for aleph_1 guests! Mathematically, one could write aleph_0 x aleph_0 = aleph_0 or 2 aleph_0 = aleph_0 (that's aleph-null squared equals aleph-null). You have to go all the way to aleph_0 2 (that's two raised to the power of aleph-null) before you find a cardinality bigger than aleph-null. (Modulo some nitpicks about the Continuum Hypothesis I mentioned earlier.) And that's a *really* big number. HTH, -Arthur ==== > > >> >> Aha. But whereas the integers merely go *on* forever, the >> irrationals also go *in* forever! Think of writing down the >> decimal representations of all the irrational numbers: once >> you start writing the digits of one irrational number, >> you'll never finish even *that* number. >> >> (Rational numbers, on the other hand, don't go *in* forever. >> Eventually, every rational number starts repeating, and we can >> just make up a notation that says, this number repeats here, >> and go on to the next one. That's why they're countable. >> Sort of. ;) >> >> This is a pretty nice explanation! > >Well, not really. At least, *I* don't think it's very good. That's >why the Sort of. and the smiley. It is a nice intuitive explanation, >but in math you have to be *very* careful about nice intuitive results. >For example, consider the set of numbers > > A = { pi } > >that is to say, the set whose only member is the number pi, 3.14159... >Now, obviously this set has cardinality (or size) 1. It's only got >one element! But if you apply the *exact* *same* *logic* that I applied >to the set of irrationals above, you'll find that once you start writing >the digits of pi, *they* never stop either! Should we conclude that >A has an infinite size? Of course not! > >But even though the members of A go in forever, you reply, they >don't go on forever anymore. So then take the set > > B = pi * Q > >that is to say, the set that you get by multiplying every rational number >by 3.14159... That's the set > > B = { 1*pi, 2*pi, 0.5*pi, 3*pi, 1/3*pi, 2/3*pi, 4*pi, ... } > >Now each of these numbers is irrational (goes in forever), *and* the >set itself is infinite (goes on forever), but the cardinality of the >set is *still* only aleph_0! > >So even though my analogy was clever and intuitive (if I do say so myself) >and I'm glad it helped you to understand cardinality a little better, >please please PLEASE remember that clever analogies are no substitute >for clever proofs! :) > >> I also read in another >> aleph-1 guests would have to arrive. > >(Or, following the notation of some of the respondants to my post, >say rather that more than aleph_0 guests would have to arrive. >The exact choice of words is all mixed up with a statement called >the Continuum Hypothesis, which is actually an axiom of certain >mathematical *systems*, not an unalienable axiom of math itself -- >kind of like Euclid's parallel postulate.) I would like to point out that that _standard_ notation is the one given by previous respondants, namely that aleph_1 is the next cardinal number after aleph_0 (by definition), aleph_2 is the next cardinal after aleph_1, etc. Every cardinal number is of the form aleph_xi for some ordinal xi. The Continuum Hypothesis only has relevance if we want to figure out _which_ xi applies for c=2^aleph_0. That is, we know c=aleph_xi for some xi, but it is independent what xi is. CH asserts that xi=1. And, as you point out, CH is independent of our usual axioms of set-theory. >> Now, following the >> logic that irrationals go *in* forever I guess I can >> hotel has infinitely many rooms, it seems that each room is >> of finite space, that's why aleph-1 guests would overflow >> the hotel. Rooms in the hotel don't provide infinite space >> (which would be required to store guests as large as an >> irrational number). > >Nope! See, you are relying too heavily on the analogy. >Suppose we had aleph_0 rooms, each room having aleph_0 beds. >Let's number the rooms with numbers 0, 1, 2,... and so on, >and let's number the beds in each room the same way. Then >we can talk about bed 0 in room 0, bed 5 in room 42, and >so on. In fact, we can abbreviate them by putting the room >number first, then a dot, and then the bed number *backwards* >(never mind why). Then > > Bed 0, room 0: 0.0 > Bed 1, room 42: 42.1 > Bed 5, room 50: 50.5 > Bed 50, room 5: 5.05 -- the digits of 50 got flipped. > >Now, how many beds do we have in total? > >Well, notice how each bed is identified by a unique number in >this system. (For example, bed 4.56 is the 65th bed in room 4.) >And each of these bed numbers looks just like a rational >number - 4.56 is 456/100, for example. And we *know* that the >size of the set of rational numbers is just aleph_0, again! > >(In this case, it doesn't matter that some rational numbers -- >for example, 10/3 -- don't correspond to beds in our hotel. >All that matters is that each bed number corresponds to a unique >rational number.) > >So even with aleph_0 beds in each of aleph_0 rooms, we *still* >have no room for aleph_1 guests! Mathematically, one could >write > > aleph_0 x aleph_0 = aleph_0 > >or > 2 > aleph_0 = aleph_0 > >(that's aleph-null squared equals aleph-null). >You have to go all the way to > > aleph_0 > 2 > >(that's two raised to the power of aleph-null) before you >find a cardinality bigger than aleph-null. (Modulo some >nitpicks about the Continuum Hypothesis I mentioned earlier.) These are not nitpicks, rather they address the fundamental _definitions_ of the alephs. And your statement is FALSE if CH is false. There may be arbitrarily many cardinal numbers between aleph_0 and 2^aleph_0. (For example, it is consistent that 2^aleph_0 = aleph_(aleph_10000000). That is, there could be aleph_10000000 different cardinal numbers between aleph_0 and 2^aleph_0.) Ciao, Apollo Hogan ==== I wonder if anyone could point me to an algorithm for finding all paths between two vertices in a non-directed, non-weighted graph. The graph is unfortunately cyclic. Paths found could be cyclic or non-cyclic, but I would find an algorithm giving only acyclic paths valuable. Xavier ==== > > >> >> >> ... stuff deleted ... >> >> > Linear versus Analytical Mechanics > >One of the really unfortunate aspects of Newton's choice of a linear >frame of reference for the analysis of mechanics is that r is poorly >defined and t is not defined at all. In other words, r is only defined >in direction and t is not defined by any consideration pertinent to >the analytical frame of reference. > >And this had a pernicious impact on the subsequent development of >angular mechanics as well as relativistic considerations and quantum >mechanics in the twentieth century. > >> >>Golly, that must be why physics doesn't work at all! And to think that I >>used to believe that we had computers and airplanes and rockets and >>satellites and all that stuff, due in large part to models based on the >>theories of physics. >> >>What's worse is that all that angular motion stuff (you know, rotating >>objects, orbital mechanics, the quantization of angular momentum, the >>whole schmear). Yet, my bike always worked just fine, and they've even >>managed to send satellites out to visit planets and all. I wonder how >>they figured out how to throw the things up there just right, so that >>they would go to the right place, and how they knew from the start, just >>how long it would take? Probably just a lucky guess, right? >> >>At any rate, it's good to be rid of those pesky wrong-brained ideas. >> >> > > I know how you feel. It's so good to have you back online critiquing > ideas. I'm sure there are more to come. > > > > As for ineffective critiques, you might note that *you* are the one who is proposing that the treatment of rotational motion has, to quote, had a pernicious impact on the subsequent development of angular mechanics as well as relativistic considerations and quantum mechanics in the twentieth century. It isn't up to anyone to prove you wrong (as I did, wrt your confused interpretation of dr/dt as being something other than velocity); you must prove your own case. So far, you have done none of that. BTW, your derivation of your identification of dr/dt with some radially-directed quantity was unacceptable: it proved nothing other than your view of mathematics as a system of random symbols to be strung together without regard to meaning or implication. Since you failed to engage in a technical discussion (apart from a stream of nonsensical formulas), I exited the discussion at at that point. My exit was in no way a recognition of any technical merit on your side. Somehow, your own lack of understanding of basic mathematics, and probably of basic physics as well, has you believing that physics is in bad shape as a model of physical reality. You assert this, without so much as a word of support, either experimental, or theoretical. One must admit that your explanations should be taken as an *attempt*, however feeble, to provide such theoretical support; if they didn't fail so thoroughly, I might be willing to allow that you had supported your case. points to the matter that physics *does* model physical reality fairly well; it makes no claims of being perfect -- it's a science, and science is, among other things, a process of continual refinement through experiment. You have not rebutted the case, even as mild a case as the one I presented, that the models of physics make predictions that can be verified by observation. Your presumed replacement for physics, so far as anyone can see, is nothing more than a soapbox for you to rant about something that evidently makes sense to you, but it is apparent that it makes sense to no one else. I'll note that I have not made any personal remarks of an insulting nature, nor do I intend to do this. I will thank you to behave in a similar manner. Dale. ==== > > >> >> > > > ... stuff deleted ... > > >> Linear versus Analytical Mechanics >> >>One of the really unfortunate aspects of Newton's choice of a linear >>frame of reference for the analysis of mechanics is that r is poorly >>defined and t is not defined at all. In other words, r is only defined >>in direction and t is not defined by any consideration pertinent to >>the analytical frame of reference. >> >>And this had a pernicious impact on the subsequent development of >>angular mechanics as well as relativistic considerations and quantum >>mechanics in the twentieth century. >> > >Golly, that must be why physics doesn't work at all! And to think that I >used to believe that we had computers and airplanes and rockets and >satellites and all that stuff, due in large part to models based on the >theories of physics. > >What's worse is that all that angular motion stuff (you know, rotating >objects, orbital mechanics, the quantization of angular momentum, the >whole schmear). Yet, my bike always worked just fine, and they've even >managed to send satellites out to visit planets and all. I wonder how >they figured out how to throw the things up there just right, so that >they would go to the right place, and how they knew from the start, just >how long it would take? Probably just a lucky guess, right? > >At any rate, it's good to be rid of those pesky wrong-brained ideas. > > >> >> I know how you feel. It's so good to have you back online critiquing >> ideas. I'm sure there are more to come. >> >> >> >> > >As for ineffective critiques, you might note that *you* are the one >who is proposing that the treatment of rotational motion has, to >quote, > had a pernicious impact on the subsequent development of > angular mechanics as well as relativistic considerations > and quantum mechanics in the twentieth century. > >It isn't up to anyone to prove you wrong (as I did, wrt your confused >interpretation of dr/dt as being something other than velocity); you >must prove your own case. So far, you have done none of that. BTW, your >derivation of your identification of dr/dt with some radially-directed >quantity was unacceptable: it proved nothing other than your view of >mathematics as a system of random symbols to be strung together without >regard to meaning or implication. Since you failed to engage in a >technical discussion (apart from a stream of nonsensical formulas), >I exited the discussion at at that point. My exit was in no way a >recognition of any technical merit on your side. > >Somehow, your own lack of understanding of basic mathematics, and >probably of basic physics as well, has you believing that physics >is in bad shape as a model of physical reality. You assert this, >without so much as a word of support, either experimental, or >theoretical. One must admit that your explanations should be taken >as an *attempt*, however feeble, to provide such theoretical support; >if they didn't fail so thoroughly, I might be willing to allow that >you had supported your case. > >points to the matter that physics *does* model physical reality >fairly well; it makes no claims of being perfect -- it's a science, >and science is, among other things, a process of continual refinement >through experiment. You have not rebutted the case, even as mild >a case as the one I presented, that the models of physics make >predictions that can be verified by observation. Your presumed >replacement for physics, so far as anyone can see, is nothing more >than a soapbox for you to rant about something that evidently makes >sense to you, but it is apparent that it makes sense to no one else. > >I'll note that I have not made any personal remarks of an insulting >nature, nor do I intend to do this. I will thank you to behave in >a similar manner. > Well, I certainly apologize if I pushed the envelope a little too far. It wasn't and isn't my intention to insult you personally or otherwise. It was merely an attempt to be cute that got a little too cute. Perhaps I'll save it for some other occasion. While you've been dismissive of what I've had to say I've never had occasion to think you've been insulting. ==== Suppose one were to go about developing Euclidean geometry analytically by defining space as R^3. In such a program, point, line, and plane would no longer be undefined terms; for example, a point would be any element of R^3, and a line would be a translate of a one-dimensional (linear) subspace. One would then go on to derive all of Euclid's axioms from properties of the reals (or, if you prefer, of your favorite axioms of set theory wherein the reals live). In such a program, would one be able to rigorously define the measure of an angle in a purely algebraic way, without an appeal to analysis? We know that dot product is very closely related to angle, but it seems to me that the concept is not amenable to a statement such as The sum of the measures of the angles of a triangle is 180 degrees. It is not clear to me that one could define angle measure without appeal to the length of the arc of a circle; for this concept, one needs integration or some other limit of polygonal paths. