September 2003 - Fractint - mailman.xmission.com

FOTD 07-09-03 (A No-Repeat Fractal [4])
by Jim Muth 07 Sep '03

07 Sep '03

FOTD -- September 07, 2003 (Rating 4) Fractal visionaries and enthusiasts: I must be developing amnesia. When I checked my e-mail box this morning I found about a dozen letters from different people telling me that I was great last night but could still use some improvement. Then there were four more people who had seen my profile online and wanted to meet me. There were four who had met me in a chat room. I had three letters that gave the information I had requested. And several letters simply said something like 'hey' or 'hi' or 'how're ya', or 'Jane told me you are too small', or the most mysterious of all -- 'dgins hebs oihndy fgotushn eod'. Finally, I wonder why the billing depart- ment at Mindspring wanted me to re-submit my credit card number to confirm that the number is correct. The number has been working as it should for 7 years. What has me worried is that I have no memory of being with twelve different people in one night, or of having my profile online. I have totally forgotten visiting any chat rooms, and I don't remember requesting any information. I have no memory at all of those many people who sent me greetings. I also wonder who this Jane is who thinks I am too small. My height is just about average. As for that message in a foreign language -- it might be important, but I can't read it. Finally, I wonder why my latest credit card statement shows a balance of $8,758.63, when last month it was $176.53 and I paid that in full and have charged nothing since then. (The first paragraph is true; the second is frivolous. Every few weeks I need a short rant about spam...) ...especially when the FOTD doesn't turn out too well. For today's image I tried the same trick that worked so well in yesterday's image. But it didn't work at all in today's image. I named the image A No-Repeat Fractal" because it's apparently impossible to repeat the same trick and get an equally fine result. I rated the image at a meager 4. The image was generated with the MandelbrotMix3a formula, which does nothing but permit 4 different exponents of Z to be combined in different portions, and permits the initial value of Z to be adjusted to a critical value of the formula. In today's case, zero is a critical value, so the (p5) parameter remains at zero. The biggest drawback is the time required to render the image -- over one hour on my machine. I strongly recommend giving Paul and Scott time to generate the image and post it to their web sites before going there to download the image. Paul's site is found at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Scott's is at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Near-perfct weather prevailed once again Saturday here at Fractal Central. The bright sun, dry grass and temperature of 79F 26C produced near-perfect cats. They had such a pleasant day in the yard that by the time they came in, they wanted to do nothing but grab a few bites of food and fall asleep. If they were still kittens, I would say they are at their best behavior when they are sleeping, but now that they are fully mature, they sleep far too much for sleep to be considered their best behavior. Today looks to be a repeat of yesterday, so I expect we'll do it all again. The duo is already romping in the yard. I'll be back in 24 hours with a new and fresh fractal, not another inferior imitation. Until then, take care, and be peaceful. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ A_NoRepeat_Fractal { ; time=1:08:11.40--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix3a function=ident passes=1 center-mag=+0.92952219950070970/+0.209587282531303\ 70/1.545435e+009 params=-2/2/1/3/0.5/4/0.25/5/0/0 float=y maxiter=1200 inside=fmod proximity=0.15 outside=tdis logmap=yes periodicity=10 colors=000Zah_eg_if`meaqdatdbpbcmadi`efZfbYg_XhXWi\ TUjQTkMSlJQmFPnCOn9NoFQpLTqQVrWYs`_q_`pZaoYbnXcmWd\ lVekUfjTfiSghRhgQifPjeOkdNlcMmbLmYJ`THOPFCXGHcGMjH\ RqHWxH`ySUzbNzmHvkGsjGphGmgGjeGgdGccG`aGY`GVZGSYGP\ XGRUETSDVPBXNAZK8`I7aF5bD4cA2d81e60e74e78f8Bg8Fi9J\ i9MkAQlAUmBXnB`pCdqCgUPpUPpUPp6ay9bwBcvEduGdtJerLf\ qOfpQgoThmVhlYik_jjajiajhajhcljenkfplhrmjtngXyhZti\ `pjbljcgkeclg_lhWmjRnlNonJooEpqAqs6qt2fkEWbQLUaBLl\ HMnMMoRMpWMr`MsfNtkNvpNwuNxzNy_mrRYnIJj94gCBiEHkGN\ mJToLZqNdrGei9e`2eS4dR5cR6bR7aR8aQ9`QA_QBZQCZQASR8\ LR7FR58R42R57S5BT5GU6KV6PW6TX7XY7aZ7e_8j`8na8rbEse\ JthOujUumZvocwrhwtetqbqo`nlYkjVigTfeQcbN``LYYIWWFT\ TDQRANO7KM5IKirIirLirN8EQ9DSACVBAXC9_D8aE7dF6fJEdM\ LcQSaTZ`XeZ_lYcsWfzVawTYtSUqQQnPMkNIhMEeKAbJBaKBaK\ BaKBaLBaLBaLBaLCaMCaMCaMCaNCaNCaNCaNEcTFdYHebIggKh\ lLiqMjvNciOXXWNlXRkYVjYYi } frm:MandelbrotMix3a {; Jim Muth z=real(p5), c=fn1(pixel), a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=real(p3), h=imag(p3), j=real(p4), k=imag(p4), l=imag(p5)+100: z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c, |z| <=l } END 20.0 PAR-FORMULA FILE==================================

