mm-222 === Subject: Super Cosmos World PictureThe World Picture of Super Cosmoshttp://focus.aps.org/ghlights of what subjectively struck me as worth reporting1. WMAP ( http://map.gsfc.nasa.gov/m_mm.html ) of CMB (Cosmic Microwave Black Body Radiation) in thermal equilibrium in slow enough adiabatic expansion of the Hubble spherical horizon of our past light cone. That is, tnk of our 3D space as effectively bounded by a slightly distorted spherical wave front that is c/H(t) away from us in our past. That is, it took light approximately 1/H(t) seconds to reach us from that initial wave front. Note, I use the word approximately.Ts Hubble sphere is one of an infinity of Level 1 parallel IT Universes on a single 3D spatially ßat post-inßationary bubble, wch is the 3D version of Euclid's globally ßat infinite plane without boundary. There are a further Level IIinfinity of these bubbles (or infinite 3D rubber sheets expanding homogeneously and isotropically on the largest of scales only) in a Cosmic Champagne Cocktail as it were perhaps from the Qabalistic Vineyard of one Ras des Troyes (1040 - Parallel Universes for the Level I and Level II classification of the chaotic inßation scenario with the Weak Anthropic Principle and Darwinian Natural Selection on the cosmic scale.Angular resolution of WMAP space probe is 40 x better than COBE space probe.Polarization of CMB is measured from last scattering of light from the time of Ôrecombination' when stable hydrogen atoms formed for the first time ~ 380,000 light years from the Big Bang, and also from ionization of atoms in space by UV photons from the first stars forming only 200 million light years from the Big Bang! The age of the Universe ~ c/Ho is 13.7 billion light years. Ts is the radius of the Hubble spherical horizon back to the Big Bang. Therefore, the Holograpc Thermodynamic Entropy S of our Universe now is of order (10^28)^2/(10)^-26 ~ 10^82 BITS.Atomic matter is about 4% of the large-scale structure of the Universe, i.e. Omega(Atomic Matter) ~ 0.04.Exotic vacuum with gravitating positive zero point pressure is about 23% of the large-scale structure of the Universe, i.e. Omega(Dark Matter) ~ 0.23.I have made a theoretical inference here that dark matter is not made symmetry partners. That is, dark matter detectors, in principle, will never click with The Right Stuff, if they click at all it will be with false positives, with Fool's Gold like Enron Stock After The Fall.Exotic vacuum with anti-gravitating negative zero point pressure is about 73% of the large-scale structure of the Universe, i.e. Omega(Dark Matter) ~ 0.73.Note that scale-dependence is crucial. You can imagine, in the sense of adaptive windowed wavelet transforms not the usual Fourier transforms of time and space, a power spectrum of different kinds of stuff plotted against scale of resolution. The above percentages are only measured at the largest scales ~ 10^26 to 10^28 cm in a coarse graining sense where the FRW metric approximation applies.Homework Problem 1Note that the Fourier transform of the temporal autocorrelation function of some disturbance is its power spectrum. What is the autocorrelation function corresponding to the adaptive windowed wavelet power spectrum where scale is plotted instead of frequency/wave numbers?2. SDSS of 200,000 galaxies + WMAP above giveho ~ 0.7 +- 0.04Omega(Dark Matter + Atomic Matter) = 0.3 +- 0.04Upper limit on rest mass of the electron neutrino ~ 0.6 evhttp://arxiv.org/abs/astro-ph/0310723SDSS 3D conformal scale-dependent wavelet-transform? map of past light cone from scale of Earth's core ->Solar System -> Milky Way Galaxy -> Clusters of Galaxies -> CMB last light.We have produced a new conformal map of the universe illustrating recent discoveries, ranging from Kuiper belt objects in the Solar system, to the galaxies and quasars from the Sloan Digital Sky Survey. Ts map projection, based on the logarithm map of the complex plane, preserves shapes locally, and yet is able to display the entire range of astronomical scales from the Earth's neighborhood to the cosmic microwave background. The conformal nature of the projection, preserving shapes locally, may be of particular use for analyzing large scale structure. Prominent in the map is a Sloan Great Wall of galaxies 1.37 billion light years long, 80% longer than the Great Wall discovered by Geller and Huchra and therefore the largest observed structure in the universe.http://arxiv.org/abs/astro-ph/03105713. Speed of Gravity?What Tom Van Flandern alleges in http://www.ldolpn.org/vanFlandern/gravityspeed.html about superluminal near field of gravity is controversial as is what he says about GPS in http://www.metaresearch.org/ But it is not my purpose here to dig into those issues. On the other hand s reference to William Walker's work is sound http://arxiv.org/abs/physics/0001063Near-field Analysis of Superluminally Propagating Electromagnetic and Gravitational FieldsA near-field analysis based on Maxwells equations is presented wch indicates that the fields generated by both an electric and a magnetic dipole or quadrapole, and also the gravitational waves generated by a quadrapole mass source propagate superluminally in the nearfield of the source and reduce to the speed of light as the waves propagate into the farfield. Both the phase speed and the group speed are shown to be superluminal in the nearfield of these systems. Although the information speed is shown to differ from group speed in the nearfield of these systems, provided the noise of the signal is small and the modulation method is known, the information can be extracted in a time period much smaller than the wave propagation time, thereby making the information speed only slightly less than the superluminal group speed. It is shown that relativity theory indicates that these superluminal signals can be reßected off of a moving frame causing the information to arrive before the signal was transmitted (i.e. backward in time). It is unknown if these signals can be used to change the past. http://arxiv.org/abs/gr-qc/0304090A mainstream experiment using gravity lensing by Jupiter of an aligned distant quasar's light give a speed of gravity 1.06 +-0.21 the speed of light http://arxiv.org/abs/astro-ph/0302294However Clifford Will disagrees with that claim inWe calculate the delay in the propagation of a light signal past a massive body that moves with speed v, under the assumption that the speed of propagation of the gravitational interaction c_g differs from that of light. Using the post-Newtonian approximation, we consider an expansion in powers of v/c beyond the leading ``Shapiro'' time delay effect, wle working to first order only in Gm/c^2, and show that the altered propagation speed of the gravitational signal has no effect whatsoever on the time delay to first order in v/c beyond the leading term, although it will have an effect to second and gher order. We show that the only other possible effects of an altered speed c_g at ts order arise from a modification of the parametrized post-Newtonian (PPN) coefficient alpha_1 of the metric from the value zero predicted by general relativity. Current solar-system measurements already provide tight bounds on such a modification. We conclude that recent measurements of the propagation of radio signals past Jupiter are sensitive to alpha_1, but are not directly sensitive to the speed of propagation of gravity. http://arxiv.org/abs/astro-ph/03011454. Gravity Waves? LIGO a giant strain gauge to measure ripples in space-time curvature. Two spiraling in neutron stars emit gravity waves that change distance between two mirrors 4km apart by less than one thousanth of the diameter of a proton! And they say that people who see UFOs are crazy? :-) Well Kip Thorne at Cal Tech tnks ts can be measured and many millions of dollars are being spent to do so. Let's hope they do better than the hot fusion boys. :-) Notng exciting to 5. The Big Rip DoomsdayThe dark energy data from type 1a supernovae requires -1/3 > w = (pressure of dark energy)/(energy density of dark energy)In my theory, dark energy and dark matter are both w = -1 zero point energy density of exotic vacuum regions that are positive and negative respectively, hence negative and positive quantum pressures respectively. Exotic vacuum with w = -1 and positive pressure will be mistaken for w ~ 0 CDM http://web.mit.edu/afs/athena.mit.edu/user/r/e/redingtn/www/ netadv/specr/012/012.htmlThe Big Rip destroying future Universe only happens if -1 > w i.e. phantom energy. There is no Big Rip in my theory where w = -1 on the nose.Appendix 1. Digression on