mm-376 > http://qedcorp.com/destiny/ > under construction > DARK MATTERS SURROUND DARK ENERGY > with my comments, additions and physics corrections > Two big stories from the world of physics may portend the arrival of > new weapons of mass destruction far more powerful and compact than > atomic bombs. In recent years it has been discovered that our universe > is being blown apart by a mysterious anti-gravity effect called dark > energy. Mainstream physicists are scrambling to explain this mysterious > acceleration in the expansion of the universe. Some physicists even > believe that the expansion will lead to The Big Rip when all of the > matter in the universe is torn asunder - from clusters of galaxies in > appears to be made of two unknowns - roughly 23% is dark matter, an > invisible source of gravity, and roughly 73% is dark energy, an > invisible anti-gravity force. Ordinary matter constitutes perhaps 4 > percent of the universe. Recently the British science news journal New > Scientist revealed that the American military is pursuing new types of > exotic bombs - including a new class of isomeric gamma ray weapons. > That was an original idea of mine in 1963 at Cornell and I discussed it > with Hans Bethe. That is one of the reasons Ron Bullough invited me to > Harwell in 1966. No doubt others thought of it but probably later. I > thought of it while at Tech/Ops in Lexington, Mass working for George > Parrant Jr. > Unlike conventional atomic and hydrogen bombs, the new weapons would > trigger the release energy by absorbing radiation, and respond by > re-emitting a far more powerful radiation. In this new category of > gamma-ray weapons, a nuclear isomer absorbs x-rays and re-emits higher > frequency gamma rays. The emitted gamma radiation has been reported to > release 60 times the energy of the x-rays that trigger the effect. > Gamma-ray weapons could trigger next arms race 19:00 13 August 03 > Exclusive from New Scientist Print Edition. Subscribe and get 4 free > issues. > An exotic kind of nuclear explosive being developed by the US Department > of Defense could blur the critical distinction between conventional and > nuclear weapons. The work has also raised fears that weapons based on > this technology could trigger the next arms race. The explosive works by > stimulating the release of energy from the nuclei of certain elements > but does not involve nuclear fission or fusion. The energy, emitted as > gamma radiation, is thousands of times greater than that from > conventional chemical explosives. > The technology has already been included in the Department of Defense's > Militarily Critical Technologies List, which says: Such extraordinary > energy density has the potential to revolutionise all aspects of > warfare. Scientists have known for many years that the nuclei of some > elements, such as hafnium, can exist in a high-energy state, or nuclear > isomer, that slowly decays to a low-energy state by emitting gamma rays. > For example, hafnium-178m2, the excited, isomeric form of hafnium-178, > has a half-life of 31 years. The possibility that this process could be > explosive was discovered when Carl Collins and colleagues at the > University of Texas at Dallas demonstrated that they could artificially > trigger the decay of the hafnium isomer by bombarding it with low-energy > X-rays (New Scientist print edition, 3 July 1999). The experiment > released 60 times as much energy as was put in, and in theory a much > greater energy release could be achieved. > http://www.newscientist.com/news/news.jsp?id=ns99994049 > I was thinking in 1963 of a gamma ray laser pumping a nuclear isomeric > transition. Bethe at the time said it wouldn't work and basically > discouraged me working on it. > Bekkum continued: > In the summer of 2000 I contacted Nick Cook, the former aviation editor > and aerospace consultant to Jane's Defence Weekly, the international > military affairs journal. Cook had been investigating black budget > super-secret research into exotic physics for advanced propulsion > technologies. > Uh Oh :) > I had been monitoring electronic discussions between various American > and Russian scientists theorizing about rectifying the quantum vacuum > for advanced space drive. Several groups of scientists, partitioned > into various research organizations, were exploring what NASA calls Breakthrough Propulsion Physics - exotic technologies for advanced > space travel to traverse the vast distances between stars. Partly > inspired by the pulp science fiction stories of their youth, and partly > by recent reports of multiple radar tracking tapes of unidentified > objects performing impossible maneuvers in the sky, these scientists > were on a quest to uncover the most likely new physics for star travel. > The NASA program was run by Marc Millis, under the Advanced Space > Transportation Program Office (ASTP). Joe Firmage, a Silicon Valley > entrepreneur, who at age 28 had found risen to CEO of a three