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[Fractint] Linked Logistic System Across Fractals, Chaos, Physics, Biology
by mdoctorow＠comcast.net 07 Sep '03

07 Sep '03

>From Osher Doctorow Ph.D. I meant to put this same subject heading on my last posting a few minutes ago, but I forgot to put a subject heading on the posting. There is one typo in the posting: the last expression should read 1/[v(1 - v)] = 1/v) + 1/(1 - v) instead of reading v(1 - v) = (1/v) + 1/(1 - v). For those who deleted the previous posting because it had no subject heading, it turns out that if, instead of taking force as the time rate of change of momentum as it has been taken since Sir Isaac Newton in the later 1700s, one takes force as proportional to the time rate of change of "logistic momentum" mv (1 - v) where m is mass and v is speed or velocity, then if the proportionality is written F = kmv(1 - v), the higher or stronger the proportionality is, that is to say the larger k > 0 is, the closer v is to 1/2 for mass approximately constant. However, 1/2 is halfway between a speed of 0 ("at rest" condition) and the speed of light which is taken as 1 (ignoring superluminal speeds here). This enables us to solve the linked logistic system: 1) dv1/dt = k1v1(1 - v1) + k2v2(1 - v2) dv2/dt = k3v1(1 - v1) + k4v2(1 - v2) since with v1 approximately 1/2, we can use partial fractions to obtain an exact solution for v2 because 1/[v(1 - v)] = 1/v + 1/(1 - v) for any v. The linked logistic system (1) was first introduced by James W. Bruno and me and Chris Kappner in a 1981 publication in population growth/migration, and reflects the interrelated effects of the two populations on each other and slightly modifies the Lotka-Volterra predator-prey equations. The latter are useful when the environment contains competitors and limitations on growth (scarcity, etc.). We did not apply the equations (1) at that time to physics or analyze the case of v1 = 1/2. I introduced the equations (1) today on stardrive1(a)yahoogroups.com to model an Alcubierre-like warp drive in space travel with v1 the rear speed and v2 the front speed of the craft which (curiously enough) are different in the Alcubierre type scenarios. Osher Doctorow Ph.D.

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by mdoctorow＠comcast.net 07 Sep '03