the Universe as a Hologram - one particular model that is probably false.The pixel resolution of the Hubble surface deteriorates as the Universe 3D space expands and accelerates. The quantum of area or cosmic pixel size isLp*^2 ~ Lp^4/3(c/H(t))^2/33D space is the holograpc image of the expanding accelerating Hubble 2D surface.The number of Bekenstein BITS in the Universe then gives the Entropy of the Universe S(t) at Cosmic Time t to beS(t) ~ k(c/H(t))^2Lp*^-2 ~ k(c/H(t)Lp)^4/3Ts explains the Second Law of Thermodynamics Arrow of Time as tied to the expansion of the Universe. That's The Good News. There is also some Bad News. That is, ts particular model may be false. I do not know as yet.k = Boltzmann's constant, Lp^2 = hG(Newton)/c^3 ~ 10^-66 cm^2, c/H(today) ~ 10^28 cm, Lp*(today) ~ 1 Gev, i.e. 1 fermi (10^-13 cm)Witten's alpha' today ~ 1/(1Gev)^2 = Universal Geometrodynamic Regge Slope of hadronic resonances of the strong force ~ (string tension)^-1All of ts is testable and ts model may be false.Rest mass of lepto-quarks m ~ (e/c)^2Lp*^-1 ~ MevTherefore, e/m ~ (c^2/e)Lp*IF e/m is to be a constant of the Universe's evolution in global FRW large-scale cosmic time ~ h/k(Absolute Temperature of the CMB), then(c^2/e)^2Lp*^2 is constantThat is,(c4/e^2)hG*/c^3 = (hc/e^2)G* is constant in cosmic time though not in scale!Note, that the only way to keep H(t) constant would beH(t) = R(t)^-1dR(t)/dt = HoR(t) = Roe^Hoti.e. a state of continuous inßation.If we exclude ts, then the e/m ratios of the lepto-quarks must be increasing if ts particular version of the world hologram idea were === confusion, but in Kaplansky's _Set Theory and Metric Spaces_ (1972),> he defines Zorn's Lemma as> Let L be a partially ordered set in wch every chain has an upper> bound. Then L contains a maximal element.> Our teacher instructed us to write in the word nonempty before> partially ordered, and in L after upper bound. To be fair,> Kaplansky notes in s remarks on the lemma that the upper bound> must be in L.Did your teacher explain why he asked you to deface your books? Asthere is no nonempty before chain, it's not needed beforepartially ordered set either; if the partially ordered set is empty,then the empty chain has no upper bound in L. As for in L, I tnkthe in wch takes care of that; the pronoun wch refers to L.Putting in wch at the front tells us that the chain and the upperbound are === for noviceGood book or source of info on Integrals for novice.I have to write a paper on integralsindefinite integralsfinite === Integrals for novice> Good book or source of info on Integrals for novice.I have to write a paper on integralsindefinite integralsfinite integralsWhat kind of paper? storical development? Theory of integrals at the level of freshman calculus? If you're more specific then people can make better suggestions. If it's supposed to be a storical type of paper, perhaps you could look at the development of integration from Eudoxus's Method of Exhaustion through Kepler's determination of the volume of a wine barrel to the way we tnk of integrals === convinced that a non-empty A can exist...>>Neither am I. But I'm also not convinced that it can't exist.>I'm not proposing an answer here, just asking a question. What goes>>wrong if A is the set of all transcendentals and B is everytng else?>I believe you lost sight of the original problem:>>A and B are two disjoint sets whose union is |R+. Both A and B are closed>>under sum and multiplication. Is it possible that neither A nor B is the>>VOID set?No, I haven't lost sight of it. I have just given two disjoint setsneither of wch is void. Maybe I didn't make it clear I want subsetsof |R+ so their union is |R+. To rephrase my question, why isn't thatan answer of yes to the question? I suppose the answer may be trivialbut I don't know that much about the algebra of === asking a question. What goes>>wrong if A is the set of all transcendentals and B is everytng else?> To rephrase my question, why isn't that an answer of yes to the === Candace Fiddler - December 17th 1979F I D D L E R6 9 4 4 12 5 18 = 58 On February 24th, 25th and 26th I went to Career Village at 229 4th Ave.South and met 6 people, Crystal was the 4th of these 6 people to providestats. Crystal is a nubile sweety and says that she is the only cld. Momis older than me by 115 days, I always consider it a positive sign when thenubile sweety has a parent that is older than me.173 Dianne 25 10 56 299/67 +115Dianne 47 Teresa 68 Fiddler 58187 Crystal 17 12 79 351/14 8338Crystal 98 Candace 31 Fiddler 58Primes Non-Primes 2 1 3 4 5 6 7 8 11 9 13 10 17 12 19 14 23 15 29 16 31 18 37 20 41 21 43 22 47 24 53 25 59 26 61 27 67 28 71 30 73 32 79 33 83 34 89 35 97 36 101 38 103 39 107 40 109 <-29th-> 42 --- ---1480 665 Mom was born on day 299, Crystal and her mom were born on days of themonth adding to 42 (29th non-prime). Crystal was born in 79 (Exodus 29).Crystal was born on the 29205th day of the century. Her first name adds to98, it's the 73rd non-prime (there are 73 verses in Bible Book 29). Hergiven names add together for 129 (the primes up to 29 add to 129). Her lastname adds to 58 (29+29), it is the 42nd non-prime wle 42 in turn is the29th non-prime. Mom and Crystal have names averaging 180, or 29 plus the29th prime (109) plus the 29th non-prime (42), or simply 29+29p+29np. Theyhave first names adding together for 145 (5x29). Their names add to the 15thprime, 49th and 42nd non-prime, the 73rd non-prime, the 11th prime and tothe 42nd non-prime, all together for 232 (8x29). In Crystal's given names,her 16 unrepresented letters exceed her 10 different letters by 109 (29thprime). The square valued letters in her given names add to 29. Herunrepresented letters exceed her represented letters by 79 (Exodus 29 and isher year of birth). The 115 days mom is older than me is a combination ofthe 73 verses of Bible Book 29 plus the 29th non-prime (42).Lucas 1 3 4 7 11 18 29 -- 73 <-the Lucas numbers up to 29 add to the 73 verses of Bible Book 29J O E L <-Bible Book 2910 15 5 12 = 42 <-29th non-primeC O P P E R <-29th element3 15 16 16 5 18 = 73 <-Book 29 and is the Lucas numbers up to 29, there is a copper riding a horse on the 1973 Canadian 25 cent pieceC E N T <-made out of 29th element3 5 14 20 = 42 <-29th non-prime Mom is 115 days older than me wle her given names add together for 115.Mom's first name adds to 47 (15th prime). Mom's common name adds to 105(Leviticus 15). Mom's initials add to 30 (15+15). Mom and Crystal were bornin years adding to 135 (9x15). Crystal's vowels add to 47 (15th prime). Imeet Crystal on the 24th (15th non-prime). Mom was born in 56 wle thereare 5x56 verses in Bible Book 15.Primes Non-Primes Lucas 2 1 1 3 4 3 5 6 4 7 8 7 11 9 11 13 10 18 17 12 29 19 14 47 23 15 76 29 16 123 31 18 199 37 20 322 41 21 521 43 22 843 47 <-15th-> 24 <-15th-> 1364 <-Jeremiah is Book 24 with 1364 verses Mom's given names add to 47 and 48 (a span of 22). Mom and Crystal wereborn in months adding to 22 and on days of the year adding to 650 (Proverbs22). Crystal was born in 79 (22nd prime). The vowels in Crystal's givennames add to 33 (22nd non-prime). The prime valued letters in Crystal'sgiven names add to 33 (22nd non-prime). Crystal was born on the 351st(117+117+117th) day of the year, pretty as there are 117 verses in BibleBook 22. The unrepeated letters in her given names add to the 117 verses ofBible Book 22. They were born on days 299 and 351, the latter is 117.39% ofthe former.Primes Non-Primes Fibonacci Lucas 2 1 0 1 3 4 1 3 5 6 1 4 7 8 2 7 11 9 3 11 13 10 5 18 17 12 8 29 19 14 13 47 23 15 21 76 29 16 34 123 31 18 55 199 37 20 89 322 41 <-13th-> 21 <-13th-> 144 <-13th-> 521 --- --- 238 154 <-Lamentations Mom was born on the 25th, corresponding to Lamentations with 154 verses(the first 13 non-primes). Crystal was born on the 17th, corresponding toEsther with 167 verses, it is the 39th (13+13+13th) prime and exceeds the154 verses of Lamentations by 13. Crystal was born in 79 (Psalm 79 is the13th chapter to contain the length of 13 verses). Her day, month and year ofbirth adds to 108 (the primes in prime positions up to the 13th prime). Shewas born on the 351st day of the year (Bible Book 13 chapter 13), it is thenumbers 1 through to 26 (13+13). Mom and Crystal were born in years addingto 135. Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 5 11 <- 11 6 13 7 17 <- 17 8 19 9 2310 2911 31 <- 3112 3713 41 <- 41 --- 108 Mom was born on the 25th wle mom and Crystal were born on days of theyear averaging 325, prettier as the numbers 1 through 25 add to 325.Primes Non-Primes 2 1 3 4 5 6 7 8 11 9 13 10 17 12 19 14 23 15 29 16 31 18 37 20 41 21 43 22 47 24 53 25 59 26 61 27 67 28 71 30 73 32 79 33 83 34 89 35 97 36 101 38 103 39 107 40 109 42 113 44 127 45 131 46 137 48 139 49 149 50 151 51 157 52 163 54 167 55 173 <-40th-> 56---- ----3087 1215 Mom was born in 56 (40th non-prime), her name adds to 173 (40th prime),prettier as I meet Crystal when I am 17173 days old. Mom was born in 56wle Crystal's first and last names add together for 156. Note that 40 plusthe 40th prime (173) plus the 40th non-prime (56) add to 269, wleCrystal's 187 valued name averages with her 351st day of birth for 269. Iwas born in 57 and I was born 269 days closer to the beginning of the yearthan to the end of the year (the 57th prime). Crystal and I were born ondays of the year adding to 399 (7x57). Mom and Crystal were born on days ofthe century adding to 49957. Mom and Crystal are an average of 35.7 yearsold (357 verses in Daniel). Crystal provided stats after Lance Lafond showedher patterns I found for m, pretty as Lance was born 19.57 weeks into theyear.389 <-77th prime104 <-77th non-prime 77 <-77---570 The Four 57'sGenesis 41 -> 41Leviticus 14 -> 104Judges 9 -> 220 <-I dreamt of 220 roofs blown 11 -> 1008 off homes in the Dakotas ---- 1373 <-220th primeChapter 57 is Exodus 7 with 25 verses Book 57 is Plemon with 25 verses -- -- 41st non-prime 16th non-prime <-together for 57->Major Books of End-Times Prophecy (Daniel and Revelation are in part about666 wle Isaiah contains 66 chapters):Daniel - 357 versesRevelation - 404 verses <-57 plus the 57th prime plus the 57th non-primeIsaiah - 1292 verses <-an average of 19.575757... verses per chapter187 Dar 17 2 57 48/317 00Daryl 60 Shawn 65 Kabatoff 62187 Marcia 6 8 80 219/147 8571Marcia 45 Veronica 87 Acevedo 55 If you people tnk that you have the right to use my abusive parents astools and arrest and torture me, then I tnk that I should have the rightto ask women to marry me, or to marry Marcia and me, our last names addtogether for the 117 verses of Song of Solomon, it's the Bible's Book ofLove. The nubile sweety was born on the 6th and has a 6 lettered first name.Isaiah is the Book with 66 chapters, pretty as it is Book 23, or the 6thprime plus the 6th non-prime (13+10=23). Isaiah 4, 12 and 20 (adds to 6x6)each contains 6 verses. Isaiah 4:1 is about Marcia (and me), and 6 otherwomen who are capable of feeling shame rather than pride, greed or lust, orwho limit their love for traditions and for people who abide by theirtraditions. You people have Egyptian penises on the roofs of your churchesand lined city streets with representations of penises, and had me torturedfor years for saying so, others just sat back in silence wle they weredoink ts to me, and similarly you remain silent and compassionless nowthat the arrests and torture have ceased. You people spent millions ofdollars having me tortured, and then annually you spend billions on yourdecorated trees, I begged and begged for assistance to ßee the country(they tortured me for years at the U of S) and you people are so cheap thatyou can't even offer to buy me a cookie when I bust my arse to show youevidence that your very name is a gift from God!!! If Crystal marries me,great, but if Marcia marries me and then Crystal marries Marcia and me, thenCrystal's mom is goink to win herself a sny new Cadillac unless she isCatholic!!! Or if Marcia and Bethany both marry me and then Crystal marriesMarcia and Bethany and me, then mom is goink to win herself a sny newCadillac even if she is Catholic!!! Or if Melinda Jarocki sleeps with me,then Crystal will win herself a used Pontiac.187 Dar 17 2 57 48/317 00Daryl 60 Shawn 65 Kabatoff 55-62249 Marcia 6 8 80 219/147 8571Marcia 45 Veronica 87 Acevedo-Kabatoff 55-62259 Bethany 28 10 76 302/64 7193Bethany 75 Ruth 67 Acevedo-Kabatoff 55-62246 Crystal 17 12 79 351/14 8338Crystal 98 Candace 31 Acevedo-Kabatoff 55-62288 Melinda 23 3 83 82/283 9530Melinda 58 Janelle 59 Elaine 46 Joyce 58 Jarocki 67 I have Scripture to support taking seven brides (Isaiah 4:1) and I haveScripture to support sleeping with Melinda Jarocki outside of wedlock (FirstKings 1:1-5), wle you people have a vast multitude of Scripturescondemning your decorated trees, phallic-capped churches and your violenceagainst me for daring to point out your pagan traditions. Good luck and mayGod bless you!!!Daryl Shawn KabatoffBox 7134Saskatoon SaskatchewanCanadaS7K 4J1Isaiah 45:4, Ephesians 3:15 - God gives you your === - September 20th 1981W H I T E F I S H23 8 9 20 5 6 9 19 8 = 107 On February 24th, 25th and 26th I went to Career Village at 229 4th Ave.South and met 6 people, Jaimie was the 5th of these 6 people to providestats. If the stats she provides are correct, then mom was just 14.28 and15.30 years old when she gave birth.107+ Dad 6107+ Mom 9 6 67 160/205 3764219 Jaimie 20 9 81 263/102 8981Jaimie 47 Lynn 65 Wtefish 107155+ James 29 9 82 272/93 9355James 48 Wtefish 107 Mom was born in 67 (19th prime), the 2460 verses of Bible Book 19 is7x19x19 minus the 19th prime (67). The kids were born in years adding to 163(the 38th or the 19+19th prime), Old Testament Book 19 and New TestamentBook 19 together contain 163 chapters (19+19th prime). Mom and the kids wereborn in years adding to 230 (Judges 19).Mom and the kids were born on an average of 77.29 years into the century(the primes up to 19 add to 77). The kids were born on days and in monthsadding together for 67 (19th prime). The kids were born on days and inmonths and years adding together for 230 (Judges 19). The kids were born ondays of the year adding to 535, corresponding to Psalm 57 (19+19+19). Thekids were born in 81 and 82, these are the 59th and 60th non-primes,together for 119. Mom and the kids were born on days of the month adding to58 (the 19th Book of the New Testament). The last name adds to 107, it's the28th prime wle 28 in turn is the 19th non-prime, Bible Book 28 is Hosea(the Mexican prophet) with 197 verses. Jaimie has 19 letters adding to 219.Her given names span a range of 19 and average 77 (the primes up to 19). Herlast two names add together for 172 (Deuteronomy 19). The Fibonacci valuedletters in her given names add to 19. Her first three vowels add to 19. Hergiven names end with letters of the alphabet that add to 19. Her repeatingletters add to 69.76% of her unrepeated letters, pretty as chapter 69 isExodus 19 wle 76 is 4x19. Her prime, square, cube and perfect valuedletters together exceed those letters that are not by 77 (the primes up to19). The kids were born on days 20 and 29, together these Bible Bookscontain 988 (19x52) verses. Brother James was born on day 272, it is twice136 (Numbers 19).Primes Non-Primes 2 1 3 4 5 6 7 8 11 9 13 10 17 12 19 14 23 15 29 16 31 18 37 20 41 21 43 22 47 24 53 25 59 26 61 27 67 28 71 30 73 32 79 33 83 34 89 35 97 36 101 38 103 39 107 40 109 42 113 44 127 45 131 46 137 48 139 49 149 50 151 51 157 52 163 <-38th-> 54 167 55 173 56 179 57 181 58 191 60 193 62 197 63 199 64 211 <-47th-> 65 Bible Book 38 (19+19) contains 211 verses (the 47th prime), wle Jaimieadds to 47 and her middle name adds to 65 (47th non-prime). Note that thereare 23 chapters in the Bible that contain the length of 19 verses and theyare in positions adding to 13519.The 23 19's 95 178 179 277 357 360 377 382 384 418 419 529 558 594 742 746 780 866 903 906 1136 1155 1178-----13519 Mom and the kids were born on days of the year adding to the 695 versesof Second Samuel. My dad is Samuel, he was born on the 355th day of theyear, pretty as The Samuels contain 55 chapters and 1505 verses. In leapyears (in years containing 366 days), Sam and I have our birthdays on daysof the year adding to the 404 verses of Revelation, Bible Book 66. Anyway,near the beginning of ts month I purchased 6 albino Corydoras catfish(wtefish).187 Dar 17 2 57 48/317 00Daryl 60 Shawn 65 Kabatoff 62187 Marcia 6 8 80 219/147 8571Marcia 45 Veronica 87 Acevedo 55 If you people tnk that you have the right to use my abusive parents astools and arrest and torture me, then I tnk that I should have the rightto ask women to marry me, or to marry Marcia and me, our last names addtogether for the 117 verses of Song of Solomon, it's the Bible's Book ofLove. The nubile sweety was born on the 6th and has a 6 lettered first name.Isaiah is the Book with 66 chapters, pretty as it is Book 23, or the 6thprime plus the 6th non-prime (13+10=23). Isaiah 4, 12 and 20 (adds to 6x6)each contains 6 verses. Isaiah 4:1 is about Marcia (and me), and 6 otherwomen who are capable of feeling shame rather than pride, greed or lust, orwho limit their love for traditions and for people who abide by theirtraditions. You people have Egyptian penises