ion > dollar internet firm, began to fund research in parallel with NASA. He > hired a NASA Ames nano-technology scientist, Creon Levit, to run the International Space Sciences Organization, > Joe did that because I suggested it. I introduced Creon to Joe. > Cook was intrigued by the apparent connections between various private > investors, defense contractors, NASA, INSCOM (American military > intelligence), and the CIA. While researching exotic propulsion > technologies Cook had heard rumors of a new kind of weapon, a sub-quantum atomic bomb, being whispered about in the dark halls of > defense research. > I think that must have come from me regarding J. P. Vigier's tight > atomic states with experiments in Beograd by Z. Maric and G. Dragic. > But how did Cook hear about that? We brought Vigier to ISSO in San > Francisco several times along with physicist Gennady Shipov from Moscow. > That story with photographs of Vigier and the group is in my > autobiography Destiny Matrix. Dragic A, Maric Z, Vigier JP; Phys. > Lett. A 265 (2000) 163. New quantum mechanical tight bound states and > 'cold fusion'. Creon Levit and Vigier met with Maric in Budapest in 2000. > Bekkum who is one of my on line students continued: > Sub-quantum physics is a controversial re-interpretation of quantum > theory, based on so-called pilot wave theories, where an information > that the predictions of ordinary quantum mechanics could be recast into > a pilot wave information theory. Recently Anthony Valentini of the > Perimeter Institute has suggested that ordinary quantum theory may be a > special case of pilot wave theories, leaving open the possibility of new > and exotic non-quantum technologies. Even thought rumors of a > sub-quantum bomb may be purely fantasy > It's not fantasy. It might not work, Maric and Dragic in Beograd, while > not ostensibly trying to make a weapon by any means, were trying to test > Vigier's basic theory of the spatially extended electron, which I think > is basically a correct idea and fits my own ideas including why the > electron appears to shrink to less than 10-16 cm under high resolution > imaging (i.e. scattering) and how the electric charge distribution is > contained by the strongly attractive zero point energy exotic vacuum dark matter core (Abraham-Becker-Lorentz-Poincare stress problem of > 100 years ago). This only works in the Bohm hidden or extra variable > interpretation. That is, a classical spatially extended electric charge > distribution is unstable. It explodes under its own self-repulsion. This > is why physicists had to postulate a point electron because they did not > understand that the strong gravity attraction of the positive zero point > pressure in a possible state of exotic vacuum would hold the charge > together. As Herbert Frohlich told me at UCSD in La Jolla in 1966 the > basic thing wrong with physics is the idea of the point electron. The > bad idea of the point electron gives the infinite energy in quantum > electrodynamics. Richard Feynman told me in his office at Cal Tech in > 1968 that infinite renormalization is a shell game, and it is a scandal > in physics that no one could do better than what he had done. They did > not know 100 years ago that 1/3 or so of the universe was this kind of > exotic vacuum. For example, there is a huge sphere of exotic vacuum of > w = -1 positive pressure that holds our galaxy together preventing our > solar system from going off into space on its own. This sphere looks > like w = 0 cold dark matter from our vantage point. What works on this > large scale also works on the small scale of the single electron (and > all the charged lepto-quarks). A neutrino has some mass and is simply a > micro-geon of pure zero point energy with positive pressure. > there is no question that physicists seriously contemplate a phase > transition in the quantum vacuum as a real possibility. The quantum > vacuum defies common sense, because empty space in quantum field theory > appear and disappear far too quickly to be detected directly, but their > existence has been confirmed by experiments that demonstrate their > influence on ordinary matter. > A major component of the physical quantum vacuum consists of virtual > electrons frothing and bubbling at the Fermi surface edge of the Dirac > negative energy sea. This is because of the Pauli exclusion principle > that only none or one electron per quantum state. A virtual electron > pops out of the vacuums Fermi surface leaving a hole behind. The hole > is the virtual positron. The result is a virtual electron-positron > pair. However, the virtual electron and the virtual positron attract > because they have opposite charges and they are exchanging virtual > photons. Therefore, some of them form a more stable bound state. An > enormous number of these virtual pairs Bose-Einstein condense into the > same center of mass quantum wave packet forming the Vacuum Coherence > Field (AKA Inflation Field). This is a