07 Sep '03

>From Osher Doctorow Ph.D. We get closer to the relationship between chaos/fractals and physics and biology by considering the linked logistic system: 1) dv1/dt = k1v1(1 - v1) + k2v2(1 - v2) dv2/dt = k3v1(1 - v1) + k3v2(1 - v2) which is a modification of the predator-pret Lotka predator-prey equations in which two speeds (2-population growth speeds) v1 and v2 are interlinked logistically with each other. James E. Bruno and I and Chris Kappner in 1981 did some publications on this and related systems in population growth/migration, but we didn't explore the physics picture or even the equations beyond the particular empirical problem studied there. The equation (1) has arisen again, this time when I introduced it today in stardrive1(a)yahoogroups.com to model a 2-velocity spacetime warp star drive of the Alcubierre-related type. What I want to emphasize here is one of the strangest outcomes of the use of equations (1) in physics. Here is the basic result: PRINCIPLE OF SPEED 1/2. The speed 1/2, on a scale from speed 0 to the speed of light taken as 1 (here I'm not considering possible superluminal speeds), is not only the average between these two extremes but is also the speed at which Force becomes very highly proportional to the time rate of change of "logistic momentum" which has the form mv(1 - v). We know since Sir Isaac Newton's time in the late 1700s that force is the time rate of change of momentum, but "logistic momentum" is more realistically the growth-related analog of momentum. For those who know calculus (for others, just notice that dv/dt cancels in the second equation below if it isn't 0 and then solve algebraically for v), force F = kd[mv(1 - v)]/dt and approximating m by a constant and incorporating it into k leads to (using the same constant k) dv/dt = kdv/dt - 2kvdv/dt which has the solution v = (1/2) - [1/(2k)] and for k very large this is close to (1/2) and gets closer the larger k is. Now if we approximate v1 in (1) by (1/2), it becomes a simple matter to solve system (1) by the same method as solving the logistic equation, that is to say by partial fractions noting that 1[v(1 - v)] = 1/v + 1/(1 - v). The logistic system of course is central to chaos and fractals and environment- limited growth in its discretized (and non-discretized) version and so on. Osher Doctorow Ph.D.

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FOTD 06-09-03 (Mandelbrot Feathers [9])
by Jim Muth 06 Sep '03

06 Sep '03

FOTD -- September 06, 2003 (Rating 9) Fractal visionaries and enthusiasts: A FOTD fractal with a rating of 9 is rare indeed, but today we have found one. It is a scene in the fractal that results when Z^(1.1) is subtracted from (1/Z) and (1/C) is added. This parent fractal consists of two large Mandeloids surrounded by scattered fractal debris. Today's scene is located at the point where a continuous filament breaks up into a series of disconnected bits. The scene is unusual in that nearly all the outside rendering options work well with it. I happened to choose the 'fmod' option with a proximity of 3, though I could just as well have chosen the 'summ' or the 'imag' options, or any of the others. It's just an easy scene from which to get a good fractal. When I was studying the image in search of a name, I thought of ostrich feathers and found the name "Mandelbrot Feathers". I was undecided as to whether the image rates an 8 or a 9. But since I have had a run of sub-par images lately, I decided on a rating of an exceptional 9, which might be stretching it a bit. But regardless, the image is still well above average. The image also benefits from a relatively fast render time of 7 minutes. And as always, the completed GIF image is posted on the internet at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Take a trip to one of the sites. You will not regret the journey. With mostly sunny skies, a high temperature of 77F 25C, and no rain, Friday was a near perfect day here at Fractal Central. The fractal cats took full advantage of the perfection by enjoying the yard from 11am until evening. They were near perfect angels, the only imperfection coming when Tippy returned with a burdock burr stuck in his tail. The burr was easily removed and the day ended peacefully with the duo's noses in their food dishes. Today is starting even more perfect. Since perfection cannot be improved upon, I hope for a repeat of Friday. To see if my ex- pectations are fulfilled, check back in 24 hours. Until then, take care, and unless I am mistaken, burdock burrs gave the inventor the idea for Velcro. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ MandelbrotFeathers { ; time=0:07:07.54--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix2 function=recip passes=1 center-mag=-0.16918139027290410/+0.463847222461736\ 50/6.731712e+011/1/90/3.88578058618804789e-016 params=1/-1/-1/1.1/0/525/0/0 float=y maxiter=900 inside=0 proximity=3 outside=fmod periodicity=10 colors=000NYNNTGON9OI3LN6JR9HWBF_EDcGE_IEXJETKEQLE\ MNFJQFFTFCVF8XF5_F2bG3`G4XH4UH5PI6MI6IZHNoRUnUTmWT\ lYTk_TjaTicTheTghTfjTelTdnTcpTbrTatTcqVdoXflYgj_ig\ `jeblbcm`enZgpWhqUjsRktPmvMnwKpxIqMhAPdCT`DVQHURHS\ TGRUGQWGPXGNZGM_FLaFJbFIcFHeFFfEEhEDiEBkEAlE9nD7oD\ 6qD5rD4sD6rE7rE9qEAqECqFDpFEpFGoFHoGJoGKnGLnGNnHOm\ HQmHRlHSlHUlIVkIXkIYjI_jJ`jJaiJciJdiKfhKghKhgKjgLk\ gLmfLnfLofLneMmdMlcMlbMkaMj`Mj_NiZNhYNhXNgWNfVNfUN\ eTOdSOdROcQObPObOOaNO`MP`LP_KPZJPZIPYHPXGPXFQWEQVD\ QVCQUBQTAQTAQUBRVCSWDSWETXETYFUZGUZHV_HV`IWaJWaKXb\ LYcLYdMZdNZeO_fO_gP`gQ`hRaiRajSbjTckUclVdmVdmWenXe\ oYfpYfpZgq_gr`hsaisaitbjucjvdkvdkwelxflygmygmxhnxh\ nxhoxhoxhoxhpxipxiqxiqxiqxirxirxirwjswjswjtwjtwjtw\ juwkuwkvwkvwkvwkwwkwwkwuiuthtsgsrfrqeqpdpoconbnmal\ l`kk_jjZiiYhhXggVffUeeTcdSbcRabQ`aP__OZZNYYMXXLWWK\ UVJTUISSVUPaZNhcLngMh`McU } frm:MandelbrotMix2 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)), k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================