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== > In such a program, would one be able to rigorously define the measure of > an angle in a purely algebraic way, without an appeal to analysis? One could define angles as isometry classes of pairs of rays, define their ordering and addition/subtraction but to show that this gives a structure isomorphic to the numbers in some interval would require some argument involving limits as far as I can see. > We > know that dot product is very closely related to angle, but it seems to > me that the concept is not amenable to a statement such as The sum of > the measures of the angles of a triangle is 180 degrees. It is not > clear to me that one could define angle measure without appeal to the > length of the arc of a circle; for this concept, one needs integration > or some other limit of polygonal paths. No, no, no! It's best to define angle measure by looking at areas of sectors, not lengths of arcs; areas are much more tractable than arclengths :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen ==== > Suppose one were to go about developing Euclidean geometry analytically > by defining space as R^3. In such a program, point, line, and plane > would no longer be undefined terms; for example, a point would be any > element of R^3, and a line would be a translate of a one-dimensional > (linear) subspace. One would then go on to derive all of Euclid's > axioms from properties of the reals (or, if you prefer, of your favorite > axioms of set theory wherein the reals live). > > In such a program, would one be able to rigorously define the measure of > an angle in a purely algebraic way, without an appeal to analysis? Note that to define distance you need the square root, which exists essentially because of a limiting argument. So it seems analysis is already being used. ==== > > Suppose one were to go about developing Euclidean geometry analytically > by defining space as R^3. In such a program, point, line, and plane > would no longer be undefined terms; for example, a point would be any > element of R^3, and a line would be a translate of a one-dimensional > (linear) subspace. One would then go on to derive all of Euclid's > axioms from properties of the reals (or, if you prefer, of your favorite > axioms of set theory wherein the reals live). > > In such a program, would one be able to rigorously define the measure of > an angle in a purely algebraic way, without an appeal to analysis? > > Note that to define distance you need the square root, which exists > essentially because of a limiting argument. So it seems analysis is already > being used. If you only care about comparing distances, and not adding them, you can use the square of the distance (norm of a vector), purely algebraic. Of course, if you can't add distances, the triangle inequality is hard to define... Similarly, one can compute cosines of angles as dot products of unit vectors. To avoid normalizing vectors (again a square root) it looks like you need to use cos^2: square of dot product divided by product of squared norms. Recovering the angle itself then involves analysis for the sqrt and inverse cos. -- David Eppstein http://www.ics.uci.edu/~eppstein/ Univ. of California, Irvine, School of Information & Computer Science ==== [About defining geometric quantities from analysis and/or algebra]: >If you only care about comparing distances, and not adding them, you can >use the square of the distance (norm of a vector), purely algebraic. Of >course, if you can't add distances, the triangle inequality is hard to >define... Well, if a,b,c are positive real numbers with squares A,B,C respectively, then a+b Good day > > Consider for given positive integer n, the sequence > > S_k = ((k+1)^n - 1)/(k^n - 1) for k = 2,3,... > > For n=5, we would have > > S_2 = (3^5-1)/(2^5-1) = 242/31 ~ 7.8064516129 > S_3 = (4^5-1)/(3^5-1) = 1023/242 ~ 4.2727272727 > S_4 = (5^5-1)/(4^5-1) = 3124/1023 ~ 3.05376344086 > ... > > Observe that as k increases, S_k decreases (in fact it diverges to 1). > > My problem is with large values of k and n, the intermediante result > k^n becomes too large to calculate with a computer--even though the > sequence element, S_k, is not. > > Is there a method to approximate S_k without calculating k^n. Or is > there perhaps some reasonable bounds that can be constructed, e.g. x < > S_k < y? > > James Since S_k seems to -> 1, look at S_k - 1 = ((k+1)^n - k^n)/(k^n - 1) = ((1+1/k)^n - 1)/(1 - 1/k^n) See what the laft formula does as k gets large. Even assuming k > n gets you close to what you want. Martin Cohen ==== > Let R be a ring and S a multiplicative subset of R (a subset of R > closed under multiplication). In the construction, the following > equivalence relation is defined on RxS: > > (r,s)~(r',s') if there exists s in S such that s(rs'-r's) = 0. > > Why is the bit about s nessecary? Why not just define To deal with rings with zero divisors. The idea is that S^{-1}R has a universal property: each ring map phi: R -> A in which phi(s) is invertible for all s in S factors through R -> S^{-1}R. For this to happen then if s is in S and st = 0 then in A, 0 = (1/s)st = t. To localize we need to annihilate all t of this form (these t form an ideal I). If you like you can first consider R/I, then use the fraction construction on this ring. > (r,s)~(r',s') if rs'-r's = 0? Try proving this relation is transitive. > Also, when/why is it nessecary for the elements of S to be regular? Regular? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen ==== > Let R be a ring and S a multiplicative subset of R (a subset of R > closed under multiplication). In the construction, the following > equivalence relation is defined on RxS: > > (r,s)~(r',s') if there exists s in S such that s(rs'-r's) = 0. > > Why is the bit about s nessecary? Why not just define > > (r,s)~(r',s') if rs'-r's = 0? > > Also, when/why is it nessecary for the elements of S to be regular? > > dan Suppose s is invertible in a ring T which contains R. Let t be the inverse of s. If sa = sb, then a = tsa = tsb = b. Hence s is regular. This may answer your question. ==== > > > > In math most equations treat time as just > another spatial coordinate, which suggests > time can be reversed and the equation will > work as well. > > Indeed, and equations of physics and quantum mechanics also. > > This contradicts the second law which suggests > that ....nothing in the universe is reversible. > > My my, an antique paradox of a past century. > > Is mathematics, as a model of reality, fundamentally flawed? > > No, math makes no such claim to model reality nor does physics even tho > their favorite pass time is to make models of reality. > > So you seem to agree that math is only self-consistent, but not > necessarily consistent with reality. > > Then that begs the question, what should we do differently in order > to have a science that ...does properly model the real world? > > Towards the end of my post I suggest that this 'ancient paradox' > has a solution and it lies with our chosen frame of reference. > > A proper model of the real world is being created > by simply inversing the frame of reference classical > science has been constructed within. Which > is a task any mathematician should find trivial. > The implications of doing so, however, are > as powerful as the task is trivial. > > Your discussion > would likely receive less boredom at sci.physics than here. > > This topic applies to ...all subjects with equal validity whether > in science, religion, arts or the humanities. > I am trying to describe a new universal theory of > organization called complexity science. > > Jonathan Too bad only the Baby finds science complex. Why he has to keep turning it into a subordinate religion. -- ------(m+ ~/:o)_| To change the world be that change. http://scrawlmark.org ==== > > >> >> >In math most equations treat time as just >another spatial coordinate, which suggests >time can be reversed in time and the equation will >work as well. >> >This contradicts the second law which suggests >that ....nothing in the universe is reversible. >> >>You seem to be mixing up mathematics and physics here. >>The second law of thermodynamics is physics, not mathematics. > > No, I'm wondering why they aren't consistent with each other. > > > Because they weren't meant to be. Math is, at best, a modeling tool. > It can be applied to physical reality, but will only be a model. For > example, there are multiple geometries. Some are more useful when > describing reality than others. > >>If a mechanical system has an inherent energy loss due to friction, >>air drag, etc., it will lose mechanical energy as time increases. >>Integrating backwards in time from a final state will not replicate the >>initial state, because energy is also lost trying to retrace the path >>backwards. A lossy acceleration term that is proportional to the square >>of the velocity dx/dt is positive for both positive and negative dx/dt. > > > > Quite right. In physics you cannot retrace world lines, in math > you can. That's a problem for me! It's a problem for the > two to reconcile, as until they do one or the other is flawed > or incomplete. > > No, they just aren't as intertwined as you think they should be. > > In reality, of course, there is no going back in time, so the > task becomes to look at the final state in order to understand > the initial conditions. > > Sometimes the same is true in math. If a function is not 1-1, then you > cannot invert it to go from a result to an initial state. > > A system that is sensitive to initial conditions tends to result > in a chaotic system. But organizing or evolving systems are > robust to initial conditions. More simply, in all of the most > interesting real world systems the initial conditions do not > matter. Yet in math they are everything. > > Again, sometimes. In some dynamical systems the initial conditions have > little to do with the result. Nowhere more true than with babies. Well, /some/ babies. Most just suck at that Great Face over them until they crap out. (Many of them even call the activity complex.) > > It is a simple frame of reference mistake that leads to the > contradiction, reverse the frame where one expands > in scale first to understand the components later, and the > contradiction disappears. > > Physics and math are seperate disciplines. Math is *a* useful tool in > physics, and physics provides *some* useful examples in math. Because > math can start with axioms that do not correspond with reality, math can > analyze things that are irrelevant to physics. Similarly, math cannot > determine some of the constants in physics, only help analyze the > measurements of those constants. > > -- > Will Twentyman -- ------(m+ ~/:o)_| To change the world be that change. http://scrawlmark.org ==== > >> > > >> >> >> >In math most equations treat time as just >another spatial coordinate, which suggests >time can be reversed in time and the equation will >work as well. >> >This contradicts the second law which suggests >that ....nothing in the universe is reversible. >> >>You seem to be mixing up mathematics and physics here. >>The second law of thermodynamics is physics, not mathematics. > >No, I'm wondering why they aren't consistent with each other. > >>Because they weren't meant to be. > > Fine, my point is that if they were consistent wouldn't > it be possible to merge them into one much more > comprehensive and simpler science? > It would be more comprehensive, but also would probably be more complicated. When you take a system and add to it, you generally end up with MORE complication rather than less. Math by itself is fairly complicated as a whole. Adding it to physics would be like adding a mountain to a hill. You end up with far more than you are likely to need. >>Math is, at best, a modeling tool. >>It can be applied to physical reality, but will only be a model. For >>example, there are multiple geometries. Some are more useful when >>describing reality than others. > > > > Shouldn't reality define the model instead? We can do this by > defining the behavior of any variable by comparing it to itself, as > opposed to some other objective criteria. The variable pr system > in question would be defined by it's relationship with it's own > possibility range. A mathematics built in this way would > still be abstract yet universally applicable, as the same math > could be used regardless of the specific nature of the > system in question. It could be applied to both living > and material systems with equal validity. > Reality defines the model when you want to model reality. There are times when there is no interest in modeling reality. Mathematicians are regularly playing games of what if.... In these cases, reality isn't a concern, just seeing what adding another axiom does, or what changing one does. Some of these changes/additions result in it being a model of NOTHING, which is called inconsistent. Sometimes they result in very interesting things that may or may not be applicable to reality. Mathematics is used as a tool for physics, but it is not bound by being a tool for physics. Rather they coexist and in some ways overlap. >> >In reality, of course, there is no going back in time, so the >task becomes to look at the final state in order to understand >the initial conditions. >> >>Sometimes the same is true in math. If a function is not 1-1, then you >>cannot invert it to go from a result to an initial state. >> >> >A system that is sensitive to initial conditions tends to result >in a chaotic system. But organizing or evolving systems are >robust to initial conditions. More simply, in all of the most >interesting real world systems the initial conditions do not >matter. Yet in math they are everything. >> >>Again, sometimes. In some dynamical systems the initial conditions have >>little to do with the result. >> >> >It is a simple frame of reference mistake that leads to the >contradiction, reverse the frame where one expands >in scale first to understand the components later, and the >contradiction disappears. >> >>Physics and math are seperate disciplines. Math is *a* useful tool in >>physics, and physics provides *some* useful examples in math. Because >>math can start with axioms that do not correspond with reality, math can >>analyze things that are irrelevant to physics. Similarly, math cannot >>determine some of the constants in physics, only help analyze the >>measurements of those constants. > > And how many different scientific disciplines are there? > The more the compartments, or reductions, the farther > one gets from reality, accuracy and truth. It is not > possible for any one person to comprehend the > totality of all the disciplines and come to a complete > and accurate view of reality. > > But if all of reality could be modeled by one universal > mathematics, then ..all could easily understand all. As a general rule, many people cannot or do not understand the mathematics that is required to understand what is being done in advanced physics. Also, there are as many subdivisions of mathematics as there are of the sciences. Different sciences can borrow from any or all of these subdivisions. As a result, you are as unlikely to have anyone understand all of mathematics as to understand all of science. Consider the number of people who have difficulty with calculus and differential equations, and then realize that these are easy compared to some of the areas of math that are being used now in the sciences. -- Will Twentyman ==== > >> >> >In math most equations treat time as just >another spatial coordinate, which suggests >time can be reversed in time and the equation will >work as well. > >This contradicts the second law which suggests >that ....nothing in the universe is reversible. >> >>You seem to be mixing up mathematics and physics here. >>The second law of thermodynamics is physics, not mathematics. > > No, I'm wondering why they aren't consistent with each other. > > > > Because they weren't meant to be. > > > Fine, my point is that if they were consistent wouldn't > it be possible to merge them into one much more > comprehensive and simpler science? No. Mathematics and science are completely different disciplines, with different aims. Mathematics is already a very successful tool for the modeling of irreversible systems. The irreversibility can be modeled mathematically as well. One of my professors explained it very well: If you have a system whose equation of state involves only even derivatives of time, then it is time reversible. If you have odd derivatives, it's not. All idealized models of physical systems have even-derivative