dynamic steady state of > detailed bce in which there is a continual inflow and outflow of > virtual pairs into and out of this giant quantum or macro-quantum > superfluid. Essentially this is a vacuum phase transition, similar to > the BCS transition from a normal metal to an electrical superconductor, > from the globally flat micro-quantum electrodynamic vacuum without any > gravity at all to the curved macro-quantum electrodynamic vacuum with > emergent gravity. Einsteins field equation of general relativity can > be derived from the phase wiggles and ripples in the robust stable > macroscopically occupied center of mass quantum wave packet of the bound > state of the virtual electron-positron pair. The exotic vacuum dark > energy and dark matter are simply the amplitude wiggles and ripples of > this same virtual pair quantum wave packet. The wave packet spreads > over the entire 3D space of the post-inflationary bubble on which our > Hubble-horizoned universe is located along with an infinity of parallel > American. If the world hologram idea is correct, take the surface area > of the expanding Hubble sphere that is the causal retarded boundary of > 3D space of our past light cone at Earth and divide it by the quantum of > area. That gives us the number of Bekenstein-Shannon c-bits and > explains the arrow of time (AKA Second Law of Thermodynamics) of > increasing thermodynamic entropy in terms of the dynamical expansion of > the 3D space of the universe. Lenny Susskind calls this DeSitter Space. Such research should be forbidden! > Too late. Pandoras Box is open. Schrodingers Cat has jumped out of it. > In the early 1970's Soviet physicists were concerned that the vacuum of > our universe was in fact only one possible state of empty space. The > fundamental state of empty space is called the true vacuum. Our > universe was considered to reside in a false vacuum, protected from > the true vacuum by the wall of our world. A change from one vacuum > state to another is known as a phase transition. This is analogous to > the transition between frozen and liquid water. Lev Okun, a Russian > physicist and historian recalls Andrei Sakharov, the father of the > Soviet hydrogen bomb, expressing his concern about research into the > phase transitions of the vacuum. If the wall between the vacuum states > was to be breached, calculations showed that an unstoppable expanding > bubble would continue to grow until it destroyed our entire universe! > Sakharov declared Such research should be forbidden! since there was > always the possibility that an experiment might accidentally trigger a > vacuum phase transition. > British Astronomer Royal, Sir Martin Rees, Master of Trinity College, > and Director of the Cambridge University Institute of Theoretical > Astronomy on Madingley Road where Stephen Hawking works discusses all > this in Chapter 9 of his important book Our Final Hour. > Could the wall of our universe be breached from within? The amount of > energy required to punch a hole through the wall appeared to be > enormous, and no known natural physical phenomena, even the most > energetic, had punched through either. A recent report commissioned to > examine potential gers at the Large Hadron Collider, one of the next > the best of our existing knowledge. Others are not so certain, however. > At least one of the Russian physicists I had corresponded with was said > to have been a former associate of Andrei Sakharov. He strongly hinted > at new theories the Russians had developed which allow for the > manipulation of the fundamental constants of nature, but he never > revealed more than a sketch of his ideas. He claimed that a breakthrough > was within reach, perhaps within five years > Who was that? Not George Ryazanov? > Recent theoretical explorations may suggest another approach to the > physics of the vacuum. The invisible gravitating dark matter could be > the other side of the invisible dark energy coin, and that suggests the > possibility of manipulating the vacuum for energy release. > Now this is my original idea that you got from our communications over > the past few years. I am the only physicist in the world today, as far > as I know who has suggested this and has already published it in my two > books of 2002 so its in the official record at the Library of Congress. > If a controllable parameter could be found to mediate the bce > between the invisible dark forces, the result would unleash the vacuum > energy of creation in all of its awful power and majesty. If it were > possible to control the dark sides of the force then spacetime, the > arena where everything we know takes place, could be bent and twisted > with infinitely greater ease than was ever suspected. This would open > Pandora's box to everything from vacuum energy weapons of mass > destruction (capable of destroying the universe!) to spacetime warp > drives and time machines. > Exactly, the above is the thesis of all my books since 2002 at