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FOTD 05-09-03 (A Very Deep Midget [5])
by Jim Muth 05 Sep '03

05 Sep '03

FOTD -- September 05, 2003 (Rating 5) Fractal visionaries and enthusiasts: The work was still hectic here at Fractal Central on Thursday, but things calmed enough in the afternoon for me to post a short reply to the list and find an 'automatic' fractal in the Mandelbrot set. Today's image takes us where many have ventured previously -- to the tip of the negative stem of the Mandelbrot set. And when I say the tip, I mean the tip. The magnitude of the image is 3x10^58, and there are 29 consecutive 9's following the 1 and the decimal point in the X-coordinate of the scene. Since the cut-off point is set to 2, today's image is almost exactly on the cut-off point. The rays radiating from the central midget are a direct result of its being so close to the cut-off point. The image rates only a 5, held down by the hackneyed nature of the area in which it lies. Fresher scenes might be found at the tips of some of the other filaments, especially near the cut-off points. I named the image "A Very Deep Midget" as a description. It is certainly a fair name, because, unless I am mistaken, the scene is the second deepest of all the FOTD images. The render time of 19 minutes is quite fast for such a high magnitude image calculated with arbitrary precision math. The X-axis symmetry helps a lot here. Whether the result is worth the wait is open to individual opinion. My opinion is that a more reasonable method of viewing the scene would be to download it from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Though the temperature on Wednesday reached a very comfortable 77F 25C, it was another disappointing day for the cats here at fractal headquarters. The sprinkles that fell off and on through the entire day culminated in a torrential downpour at 7pm, accompanied by thunder. But by that time the cats had already given up all hope of outdoor fun and satisfied them- selves with a tuna treat. When an even stronger storm arrived around midnight, they watched the rain cautiously, until a close lightning strike and a loud bang of thunder sent them scurrying for cover. Thursday began with steady rain, which ended at midday, leaving the skies leaden gray for the rest of the day. The temperature of 75F 24C was fair enough, but fractal cats need dry grass in which to romp, so more tuna was needed. Today is starting much better. It is sunny, cool and dry. Can the dynamic duo actually have a good day after a long string of bad ones? Check back tomorrow to see. As for me, the load of work is still heavy, so I'd best get busy. Until tomorrow, take care, and keep your spirits up when the rain comes down. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= A_Very_Deep_Midget { ; time=0:19:23.10--SF5 on a p200 reset=2002 type=mandel passes=1 center-mag=-1.9999\ 99999999999999999999999997080345601743762432509639\ 57868601/-1.6623896e-61/3.139258e+058 params=0/0 float=y maxiter=1500 inside=0 symmetry=xaxis periodicity=10 colors=000iHpmEorBnv8mvMgvZbvkXvxSp\ lWka_eQc`FgW4k`9adESiAGn75q3GlCMgKSYSYR`cLhiEpo8pu\ DoiHkiKk7MmDPqJRtPTyUhxFxr0vkCqcOe_TibXmg`qfdhda`Y\ ZTUWLATD1RM6OVBMjGKtLIzYGaN89517746874993AC2BE8EFD\ HGJKHONITQJZTKcWLhZLeVQcRV`N_ZKdWGiUCnR8sP5xHYyYdR\ dTPkINP_K2qI7jMCdPHZSMSVRMYWG``AcfDdlFeqHekNaeSY_X\ VUaROfNIkKHfNGaPFXRFSTENVDIXDEZkEUfCYbB`ZAcV9fQ7iM\ 6lI5oE4rCBnBIjAPf9Wb8bZ6dS4eM3gF1h90i37aMDVcJOuQHd\ XBPb9Vh7_n6dt4iy3nd7qLBt1Fw6MpATiE_bJfXNmQRtJVzDPq\ NKhWF`eASn5Kw7Il8Ha9FRAEGBD5DG6EI6FK7GM7HO8IQ8HRBH\ SEHTHHUKHVNKSNMPNPMNRKNTHNWENYBN_9NfT_`PWWLSRHOMDK\ H9GC5C71969H6GO5NW5Vb4aj4hq4oxBmtIlqPknVjkahhhgeof\ bue_tiXsmUsqStlOugKvbGwZCxU8yP4zL1pW3fe4Xo6Oy7AkoB\ UYnhTVTMl6vb8kUA`KCQLpcHbWEQOc3GU7GKAG5sY8YPkwYakT\ T_PKOKzvwNm8KcAHVCEMEmzVUaNgOaRIRtxqdhdQTSFBHDCGeD\ pQDYn`9`TCOLEZksv2kZ8WaDTVDPODMHDJdKr107 } END PARAMETER FILE=========================================