on the roofs of your churchesand lined city streets with representations of penises, and had me torturedfor years for saying so, others just sat back in silence wle they weredoink ts to me, and similarly you remain silent and compassionless nowthat the arrests and torture have ceased. You people spent millions ofdollars having me tortured, and then annually you spend billions on yourdecorated trees, I begged and begged for assistance to ßee the country(they tortured me for years at the U of S) and you people are so cheap thatyou can't even offer to buy me a cookie when I bust my arse to show youevidence that your very name is a gift from God!!! If Jaimie marries me,great, but if Marcia marries me and then Jaimie marries Marcia and me, thenbrother James is goink to win mself a sny new Cadillac unless he isCatholic!!! Or if Marcia and Bethany both marry me and then Jaimie marriesMarcia and Bethany and me, then James is goink to win mself a sny newCadillac even if he is Catholic!!! Or if Melinda Jarocki sleeps with me,then Jaimie will win herself a used Pontiac.249 Marcia 6 8 80 219/147 8571Marcia 45 Veronica 87 Acevedo-Kabatoff 55-62259 Bethany 28 10 76 302/64 7193Bethany 75 Ruth 67 Acevedo-Kabatoff 55-62229 Jaimie 20 9 81 263/102 8981Jaimie 47 Lynn 65 Acevedo-Kabatoff 55-62288 Melinda 23 3 83 82/283 9530Melinda 58 Janelle 59 Elaine 46 Joyce 58 Jarocki 67 I have Scripture to support taking seven brides (Isaiah 4:1) and I haveScripture to support sleeping with Melinda Jarocki outside of wedlock (FirstKings 1:1-5), wle you people have a vast multitude of Scripturescondemning your decorated trees, phallic-capped churches and your violenceagainst me for daring to point out your pagan traditions. Good luck and mayGod bless you!!!Daryl Shawn KabatoffBox 7134Saskatoon SaskatchewanCanadaS7K 4J1Isaiah 45:4, Ephesians 3:15 - God === ellipse from 4 pointsI have the position of 4 points on a 2D plane. The points are> unequally spaced.> Is there anyway I could fit an ellipse (or any other circular shape)> to these points (it has to pass through the 4 points)?FYI, ts is for an image processing algorithm. I have tried to use> Hermite Interpolation, but I can't seem to find a way to get the> tangent values at each of the 4 points so the curve looks like an> ellipse/circle.> Any help would be appreciated greatly. ,TejasYou can put a conic through any _five_ points (admittedly sometimes a> degenerate conic). The idea is to take the equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,plug in your n points, and get n linear equations for A,B,C,D,E,F. So> for four more-or-less random points in the plane, you'll get 4> homogeneous equations for the six variables, yielding a one> dimensional family of solutions.These will be (usually) either ellipses or hyperbolas. Since you want> an ellipse, you'll need to check wch of the equations in your> family, if any, are ellipses. You can write down a general condition,> or simply complete the squares to get an equation of the form aX^2 + bY^2 = c.If a,b,c all have the same sign, voila, an ellipse. If not, you have a> hyperbola. (You won't end up with the empty set from an equation like> X^2+Y^2=-1, since you started with some points on your curve.)JoeSThe test condition, on equations in the general form of formAx^2 + Bxy + Cy^2 + Dx + Ey + F = 0, is the value of B^2-4*A*C.Ts expression is negative for ellipses, zero for parabolas and positive for hyperbolas, though it does not guarantee existence of a non-degenerate conic. For example, x^2 + y^2 + 1 = 0 is a === from 4 pointsIn sci.math, Tejas unequally spaced.> Is there anyway I could fit an ellipse (or any other circular shape)> to these points (it has to pass through the 4 points)?> FYI, ts is for an image processing algorithm. I have tried to use> Hermite Interpolation, but I can't seem to find a way to get the> tangent values at each of the 4 points so the curve looks like an> ellipse/circle.> Any help would be appreciated greatly. > ,> TejasWell, the most straightforward (and hardest!) method would be totake the general equation of a conic section:Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0and come up with 8 equations in 6 unknowns. Not exactly the prettiest.It gets weirder as one should be able to use 5 factors:ax^2 + bxy + cy^2 + dx + ey + 1 = 0(where a = A/F etc.)but one requires 3 points to define a circle, so I'm obviouslymissing sometng degrees-of-freedom === Tejas> unequally spaced.> Is there anyway I could fit an ellipse (or any other circular shape)> to these points (it has to pass through the 4 points)? FYI, ts is for an image processing algorithm. I have tried to use> Hermite Interpolation, but I can't seem to find a way to get the> tangent values at each of the 4 points so the curve looks like an> ellipse/circle.> Any help would be appreciated greatly. , Tejas> Well, the most straightforward (and hardest!) method would be to> take the general equation of a conic section:> Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0> and come up with 8 equations in 6 unknowns. Not exactly the prettiest.> It gets weirder as one should be able to use 5 factors:> ax^2 + bxy + cy^2 + dx + ey + 1 = 0> (where a = A/F etc.)> but one requires 3 points to define a circle, so I'm obviously> missing sometng degrees-of-freedom wise.Pascal's theorem (about an inscribed hexagon) can be rephrasedto give a parametric expression for one point on the conic, interms of five others. Say A,B,C,X,Y are five points. The generalpoint Z is(A(w(CX)))(B(w(CY))where w is a variable line through (AY)(BX). In the improvisednotation here, the product of two lines is their point of intersection,and the product of two === from 4 points> a single circle requires 3 points, and ellipse requires 3 also.> with 4 points you can have a set of 3 such circles and average between them> to smooth it> same with ellipses.It takes 5 points to determine a general conic. If the four points areconsistent with an ellipse, there will be an infinity of ellipses goingthrough those points. Consider the four points (+/- 1, +/- 1).-- Daniel W. sonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/039 === spaces> 1. Let f,g be in L^p. Let ||_p be the L^p norm. How can I show that> |f+g|_p = |f|_p + |g|_p only if f or g are zero almost everywhere or they> are multiples of each other? I really don't know where to start.That | |_p is a norm is often derived from Holder's Inequality. When does equality hold in the latter?> 2. Let f be in L^infinity and L^p for some finite p. I'm trying to show> that lim as p -> infinity of |f|_p = |f|_infinity. I can show thatlim sup |f|_p <= |f|_infinityso all I need now is that lim inf |f|_p >= |f|_infinity. Is there a quick> way to do ts? Should I be tnking about ts in another way?nt: If t < |f|_infinity, then the integral of |f|^p is greater than (t^p)*(measure of {|f| > t}).> 3. Ts one uses Lebesgue measure on R. Let p be in (1, infinity). Find a> function f in L^p but not in L^q for q in [1,infinity) not equal to p. I'm> having trouble even believing that ts is possible. For some function f,> if |f(x)| > 1 for all x, then |f|^q <= |f|^p for q <= p so if f is in L^p,> then it will be in L^q, right? So the function we're looking for must be> one whose absolute value jumps back and forth above 1 and below 1 I> conclude. I also know that it can't be essentially bounded, right? Any> nts on what the function f I'm looking for would look like?nt: For q > p, work on (0,1). For q < p, work on (1,oo). On (0,1) it looks like sum (n=1,oo) c_n*[1/x^(p_n/p)] will work if p_n increases to 1 and the c_n are properly === anticlassicalist }{ ii: the spectre continues|: | > I know many models whose Heyting structure is far more simplistic|: | > than the corresponding Boolean embedding.|: ||: | Can you name them? Heyting algebras are always infinite, afaik.|:|: Note that simplistic means excessively simplified.|:|: Boolean algebras are a special case of Heyting algebras, and there are|: plenty of finite Heyting algebras even excluding finite boolean algebras.||Yes, additional structure can make these potential infinities collapse.Eh? It's not a matter of infinities collapsing.|I wanted to stress that both algebras are finitely generated, however,Both of what algebras? The Heyting algebras you describe as simpler thanthe Boolean algebras they embed in? I would guess that a finitelygenerated Heyting algebra need not embed in any finitely generated booleanalgebra.|and it|is only through the potential application of axioms over an infinity that|any infinities arise.I don't tnk the phrase application of axioms over an infinity carriesany specific meaning.