equations of state, and idealized models are extremely important for understanding the behavior of systems, even though they are not exact models of anything we will ever actually encounter. F = ma worked well for Newton, even though he had no frictionless systems to work with. We HAVE mathematical models of the irreversible aspects of the universe, and they work quite well. It's not that these are outside of the bounds of mathematics. It's that they are outside the bounds of simplified models. >Math is, at best, a modeling tool. > It can be applied to physical reality, but will only be a model. For > example, there are multiple geometries. Some are more useful when > describing reality than others. > > Shouldn't reality define the model instead? We know Newton's equations are not proper for modeling the motion of the planets, especially if we take a two-body model of those motions. We know there are tiny relativistic corrections. Yet there is little utility for most purposes in using highly-accurate equations. The mathematical model's job is to enable you to calculate what you need to calculate, to the accuracy you need and no more. It's job is not to be exact, since we know there are physical limitations to how exact we can be anyway. - Randy ==== >> >>In math most equations treat time as just >>another spatial coordinate, which suggests >>time can be reversed in time and the equation will >>work as well. >> >>This contradicts the second law which suggests >>that ....nothing in the universe is reversible. >> >> You seem to be mixing up mathematics and physics here. >> The second law of thermodynamics is physics, not mathematics. >No, I'm wondering why they aren't consistent with each other. Take the example below of a mechanical system with friction. The system loses energy forward in time. In order to retrace the path backwards in time, you would need to magically add energy to the system. That can be done mathematically, but it doesn't represent physical reality, in which friction still causes energy loss. Time reversal works fine for conservative systems. Physicists and engineers tend to think of mathematics as a tool that can, when used properly, describe physical reality. But if one has a bad mathematical model, or uses a model beyond its validity, one will not get a description of physical reality no matter how good the mathematics is. >> >> If a mechanical system has an inherent energy loss due to friction, >> air drag, etc., it will lose mechanical energy as time increases. >> Integrating backwards in time from a final state will not replicate the >> initial state, because energy is also lost trying to retrace the path >> backwards. A lossy acceleration term that is proportional to the square >> of the velocity dx/dt is positive for both positive and negative dx/dt. >Quite right. In physics you cannot retrace world lines, in math >you can. That's a problem for me! It's a problem for the >two to reconcile, as until they do one or the other is flawed >or incomplete. >In reality, of course, there is no going back in time, so the >task becomes to look at the final state in order to understand >the initial conditions. >A system that is sensitive to initial conditions tends to result >in a chaotic system. But organizing or evolving systems are >robust to initial conditions. More simply, in all of the most >interesting real world systems the initial conditions do not >matter. Yet in math they are everything. >It is a simple frame of reference mistake that leads to the >contradiction, reverse the frame where one expands >in scale first to understand the components later, and the >contradiction disappears. >Jonathan >s >> >> >> >> >> >> >> >> >> >> >> >> >> -- >> John E. Prussing >> University of Illinois at Urbana-Champaign >> Department of Aerospace Engineering >> http://www.uiuc.edu/~prussing -- John E. Prussing University of Illinois at Urbana-Champaign Department of Aerospace Engineering http://www.uiuc.edu/~prussing ==== > > >> In . >> >>In math most equations treat time as just >>another spatial coordinate, which suggests >>time can be reversed in time and the equation will >>work as well. >> >>This contradicts the second law which suggests >>that ....nothing in the universe is reversible. >> >> You seem to be mixing up mathematics and physics here. >> The second law of thermodynamics is physics, not mathematics. > > > >No, I'm wondering why they aren't consistent with each other. > > Take the example below of a mechanical system with friction. The system > loses energy forward in time. In order to retrace the path backwards in > time, you would need to magically add energy to the system. That can > be done mathematically, but it doesn't represent physical reality, in > which friction still causes energy loss. Not true, you are simply considering your model as a function of time not space. If you reverse your path, it represents a specific temporal location, which makes the model very useful for representing systems in a state that can no longer be observed (the past). ==== > > Too bad only the Baby finds science complex. Why he has to keep > turning it into a subordinate religion. That depends on which definition of complex you are using, the old or the new. Science is a system of highly disconnected components with each part obeying different and mostly arbitrary rules. Collectively 'science' is a chaotic system ..a gas which is as far from simplicity, elegance and truth that can be found. Congratulations~ You can continue, if you find it useful, to smash everything into the ground, and each time harder than the last. Perhaps when you've ground It All into dust then the sculpture will at last be revealed. You're insane to keep trying, so deep you've cut yourself into the Strip-Mine that even trying to climb out is incomprehensible to you. It's like watching someone dig their own grave in search of Heaven. The Farthest thunder that I heard Was nearer than the sky, And rumbles still, though torrid noons Have lain their missiles by. The lightning that preceded it Struck no one but myself, But I would not exchange the bolt For all the rest of life. Indebtedness to oxygen The chemist may repay, But not the obligation To electricity. It founds the homes and decks the days, And every clamor bright Is but the gleam concomitant Of that waylaying light. The thought is quiet as a flake, A crash without a sound; How life's reverberation Its explanation found! s > -- > ------(m+ > ~/:o)_| > To change the world > be that change. > http://scrawlmark.org ==== > > >> > > >> >> >> >In math most equations treat time as just >another spatial coordinate, which suggests >time can be reversed in time and the equation will >work as well. >> >This contradicts the second law which suggests >that ....nothing in the universe is reversible. >> >>You seem to be mixing up mathematics and physics here. >>The second law of thermodynamics is physics, not mathematics. > >No, I'm wondering why they aren't consistent with each other. > >>Because they weren't meant to be. > > Fine, my point is that if they were consistent wouldn't > it be possible to merge them into one much more > comprehensive and simpler science? > > > It would be more comprehensive, but also would probably be more > complicated. When you take a system and add to it, you generally end up > with MORE complication rather than less. But the variable, pi, indicates that you can only run halfway into the woods. -- ------(m+ ~/:o)_| To change the world be that change. http://scrawlmark.org ==== > > >> >> >In math most equations treat time as just >another spatial coordinate, which suggests >time can be reversed in time and the equation will >work as well. > >This contradicts the second law which suggests >that ....nothing in the universe is reversible. >> >>You seem to be mixing up mathematics and physics here. >>The second law of thermodynamics is physics, not mathematics. > > No, I'm wondering why they aren't consistent with each other. > > > > Because they weren't meant to be. > > > Fine, my point is that if they were consistent wouldn't > it be possible to merge them into one much more > comprehensive and simpler science? > > No. Mathematics and science are completely different > disciplines, with different aims. > > Mathematics is already a very successful tool for the > modeling of irreversible systems. The irreversibility can > be modeled mathematically as well. One of my professors > explained it very well: If you have a system whose > equation of state involves only even derivatives of time, > then it is time reversible. If you have odd derivatives, > it's not. > > All idealized models of physical systems have even-derivative > equations of state, and idealized models are extremely > important for understanding the behavior of systems, even > though they are not exact models of anything we will > ever actually encounter. F = ma worked well for Newton, > even though he had no frictionless systems to work with. > > We HAVE mathematical models of the irreversible aspects > of the universe, and they work quite well. It's not that > these are outside of the bounds of mathematics. It's > that they are outside the bounds of simplified models. > >Math is, at best, a modeling tool. > It can be applied to physical reality, but will only be a model. For > example, there are multiple geometries. Some are more useful when > describing reality than others. > > Shouldn't reality define the model instead? > > We know Newton's equations are not proper for modeling > the motion of the planets, especially if we take a two-body > model of those motions. We know there are tiny relativistic > corrections. Yet there is little utility for most purposes > in using highly-accurate equations. The mathematical model's > job is to enable you to calculate what you need to calculate, > to the accuracy you need and no more. It's job is not to be > exact, since we know there are physical limitations to how > exact we can be anyway. > > - Randy What he said. -- ------(m+ ~/:o)_| To change the world be that change. http://scrawlmark.org ==== > > > >> In . >> >>In math most equations treat time as just >>another spatial coordinate, which suggests >>time can be reversed in time and the equation will >>work as well. >> >>This contradicts the second law which suggests >>that ....nothing in the universe is reversible. >> >> You seem to be mixing up mathematics and physics here. >> The second law of thermodynamics is physics, not mathematics. > > > >No, I'm wondering why they aren't consistent with each other. > > Take the example below of a mechanical system with friction. The system > loses energy forward in time. In order to retrace the path backwards in > time, you would need to magically add energy to the system. That can > be done mathematically, but it doesn't represent physical reality, in > which friction still causes energy loss. > > Not true, you are simply considering your model as a function of time not > space. If you reverse your path, it represents a specific temporal > location, which makes the model very useful for representing systems in a > state that can no longer be observed (the past). Pf. There are no negatives in metaphysics. -- ------(m+ ~/:o)_| To change the world be that change. http://scrawlmark.org ==== > > > > Too bad only the Baby finds science complex. Why he has to keep > turning it into a subordinate religion. > > That depends on which definition of complex you are > using, the old or the new. Science is a system of > highly disconnected components with each part > obeying different and mostly arbitrary rules. Science isn't, kid; /you/ are. And you keep complaining that you are, and that you don't get it. And that it's all Science's Fault that you don't. And that you're gonna emit some chaos that explains your complexity. Kid, a communication channel cannot transmit a communication /more ordered/ than itself. /Or/ exceeding its own data rate. > Collectively 'science' is a chaotic system ..a gas > which is as far from simplicity, elegance and > truth that can be found. > > Congratulations~ > > You can continue, if you find it useful, to smash everything > into the ground, and each time harder than the > last. Perhaps when you've ground It All > into dust then the sculpture will at last > be revealed. > > You're insane to keep trying, so deep you've cut > yourself into the Strip-Mine that even trying to > climb out is incomprehensible to you. It's > like watching someone dig their own grave > in search of Heaven. > > The Farthest thunder that I heard > Was nearer than the sky, > And rumbles still, though torrid noons > Have lain their missiles by. > The lightning that preceded it > Struck no one but myself, > But I would not exchange the bolt > For all the rest of life. > Indebtedness to oxygen > The chemist may repay, > But not the obligation > To electricity. > It founds the homes and decks the days, > And every clamor bright > Is but the gleam concomitant > Of that waylaying light. > The thought is quiet as a flake, > A crash without a sound; > How life's reverberation > Its explanation found! > > s > > -- > ------(m+ > ~/:o)_| > To change the world > be that change. > http://scrawlmark.org -- ------(m+ ~/:o)_| To change the world be that change. http://scrawlmark.org ==== > >> > > >> >> > > > >> >> >> >> >In math most equations treat time as just >another spatial coordinate, which suggests >time can be reversed in time and the equation will >work as well. >> >This contradicts the second law which suggests >that ....nothing in the universe is reversible. >> >>You seem to be mixing up mathematics and physics here. >>The second law of thermodynamics is physics, not mathematics. > >No, I'm wondering why they aren't consistent with each other. > >> >>Because they weren't meant to be. > >Fine, my point is that if they were consistent wouldn't >it be possible to merge them into one much more >comprehensive and simpler science? > >> >>It would be more comprehensive, but also would probably be more >>complicated. When you take a system and add to it, you generally end up >>with MORE complication rather than less. > > > But the variable, pi, indicates that you can only run halfway into > the woods. Halfway into a set of woods of infinite radius could take a while. -- Will Twentyman ==== > > > > Too bad only the Baby finds science complex. Why he has to keep > turning it into a subordinate religion. > > That depends on which definition of complex you are > using, the old or the new. Science is a system of > highly disconnected components with each part > obeying different and mostly arbitrary rules. > > Science isn't, kid; /you/ are. Of course, I must be wrong since you don't understand what I am saying. If I am wrong then why would two of the founders of these ideas be awarded last year perhaps the most prestigious scientific honor that exists? Past winners of the Japan prize include Robert Gallo for the co-discovery of the HIV virus, Timothy Berners-Lee, the inventor of the World Wide Web, MIT's Marvin Minsky for his seminal artificial intelligence work, and Johns Hopkins' Donald Henderson (with others) for the eradication of smallpox. The two concepts--chaos and fractal, --have been established as universal concepts underlying such phenomena, irrespective of specific fields. Their applicability has been extended even to modern technology, the arts, economics and the social sciences. Dr. Mandelbrot and Dr. Yorke found, respectively, that fractals and chaos are the universal structures existing in complex systems, and they elucidated their fundamental properties. They have furnished us with new frameworks for understanding complex phenomena, and they have contributed both by establishing fundementals and by providing us with