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Re: [Fractint] FOTD 03-09-03 (Spiral [3])
by Jim Muth 04 Sep '03

04 Sep '03

At 06:24 PM 9/3/03 -0700, John Wilson wrote: >Hey, I didn't *state* that 7-armed spirals were rare! >I *asked* if this were so. My fault. I was rushed. I should have taken time to check exactly what you had said. >Since you were able to produce one on demand, I guess >the answer is that they are *not* rare. A spiral of any number of arms can be produced on demand. >My 7up, however is a much nicer spiral, with another >somewhat intriguing feature...I thought. If I back >away some distance from my screen, I see that the sub- >elements of the slowly spiralling arms form another >spiral...a much faster spiral, (shorter pitch). It's a fine spiral. In no way did I intend to criticize it or the formula that created it. >Something that was discussed in Philofractal back in >May and June...I have been surprised by the lack of >comment in "Scientific American" regarding the multiple >universe article by Max Tegmark. Well a few letters >have finally surfaced. It was my own belief that this >whole thing was built upon flawed reasoning which led >to paradox. I never saw a paradox. I saw a hypothesis which is so vast as to be almost mystical. It is not surprising that the more rational thinkers would search for flaws in the reasoning. >One reader pointed out an omission in the boundless >possibilities suggested by Tegmark...a theism. I guess >that this reader is thinking along the same lines that I >followed; "Space appears to be infinite in size. If >so, then somewhere out there, everything possible becomes >real, no matter how improbable it is". Therefore there's >a universe somewhere which is built and managed by "God". In fact, if Tegmark's theory is true, there are an infinity of such universes. In an infinite sub-set of these, the creating God is hidden. In another sub-set, God can be openly seen doing his divine work. And there are an infinity of other universes such as materialists consider ours to be -- accidents that by the unlikeliest of chance came into existence with parameters balanced just right to make the development of conscious intelligent life possible. Since, in our universe, God cannot be seen doing his divine work, our universe must be of the first or last variety. And now the question becomes, what of the multiverse that holds all these infinite universes -- does it have a God who created and is watching over it? If so, that God is also God of the subset universes and their lesser Gods, and even the accidental universes with no lesser Gods still have that greater God in the higher world. God just won't go away. Jim M.