|Ts, I feel, is why the notion of potential|infinity is so stressed in concstructive circles, because much of the|distinction in concepts only occurs over ts realm, the finite sharing most|properties with the boolean.The notion of potential infinity is not often stressed byconstructivists, in my experience. It's stressed by some people talking*about* constructivism, in an attempt to characterize the differencebetween constructivism and classical plosophy of mathematics.It sounds like you mean to say that finite Heyting algebras share somenatural properties with Boolean algebras. Perhaps you'd like to statesome?[...]|: | Constructivism has so many variants that, in order to study|: | them well, most systems are defined and studied using classical|: | means and classical tnking habits. In more vague terms:|: | Ôreasoning on the meta level is still classical'.|:|: I don't tnk pluralism is much of a reason, here. People talk about|: constructivism as though there were a lot of constructivists around,|: but really it's quite a small enterprise. Nearly all constructive|: mathematics (by wch I mean, mathematics that's done intentionally|: constructively, not merely mathematics that happens to be constructive)|: that's done is done in Bishop's constructivism plus perhaps a few added|: axioms. Markov's school is alleged to have used the assumption that all|: functions from N to N are computable, for example.||My first intention has not been to get mathematicians to work|constructively, only to teach the logical structure in an order of|increasing specialisation.I don't tnk ts project will get very far unless they see some kindof concrete payoff.|Heyting algebras are much more general than|merely models for constructive mathematics. They also underly a huge number|of models in the sciences, wch has been why I have listed so many of them.|In fact, with the theory of causal sets, we can attach a natural Heyting|structure to many theories that share the underlying causal structure.There are plenty of partial orderings you could cite, but it doesn't addup to much of a motivation to tnk Heyting with them.|: Part of what makes the situation confusing is that the ratio between|: ordinary mathematics done constructively and metamathematics about|: it is much lower than the ratio between ordinary mathematics done not|: bothering with constructivity and the metamathematics of that. It's to|: the point that Mathematical Reviews places constructive mathematics under|: the 03 (logic) category. You might, for example, wonder whether such|: tngs as -Lof's type theories count as counterexamples to my claim|: above. It's possible that somebody out there has been actually doing|: mathematics in them, but not as far as I know.||The Heyting structure extends to untyped lambda calculi as well, through|Curry-Howard, and CS students regularly study computability and mathematical|frameworks. Obviously, turing-completeness is often a requirement for any|new language proposed, and discovering turing-completeness in c++'s template|metaprogramming mechanism was a crucial step to modern generative|programming paradigms, so much of the focus is on the cut-elimination|operations and similar reduction theorems. However, illustrating reasoning|in terms of the logical structure is not as well taught. I liken ts to|the fact that the logical reasoning in quantum mechanics is rarely taught in|terms of orthomodular lattices, though doing so obviously prevents a lot of|the conceptual difficulties associated with quantum mechanics.Apparently you have some liking for theoretical computer science.You might want to have a look at Mackey's book on quantum mechanics inwch he presents it in terms of axioms about questions, wch amountto the closed subspaces of a lbert space.I'm afraid using orthomodular lattices doesn't really keep people fromone slit, or through both at the same time?, nor should it. As Feynmanonce said, all the weirdness boils down to just that one simple case. Ifyou can make coherent sense of that situation, then the rest falls intoplace. I don't doubt that orthomodular lattices are a useful concept, butI don't tnk it has much to do with the conceptual difficulties peopletend to have with quantum mechanics.|: If you were to count as schools of classical mathematics all the|: different nonconstructive formal systems in wch one could do|: mathematics, the number would hugely exceed the number of constructive|: formal systems. Even if you were to restrict yourself just to classical|: theories in wch _some_ mathematics has actually been done, you can find|: set theorists who've taken as their starting points initial assumptions|: of varying strengths.||I believe ts narrows the applicability of the logical structure far too|much.Please remember that ts was all in response to the remark that if youuse a constructive foundation, you suffer from a wealth of alternativeconstructivisms. Obviously applying a logic foundationally is not theonly way to apply it, or else logicians would have little to talk about.There's a world of difference between saying you have some kind of cute,simple algebraic structure that appears often, and saying that you havea fundamental concept.|There is some type of universality in Heyting structures not shared|by the Boolean that allows it to model propositions of a huge variety. Many|of the early foundationalists saw ts and did much research in the area|(Kleene, Tarski, etc.). With the semantical identification with S4, one|finds a deep identification with notions of possible worlds, descriptions of|necessity, and the basic modality of science, computability, and proof.|Certainly, as well, proof theory today is ghly inßuenced by its Heyting|structure, and I am looking for why ts is not so in more fields that|implicitly have the structure ding away in their analyses.Best of luck with your investigation. But it seems to me that you'rebeing a little too glib with the attribution of deep identification.I once saw a book that argued for the existence of God on the basis thatthe golden ratio (sqrt(5)+1)/2 occurred in a supernatural profusion. Itnk they failed to appreciate the extent to wch ts occurred simplybecause it's a root of a very simple equation, x^2=x+1.In your examples, you mention a number of cases where there's an obviouspartial ordering in the background, and the Heyting algebra structurecomes from it. Sure, ts is commonplace, but where does that reallyleave us? For a concept to be really fruitful or deep, it has to bemore than just frequent in occurrence. If you err too far on the side ofgenerality, you wind up with many examples, but not being able to addmuch content to any of them.On the other hand, if you managed to uncover a good constructive theoremor a few, you might really have sometng.[...]|: Nevertheless, I'd just as soon not have someone trying to get people|: interested in it in the manner Galathaea has been trying to.|:|: The excessive cross-posting is a bad sign. The fact alone that one has|: had to try to justify it almost always means that one has gone too far.|: And excessive cross-posting usually means that someone feels entitled to|: grab attention at others' expense.||Do you disagree with any of the points I have made in the towards a|constructive education or more focus... posts? Or do you believe that I|have in some other way violated the constraints of the groups' topicalities?I consider them only very poorly topical in the majority of the groups towch you posted them. I tnk perhaps you are rather generous in generalin attributing mutual relevance to ideas you like, for one tng.Remember that the intended purpose of these rules is to make it easier forpeople to read what they want to and not be bothered with what they don'twant to. I tnk you've approached ts in a manner much like the typicalexcessive crossposter does. They usually hugely overestimate the interestthat their particular message will have for the readersp of the groupsthey're posting to (or don't care).People post stuff to sci.logic about arguments for or against theexistence of God, with the explanation that they're interested indiscussing the logic of the argument or seeing experts in logicexpose the fallacies of their opponents. People will likewise startcrossposting threads to sci.math in wch numbers have some relevance,when pretty clearly that's not enough.The important question is not simply are there questions of logic here?or are there mathematical questions here? and so on. The question is,is ts the kind of tng that a person who would read that group wouldbe especially likely to be interested in reading?. It's not some abstractconceptual problem; it's a practical matter of trying to keep some kindof loose order in order to make it easier for people to read what theywant and avoid what they don't want.Whether you realize it or not, the fact that a person reads sci.lang isreally poor evidence that they'd find ts kind of thread interesting.