applications. Therefore, Dr. http://www-chaos.umd.edu/japanprize.html > And you keep complaining that you are, and that you don't get > it. > And that it's all Science's Fault that you don't. > And that you're gonna emit some chaos that explains your > complexity. > Kid, a communication channel cannot transmit a communication /more > ordered/ than itself. > /Or/ exceeding its own data rate. > > Collectively 'science' is a chaotic system ..a gas > which is as far from simplicity, elegance and > truth that can be found. > > Congratulations~ > > You can continue, if you find it useful, to smash everything > into the ground, and each time harder than the > last. Perhaps when you've ground It All > into dust then the sculpture will at last > be revealed. > > You're insane to keep trying, so deep you've cut > yourself into the Strip-Mine that even trying to > climb out is incomprehensible to you. It's > like watching someone dig their own grave > in search of Heaven. > > The Farthest thunder that I heard > Was nearer than the sky, > And rumbles still, though torrid noons > Have lain their missiles by. > The lightning that preceded it > Struck no one but myself, > But I would not exchange the bolt > For all the rest of life. > Indebtedness to oxygen > The chemist may repay, > But not the obligation > To electricity. > It founds the homes and decks the days, > And every clamor bright > Is but the gleam concomitant > Of that waylaying light. > The thought is quiet as a flake, > A crash without a sound; > How life's reverberation > Its explanation found! > > s > > -- > ------(m+ > ~/:o)_| > To change the world > be that change. > http://scrawlmark.org > > > -- > ------(m+ > ~/:o)_| > To change the world > be that change. > http://scrawlmark.org ==== I am interested in axiomizing Number Theory. I'm not talking about some bogus list of properties of addition and multiplication, but rather a set of formal axioms and rules of inference that allows us (a program) to derive theorems from Number Theory. It seems to be a natural branch of mathematics to axiomize, since it figures so heavily in proofs concerning Logic (e.g. Godel's Incompleteness Theorems) and Logic is so easily axiomized. I have axiomized the Theory of Computation (a.k.a. Computability) and Program Synthesis of Number Theoretic functions (prime listing or checking, factoring, etc.) in http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1 and http://www.arxiv.org/html/cs.lo/0003071 . The first step is to gather together a few dozen of the very simplest theorems from Number Theory. Then we look for primitives etc. The effort can be done here, or in private collaboration with me interested, with periodic progress reports posted here, culminating with a published paper. Charlie Volkstorf Cambridge, MA axiomize at aol dot com ==== It is axiomatized! It follows from the field axioms. Lurch > > I am interested in axiomizing Number Theory. I'm not talking about > some bogus list of properties of addition and multiplication, but > rather a set of formal axioms and rules of inference that allows us (a > program) to derive theorems from Number Theory. > > It seems to be a natural branch of mathematics to axiomize, since it > figures so heavily in proofs concerning Logic (e.g. Godel's > Incompleteness Theorems) and Logic is so easily axiomized. I have > axiomized the Theory of Computation (a.k.a. Computability) and Program > Synthesis of Number Theoretic functions (prime listing or checking, > factoring, etc.) in http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1 > and http://www.arxiv.org/html/cs.lo/0003071 . > > The first step is to gather together a few dozen of the very simplest > theorems from Number Theory. Then we look for primitives etc. > > The effort can be done here, or in private collaboration with me > interested, with periodic progress reports posted here, culminating > with a published paper. > > Charlie Volkstorf > Cambridge, MA > axiomize at aol dot com ==== <3df1e59f.0307250955.79ee3a83@posting.google.com>... > >I am interested in axiomizing Number Theory. I'm not talking about >some bogus list of properties of addition and multiplication, but >rather a set of formal axioms and rules of inference that allows us (a >program) to derive theorems from Number Theory. Huh?? Peano arithmetic is the most well-understood axiom system on Earth. It is to logic what Maxwell's equations are to physics, the periodic table is to chemistry and E coli is to genetics. The axioms are: PA1: s(n) =/= 0 PA2: s(n)=s(m) -> n=m PA3: n + 0 = n PA4: n + s(m) = s(n+m) PA5: n x 0 = 0 PA6: n x s(m) = n x m + n IND: [Phi(0) & forall n(Phi(n) -> Phi(s(n)))] -> forall n Phi(n) The rules are modus ponens and generalization (or any sound and complete deductive system for FOL). In what sense are these axioms, rules and the induction scheme bogus? They're clearly true of the intended structure (N, 0, s, +, x). As exercises, why don't you see if you can prove: For all n, m: n+m = m+n For any prime p, there is a prime p' > p, from these axioms? >The first step is to gather together a few dozen of the very simplest >theorems from Number Theory. Then we look for primitives etc. Don't be silly. This is done (at least sketched) in any intermediate mathematical logic course. Can you give an example of a natural arithmetic statement Phi which itself is correct, but is not provable in PA? Goedel discovered in 1930 that there are true but quasi-meta-linguistic Pi_1 statements not provable in PA. It took approximately a century from the formulation of the Peano-Dedekind axioms for anyone to come up with such a natural arithmetic statement---which Paris and Harrington did in 1976, with an example from Ramsey Theory. --- Jeff ==== It seems like I have heard of this before. And the idea is obvious. But the next-highest-order sequence above the standard Bell numbers is not in the on-line Encyclopedia of Integer Sequences, IF I have calculated the first few terms correctly (by hand). It is known that, if B(k) is the k_th (standard) Bell number, sum{k=0 to oo} B(k) x^k /k! = exp(exp(x)-1). But we can generalize this sequence. For a fixed m >= 2, {B(m,k)}, where sum{k=0 to oo} B(m,k) x^k /k! = exp(exp(...exp(exp(x)-1)..)), with m 'exp's, (therefore, B(2,k) is a standard Bell #) then: B(1,k) = 1 for all k >= 0 (violating the generating-function, due to B(1,0) = 1, and not 0); B(m,0) = 1; and: B(m,j+1) = sum{k=0 to j} binomial(j,k) B(m-1,k) B(m,j-k) (If I have not erred) By the way, {B(3,k)} to start: 1, 1, 2, 6, 23, 106,... Am I even right about any of this? (If not, this would explain why {B(3,k)} is not in the EIS.) Leroy Quet ==== This is wrong! But.... (see below) >.... > > But we can generalize this sequence. > For a fixed m >= 2, {B(m,k)}, where > > sum{k=0 to oo} B(m,k) x^k /k! = > > exp(exp(...exp(exp(x)-1)..)), > > with m 'exp's, > > (therefore, B(2,k) is a standard Bell #) > > then: > > B(1,k) = 1 for all k >= 0 (violating the generating-function, due to > B(1,0) = 1, and not 0); > > B(m,0) = 1; > > and: > > B(m,j+1) = > > > sum{k=0 to j} binomial(j,k) B(m-1,k) B(m,j-k) > > > > (If I have not erred) > > By the way, {B(3,k)} to start: > > 1, 1, 2, 6, 23, 106,... > > Am I even right about any of this? > (If not, this would explain why {B(3,k)} is not in the EIS.) I had several people tell me that I have made a mistake regarding the Bell number generalization. First, they pointed out that the generating function is REALLY exp(exp(...exp(exp(x)-1)-1..)-1) And then I figured out that the recursion-sum should have been: sum{k=0 to j} binomial(j,k) B(m-1,k+1) B(m,j-k) {with a 'k+1' replacing a 'k' as an index} (unless I am again incorrect) But... The original erroneous recursion-sum gives the sequences where {B(r,k)} has an exponential-generating-function of b_r(x) = exp(integral{0 to x} b_{r-1}(y) dy) (I think), where b_1(x) = exp(x). (b_2(x) is still e^(e^x-1), in any case.) Now THIS original/erroneous sequence, at least {B(3,k)}, is not yet in the EIS. :) (The sequence I give the 1st few terms of in the original post is B(3,k) as calculated by: sum{k=0 to oo} B(3,k) x^k/k! = exp(integral{0 to x} e^(e^y -1) dy).) Leroy Quet ==== What is the number of length n binary vectors modulo cyclic permutations ? (i.e. 110 and 011 are equivalent) ==== I was looking at this problem some time ago Given u,v,x,y being integers all greater than or equal to 2, when is C(u,v) = C(x,y) ? Here C(u,v) denotes the binomial coefficient corresponding to choosing a subset of size v from the set of u elements. Did any of you seen this problem before, and if so, is there a solution somewhere ? ==== >Given u,v,x,y being integers all greater than or equal to 2, when is >C(u,v) = C(x,y) ? Here C(u,v) denotes the binomial coefficient >corresponding to choosing a subset of size v from the set of u >elements. You mean like 120 = C(10,3) = C(16,2) 210 = C(10,4) = C(21,2) 1540 = C(22,3) = C(56,2) 3003 = C(14,6) = C(15,5) = C(78,2) (I got those just by looking at all C(u,v) with u < 100.) >Did any of you seen this problem before, and if so, is there a >solution somewhere ? Not sure. I know I've seen people looking to see when the C's can be perfect powers. Obviously if you impose upper bounds on u and x there are only finitely many cases to check. If you impose limits on v and y there will also be only a finite number of solutions but that's not obvious and as far as I know it's not effective, either (that is, you usually can't tell in advance how many solutions there can be, nor how large they can be (so that you can quit looking at some point) ). dave ==== > >I was looking at this problem some time ago > >Given u,v,x,y being integers all greater than or equal to 2, when is >C(u,v) = C(x,y) ? Here C(u,v) denotes the binomial coefficient >corresponding to choosing a subset of size v from the set of u >elements. Some obvious cases: C(u,v) = C(u,u-v) C(u,u) = C(v,v) = 1 C(u,v) = C(C(u,v),C(u,v)-1) Let's say a nontrivial case is where 2 <= v <= u/2 and 2 <= y <= x/2. The only nontrivial cases for u <= x <= 200 are C(10, 3) = C(16, 2) = 120 C(10, 4) = C(21, 2) = 210 C(22, 3) = C(56, 2) = 1540 C(14, 6) = C(15, 5) = C(78, 2) = 3003 C(36, 3) = C(120, 2) = 7140 C(19, 5) = C(153, 2) = 11628 C(103, 40) = C(104, 39) = 61218182743304701891431482520 Let's see: if C(u,v) = C(u+1,v-1) it means (u+2-v)(u+1-v)=v(u+1) or u^2 - 3uv + v^2 + 3u - 4v + 2 = 0 i.e. 5 (2u - 3v + 3)^2 - (5 v - 1)^2 = 4 The Pell equation 5 X^2 - Y^2 = 4 has positive integer solutions (X_n, Y_n) where [ 9 4 ] [ X_n ] [ X_{n+3} ] [ 20 9 ] [ Y_n ] = [ Y_{n+3} ] and for n=0,1,2 we have [ 1 ] [ 2 ] [ 5 ] [ 1 ], [ 4 ], [ 11 ] We may also need solutions with X<0. Now 2u-3v+3=X and 5v-1=Y implies u = -6/5 + X/2 + 3/10 Y, v = (Y+1)/5 We need Y=-1 mod 5 and X = Y mod 2 to get an integer solution for u and v. Now Y_{n+3} = - Y_n mod 5, and X_{n+3}-Y_{n+3} = X_n - Y_n mod 2. The solutions [X_n, Y_n] of the Pell equation will give us positive integer solutions for u and v if n is odd. So do [-X_n, Y_n] for n>1, but it turns out that these have u>I think I see the problem through all this haze. The change in the >>vector v (may we speak of the rate of change of the v vector, i.e., >>acceleration?) is due to an acceleration component, not a velocity >>component. There is indeed a radial acceleration (dv/dt) directed toward >>the center of rotation for the circular motion you describe. It's called >>centripetal acceleration. > >Do tell. So, what is it we do exactly with this centripetal >acceleration? We combine it vectorially directly with tangential >velocity? My, my. I've heard tell of vector addition. I may even have >used it once or twice. But this is the first I've ever heard of the >vector addition of a force with a velocity. Very close. You garbled the actual description. The centripetal acceleration gives rise to a centripetally directed change vector. This change vector is a velocity. It is added vectorially to the original velocity to give a new velocity. Did you not realize that vector addition of velocities has been part of this conversation from the beginning? Don't you realize that when we calculate (v2-v1)/(t2-t1) we are looking at two different vectors v2 and v1 and doing the vector difference? Let v2 - v1 = D. We're telling you that a*(t2-t1) = D, a vector with units of velocity. And that v1 + D = v2. This is VECTOR addition. >Are you sure you have academic credentials? You certainly belong in an >institution. For believing that acceleration is a vector? For believing that a vector change in a velocity gives rise to a new velocity by vector addition? - Randy <3f156fe7.34641502@netnews.att.net> <585ab5d8.0307161343.c0ba484@posting.google.com> <3f15fcdc.37862304@netnews.att.net> <585ab5d8.0307171038.3435db69@posting.google.com> <3f1751e3.41708891@netnews.att.net> <585ab5d8.0307180501.fafaafc@posting.google.com> <3f18a02a.547318@netnews.att.net> <3f19abe4.5169345@netnews.att.net> <2a0cceff.0307220427.13bc8570@posting.google.com> <3f1d5002.16373751@netnews.att.net> <3f1db75b.18968483@netnews.att.net> <8vdDM2c71kH$EwZ7@baesystems.com> <3f1f13a5.26605449@netnews.att.net> ==== In message <3f1f13a5.26605449@netnews.att.net>, Lester Zick > >>In message <3f1db75b.18968483@netnews.att.net>, Lester Zick > >Ah, well, you know that isn't quite true. >> >>For what circular orbit can you not define an origin such that d|r|/dt >>is always zero? > >For any circular orbit. >> >>Well, that's a categorical statement for once. Also false. >> >>How do you define circular? > >Ah well, methinks this is a trick question. No. I want to know what screwy definition of circular you are using whereby |r| = constant, r.n = 0 does not define a circle in your world. > >If you cannot show what causes the circular rotation of v there can be >no circle. > >Please, please tell me this is not just a red herring. >> You are confusing the defining property of a circle with the means of constructing it. -- Richard Herring ==== >If you cannot show what causes the circular rotation of v there can be >no circle. HUH????? You don't need rotation to get a circle. A circle is just a set of points. You specify a circle by specifying the conditions that must be met by that set of points, is all. It just so happens the motions of certain objects lie on a circle (or can be modelled by using a circle, wh/ comes to the same thing.) Actually, no real world object actually moves in a circle. Some come close - but close is all you get. -- Best Wishes, Wolf Kirchmeir, Blind River ON Not that brains are everything -- you'll also need a skull to put them in. (Nancy Franklin, 1997) ==== >In message <3f1f13a5.26605449@netnews.att.net>, Lester Zick >> >In message <3f1db75b.18968483@netnews.att.net>, Lester Zick >> >>Ah, well, you know that isn't quite true. > >For what circular orbit can you not define an origin such that d|r|/dt >is always zero? >> >>For any circular orbit. > >Well, that's a categorical statement for once. Also false. > >How do you define circular? >> >>Ah well, methinks this is a trick question. > >No. I want to know what screwy definition of circular you are using >whereby |r| = constant, r.n = 0 does not define a circle in your world. >> >>If you cannot show what causes the circular rotation of v there can be >>no circle. >> >>Please, please tell me this is not just a red herring. > >You are confusing the defining property of a circle with the means of >constructing it. > And you're confusing the existence of circles with the inability to say what causes v to rotate. ==== > >>If you cannot show what causes the circular rotation of v there can be >>no circle. > >HUH????? > >You don't need rotation to get a circle. A circle is just a set of points. >You specify a circle by specifying the conditions that must be met by that >set of points, is all. It just so happens the motions of certain objects lie >on a circle (or can be modelled by using a circle, wh/ comes to the same >thing.) Actually, no real world object actually moves in a circle. Some come >close - but close is all you get. > > Actually you do need rotation to get a circle. V has to rotate whether you construct a circle with a compass or in your mind. What the coordinates specify is just the path of rotation. But the rotation is there. I'm of mixed opinion whether there are any real world objects that move in circles. Or for that matter in straight lines. But I would think that the rotation of protons and electrons at rest would come the closest. ==== > ><...