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No FOTD for 04-09-03
by Jim Muth 04 Sep '03

04 Sep '03

Fractal fans: Rather than post another FOTD image with compromised quality, I will post no FOTD today, September 04. I must get the work backlog caught up before I can concentrate on fractals. The FOTD will resume on September 05. Jim M.

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FOTD 03-09-03 (Spiral [3])
by Jim Muth 04 Sep '03

04 Sep '03

FOTD -- September 03, 2003 (Rating 3) Fractal visionaries and enthusiasts: When I opened my e-mail box this morning, I found, among all the offers of more length, a letter informing me that I had just won 13-million dollars, or was it euros, in the European Lottery. This is strange, since I never bought a ticket. But I'm not questioning that aspect. The letter then asked for my bank account number so that the money I had won could be deposited in my account. I replied, requesting that the lottery commission send a check instead. As soon as the check arrives, I shall retire and devote the rest of my days to exploring the world of fractals. I wonder how long it will be before the check arrives. As for today's fractal, it is worth about as much as I am going to be enriched by winning a lottery I never entered. (Could anyone be gullible enough to actually reply to such letters?) I gave today's image a courteous rating of a 3. Actually, it might be closer to a 2. The image is so poor because I had no time to work on it. An unexpected rush job came in on Monday and kept me busy until after midnight. I'm still not quite caught up. Having no time to find a new fractal, I turned to one I generated yesterday, when I saw a message on the Fractint list that 7-armed spirals are a rarity in fractals. I had not realized that such spirals were rare, and to see if this were so, I went to a period-7 bud of the Mandelbrot set and drew the Julia set of that area. Then, for a bit of added interest, I double rotated the image 0.1 degree from the julia direction and got my 7-armed spiral, with a vague hint of Mandelbrot buds in the background. The picture had potential, but then the work came in and I had no time to color it properly. I leave the colors as they were when I had to stop working on it. I named the image "Spiral", and used it as FOTD for today. I might have done better not having a FOTD for today at all, but having gone this far, I am not going to scrap it now. The render time of 18 minutes is too slow for such a question- able image. I advise downloading the finished GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The Tuesday temperature of 84F 29C here at Fractal Central was fair enough, but there was little sun. The occasional sprinlkes of rain were not enough to wet a person, but made the grass too wet for the sensitive paws of the dynamic cats. As a result, they had a bad day and needed their evening treat of tuna to restore their good moods. Today is starting cloudy, cool, foggy, drizzly, with occasional sprinkles of rain. It is sure to be another bad day for the duo. For me, I've still got that left-over work to catch up on. So until next FOTD, take care, and don't get twisted up in an endless spiral. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Spiral { ; time=0:18:49.82--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=ident/flip passes=1 center-mag=-0.344854/0.856196/4.523577/1/\ 70/-5.05012698326368081e-014 params=89.9/89.9/2/0/\ 0/0/0.36837/0.14068 float=y maxiter=50000 inside=0 logmap=yes periodicity=10 colors=000UFAUPAUFAUPAUFAUQAUFAUQAUGAUUAUJAUXAULAU\ ZAUNAUaAUOAUbAUPAUdAUQAUeAURAUcAUSAUbAUTBUaCUUDU`E\ UVFU_GUWHUZIUXJUYKUYMUZNUZOUZPUZQUZRUZSVZTWZUXZVYZ\ WZZX_ZY`ZZaZ_bZ`cZ`dZ_eZ_f_ZgZZh_YiZYj_XkZXl_WmZWm\ _VlZVl_UlZTk_TkZSj_SjZRj_RiZQi`QiZPh`PhZOg`OgZNg`M\ fZMf`LeZLeaKe_KdaJd`JcbIc`IccHbaHbdGacGaeFadE`fE`f\ D_fD_fC_fCZfBZfBYfAYfAYf9Xf9Xf8Yg7Xf8Xf9Xe9XeAXdAX\ dBXdBXcCXcCXbDXbDXbEXaEXaFX`FX`GX`GX_HX_IXZJXZKXZL\ XYMXYNXXOXXPXXQXWRXWSXVTXVUXVVXUWXUXXTYXTZXT_XS`XS\ aXRbXRcXRdXQeXQfXPgXPhXPiXOjXOkXNlXNmXNmXMmXMmXLmX\ LmXLmXKmXKmXJmXJmXJmXImXImXHmXHmXHmXGmXGmXFmXFmXFm\ XEmXEmXDmXDmS8mVBmXDmZFm`HmbJmdLmfNmhPmiQmjRmjRnjR\ ojRpjRqjRrkRskRtkRukRvkRwkRxlRylRzlRzlRzlSzlSzlSzm\ SzmSzmSzmSzmSzmSznSznSznSznSznSznSznSzoSzoTzoTzpUz\ pVzpVzqWzqWzqXzrYzrYzrZzsZzs_zs`zt`ztaztazsbzrbzrb\ zqczqczpczpdzodzndznezmez } frm:multirot-XY-ZW-new {; draws 6 planes and rotations ;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J ;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994, z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p3, c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================