least. > A quick survey of the international electronic archive of physics > papers at www.arXiv.org shows that research into the vacuum of spacetime > for energy production is alive and well. > I do not think that is true. You need to cite specifics here. There are > lots of flakey new age papers on free energy on the Web written by > people without any real credentials but they are not on www.arXiv.org > which is not even allowing competent fringe papers in controversial > topics like cold fusion. So what exactly are you thinking of here? > Indeed, Paul Ginsparg, who controls the archive, does not even allow > Carlos Castro to publish conservative competent papers on Clifford > Algebras which are not fringe at all! > Most authors are independent researchers struggling with limited > funding and resources, yet their theoretical results suggest that > somewhere in Nick Cook's black world, a major breakthrough has already > taken place. Most likely the United States and Russia are in the lead, > but China, France, Ukraine, Iran, India and Saudi Arabia all have > scientists actively pursuing the fundamental physics that determine the > fabric of our reality, and are seeking the theory and the means to > access the enormous energies locked inside of the vacuum since the > creation of the universe. Even if the black budget world has yet to > unleash the enormous potential of vacuum energy, there are signs that > those in power may have begun to take notice. Dr. Harold Puthoff, a > scientist with strong government connections, who has previously worked > on classified projects for the CIA, is a major proponent of vacuum > energy physics. Nick Cook's book, The Hunt for Zero Point, and his > recent stories on zero point energy in Jane's Defence Weekly have also > brought attention to the gers and military potential of vacuum > research. The American intelligence community financed so-called psychic > spies for over twenty years and through four presidential > administrations. It is highly unlikely that they would ignore the > potential of the quantum vacuum. Dr. George Chapline, of the Lawrence > Livermore National Laboratory, and Dr. Jack Sarfatti in San Francisco, > knew each other in the sixties in La Jolla, have independently proposed > that the quantum vacuum may unstable to the formation of coherent > virtual processes. Sarfatti suggests that gravity is an emergent > property determined by the physics of the vacuum. His idea is to find a > means of directly interacting with the physics of the vacuum that > controls the shape of spacetime. Such a possibility would be consistent > with the reported success of Evgeny > Podkletnov, the Russian scientist who is experimenting with spinning > superconducting disks. Podkletnov's most recent papers report the > appearance of a mysterious coherent beam of gravity like radiation > with a measured force of 1000 G. In an interview on BBC radio, Nick > Cook pointed out one immediate application of the Podkletnov beam - the > destruction of missiles and satellites in flight or in orbit around the > earth. Cook showed the BBC internal documents from Boeing, the American > aerospace contractor, proving their interest in Podkletnov's research. > This beam stuff I am suspicious of. Of course if the experiment is good, > I have to think more about it. I am not so sure if Podkletnovs > experiment is any good and has been replicated. > The connections between Podkletnov's results, and the kind of vacuum > research explored by Sarfatti, beginning in 1999 at the International > Space Sciences Organization are the latest threads in a trail that most > likely originates in cold war disinformation, a game played by East and > West against each other. Glasnost has shifted the bce of > partnerships and the positions of the players, but not the stakes of an > outcome that would leave the world with even more prolific and powerful > weapons of mass destruction. > That is true, as shown in Destiny Matrix, however you leave out the > most important evidence -UFOs! > whole business are the connections. Although Nick Cook never revealed > the identity of his deep throat contact called Dr. Marckus in > the book The Hunt for Zero Point, there was no question that the > Podkletnov results had played a major part in fitting together the > pieces of the puzzle. The amount of interest was in Podkletnov's > reports by NASA, Boeing, and others in the international arena of > aerospace and military research communities was evidence that there was > more here to explore than the latest musings of the intellectual elite. > The truth is that a fundamental theory of gravity at the scales of > subatomic nuclear physics does not exist. The fact is that no one > understands the nature of the gravitational field at very small scales. > In fact gravity has barely been probed much below one millimeter. Every > attempt to unify the physical theories of gravity with the well-known > standard model physics of