> > >>If you are going to use expressions like radial or tangential >>dr/dt, you are implicitly adopting the vector meaning. In this case, >>for circular orbits, dr/dt is purely tangential ... OTOH, if you say >>the value of r never changes, you are clearly using the first, >>scalar, sense of r. Cunningly, you move between the two senses of r >>in the same paragraph! Now, why don't you behave? >> >> Cunningly? You're suggesting I have the guile to confuse the issue but >> not to analyze it correctly? Do you understand what the definition of >> is is? Perhaps I should run for president. I could use the job. > >You definitely have some political instincts, as well as a thick skin. > And probably as good an understanding of calculus as any president >after Eisenhower. Why not. And you have a thick head. So what? Let's trade some more insults. I'm sure that'll elevate the conversation considerably. > >>I must take a seminar offered by the Learning Annex: the one on >>self-empowerment (what else?) where one of the deadly errors you are >>taught to avoid is arguing with the irrational. >> >>That wouldn't be _you_, would it, Mr. Zick? Ah yes ... I'm an old >>hypocrite ... as Groucho Marx said, we need the eggs. Sigh. >> >>answer, do ... >> >> Yeah, I have a hard time telling whether you're part of the thread or >> not. You come and go at various times with next to no special insight >> on the basic analysis of circular rotation. I could and have found the >> same explanations in any textbook or in the posts of any of a hundred >> other posters. Of course you need the eggs. In your position who >> wouldn't? > >Ah ... one jib deserves another. > >> If you want to participate in the conversation, I strongly suggest you >> do so instead of acting like some deus ex dropped on a confused and >> bewildered group of analytical amateurs and miscreants. > >In my position? At least I didn't drop a sandbag on you. I'm not sure which sandbag you're referring to. Not that it matters. > >OK ... some meat at least: > >> I only have one question to answer yours. If there is no radial dr/dt, >> what causes tangential v to rotate? > >I'm not entirely sure, but I begin to think you've arrived by some >heroic act of will at your own concept of r, which is somewhere >between a scalar and a vector. Not unlike the time I left the normal >bicycle control envelope to prevent a fall -- and deranged both wheels >in the process. I respect your will -- but in the process you've >arrived at some weird monster of the imagination. > >The short answer to your question is centripedal acceleration -- >that's what causes v -- which happens to be tangential for a circular >orbit -- to rotate. > >Now ... your expression dr/dt suggests that you are thinking of >dr/dt as a vector, which you have decomposed at a point into radial >and tangential components. Your tangential v suggests the same >thing, except the one wonders why you use v and dr/dt in almost >the same breadth. > >The answer seems to be that you don't consider them the same kind of >object. This is an extremely non-standard view which creates many >communication faults. But, as we all know, no word-hoard in existence >will force the painful process of realignment to occur without your >will. > >This is actually an old fear of mine: the only way I want to learn >anything these days is under tutorship ... in the full sense. A tutor >is there to steer you on the right course, but also to be constantly >challenged. But the fear is, without a tutor, one will generate a >path which acquires the force of habit into the thicket ... which is a >complete blind end. > >Of course that is the common hazard of the researcher ... but I guess >at least we should try to avoid this failure over well-trodden ground. Well, all you're doing here is trying to psychologize me in naive terms, trying to second guess motives and methods instead of concentrating on answers. Joe Legris has already suggested that the only kind of analysis I'm suited for is psycho. So I think the two of you might get together. > >> Do you tie an ethereal string to >> it? Does centripetal force somehow interact with v directly without >> the intercession of radial dr/dt? > >It's hard to know what you mean by dr/dt here, except that it is, >again, non-standard. I could hazard a guess, based on your mention of >the things falling in circular orbit trope -- always falling, never >losing altitude -- that your thinking could run something like this: > >Consider the _scalar_ r. You now form what might be called a total >derivative of r: dr/dt = (dr/dt)_1 + (dr/dt)_2 . You know that >total derivative is zero (r doesn't change) ... or do you? You want >to insist it does, or at least dr/dt does not equal 0. Of course radial dr/dt does not equal zero. This is one of the hazards of only occasionally dropping by. > >Anyway, it seems as if your dr/dt is one of two components, the >instantaneous always falling component, which is just nulled out in >the total derivative by the effect of tangential v: the moving a >same altitude. Well, at least you understand the concept. Now, if I could just get you to stop rolling around on the floor laughing. > >The more compact way of saying all this is closer to centripetal >force somehow interact with v directly. In particular dv = a dt = >F/m dt -- that is, the little change in v is a given by the >acceleration acting over a small interval, itself parallel to and >given by the force. The fact that in a circular orbit force F is >always perpedicular to v, which is purely tangential, accounts for the >rotation of v. How? Are we back to the vector sum of F and V without the intercession of radial dr/dt or what? > >> It's true of course that I have >> heard of vector addition of velocities before. I've just never heard >> of the vector addition of a force and velocity before. > >Nor has anybody else: v -> v + dv; dv = F/m dt Yes, but what produces dv? It appears here from what I can see that you've got centripetal force producing a tangential dv. Peculiar indeed. Perhaps I need to refine the question to ask how centripetal force produces tangential dv and then having produced tangential dv causes v to rotate in a radial direction? On the other hand if you're suggesting that dv acts radially then I still have to ask how v and dv combine vectorially without the intercession of some radial dr/dt? > >Force is divided by a mass to give an acceleration, multiplied by a >small time to given a small increment in velocity: velocity is added >to velocity, demivelocity to demivelocity, acceleration without end. >Amen. So is this incremental velocity oriented tangentially or radially? Whichever we're back to the same questions. Either we have radial dr/dt resulting from centripetal acceleration or we're left trying to explain the rotation of v. > >> And if dL/dt = 0 because dr/dt lies in the direction of v, why does p >> rotate? > >You know ... I'm going to give up on trying to interpret that oracle. >There are limits, and that passes mine ... though, wait, e'en so ... >as a protracted student of Zick, I begin to make headway: > >dL/dt = 0 > >Ok. Whatever you understand by that, which I have no way of knowning >because of singularity of your approach and notation, I cannot object. > >dr/dt lies in the direction of v (apparent contra-factual) > >This is in Zickian notation, and the translation table is not >complete. > >Taking r as a vector, dr/dt == v, as you have been told, not lies in >the direction of. > >Taking r as a scalar (perhaps |r|, the magnitude of vector r), your >assertion is technically meaningless -- scalars lack a direction. > >But, bending a little, trying to Grok you, perhaps you mean something >like the above discussion of always falling but never losing >altitude, which I will not reprise here. > >because, in > >if dL/dt = 0 because dr/dt lies in the direction of v > >??? > >why does p rotate, in > >if dL/dt = 0 ... why does p rotate? > >ITYM by p the linear momentum of the orbiting body. > >If we may use semi-modern notation a second, note that L == m (r x p), >where r is a vector wrt the origin, p is the vector momentum, and >x is the vector cross product. It is is possible to hold L constant >while r and p both change -- if that what you mean. A rotating p does >not imply dL/dt /= 0. > >> The quality of your prose is high. Not quite up to my >> standards but certainly good enough for government work. > >Proseyard wars! Or should I say, the war of the proses? Well that's not bad. Except it's not much of a war. You need considerable work on forensic conic sections, you know, hyperbolic and elliptical expressions. I wonder if there is a conic section called irony? > >> Yet you don't >> even apparently understand what you're saying much less what I'm >> saying. > >Oh boy, do you leave yourself wide open. It's true I'm having trouble >understanding what you are staying -- though I'm trying to interpret >the oracle. I would be tempted and maybe not unjustified to turn your >comment around and suppose that you don't understand what you are >staying; at any rate, your refusal to adopt conventional concepts and >usage all but assures nobody _else_ will understand what you are >saying. Yes, well you see it's called a bait and switch gambit - a fiendishly clever ploy designed to entice an opponent into naively hysterical recrimination and denunciation. It usually works quite well too until the opponent begins to understand that he's really trying to explain things more to himself than to me. Then he begins to settle down and becomes more methodical. Or he just keeps on ranting and raving until he gets exhausted. Either way he winds up doing most of the work. > >As for not understanding what I am saying ... you are making reducing >spiral orbits into the crank attractor. Your other straight men -- >you've accumlated quite a coterie, Gracie -- seem to have no problem >understanding what I am saying in my elementary prosody. Only you do. > And you suppose therefore that nobody else but you understands >anything? Careful ... the attraction is strong, the sands are >shifting, and the crank-door spider is waiting ... I think I already >see a hairy tarsus reaching out to grab your ankle! Flee, Lester, >flee while you still can !!! And would that hairy tarsus grab my ankle with radial dr/dt such as to alter my v? Or would it just alter my v without the intercession of any radial dr/dt like classical angular mechanics states categorically is all that is needed to produce circular rotation in me? > >I can't watch. Then just listen and learn. Apparently I don't need a tutorial any more than you need to give a tutorial. More eggs anyone? ==== > > Well, all you're doing here is trying to psychologize me in naive > terms, trying to second guess motives and methods instead of > concentrating on answers. Joe Legris has already suggested that the > only kind of analysis I'm suited for is psycho. So I think the two of > you might get together. > Hey, wait a minute. You are taking my jabs too seriously. I just figured that as long a you persist in cross posting your goofy physics threads to sci.cognitive you remain fair game for on-topic ridicule. For what it's worth, I think your patient tutors are just as silly for taking all this nonsense even semi-seriously. May I propose a last word (ha! good luck!)? Lester's position may be untenable, but it is also immutable and, alas, unresolvable. Let each correspondent retire to his corner of the universe and be done with this. -- Joe Legris ==== >> >> Well, all you're doing here is trying to psychologize me in naive >> terms, trying to second guess motives and methods instead of >> concentrating on answers. Joe Legris has already suggested that the >> only kind of analysis I'm suited for is psycho. So I think the two of >> you might get together. >> > >Hey, wait a minute. You are taking my jabs too seriously. I just figured >that as long a you persist in cross posting your goofy physics threads >to sci.cognitive you remain fair game for on-topic ridicule. Of course. I don't mind. > >For what it's worth, I think your patient tutors are just as silly for >taking all this nonsense even semi-seriously. > >May I propose a last word (ha! good luck!)? Lester's position may be >untenable, but it is also immutable and, alas, unresolvable. But ultimately differentiable and infinitesimally integratable. > >Let each correspondent retire to his corner of the universe and be done >with this. > No harm, no foul. ==== > > >>If you cannot show what causes the circular rotation of v there can be >>no circle. > >HUH????? > >You don't need rotation to get a circle. A circle is just a set of points. >You specify a circle by specifying the conditions that must be met by that >set of points, is all. It just so happens the motions of certain objects lie >on a circle (or can be modelled by using a circle, wh/ comes to the same >thing.) Actually, no real world object actually moves in a circle. Some come >close - but close is all you get. > > > Actually you do need rotation to get a circle. V has to rotate whether > you construct a circle with a compass or in your mind. What the > coordinates specify is just the path of rotation. But the rotation is > there. Your perverse claim is like saying that since we can hopskotch in a certain pattern that when this pattern of number exists, there must be a hopping -- whether with your feet or in your mind. Besides describing a circle as the path taken at constant radius around an origin, I could describe a circle as formed of segments or arcs interspersed with hops across the center landing on disjoint segments or arcs. Does that mean the hops are always there too? Or I could describe a circle as the set of all point reachable by taking a hop of fixed length |r| from a given center in a random direction. Now where is your v, and where is your rotation? ==== > >> In message <3f1d5002.16373751@netnews.att.net>, Lester Zick > >answer, do ... >> >>Yeah, I have a hard time telling whether you're part of the thread or >>not. You come and go at various times with next to no special insight >>on the basic analysis of circular rotation. > >I stop by from time to time to add my 2 cents. > > You can't afford it. Maybe -- but since you are intellectually destitute, I feel it's the least I can do. ==== > > >>I think I see the problem through all this haze. The change in the >>vector v (may we speak of the rate of change of the v vector, i.e., >>acceleration?) is due to an acceleration component, not a velocity >>component. There is indeed a radial acceleration (dv/dt) directed toward >>the center of rotation for the circular motion you describe. It's called >>centripetal acceleration. > >Do tell. So, what is it we do exactly with this centripetal >acceleration? We combine it vectorially directly with tangential >velocity? My, my. I've heard tell of vector addition. I may even have >used it once or twice. But this is the first I've ever heard of the >vector addition of a force with a velocity. > > Very close. You garbled the actual description. The centripetal > acceleration gives rise to a centripetally directed change vector. > This change vector is a velocity. It is added vectorially to the > original velocity to give a new velocity. > > Did you not realize that vector addition of velocities has been part > of this conversation from the beginning? Don't you realize that when > we calculate (v2-v1)/(t2-t1) we are looking at two different vectors > v2 and v1 and doing the vector difference? You know, I saw in the gloming the same shape that John Prussing saw: a weird, mishappen creature shambling in the mist; part scalar, part vector. The villagers shudder, cross themselves, and hurry home. The kindly parish priest approaches it and raises a hand in friendship -- he's probably ripped to shreds for his trouble: same old story. > Let v2 - v1 = D. We're telling you that a*(t2-t1) = D, a vector with > units of velocity. And that v1 + D = v2. This is VECTOR addition. > >Are you sure you have academic credentials? You certainly belong in an >institution. > > For believing that acceleration is a vector? For believing that a > vector change in a velocity gives rise to a new velocity by vector > addition? Lester's frustration is growing, his insult level apace. ==== > > Well, all you're doing here is trying to psychologize me in naive > terms, trying to second guess motives and methods instead of > concentrating on answers. Joe Legris has already suggested that the > only kind of analysis I'm suited for is psycho. So I think the two of > you might get together. > > > Hey, wait a minute. You are taking my jabs too seriously. I just figured > that as long a you persist in cross posting your goofy physics threads > to sci.cognitive you remain fair game for on-topic ridicule. > > For what it's worth, I think your patient tutors are just as silly for > taking all this nonsense even semi-seriously. > > May I propose a last word (ha! good luck!)