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The Hyper-Logistic Equation
by mdoctorow＠comcast.net 03 Sep '03

03 Sep '03

>From Osher Doctorow Ph.D. The logistic equation is in calculus terminology: 1) dx/dt = c1x(c2 - x) where c1, c2 > 0. When x and t are taken to be in the interval [0, 1] (as for example when x is taken to be a percentage of the population), then c2 can be replaced by 1: 2) dx/dt = c1x(1 - x) This is in fact the usual Rare Event scenario on [0, 1] scales. To extend and generalize the logistic equation, which is very useful for biological growth, to growth in physics and psychology and astrophysics and so on, the simplest general way would be to replace (2) by: 3) dx/dt = c1xf(x, t)(h(x, t) - xg(x, t)) where f(x, t) and g(x, t and h(x, t) are functions that depend on x and t. The case where f(x, t) = g(x, t) = h(x, t) = 1 is the ordinary logistic equation. Since x is some arbitrary variable that grows and/or contracts, why not replace x in a particular scenario by dx/dt? Then we get: 4) d^2x/dt^2 = c1(dx/dt)f(x, t)[h(x, t) - dx/dt g(x, t)] Since the equation has both enough dx/dt's and even has a second derivative d^2x/dt^2, the next simplest choice for f(x, t), h(x, t), and g(x, t) would be either a constant or a linear function of x or a linear function of t. In the bracket, RET tells us that the form of a difference suggests that one term is causal and the other is caused (with probability), so since having t or x in both terms would factor them out into f(x, t), the simplest case is to assign x to one of the two terms in the brackets and t to the other. At this stage, let us compare this picture in (4) with the difference k3 from xp and Et in quantum theory, remembering that xp = k1, Et = k2, xp - Et = k1 - k2 and we define k3 = k1 - k2: 5) mx(dx/dt) - (1/2)m(dx/dt)^2 t = k3 which factors to: 6) m(dx/dt)[x - (1/2)dx/dt] = k3 Equations (4) and (6) are identical except for k3 and d^2x/dt^2 if we set the constant m = f(x, y) and h(x, t) = x and g(x, t) = (1/2)t. This is one of the two ways that we wanted to select h and g (we could have put t and x in opposite terms) except for the constant 1/2, and it is certainly satisfactory to have f as a constant m. We also have the added "bonus" that either space (x) exerts a probable causal influence on time t or vice versa. What do we do about k3 versus d^2x/dt^2, or using differential operator notation, Dtt(x)? It certainly is not unimaginable that the acceleration Dtt (x) of x is constant. It is roughly similar to a constant applied force. We could propose to generalize the resulting equation beyond constant accelerations and just consider that constant k3 holds in certain scenarios, but like special relativity we might as well begin with something constant. So we have another way of deriving a very general "hyper-logistic" equation of growth across several domains. Osher Doctorow Ph.D.