electromagnetism, and the strong and weak > nuclear forces, has failed. More importantly there has been recent > progress in the exotic areas of mainstream research, such as superstring > theory, which suggest new kinds of physics, which might support > explanations for Podkletnov's impulse gravity effect. One of the > current fads in theoretical physics involves large extra dimensions of > space that allow a much stronger version of gravity to leak off the > membrane world of our ordinary three dimensions. The large dimensional > picture allows for the well known forces of electromagnetism, and the > strong and weak nuclear forces, to be confined to a three dimensional brane-world floating in a higher dimensional space. Gravitons, the > are able to slip off of our brane-world, which explains why the > gravitational force is so much weaker than the other forces that hold > matter together. Gravitons could be exchanged between our brane-world > and another brane floating nearby in the same higher dimensions. > The Sarfatti picture offers a more direct interaction with the new > physics than the brane world ideas. Sarfatti's vision is to find a > means of using electromagnetic fields in the Josephson effect to couple > to the virtual electron-positron pair giant coherent condensate > inflation field inside the vacuum that controls the shape of spacetime > to the real electron pair giant coherent condensate of a control high > temperature superconductor. UCBs Ray Chiao has a similar idea using a > superconductor to transduce electromagnetic far field waves to gravity > waves with high efficiency conversion. Sarfatti wants to do the same > thing with non-propagating electromagnetic and gravity near fields. One > wonders if the black budget world may have already produced some of the > technology needed to explore and test these new realms. > Not a chance. They are clueless about the theory. They are still stuck > in Hal's PV model and Bernie Haisch's zero point ideas, which will > never, in my opinion fly. They are not asking the right questions and do > not have the right idea in their minds. It is my belief, until I see > evidence to the contrary, that I am the only physicist on the planet > today who is doing real theoretical work directly relevant to the > achievement of practical metric engineering anchored in the now observed > reality of dark energy. All my work is public. I would love to be > proved wrong on this especially by Hal Puthoff, but I am not holding my > breath.;-) Extraordinary claims require extraordinary proof. Everything > else I have seen is either on the wrong track asking the wrong questions > like the work of Puthoff, Haisch, Ibison & Rueda for example, which at > least is real physics that has proved itself wrong in Ibison's PV > cosmology paper, or else the claims are patently obvious nonsense that > Feynman called Cargo Cult Science. There is also the Russian torsion > work of Akimov and Shipov and I am not prepared to make a definitive > statement on that, as the issue is not simple because of several factors > some political. One must be careful there to separate Shipov's > theoretical work from claims made about practical devices including > weapons applications. I do, however, agree with you that there is a real > issue here as defined in Ch. 9 of Martin Rees's Our Final Hour. > Hal Puthoff coined the term metric engineering for flying saucer > technology. Hal has been working on this problem for many decades and > has held high USG security clearances and has been privy to reliable > information that the saucers are real and are alien. Otherwise he > would not be working on the problem. However, Hal's theories, both of > the zero point energy and of the gravity field will not solve the > problem because they are too naively based and do not ask the right > questions. The basic physics required for this task is way beyond the > depth of Puthoffs self-described engineering approach and can be > found in Rovelli's new book on quantum gravity. Hey, I just started reading your stuff, and I find it all very talk about here, from the New York Times, written so the average joe could also enjoy some of these ideas, for anyone who might interested: New Data on 2 Doomsday Ideas, Big Rip vs. Big Crunch === Subject: Re: Infinite Galois Groups >Let K be the field generated by the square roots of the prime numbers. >What is G(K/Q)? What is G(F/K)? > This is the increasing union of > Q < Q(sqrt(2)) < Q(sqrt(2),sqrt(3)) < Q(sqrt(2),sqrt(3),sqrt(5)) < ... > In each case, the relative Galois group is Z/2Z, and the Galois group > of the extension Q(sqrt(2),...,sqrt(pn)) over Q > where pn is the n-th prime is equal to (Z/2Z)^n (you would need to > prove this; see for example Section 6.7 in Lang's Algebra, 3rd edition). > The maps going down are > ... --> Z_2 x Z_2 x Z_2 --> Z_2 x Z_2 --> Z_2 --> 1 > where each map chops off the last coordinate. The inverse limit is an > infinite product of copies of Z_2. Of course here Galois theory provides the most elegant proofs, but it is worth mentioning that elementary proofs are available, e.g. THEOREM Let Q be a field with 2 != 0 and L = Q(S) be an extension of Q generated by n square roots S = {:/a, :/b, ...