? Lester's position may be > untenable, but it is also immutable and, alas, unresolvable. > > Let each correspondent retire to his corner of the universe and be done > with this. A just-as-silly-tutor responds: There is something in what you say, and I have already suggested that troller and trollee are locked in a kind of symbiotic relationship: like drugie and narc. No ... not even that ... the drugie doesn't need the narc, just the other way around. But this is true symbiosis. If Lester is a troll, he is an adept one: he skirts meaning just closely enough that one hopes to identify his delusions if only taxonomically, and not therapeutically. Viva la trollerance. ==== >> >> Well, all you're doing here is trying to psychologize me in naive >> terms, trying to second guess motives and methods instead of >> concentrating on answers. Joe Legris has already suggested that the >> only kind of analysis I'm suited for is psycho. So I think the two of >> you might get together. >> >> >> Hey, wait a minute. You are taking my jabs too seriously. I just figured >> that as long a you persist in cross posting your goofy physics threads >> to sci.cognitive you remain fair game for on-topic ridicule. >> >> For what it's worth, I think your patient tutors are just as silly for >> taking all this nonsense even semi-seriously. >> >> May I propose a last word (ha! good luck!)? Lester's position may be >> untenable, but it is also immutable and, alas, unresolvable. >> >> Let each correspondent retire to his corner of the universe and be done >> with this. > >A just-as-silly-tutor responds: > >There is something in what you say, and I have already suggested that >troller and trollee are locked in a kind of symbiotic relationship: >like drugie and narc. No ... not even that ... the drugie doesn't >need the narc, just the other way around. But this is true symbiosis. > >If Lester is a troll, he is an adept one: he skirts meaning just >closely enough that one hopes to identify his delusions if only >taxonomically, and not therapeutically. Viva la trollerance. Well at least you've picked up the level of your trollery. Droll on. ==== >> >> >If you cannot show what causes the circular rotation of v there can be >no circle. >> >>HUH????? >> >>You don't need rotation to get a circle. A circle is just a set of points. >>You specify a circle by specifying the conditions that must be met by that >>set of points, is all. It just so happens the motions of certain objects lie >>on a circle (or can be modelled by using a circle, wh/ comes to the same >>thing.) Actually, no real world object actually moves in a circle. Some come >>close - but close is all you get. >> >> >> Actually you do need rotation to get a circle. V has to rotate whether >> you construct a circle with a compass or in your mind. What the >> coordinates specify is just the path of rotation. But the rotation is >> there. > >Your perverse claim is like saying that since we can hopskotch in a >certain pattern that when this pattern of number exists, there must be >a hopping -- whether with your feet or in your mind. Besides >describing a circle as the path taken at constant radius around an >origin, I could describe a circle as formed of segments or arcs >interspersed with hops across the center landing on disjoint segments >or arcs. Does that mean the hops are always there too? > >Or I could describe a circle as the set of all point reachable by >taking a hop of fixed length |r| from a given center in a random >direction. Now where is your v, and where is your rotation? So, which came first, the chicken or the egg? I think you'll find it difficult to describe set of points without describing the circle to begin with and the rotation of v. Of course one never knows. I daresay that a hop of fixed length from a given center might also include those normal to a plane? Hard to tell what you mean by a circle from this kind of definition. I don't think for present purposes that playing hopscotch will help very much unless you choose to do it with real scotch. ==== > > >>If you cannot show what causes the circular rotation of v there can be >>no circle. > >HUH????? > >You don't need rotation to get a circle. A circle is just a set of points. >You specify a circle by specifying the conditions that must be met by that >set of points, is all. It just so happens the motions of certain objects lie >on a circle (or can be modelled by using a circle, wh/ comes to the same >thing.) Actually, no real world object actually moves in a circle. Some come >close - but close is all you get. > > > Actually you do need rotation to get a circle. Actually, you don't. > V has to rotate whether > you construct a circle with a compass or in your mind. This discussion is about circles in general, not those traced by a moving object. For instance, the shape of ripples from a drop hitting a pond. Nothing is rotating to make those circles. The cross section of a laser beam. Nothing is rotating to make that circle. The shape of a soap bubble. Nothing is rotating to make that shape. The halo effect you see sometimes around the moon. Nothing is rotating to make that circle. - Randy <3f156fe7.34641502@netnews.att.net> <585ab5d8.0307161343.c0ba484@posting.google.com> <3f15fcdc.37862304@netnews.att.net> <585ab5d8.0307171038.3435db69@posting.google.com> <3f1751e3.41708891@netnews.att.net> <585ab5d8.0307180501.fafaafc@posting.google.com> <3f18a02a.547318@netnews.att.net> <3f19abe4.5169345@netnews.att.net> <2a0cceff.0307220427.13bc8570@posting.google.com> <3f1d5002.16373751@netnews.att.net> <3f1db75b.18968483@netnews.att.net> <8vdDM2c71kH$EwZ7@baesystems.com> <3f1f13a5.26605449@netnews.att.net> <3f1ff353.29105836@netnews.att.net> ==== In message <3f1ff353.29105836@netnews.att.net>, Lester Zick > >>In message <3f1f13a5.26605449@netnews.att.net>, Lester Zick > >>In message <3f1db75b.18968483@netnews.att.net>, Lester Zick > >In the first case dr/dt is the rate of change of the radius, and is >... among a myriad of other possibilities ... zero for a circular >orbit. If we are talking about orbits. >> >Ah, well, you know that isn't quite true. >> >>For what circular orbit can you not define an origin such that d|r|/dt >>is always zero? > >For any circular orbit. >> >>Well, that's a categorical statement for once. Also false. >> >>How do you define circular? > >Ah well, methinks this is a trick question. >> >>No. I want to know what screwy definition of circular you are using >>whereby |r| = constant, r.n = 0 does not define a circle in your world. No answer? >If you cannot show what causes the circular rotation of v there can be >no circle. > >Please, please tell me this is not just a red herring. >> >>You are confusing the defining property of a circle with the means of >>constructing it. >> >And you're confusing the existence of circles with the inability to >say what causes v to rotate. > No, I'm asking you what kind of a circle in your world doesn't obey |r| = constant for a suitable choice of origin. Rotation can wait until we understand what you mean by circular orbit. -- Richard Herring ==== > >> >> >If you cannot show what causes the circular rotation of v there can be >no circle. >> >>HUH????? >> >>You don't need rotation to get a circle. A circle is just a set of points. >>You specify a circle by specifying the conditions that must be met by that >>set of points, is all. It just so happens the motions of certain objects lie >>on a circle (or can be modelled by using a circle, wh/ comes to the same >>thing.) Actually, no real world object actually moves in a circle. Some come >>close - but close is all you get. >> >> >> Actually you do need rotation to get a circle. V has to rotate whether >> you construct a circle with a compass or in your mind. What the >> coordinates specify is just the path of rotation. But the rotation is >> there. > >Your perverse claim is like saying that since we can hopskotch in a >certain pattern that when this pattern of number exists, there must be >a hopping -- whether with your feet or in your mind. Besides >describing a circle as the path taken at constant radius around an >origin, I could describe a circle as formed of segments or arcs >interspersed with hops across the center landing on disjoint segments >or arcs. Does that mean the hops are always there too? > >Or I could describe a circle as the set of all point reachable by >taking a hop of fixed length |r| from a given center in a random >direction. Now where is your v, and where is your rotation? > > So, which came first, the chicken or the egg? > > I think you'll find it difficult to describe set of points without > describing the circle to begin with and the rotation of v. Of course > one never knows. I daresay that a hop of fixed length from a given > center might also include those normal to a plane? Hard to tell what > you mean by a circle from this kind of definition. Which would generate a sphere. Of course, if you pretend that you didn't understand the tacit restriction in a plane, that defect if readily corrected. I just did. Now perhaps you will object that one can't hop in a plane since the hop itself takes one out of a plane!? I don't doubt it. Maybe I should object that if one is free to move in space it is hard to tell what you mean by a circle by your definition involving constantly rotating v -- what if it rotates out of the plane? Or maybe I should affect that since you didn't dot all the i's about your rotation of v that this covers the paths of wandering drunks also? Randy Poe gave a better set of examples of rotation free physically examples realize the idea of moving outward from a point -- in a plane -- a distance independent of direction, in all directions simultaneously. > I don't think for present purposes that playing hopscotch will help > very much unless you choose to do it with real scotch. Now there at least is a good idea. ==== > >> For what it's worth, I think your patient tutors are just as silly for >> taking all this nonsense even semi-seriously. >> >> May I propose a last word (ha! good luck!)? Lester's position may be >> untenable, but it is also immutable and, alas, unresolvable. >> >> Let each correspondent retire to his corner of the universe and be done >> with this. > >A just-as-silly-tutor responds: > >There is something in what you say, and I have already suggested that >troller and trollee are locked in a kind of symbiotic relationship: >like drugie and narc. No ... not even that ... the drugie doesn't >need the narc, just the other way around. But this is true symbiosis. > >If Lester is a troll, he is an adept one: he skirts meaning just >closely enough that one hopes to identify his delusions if only >taxonomically, and not therapeutically. Viva la trollerance. > > Well at least you've picked up the level of your trollery. Droll on. No ... I'm no troll. I realize I'm probably being trolled, but reply anyway, for whatever personal anodyne amusement. The definition of a troll is a person who makes delibrately provocative statements in order to enjoy the reaction. The only question is whether your idiosyncratic dionysian musings are delibrately provocative or only accidentally so -- reflecting your sincere convictions of the moment. I said if you are a troll you are a good one -- the best keep that edge of ambiguity in their provocation. Although even fairly obvious ones can enjoy long runs -- it's the symbioses, don't you know. Where is Spaceman today ... who cares? ==== > >> >> > > >>If you cannot show what causes the circular rotation of v there can be >>no circle. > >HUH????? > >You don't need rotation to get a circle. A circle is just a set of points. >You specify a circle by specifying the conditions that must be met by that >set of points, is all. It just so happens the motions of certain objects lie >on a circle (or can be modelled by using a circle, wh/ comes to the same >thing.) Actually, no real world object actually moves in a circle. Some come >close - but close is all you get. > > >> >>Actually you do need rotation to get a circle. > > > Actually, you don't. > > >>V has to rotate whether >>you construct a circle with a compass or in your mind. > > > This discussion is about circles in general, not those traced > by a moving object. For instance, the shape of ripples from > a drop hitting a pond. Nothing is rotating to make those > circles. The cross section of a laser beam. Nothing is rotating > to make that circle. The shape of a soap bubble. Nothing is > rotating to make that shape. The halo effect you see sometimes > around the moon. Nothing is rotating to make that circle. > > - Randy Nice examples. And a non-parametric definition of a circle does not imply motion because it is independent of time. Nevertheless, this unfortunate thread continues to revolve around the singularity at (L,Z). I think we're getting sucked in. -- Joe Legris ==== >> > > For what it's worth, I think your patient tutors are just as silly for > taking all this nonsense even semi-seriously. > > May I propose a last word (ha! good luck!)