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Geometric Physics and Growth
by mdoctorow＠comcast.net 02 Sep '03

02 Sep '03

>From Osher Doctorow Ph.D. The simplest population growth equation is one that works for bacteria: 1) dx/dt = kx where k > 0. The simplest population decay equation is one that works for radioactive decay: 2) dx/dt = -kx (k > 0) The first has an exponential solution x(t) = x(0)exp(kt), the second a negative exponential solution x(t) = x(0)exp(-kt) I remarked some years ago that Logic-Based-Probability (LBP) and Rare Event Theory proceed somewhat differently from Category Theory, which latter is a branch of very advanced Algebra pioneered by Saunders MacLane and William Lawvere in which the "composition" operation is the main operation. For example, suppose that y = exp(t) and z = y^2, then y o z is the composition of y and z and is equal to exp(y^2), so y o z tells you to put z inside the parentheses of exp(t) instead of t (where y is exp(t)). In LBP and RET and Geometric Physics, we're mostly interested in much simpler operations. For example, we use y * z which is simply (exp(t))*(y^2) where * is just ordinary multiplication. Or we use y + z which is exp(t) + y^2. Since dx/dt is the instantaneous speed or velocity (rate of change of x as t changes), let's write it as an Operator Dt(x): 3) Dt(x) = dx/dt or more generally: 4) Dt(f(x)) = df(x)/dt Since dx/dt = kx is the simplest growth equation, our orientation toward multiplication rather than composition would suggest looking at: 5) Ut(x) = kx 6) Vt(x) = 1/(kx) because: 7) Ut(x) * Vt(x) = 1 so that Vt(x) is a multiplicative inverse of Ut(x). Now let's go one step further and define: 8) W(x, t) = 2x/t which has both a numerator and denominator variable of its arguments. So this leads us to: 9) dx/dt = 2x/t The solution of (9) is rather simple because dx/x = 2dt/t when integrated on both sides leads to log(x) = 2log(t) + c so taking exponentials to both sides leads to x = x(0)t^2. However, keep track of the inequality: 10) dx/dt < 2x/t while I remark about the Heisenberg Uncertainty Principles (HUPs) in the form: 11) U(x)U(p) > = h/(2pi) 12) U(E)U(t) > = h/(2pi) where x is position or distance/displacement (from the origin), p is momentum, E is energy, t is time, and this time U( ) means "Uncertainty of" or "Uncertainty in," so that U(x) means "uncertainty in x". If speeds are not too close to that of light and also potential or potential energy is small, we have approximately: 13) p = mv 14) E = (1/2)mv^2 where v is velocity or speed. In most applications in physics, for example to cosmology and quantum entanglement, a rather curious "oversight" is made - instead of using U( ) in (12), as in U(x), the origin variable is really used, that is to say x. For example, when physicists say that high energy forces act for short times or low energy forces act for long times or that the vacuum seethes with particle- antiparticle pairs coming into existence because short distances are associated with high momenta, they are using (for the latter) xp > h/(2pi) and for the former Et > h/(2pi) and not uncertainties. What happens if we replace U(E) by E, U(x) by x, U(v) by v, U(t) by t in equations (11) and (12)? We get: 15) x(mv) > h/(2pi) 16) (1/2)mv^2 t > h/(2pi) Now it's common to replace the inequalities in the last two equations by equalities and in that case we do have a choice of selecting slightly different quantities from h/(2pi) on the right hand sides. Let's call them constants k1 and k2 respectively: 17) xmv = k1 18) (1/2)mv^2 t = k2 Then we have, by subtracting (18) from (17) and defining k3 = k1 - k2: 19) xmv - (1/2)mv^2 t = k1 - k2 = k3 = mv(x - (1/2)vt) = k3 and therefore if (1/2)vt < x or v = dx/dt < 2x/t, then for nonnegative v the expression equal to k3 will be nonnegative. Readers will recognize this expression from the previous postings. Why would we want k3 to be nonnegative? Well, in RET and LBP and Geometric Physics, the probable influence of x on y can only be calculated if x - y is nonnegative, that is to say y < = x. So if k3 in (19) is nonnegative, then x influences t in x - (1/2)vt. I'll try to continue this shortly. Osher Doctorow

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