|What I have tried to find is people in all of these groups from several|directions (wch I have worked hard to detail), to see what that|communities ideas are concerning the education proposal, because it is a|fractured and disparate community wch I felt might share a common goal.Often the best approach is to make a brief mention in various groups thatyou intend to start a discussion somewhere else, and then leave thosegroups alone. If anybody actually is keenly interested in reading it, theywill find it easy to subscribe to the group in wch the discussion is nowtaking place.|I have seen many pleas against the cross posting. None of them have been|very convincing in my opinion, since I have made it quite clear the points|of topicality I want to discuss.Yes, what *you* want to discuss. If they don't have much interest, theyget to keep deleting it as postings generated from other groups keepshowing up.|Often these have been from people who|admitted they were unfamiliar with the actual work in the topic they were|attempting to defend, and usually they were uniterested in making any effort|to learn about it. All of my main posts have worked to make ts absolutely|clear.|Massive crossposting has been dubbed a message around more than ten timesthe number of places where it really belongs, though, falls into a commoncategory, however you want to term it.|: I would generally advise against being a self-proclaimed liar, even if|: ts is meant in a humorous way (wch I don't know).||Its just a fact that many psychologists have verified that most people|(percentages close to unity) lie in their life. I've done it.Calling someone who lies to only the usual degree a liar without morecontext is a little bit like calling someone a tennis player becausethey have, at some point, played tennis. I am not a badminton player.I am someone who's at some time played badminton.If there's notng more to it than ts, then I'd have to say it'sa bit of not especially amusing wmsy.| I've written|stories. Fiction. I like the idea of a fiction lying itself into reality,|much like the mythology of galathaea. But also like a scientific model,|wch never knows itself to be true but seeks justification.Do you actually have a scientific model, or just a sketch?|I write it in my signature to annoy those who cannot get over it. Its an|annoyance they will have to carry with them until they forget my signature|(rather transitory unless they keep reading my threads), or until they|accept at a much more fundamental level the metaphor and fiction that|underlies their entire perception of the world and methods of modeling it.Count on their either (a) ignoring it, or (b) deciding you're sort of anannoying person but not otherwise bothering with it.|: Galathaea seems to me to be one of the people Barabara Sher, the career|: counsellor, calls a skimmer, as opposed to a diver. A skimmer deals|: with more tngs in less depth; a diver deals in fewer at greater depth.|: There's notng necessarily wrong with being a skimmer, but it seems to me|: that there's a kind of effort required to be a skimmer who makes an actual|: contribution, rather than just being the dilettant and tossing around|: stuff you've heard about. As someone who's more of a diver, I'm not all|: that good at advising someone how to be a good skimmer. The advice I'm|: tempted to give is basically to be more like a diver: stick to specific|: topics long enough to be sure you actually have sometng in your hands!||Be careful here. Ts is not a very good distinction for how I explore|topics. I am an obsessive reader, going through anywhere from 800 to 1500|pages a week, with copious notes, etc.Yes, very much the sort of person described by ts author.|A lot of my research has focused on|structural analyses of topics and foundationalist approaches,Vague.| and so I have|had to skim huge pantheons of objects to become familiar with the various|territories. However, my learning model includes going deeper and deeper|into the topics I feel need most exploring to understand the structural|questions I want to answer. For example, I bohminised Witten's cubic|bosonic string model during my analysis of extensions of realist ontologies|of quantum mechanics in order to demonstrate to myself that some of my|notions concerning the isomorpsm of Bohm and its relation to quantisation|could carry over to some modern theories. I have done original research in|the study of functors from the category of Poisson manifolds to the category|of lbert spaces. I have derived results on the combinatorial enumeration|of certain magma types. Many would not consider these types of calculations|to be that of the skimmer type, and the classification is often used in a|derogatory way.I mention the source partly to indicate that it isn't meant asnecessarily derogatory. Her point was to help people realize that theymight be more one way or the other, and that they should quit trying toapply a style that doesn't really suit them, and instead develop whatevertheir own appropriate style is. On the axis between believing in one rightway to do everytng, and believing we each should do our own tng,ts author certainly would fall well toward the liberal end.I mention it as a way of indicating that I also accept such differencesas naturally present, and don't intend to count it against someone ifthey merely have a different style from mine.On the other hand, your postings I've seen so far have been far fromdense in content. I suspect you tnk they are, because you count ascontentual remarks that are only suggestive.|: It seemed to me that a lot of the examples, and maybe all of them, of|: tngs whose logic is constructive (whatever that is supposed to mean,|: specifically) that we've seen here, are just special cases of the|: topological interpretation. Ts business about perception of the letter|: W, for example; you can dress it up in the language of basins of|: attraction for the dynamics of your visual cortex or whatever, but it|: still boils down to talking about open sets of stimuli that get perceived|: as W. Idealizing tngs a bit, one could say that the complementary|: perception, that sometng is not a W, also corresponds to an open set.|: Then since there are borderline cases, perception either as W or not W|: doesn't cover all possible cases. It seems to me that Galathaea's|: description made it sound rather more mysterious than it is.||The region-connection calculus is more developed than that, as are analyses|of pattern recognition and the classification problem, so I don't quite|agree here.But is it developed on the basis of its properties as a Heyting algebra?It's easy to name rich structures that are also examples of some simplealgebraic structure like a commutative ring. Whether they actually serveas examples in wch the fact of its being a ring is interesting is adifferent story.[...]|: A lot of the discussion I've stayed out of just because there doesn't seem|: to be all that much content in it. Let's please knock it off with the|: massive cross-posting and deal more patiently with the various topics one|: at a time.||Most of own|newsgroups, not asking for intelligent discussion, just beyond all sense. Spammingimplies wide distribution.It's often a convenient excuse by massive crossposters that they'renot to blame for the resulting ßood of irritation.|I am always eager to go into more depth as time|permits me, and I have been struggling to give myself more and more time as|the questions turn more and more to a technical nature.||: If someone wants to cp in on the mathematical side of constructivism,|: try helping me satisfy some of my curiosity. I've had the question of the|: degree to wch the Jordan-Holder theorem is constructive on the back|: burner for a long time. It's easy to see that the fact that any two|: decomposition series have a common refinement is constructive. But then|: given two decompositions with simple quotients, it's not clear to me that|: we should be able to get isomorpsms between them in some order. We can|: get a common refinement where not all the quotients are nontrivial, but|: we have no way in general to determine whether a quotient group is|: trivial. On the other hand, I haven't thought of a good counterexample,|: either.