? Lester's position may be > untenable, but it is also immutable and, alas, unresolvable. > > Let each correspondent retire to his corner of the universe and be done > with this. >> >>A just-as-silly-tutor responds: >> >>There is something in what you say, and I have already suggested that >>troller and trollee are locked in a kind of symbiotic relationship: >>like drugie and narc. No ... not even that ... the drugie doesn't >>need the narc, just the other way around. But this is true symbiosis. >> >>If Lester is a troll, he is an adept one: he skirts meaning just >>closely enough that one hopes to identify his delusions if only >>taxonomically, and not therapeutically. Viva la trollerance. >> >> Well at least you've picked up the level of your trollery. Droll on. > >No ... I'm no troll. I realize I'm probably being trolled, but reply >anyway, for whatever personal anodyne amusement. The definition of a >troll is a person who makes delibrately provocative statements in >order to enjoy the reaction. The only question is whether your >idiosyncratic dionysian musings are delibrately provocative or only >accidentally so -- reflecting your sincere convictions of the moment. You're actually more of a droll than a troll. > >I said if you are a troll you are a good one -- the best keep that >edge of ambiguity in their provocation. Although even fairly obvious >ones can enjoy long runs -- it's the symbioses, don't you know. Where >is Spaceman today ... who cares? Ah, well, you know that trolls of the type you're describing only enjoy runs because of the reactionaries they troll for. I've never had any communication with them or vice versa because all they discuss are their hurt feelings. I think there is some form of compensation involved that hard heads feel compelled to assuage. So far as I can tell I'm only discussing ideas. I don't see anyone else explaining anything. So I thought I'd give it a try. ==== >In message <3f1ff353.29105836@netnews.att.net>, Lester Zick >> >In message <3f1f13a5.26605449@netnews.att.net>, Lester Zick >> >In message <3f1db75b.18968483@netnews.att.net>, Lester Zick >> >>In the first case dr/dt is the rate of change of the radius, and is >>... among a myriad of other possibilities ... zero for a circular >>orbit. If we are talking about orbits. > >>Ah, well, you know that isn't quite true. > >For what circular orbit can you not define an origin such that d|r|/dt >is always zero? >> >>For any circular orbit. > >Well, that's a categorical statement for once. Also false. > >How do you define circular? >> >>Ah well, methinks this is a trick question. > >No. I want to know what screwy definition of circular you are using >whereby |r| = constant, r.n = 0 does not define a circle in your world. > >No answer? No interest. What defines a circle in my world and everyones world is the rotation of v. > >>If you cannot show what causes the circular rotation of v there can be >>no circle. >> >>Please, please tell me this is not just a red herring. > >You are confusing the defining property of a circle with the means of >constructing it. > >>And you're confusing the existence of circles with the inability to >>say what causes v to rotate. >> >No, I'm asking you what kind of a circle in your world doesn't obey |r| >= constant for a suitable choice of origin. Every kind of circle obeys this dictum provided the rotation of v is properly described. You're confusing the definition of a circle with its properties and the properties of points on it. > >Rotation can wait until we understand what you mean by circular orbit. > Or what you mean by a circular orbit. So far all you've described are the properties of equidistant points. The properties of a circle in general depend on the definition of a circle. ==== > > >I think I see the problem through all this haze. The change in the >vector v (may we speak of the rate of change of the v vector, i.e., >acceleration?) is due to an acceleration component, not a velocity >component. There is indeed a radial acceleration (dv/dt) directed toward >the center of rotation for the circular motion you describe. It's called >centripetal acceleration. >> >>Do tell. So, what is it we do exactly with this centripetal >>acceleration? We combine it vectorially directly with tangential >>velocity? My, my. I've heard tell of vector addition. I may even have >>used it once or twice. But this is the first I've ever heard of the >>vector addition of a force with a velocity. > >Very close. You garbled the actual description. The centripetal >acceleration gives rise to a centripetally directed change vector. >This change vector is a velocity. It is added vectorially to the >original velocity to give a new velocity. So? It seems like you're agreeing with me. > ==== >> > > >>If you cannot show what causes the circular rotation of v there can be >>no circle. > >HUH????? > >You don't need rotation to get a circle. A circle is just a set of points. >You specify a circle by specifying the conditions that must be met by that >set of points, is all. It just so happens the motions of certain objects lie >on a circle (or can be modelled by using a circle, wh/ comes to the same >thing.) Actually, no real world object actually moves in a circle. Some come >close - but close is all you get. > > > Actually you do need rotation to get a circle. V has to rotate whether > you construct a circle with a compass or in your mind. What the > coordinates specify is just the path of rotation. But the rotation is > there. >> >>Your perverse claim is like saying that since we can hopskotch in a >>certain pattern that when this pattern of number exists, there must be >>a hopping -- whether with your feet or in your mind. Besides >>describing a circle as the path taken at constant radius around an >>origin, I could describe a circle as formed of segments or arcs >>interspersed with hops across the center landing on disjoint segments >>or arcs. Does that mean the hops are always there too? >> >>Or I could describe a circle as the set of all point reachable by >>taking a hop of fixed length |r| from a given center in a random >>direction. Now where is your v, and where is your rotation? >> >> So, which came first, the chicken or the egg? >> >> I think you'll find it difficult to describe set of points without >> describing the circle to begin with and the rotation of v. Of course >> one never knows. I daresay that a hop of fixed length from a given >> center might also include those normal to a plane? Hard to tell what >> you mean by a circle from this kind of definition. > >Which would generate a sphere. Of course, if you pretend that you >didn't understand the tacit restriction in a plane, that defect if >readily corrected. I just did. Now perhaps you will object that one >can't hop in a plane since the hop itself takes one out of a plane!? > I don't doubt it. > >Maybe I should object that if one is free to move in space it is hard >to tell what you mean by a circle by your definition involving >constantly rotating v -- what if it rotates out of the plane? Or >maybe I should affect that since you didn't dot all the i's about your >rotation of v that this covers the paths of wandering drunks also? Well, well. Ones nose does get so easily bent out of shape when one makes a rookie mistake. Please, let's not get into the definition of spheres, planes, straight lines, points, etc. It seems you can't even handle circles. Maybe Spaceman can help. I'm not quite sure how v can hop out of a plane as you seem wont to do. V is a constant vector and I imagine that centripetal a is a constant vector and I imagine they define a plane between them. But that's only my imagination. > >Randy Poe gave a better set of examples of rotation free physically >examples realize the idea of moving outward from a point -- in a plane >-- a distance independent of direction, in all directions >simultaneously. Sure. The only problem is that they aren't circles. > >> I don't think for present purposes that playing hopscotch will help >> very much unless you choose to do it with real scotch. > >Now there at least is a good idea. ==== >> >> >If you cannot show what causes the circular rotation of v there can be >no circle. >> >>HUH????? >> >>You don't need rotation to get a circle. A circle is just a set of points. >>You specify a circle by specifying the conditions that must be met by that >>set of points, is all. It just so happens the motions of certain objects lie >>on a circle (or can be modelled by using a circle, wh/ comes to the same >>thing.) Actually, no real world object actually moves in a circle. Some come >>close - but close is all you get. >> >> >> Actually you do need rotation to get a circle. > >Actually, you don't. > >> V has to rotate whether >> you construct a circle with a compass or in your mind. > >This discussion is about circles in general, not those traced >by a moving object. For instance, the shape of ripples from >a drop hitting a pond. Nothing is rotating to make those >circles. The cross section of a laser beam. Nothing is rotating >to make that circle. The shape of a soap bubble. Nothing is >rotating to make that shape. The halo effect you see sometimes >around the moon. Nothing is rotating to make that circle. > The only problem here is that nothing you're describing is a circle. A circle is a mathematical construct generated by the rotation of v. Under sufficient enlargement all the things you note resolve into non circular components. A circle can never be enlarged into constituent points. ==== >> > > >> >> >If you cannot show what causes the circular rotation of v there can be >no circle. >> >>HUH????? >> >>You don't need rotation to get a circle. A circle is just a set of points. >>You specify a circle by specifying the conditions that must be met by that >>set of points, is all. It just so happens the motions of certain objects lie >>on a circle (or can be modelled by using a circle, wh/ comes to the same >>thing.) Actually, no real world object actually moves in a circle. Some come >>close - but close is all you get. >> >> > >Actually you do need rotation to get a circle. >> >> >> Actually, you don't. >> >> >V has to rotate whether >you construct a circle with a compass or in your mind. >> >> >> This discussion is about circles in general, not those traced >> by a moving object. For instance, the shape of ripples from >> a drop hitting a pond. Nothing is rotating to make those >> circles. The cross section of a laser beam. Nothing is rotating >> to make that circle. The shape of a soap bubble. Nothing is >> rotating to make that shape. The halo effect you see sometimes >> around the moon. Nothing is rotating to make that circle. >> >> - Randy > >Nice examples. And a non-parametric definition of a circle does not >imply motion because it is independent of time. > >Nevertheless, this unfortunate thread continues to revolve around the >singularity at (L,Z). I think we're getting sucked in. > Aha. Traduced by the truth at last. The problem everyone seems to be having is that there is a vast distinction between the points on a circle and the circle itself. There would seem to be confusion between coplanar equidistant points and a circle. Coplanar equidistant points have a set of properties but the circle itself has a separate and distinct definition apart from those points. The points on a circle have certain properties that have nothing to do with the definition of the circle on which those points lie. ==== >The problem everyone seems to be having is that there is a vast >distinction between the points on a circle and the circle itself. >There would seem to be confusion between coplanar equidistant points >and a circle. Lester, Lester, Lester.... Sigh. You really don't get it, do you. A circle is as set of points. Period. Define the set, and you've defined a circle. There are many ways of generating a circle - rotating a vector V around a point is just one of them. (Footnote) But the method of generating a circle isn't what makes it a circle. BTW, there are several possible definitions of a circle. One of the goals of teaching mathematics is to get the student to understand when differing definitions are logically equivalent. One of the loveliest properties of mathematics is precisely its ability to show that apparently different definitions describe the same object, that differing concepts are in fact equivalent. A proof of a theorem may be no more than a demonstration of equivalence. - for if we have already demonstrated that theorem X is true, and we then show that theorem Y is the equivalent of X, then we have shown that Y is true also. Footnote: Actually, rotating V around a point generates an infinite number of circles, since every point on V will generate a circle. Moreover, the centre of rotation can be anywhere. And if the rotation itself isn't circular, then of course the objects generated by rotating V won't be circular either. IOW, your definition begs the question, since it's equivalent to If you rotate V in a circular motion around a point, then you get a circle. That definition of a circle assumes the definition of a circle. -- BTW, what path will be traced by an endpoint of V if the center of rotation O itself moves in a circle? -- Best Wishes, Wolf Kirchmeir, Blind River ON Not that brains are everything -- you'll also need a skull to put them in. (Nancy Franklin, 1997) ==== > >>The problem everyone seems to be having is that there is a vast >>distinction between the points on a circle and the circle itself. >>There would seem to be confusion between coplanar equidistant points >>and a circle. > >Lester, Lester, Lester.... > >Sigh. > >You really don't get it, do you. A circle is as set of points. Period. Define >the set, and you've defined a circle. There are many ways of generating a >circle - rotating a vector V around a point is just one of them. (Footnote) >But the method of generating a circle isn't what makes it a circle. > >BTW, there are several possible definitions of a circle. One of the goals of >teaching mathematics is to get the student to understand when differing >definitions are logically equivalent. One of the loveliest properties of >mathematics is precisely its ability to show that apparently different >definitions describe the same object, that differing concepts are in fact >equivalent. A proof of a theorem may be no more than a demonstration of >equivalence. - for if we have already demonstrated that theorem X is true, >and we then show that theorem Y is the equivalent of X, then we have shown >that Y is true also. > >Footnote: Actually, rotating V around a point generates an infinite number of >circles, since every point on V will generate a circle. Moreover, the centre >of rotation can be anywhere. And if the rotation itself isn't circular, then >of course the objects generated by rotating V won't be circular either. IOW, >your definition begs the question, since it's equivalent to If you rotate >V in a circular motion around a point, then you get a circle. That >definition of a circle assumes the definition of a circle. -- BTW, what path >will be traced by an endpoint of V if the center of rotation O itself moves >in a circle? > Wolf, Wolf, Wolf, You just don't get it, do you? I don't really care one way or the other about the definition of a circle. Somehow we seem to have wandered off into neverland. What I asked and care about is the rotation of v in circular rotation. A circle is not just a set of points. Points lie on a circle. The circle reflects a different definition that the points on a circle. People have suggested that dr/dt in dL/dt lies parallel to v. All I ask is how v rotates if this is true. The rest is nonsense. I could argue with you and everyone else about the definition of circles etc. but it wouldn't explain the rotation of v. So far only Edward Green and Randy have bothered to address the problem in specific terms. ==== > >> >> >>I think I see the problem through all this haze. The change in the >>vector v (may we speak of the rate of change of the v vector, i.e., >>acceleration?) is due to an acceleration component, not a velocity >>component. There is indeed a radial acceleration (dv/dt) directed toward >>the center of rotation for the circular motion you describe. It's called >>centripetal acceleration. > >Do tell. So, what is it we do exactly with this centripetal >acceleration? We combine it vectorially directly with tangential >velocity? My, my. I've heard tell of vector addition. I may even have >used it once or twice. But this is the first I've ever heard of the >vector addition of a force with a velocity. >> >>Very close. You garbled the actual description. The centripetal >>acceleration gives rise to a centripetally directed change