} of elts a,b,...in Q. If every nonempty subset of S has product not in Q then each successive adjunction Q(:/a), Q(:/a,:/b),...doubles degree, so totally [L:Q] = 2^n Hence all of the 2^n such subset products comprise a basis of L over Q. PROOF: by induction on the tower height n = number of root adjunctions. Lemma below => [1,:/a] [1,:/b] = [1,:/a,:/b,:/ab] is a Q-vector space basis of Q(:/a,:/b) if (and only if) 1 is the only basis element in Q. Our job is: lift this to n > 2: [1,:/a] [1,:/b] [1,:/c] ... (2^n elts) n = 1: L = Q(:/a) so [L:Q] = 2 since :/a not in Q by hypothesis. n > 1: L = K(:/a,:/b), K of height n-2. By induction [K:Q] = 2^(n-2) so we need only show [L:K] = 4 since then [L:Q] = [L:K][K:Q] = 4 2^(n-2). The lemma below shows [L:K] = 4 if r = :/a, :/b, :/ab all aren't in K, true since induction on K(r) of height n-1 yields [K(r):K] = 2. QED LEMMA [K(:/a,:/b) : K] = 4 if :/a,:/b,:/ab all aren't in K, & 2 != 0. Proof: Let L = K(:/b). [L:K] = 2 via :/b not in K, thus it is sufficient to prove [L(:/a):L] = 2. It fails only if :/a in L = K(:/b) and then :/a = r + s :/b for r,s in K; but that's not possible since above^2 -> a = rr + ss b + 2rs :/b contradicts hypotheses as follows: rs != 0 => :/b is in K via solving above for :/b, using 2 != 0 s = 0 => :/a is in K via :/a = r in K r = 0 => :/ab is in K via :/a = s :/b, multiplied by :/b. QED In the classical case Q is the field of rationals and the square roots have radicands being distinct primes. Here it is quite familiar that a product of any nonempty subset of them is irrational since, over a UFD, a product of coprime elements is a square iff each factor is a square (mod units). So the classical case satisfies the theorem's hypotheses Elementary proofs like that above are often credited to Besicovitch (see below). But I haven't seen his paper so I can't say for sure whether or not his proof proceeds in a similar way as that above. If a reader has his proof handy, I'd be very grateful if they could summarize it here. Finally, see below for some stronger results. -------------------------------------------------------------- -------------- -- 2,33f 10.0X Besicovitch, A. S. On the linear independence of fractional powers of integers. J. London Math. Soc. 15, (1940). 3--6. -------------------------------------------------------------- -------------- -- Let a_i = b_i p_i, i=1,...,s , where the p_i are s different primes and the b_i positive integers not divisible by any of them. The author proves by an inductive argument that, if x_j are positive real roots of x^{n_j} - a_j = 0, j=1,...,s , and P(x_1,...,x_s) is a polynomial with rational coefficients and of degree not greater than n_j - 1 with respect to x_j, then P(x_1,...,x_s) can vanish only if all its coefficients vanish. Reviewed by W. Feller -------------------------------------------------------------- -------------- -- 15,404e 10.0X Mordell, L. J. On the linear independence of algebraic numbers. Pacific J. Math. 3, (1953). 625--630. -------------------------------------------------------------- -------------- -- Let K be an algebraic number field and x_1,...,x_s roots of the equations x_i^n_i = a_i (i=1,2,...,s) and suppose that (1) K and all x_i are real, or (2) K includes all the n_i th roots of unity, i.e. K(x_i) is a Kummer field. The following theorem is proved. A polynomial P(x_1,...,x_s) with coefficients in K and of degrees in x_i , less than n_i for i=1,2,...,s , can vanish only if all its coefficients vanish, provided that the algebraic number field K is such that there exists no relation of the form x_1^m_1 x_2^m_2 ... x_s^m_s = a, where a is a number in K unless m_i = 0 mod n_i (i=1,2,...,s) . When K is of the second type, the theorem was proved earlier by Hasse [Klassenkorpertheorie, Marburg, 1933, pp. 187--195] by help of Galois groups. When K is of the first type and K also the rational number field and the a_i integers, the theorem was proved by Besicovitch in an elementary way. The author here uses a proof analogous to that used by Besicovitch [J. London Math. Soc. 15b, 3--6 (1940); these Rev. 2, 33]. Reviewed by H. Bergstrom -------------------------------------------------------------- -------------- -- 46 #1760 12A99 Siegel, Carl Ludwig Algebraische Abhaengigkeit von Wurzeln. (German) Acta Arith. 