||So are you questioning the second isomorpsm theorem of groups as not being|constructive?The second isomorpsm theorem says that if H and K are subgroups of agroup, with K normal, then HK/K is isomorpc to H/(H^K). That's the oneused in the theorem that any two decompositions (whether with simplequotients or not) there's a common refinement. Those are constructive.Given an element of HK/K, it's represented by an element of HK, and thefactor in H is a representative of the corresponding element of H/(H^K).Converly an element in H representing a class in H/(H^K) represents theclass in HK/K that maps to it. There's no problem there.|Although I haven't explored ts before, I never noticed anytng ding in|there that wasn't extendible to constructive definitions of groups or made|use of bivalence. I thought it was a simple application of intersections|and joins, but now you have me intrigued. Would you like to expand on ts?Take the special case of a group with two decomposition series of length2: suppose I have a group G with two simple normal subgroups N1 and N2with simple quotients G/N1 and G/N2. I can write a common refinement ofthe two decompositions either as 11, then y can be expressed as a product of conjugates of x and x^{-1}.|: I tnk that's a pretty strong assumption, but not crazy. I'm not sure|: for instance whether it holds (constructively, of course) for the|: classical simple Lie groups.||Usually, I find it is more natural to approach problems like ts in the|opposite direction when looking for constructive deductions.After reading the following, I wonder in what sense you tnk ts isopposite to my approach. A sort of generators-and-relations approachto constructing examples is often good. I tnk it makes sense also totry examples based on infinite alternating groups and the like.|In other|words, I would define those groups first with a distinguished element x and|all elements that can be constructed as products of conjugates (through|other constructed or defined elements) of x and its inverse. Ts class of|groups is quite large. Then look at the structure required to|constructively prove elements apart from x that have ts conjugation|construction equal to the entire group as well. For finite groups, of|course, ts can be carried out to completion constructively. For infinite|groups, of course, its much more difficult, although the relationsp being|implied is finite between all elements. Constructing the elements to|conjugate through would get you there, though.Ts is fine; I just don't know as much about infinite simple groups asI'd like. One really wants sometng like the possibility of makingslight deformations in the subgroups, so that they're not too easilydistinguished up to isomorpsm.|I tnk ts approach is very basic to the logic that I would desire being|taught more. Ts is the computational approach so inherent to|construction, that you build the structures you desire to study through|finitely axiomatising the definitions and deduce constructive consequences,|wch does oppose the infinite axiomatics underlying certain classical|constructions. It is very much the difference between bottom-up and|top-down approaches.I'm not convinced there's a serious contrast. I tnk serious researchersin these areas do combinations of what you would call top down and bottomup.|I know ts answer is kind of vague, but I would need to figure out better|what the obstructions are to such constructions, and I have not been|tnking of Jordan-Holder and similar theorems in a constructive light yet,|so I will need to revisit some of my materials.|| for your interest, by the === majors?> If you haven't already, get some of the easier topics out of the way by> studying on your own: linear algebra, probability, number theory, set> theory, as much calculus as you can stand. If you don't find these topics> easy, change to accounting.I don't tnk accounting is easy--don't accountants have to study quite abit of tax law? Tax law can be very confusing. I've never been so confusedby a handful of numbers as when I took a course in partnersp taxation inlaw school.-- === you haven't already, get some of the easier topics out of the way by> studying on your own: linear algebra, probability, number theory, set> theory, as much calculus as you can stand. If you don't find thesetopics> easy, change to accounting.> I don't tnk accounting is easy--don't accountants have to study quite a> bit of tax law? Tax law can be very confusing. I've never been soconfused> by a handful of numbers as when I took a course in partnersp taxation in> law school.Of course. I should have mentioned logic. If you are have no trouble withlogic (the mathematical kind, at least) then avoid === finite trig sumIgnacio Larrosa Ca.96estro> Hammick escribi.97:> If r is an odd number >=3 then> (*) sum[k=1 to r-1] 1/sin^2 (2pi k/r) = (r^2 -1)/3....> S = Sum(1/sin^2(2pi k/r), k, 1, r-1) = 2Sum(1/sin^2(2pi k/r), k, 1,(r-1)/2)> (r is odd)> Let x = 2pi/r and t = sin^2(x). You has that> 1 + 0*i = (cos(rx) + i*sin(rx)) = (cos(x) + i*sin(x))^r> Taking the imaginary part on both sides,> 0 = C(r, 1)cos^(r-1)(x)*sin(x) - C(r, 3)cos^(r-3)sin^3(x) + ... +> (-1)^((r-1)/2)sin^r(x)> dividing by sin(x),> 0 = C(r, 1)cos^(r-1)(x) - C(r, 3)cos^(r-3)sin^2(x) + ... +> (-1)^((r-1)/2)sin^(r-1)(x) (#1)> All exponents of cos(x) and sin(x) are even. Then the RHS can be write asa> polynomial in t. Let s = (r-1)/2. Then you get> 0 = A(0) + A(1)t + A(2)t^2 + .... + A(s)t^s> The roots of ts polynomial are the values of sin^2(2pik/r), with k = 1,> ..., s = (r-1)/2. Note that sin^2(2pk/r) = sin^2(2pi(r-k)/r).> Let t = 1/u and multiply by u^s> 0 = A(0)u^s + A(1)u^(s-1) + A(2)u^(s-2) + ...+ A(s-1)u + A(s)> The roots of ts polynomial are the values of 1/sin^2(2pik/r), with k =1,> ..., s. By Cardano-Vieta relations, the sum of all roots is> S = 2*(- A(1)/A(0))> A(0) = C(r, 1) = r> A(1) is the exponent of sin^2(x) in #1> A(1) = - C(r,1)(r-1)/2 - C(r, 3) = - r(r + 1)(r - 1)/6> Then S = (r + Larrosa Ca.96estro Iganacio. Nice piece. Sorry I was slow to respond; ts computerhas SARS or Bird Flu or === ontology: (was: the anticlassicalist }{ i: linguistic negation)> No buddy no.You need to learn when to use commas: No, buddy, no.> [...]> There you goComma, dammit, comma!sczophreniaComma, dammit, comma!What is it with you krauts and punctuation? What is the use of commasif your everytng-must-have-its-order-don't-mess-with-the-rules-perfec tl y-squarebrains already know where they are going to go? Why bother to put themin at all? Punctuation can be very useful for controlling the ßow ofverbalization.> lowers mself to talking with a Cro-Magnon like you. I mean, you'reHey! I'm not the French guy here, that was Jaques. I'll acceptPiltdown though.> Caps, cabr.97n! First Amendment.Cabr.97n el puto de tu padre, juelacienmilputa. > It's notng personally, Herr von Kackpott. I just didn't take my> anti-misanthropy pills ts week and see all these assholes in variousOne tng in common at === ontology: (was: the anticlassicalist }{ i: linguistic negation)[...]> Cabr.97n el puto de tu padre, juelacienmilputa.Your Spanish sucks.[...] > man sieht === Re: Clean coordinates on a unit sphere> Your mistake is with your infinite tight winding idea. I don't tnkthat> the idea can be made rigorous enough to work. Either it goes north byZERO> from one turn to the next... wch is not very useful. Or, it goes northby> some positive amount... and we miss some points.> If it goes north by an infinitesimal amount, do we miss any points?> I've also been banging my head against the wall for the 2d analoge> problem of filling a rectangle with a single thread. Imagine the thread> starts in the top left corner, then wraps in a zig zag over the page. It> finally ends in the bottom right. The parameter value at the top left is> 0, bottom right 1. Centre of the rectangle is 0.5. In fact, all points> corresponding to 1/(a*2 + b*2^2 + c*2^3 + ...) where a, b, c... are> either 0 or 1, lie on the diagonal line.I do not understand that.> Shouldn't there then be a mapping that associates each real number with> a 2d coordinate? Is ts effectively the same as interleaving the two> parameters of cartesian coordinates into a single parameter (or> concatenating, or any other jumbling of digits)? If yes, why?You may want to have a look at some of the links athttp://mathworld.wolfram.com/Space-FillingFunction.html-- === coordinates on a unit sphere> You said that bijection was exactly what you meant. Cardinality is> defined in terms of bijections.I tnk I understand now - yes, I wanted a bijection = conservation of cardinality.