vector. >>This change vector is a velocity. It is added vectorially to the >>original velocity to give a new velocity. > >So? It seems like you're agreeing with me. And it seems to you like up is down, and it seems to you that constants aren't constant. But none of those things is true. I'm not agreeing with you. You seem appalled at the idea that the description of what a force does to a velocity is to add a vector to it. I'm saying that's exactly what a force does. How does that constitute agreeing with you? You just got finished telling somebody they have no academic credentials and should be confined to an institution for saying exactly the same thing I'm saying. Do you even have any idea what you or anybody else is saying? - Randy ==== > > The rest is nonsense. I could argue with you and everyone else about > the definition of circles etc. but it wouldn't explain the rotation of > v. So far only Edward Green and Randy have bothered to address the > problem in specific terms. > I think it's time to bring in an acknowledged heavy-weight. No, not Uncle Al, but his arch nemesis, Archimedes Plutonium. Here's his calling card: whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies -- Joe Legris ==== > >This discussion is about circles in general, not those traced >by a moving object. For instance, the shape of ripples from >a drop hitting a pond. Nothing is rotating to make those >circles. The cross section of a laser beam. Nothing is rotating >to make that circle. The shape of a soap bubble. Nothing is >rotating to make that shape. The halo effect you see sometimes >around the moon. Nothing is rotating to make that circle. > > The only problem here is that nothing you're describing is a circle. A > circle is a mathematical construct generated by the rotation of v. > Under sufficient enlargement all the things you note resolve into non > circular components. Well, I'm not sure how you do sufficient enlargement of a rainbow, nor what the components would be. But at any rate, so what? The edge of a circular disk, the path taken by anything in circular motion, the orbits of the planets, are also not perfect circles or conic sections and break down under enlargement into perturbed versions of those things. A perfect circle is a mathematical idealization. So what? It's an idealization of both the stuff I was talking about and of the circular paths you've been talking about. Neither one is actually a geometrically perfect infinitely thin circular. These approximate circles do not require a rotation, that's the point. Are you changing the subject deliberately? - Randy ==== > Wolf, Wolf, Wolf, > > You just don't get it, do you? I don't really care one way or the > other about the definition of a circle. Well, apparently you do. Because you offer your own definition below, one which differs from the standard. > Somehow we seem to have > wandered off into neverland. What I asked and care about is the > rotation of v in circular rotation. > > A circle is not just a set of points. This is your assertion. However, a circle IS a set of points. In particular, it is the set of points which lie in a plane and which are equidistant from a specified point. You can't have it both ways. You can't say you don't care about the definition, then make bald assertions about what a circle is which are at odds with the definition. > Points lie on a circle. And nothing but those points lie in the circle. - Randy ==== > > The rest is nonsense. I could argue with you and everyone else about > the definition of circles etc. but it wouldn't explain the rotation of > v. So far only Edward Green and Randy have bothered to address the > problem in specific terms. > > > > I think it's time to bring in an acknowledged heavy-weight. No, not > Uncle Al, but his arch nemesis, Archimedes Plutonium. Here's his calling > card: > > whole entire Universe is just one big atom where dots > of the electron-dot-cloud are galaxies _Now_ who's being just as silly as somebody else, Mr. You Patient Tutors Bre Being Just as Silly As a Silly Ass? ==== > >> >The rest is nonsense. I could argue with you and everyone else about >the definition of circles etc. but it wouldn't explain the rotation of >v. So far only Edward Green and Randy have bothered to address the >problem in specific terms. > >> >> >>I think it's time to bring in an acknowledged heavy-weight. No, not >>Uncle Al, but his arch nemesis, Archimedes Plutonium. Here's his calling >>card: >> >>whole entire Universe is just one big atom where dots >>of the electron-dot-cloud are galaxies > > > _Now_ who's being just as silly as somebody else, Mr. You Patient > Tutors Bre Being Just as Silly As a Silly Ass? Are you referring to me? Zick and Plutonium appear to employ similar investigative techniques. There are many things they cannot teach each other, so a collaboration -- Joe Legris ==== > > The only problem here is that nothing you're describing is a circle. A > circle is a mathematical construct generated by the rotation of v. > Under sufficient enlargement all the things you note resolve into non > circular components. A circle can never be enlarged into constituent > points. > Circle http://mathworld.wolfram.com/Circle.html A circle is the set of points equidistant from a given point O. The distance r from the center is called the radius, and the point O is called the center. ==== > >>The only problem here is that nothing you're describing is a circle. A >>circle is a mathematical construct generated by the rotation of v. >>Under sufficient enlargement all the things you note resolve into non >>circular components. A circle can never be enlarged into constituent >>points. >> > > > > Circle > http://mathworld.wolfram.com/Circle.html > > A circle is the set of points equidistant from a given point O. > The distance r from the center is called the radius, and the > point O is called the center. Authoritative references? Straightforward explanations? Well, lah-di-dah. Those of us down here in the trenches of *real* philosophy have to make do with uniformed guesses backed up by the solid evidence of misconceived thought experiments. Multiple choice question: If a circle is just a collection of points, and if points have no size, then what is in between them? a) an explanatory gap b) the missing link c) a circular argument d) the unexcluded middle Answer: e) I've changed my mind - let's talk about physics. -- Joe Legris ==== > > >>The only problem here is that nothing you're describing is a circle. A >>circle is a mathematical construct generated by the rotation of v. >>Under sufficient enlargement all the things you note resolve into non >>circular components. A circle can never be enlarged into constituent >>points. >> > > > > Circle > http://mathworld.wolfram.com/Circle.html > > A circle is the set of points equidistant from a given point O. > The distance r from the center is called the radius, and the > point O is called the center. > > Authoritative references? Straightforward explanations? Well, > lah-di-dah. Those of us down here in the trenches of *real* philosophy > have to make do with uniformed guesses backed up by the solid evidence > of misconceived thought experiments. Become informed. You cannot converse meaningfully on a subject if you know nothing about it. > Multiple choice question: If a circle is just a collection of points, > and if points have no size, then what is in between them? > > a) an explanatory gap > b) the missing link > c) a circular argument > d) the unexcluded middle None of the above. Any reasonable calculus text explains convergence which is the key to your question. Might I recommend Calculus by George F. Simmons? I'm not related but I liked the book when I taught from it. For a deeper insight, I think the title is Topology and Analysis also by George F. Simmons. The book covers point set topology in great detail. You can also learn the fundamentals of function spaces from the book. > Answer: e) I've changed my mind - let's talk about physics. But you need to know a bit of mathematics because theories in physics are generally expressed in mathematical language. Chuck -- ... The times have been, That, when the brains were out, the man would die. ... Macbeth Chuck Simmons chrlsim@webaccess.net ==== > >> > > >>The only problem here is that nothing you're describing is a circle. A >>circle is a mathematical construct generated by the rotation of v. >>Under sufficient enlargement all the things you note resolve into non >>circular components. A circle can never be enlarged into constituent >>points. >> > > > >Circle > http://mathworld.wolfram.com/Circle.html > > A circle is the set of points equidistant from a given point O. > The distance r from the center is called the radius, and the > point O is called the center. >> >>Authoritative references? Straightforward explanations? Well, >>lah-di-dah. Those of us down here in the trenches of *real* philosophy >>have to make do with uniformed guesses backed up by the solid evidence >>of misconceived thought experiments. > > > Become informed. You cannot converse meaningfully on a subject if you > know nothing about it. > > >>Multiple choice question: If a circle is just a collection of points, >>and if points have no size, then what is in between them? >> >> a) an explanatory gap >> b) the missing link >> c) a circular argument >> d) the unexcluded middle > > > None of the above. Any reasonable calculus text explains convergence > which is the key to your question. Might I recommend Calculus by > George F. Simmons? I'm not related but I liked the book when I taught > from it. For a deeper insight, I think the title is Topology and > Analysis also by George F. Simmons. The book covers point set topology > in great detail. You can also learn the fundamentals of function spaces > from the book. > > >>Answer: e) I've changed my mind - let's talk about physics. > > > But you need to know a bit of mathematics because theories in physics > are generally expressed in mathematical language. > > Chuck You know something's gone terribly wrong when parody is indistinguishable from the real thing. Lester!! Stop it now or I'll report you to the authorities. If the universe crumbles and ... what's that sound? ... EGAD ... bo.... ==== If you develop an equation and publish it, can it still be subject to copyright so others can't republish? John ==== > > If you develop an equation and publish it, can it still be subject to > copyright so others can't republish? > > > John > Unless your publication was extremely brief, so that the single equation formed a substantial portion of the entire copyrighted publication, it is doubtful that any copyright you obtain could be used to prevent others from republishing. This is because of the doctrine of Fair Use. Others would, assuming the equation forms some small percentage of your publication, be allowed to quote from your paper that equation and make their own observations upon it. Of course if the equation were not properly attributed to you (assuming you have indeed managed to invent something truly original), then that would be plagiarism. But plagiarism and copyright violation are not at all the same thing. ==== > > > If you develop an equation and publish it, can it still be subject to > copyright so others can't republish? > > > John > > > > Unless your publication was extremely brief, so that the single equation > formed a substantial portion of the entire copyrighted publication, it is > doubtful that any copyright you obtain could be used to prevent others > from republishing. This is because of the doctrine of Fair Use. Others > would, assuming the equation forms some small percentage of your > publication, be allowed to quote from your paper that equation and make > their own observations upon it. > > Of course if the equation were not properly attributed to you (assuming > you have indeed managed to invent something truly original), then that > would be plagiarism. But plagiarism and copyright violation are not at > all the same thing. > I wonder if it could be patented? cheers hanford ==== > If you develop an equation and publish it, can it still be subject to > copyright so others can't republish? Patent claim applies to algorithms because the process of the algorithm can be changed algebraically and still be the same process and because the algorithm can be described without reliance on copy. (Consider that large amounts of source code could not be put in a derived form without using copy as a guide...and so large amounts of source code can benefit from copyright law and protection against derived forms.) Also, public description by the developer of the algorithm is a patent claim. This is obvious if the developer is claiming something unique, original, or proprietary. (This is fundamental or natural law but structured law adds a twenty year limit...and also recommends actual patent registration.) The next issue is discovery of pure math. The pure math can not be a patent claim but application of the pure math can be a patent claim and application could simply be computational efficiencies or coding efficiencies. In other words the discover can have no claim on the pure math itself but can have patent claim on the use of the pure math when implemented in a computer program. ==== > If you develop an equation and publish it, can it still be subject to > copyright so others can't republish? > > Patent claim applies to algorithms because the process of the algorithm can > be changed algebraically and still be the same process and because the > algorithm can be described without reliance on copy. (Consider that large > amounts of source code could not be put in a derived form without using copy > as a guide...and so large amounts of source code can benefit from copyright > law and protection against derived forms.) Patent claims were never intended to apply to algorithms. That they presently do (in the Wild Wild USA) is a historical accident. See http://www.softwarepatents.co.uk/past/how_the_us_got_there.html. > Also, public description by the developer of the algorithm is a patent > claim. This is obvious if the developer is claiming something unique, Is it? It could certainly be used as prior art to defeat a subsequent patent claim, but public description is not a patent claim in and of itself. > original, or proprietary. (This is fundamental or natural law but structured > law adds a twenty year limit...and also recommends actual patent > registration.) Rubbish. > The next issue is discovery of pure math. The pure math can not be a patent > claim but application of the pure math can be a patent claim and application > could simply be computational efficiencies or coding efficiencies. In other > words the discover can have no claim on the pure math itself but can have > patent claim on the use of the pure math when implemented in a computer > program. This is correct. Does it seem reasonable? We are led to believe that patents are an inalienable right -- much like taxation -- but -- again like taxation -- they were only introduced as a formal system fairly recently. The intention of a patent is to protect the poor [inventor] against the rich [manufacturer]. In practice it does no such thing - the primary beneficiaries of patents are those with enough money to apply for many *and* use them in litigation. David Turner ==== > If you develop an equation and publish it, can it still be subject to > copyright so others can't republish? I cannot _copy_ it and publish it (apart from fair use etc. etc. ). However, I can develop exactly the same equation independently and publish it. Maybe finding the equation required some stroke of genius that you had and I didn't. I can still read your paper, understand it, and _then_ develop the same equation and publish it. ====