> You were the one who objected that the representation of the poles is> non-unique. That is, the north pole is represented by (x,pi/2) for any x> in [0,2pi). So there are infinitely many points on the edge of the> rectangle that all map to a single point at the north pole. That is, the> mapping from edge to pole is not a bijection, and in fact the sets have> different sizes (cardinalities). The fact that cardinality is is not> preserved by ts mapping is exactly what you were objecting to. You could restrict the mapping to [0,2pi) x (-pi,pi), and throw in the> single points at (0, +/- pi). That would make the mapping a bijection,> but somehow I have the feeling that it still won't satisfy you.Yes, I'm difficult to please... the exception at -pi and pi just doesn't meet my gh expectations ;)I tnk Glynne's posting is more what I'm after... but a bunch === unit sphere> - I'm a computer science student working in the field of vision and> grapcs. I'm looking for a clean way to represent a _direction_ in 3d> space (ts can also be imagined as a position on a unit sphere). I don't like the traditional methods of using a 3d vector or spherical> coordinates for the following reasons: - 3d Vector e.g. (x, y, z): Ts is of course very common - can be> rotated with matrices, quaternions... but it also describes length,> wch I don't need. - Spherical coordinates e.g. (theta, p): Ts is more appropriate as> it does not describe length. But it still appears ugly to me since> multiple coordinate pairs can describe the same point e.g. (x, pi/2). Is there a better way? I'm sure I'm not the first who asked ts> question. It seems similar to the problem of creating a 2d map of earth,> so it's probably been thought about before... And since there doesn't> seem to be a _popular_ solution, it's probably hard - but that's what> makes it FUN !!! My ideal solution would map a unit sphere with a uniform density (if> that makes any sense to any one). Could a person use some type of> tesselation of a sphere with an arbitrary density??? Any ideas are good ideas! Please keep the language simple - I'm not as> smart as you. in advance, Nathan> for all your comments so far! My density criterion is not well> defined obviously, mostly because I'm not even sure what exactly I want.> One way I was tnking about it was, that by increasing the parameter(s)> with a constant step, and drawing the according points on the sphere,> the resulting points should be uniformly distributed.> A strange, possibly dumb question: Are at two or more parameters> required? Is it possible to map the surface with a single parameter (or> maybe 1.5 ?) Maybe real numbers are not appropriate for what I'm> striving towards?> Here's a _crazy_ idea that I had along the way (trying to map the sphere> surface with single parameter):> Imagine the south pole being represented by 0, the north pole being> represented by 1. Now wind a path from one pole to the other around the> sphere in a spiral shape. With an infinite tight winding, every point on> the sphere is covered. The point corresponding to the parameter value of> 0.5 is on the equator and so on. _Obviously_ we can now map all points> on the sphere with a single parameter. Where's my mistake?> Hope you have some fun with my insanity,> NathanHave you considered a erarccal Triangular === coordinates on a unit sphere> Have you considered a erarccal Triangular Mesh?> http://www.sdss.jhu.edu/htm/> ~GlynneThere - that's the picture I had in the back of my head when I was talking about Could a person use some type of tessellation of a sphere with an arbitrary density???But now I have ts strange suspicion that the mapping from cartesian or spherical coordinates to the HTM might not be as easy as the === rightward strings tacked-on to p-adics serves as Orthogonality and makes Doubly-Infinites the points of Lobachevskian GeometryWhat I am trying to convey is that P-adics are infinite leftward> strings and they are all positive numbers. But in Doubly Infinites,> because you have a infinite rightward string attached to a infinite> leftward string, that the rightward string concaves the path. So that> where the P-adic would form a convex lens surface, the Doubly-Infinite> forms the opposite surface of concave and thus the opposite sign of> negative vice positive.> In p-adics of 10-adics take for example:......999997 and .......99998both are positive numbers and both have a Riemannian curvature because the entire 10-adics bendback around joining up at 0. (Gee, I wish Abian were still around forI would love to see s input).But now let us look at several Doubly-Infinites near the two 10-adicslisted above.Consider these Doubly-Infinites:.....99997.34343434..........99997.5555555.... ......99997.6700000.........99997.75000..........99997.888888. ........999997.900000.........999998.0000000010000.....The talk of convex curvature as Riemannian formed by the P-adics andthe talk of concaveness that is Lobachevskian geometry and wch Iintuit that the Doubly-Infinites form. Since p-adics are only infinitestrings leftward they form convexity shape. Doubly-Infinites also havea leftward infinite string but it is the rightward infinite stringthat creates an Orthogonality to the finality of where the point endsup. When you tack on a infinite rightward string to a p-adic itcreates an Orthogonality and changes the convexity into concaveness.It thus alters the positive sign value of what the p-adic possessedand transforms it into a negative sign value because sign value ismerely a means of signifying a 180 degree change of direction.Concaveness is 180 degree change of directionSo if we consider purely the P-adic of its infinite leftward string itforms convexness like a convex lens. But if you take the same p-adicsand tack-on infinite rightward strings then they serve asOrthogonality and the entire group of Doubly-Infinites become aLobachevskian surface or curve.Arcmedes Plutoniumwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are galaxies(www.iw.net/~a_plutonium) website of the science of AP under revisionwhat used to be my old science websitewww.newphys.se/elektromagnum/physics/LudwigPlutonium === the perception of colour?... One can be looking> at a green whose wavelength mix stimulates mostly your green receptor cones,> or alternatively at one wch stimulates just the correct mixture of red> cones and blue cones, and get the same effect. Side by side you'd swear> they were the same hue, saturation, luminosity, everytng.Bill's posts are usually very wise, but I'm afraid s CRT needs degaussingor sometng. Red + Blue gives Purple. . . . . . .Here's an interesting fact about human vision. Before leaving retina,the three cone responses are transformed to three other coordinates: Luminance <-- Red_cone plus Green_cone (plus Rods if dark) Red vs Green <-- Red_cone minus Green_cone Blue vs Yellow <-- Blue_cone - (Red plus Green)Ts helps explain why many applications deal with 4 primary colors(a plane has four compass points: R,G,B,Y) wle others dealwith 3 primaries (3 points define a plane).Note that blue cones do not contribute to perceived luminance.(Blue-light === Re: Bayesian Class and Math/Stat Teacng Techniques> [...] Why is a degree called Masters in Applied Statistics> so heavy in theory? Ideally, you study the theory so that you can invent the practicesas they are needed for particular problems.I'm a big fan of theory, in general, although I have to say that the theory of statistics as it is conventionally taught isthe kind of stuff that gives theory a bad name.> I'm not interested in theory in the absence of application. > I enrolled in Masters in Applied Statistics to learn how to> use statistical techniques to solve real world problems, how > to use statistical software to solve real world problems, etc. [...]My advice to you is to get a book you're comfortable with and maybesome software you're comfortable with, and start working throughsome problems on your own. Working on problems will help motivate later study -- working on a problem that you can't solveis actually very good, since you'll be able to recognize a solution when you see it.It's better if you make up the problems yourself -- textbookproblems are phrased in a way that supposes one will apply theright technique, i.e., whatever was covered in the chapter.You'll learn more by finding & stating problems in the wild.For what it's === Math/Stat Teacng Techniques> One of my avocations is evaluating the work of bozos > like you in lawsuits.> Why do tnk I'm a bozo?It looks like he just felt the need to call -somebody- a bozo.You just happened to get in the way.It makes a man feel so good -- BOZO, BOZO, BOZO! There, I said it! I feel better already.-- Robert Doctor of Bozology Dodier