21 (1972), 59--64. -------------------------------------------------------------- -------------- -- Two nonzero real numbers are said to be equivalent with respect to a real field R if their ratio belongs to R . Each real number r != 0 determines a class [r] under this equivalence relation, and these classes form a multiplicative abelian group G with identity element [1]. If r_1,...,r_h are nonzero real numbers such that r_i^n_i in R for some positive integers n_i (i=1,...,h) , denote by G(r_1,...,r_h) = G_h the subgroup of G generated by [r_1],...,[r_h] and by R(r_1,...,r_h) = R_h the algebraic extension field of R = R_0 obtained by the adjunction of r_1,...,r_h . The central problem considered in this paper is to determine the degree and find a basis of R_h over R . Special cases of this problem have been considered earlier by A. S. Besicovitch [J. London Math. Soc. 15 (1940), 3--6; MR 2, 33] and by L. J. Mordell [Pacific J. Math. 3 (1953), 625--630; MR 15, 404]. The principal result of this paper is the following theorem: the degree of R_h with respect to R_{h-1} is equal to the index j of G_{h-1} in G_h , and the powers r_i^t (t=0,1,...,j-1) form a basis of R_h over R_{h-1} . Several interesting applications and examples of this result are discussed. Reviewed by H. S. Butts === Subject: Re: the anticlassicalist }{ vi: into the quantum For the term to exist to mean anything at all, there must be > something that doesn't exist. That what doesn't exists, exists as a concept. The non-existence does not exist, only if I don't know it. But as I know about non-existence, it exists. > It does not matter how the one is distinguished from the other or who > is doing the distinguishing; it does not matter where the line is > drawn between that which exists and that which doesn't exist, how it > is drawn, who is drawing it, whether it is sharp or fuzzy. > What matters is that for the phrase X exists to mean anything at > all, there must be some entity Y for which it is valid to state Y > does not existAll exists and that what doesn't ain't. > Otherwise the word exists in the phrase X exists would express > nothing whatsoever. Take a deep breath. > Or (in light of the above): ontology is about that which is > distinguished from that which does not exist. Ontology imagines reality knowledge. > For a distinction to take place requires that there is more than > possible outcome. Would you explain in view of membership in the universal class? > Q: What is red and invisible? > A: No tomatoes. > Q: But what if they're green and invisible? > A: Then they aren't ripe yet. What if I'm not invited to lunch, do I still enjoy the existence shifted, tossed salad? Riddle of the day: are the unconceived existence challenged? === Subject: Re: ATTENTION: Open dispute with my college about Procedures. Please read. > Why don't you put all this creative energy into studying? I thought i mastered the subject. I am thinking of withdrawing from school completely. I am stupid; my teacher has emphasized this to me. I do not know what to do anymore; everything is not as great as it used to be. Life is not enjoyable anymore it seems. === Subject: Re: x^2 + y^4 = z^4 <4034C4F2.3090700@free.invalid > descent methods for both x^4 + y^4 = z^2 and > x^4 - y^4 = z^2 can be found in : > http://www.mathpages.com/home/kmath288.htm === Subject: Re: Math notation question: '(' > I just ran across a math question in a review and I'm totally stumped > by the answer. Then I realized that I may have been misreading the > notation. > The statement was: > There exist x, y, and z that are nonzero. Such that 1 ( y>x and xy=z. The period after nonzero looks wrong. Is it a comma? > Does 1 ( y>x mean that both y & x are less than 1? Looking at the > answer that is the only thing that makes sense to me. I just don't > understand the notation. Neither do I. Have you accurately transcribed the _complete_ sentence? === Subject: Re: Ideas for course on great ideas in (theoretical) CS? > I have been coerced into teaching a Honors course the Fall > (mostly for non-CS freshman/sophomore Honors students). My > idea was to do some Great Ideas/Problems/Puzzles etc. from > computer science -- emphasis on theory/algorithms > and related areas like graph theory/combinatorics. > Of course, the honors college wants a syllabus in one week! > I looked at the book, Great Ideas in CS and though a nice > book, seems a bit light on the theory side ... given that > I want to focus on theory to keep me interested. I saw the > course/web site at CMU Great Ideas in Theoretical Computer Science > and may use that as a guide for some of the course. Examples > of some things I might discuss (besides a couple weeks on > basics/definitions/history) include Towers of Hanoi, Byzantine > Generals, voting problems, maybe a gentle discussion > of interactive proofs, prisoner's dilemma, game of life, primality > testing, graph coloring ... anything that can be discussed in a > day or so to folks with no CS background, yet which has some > theory component to it ... stuff that is surprising or counter-intuitive > is all the better ;) > Anyway, if anyone has any suggestions for material/topics > that I might cover, I would most appreciate it. Any pointers > would be accessible to students would be great (I have a couple) > would be great. > (or a link to one) to comp.theory in a week or so. > Chip Klostermeyer You might look through the MIT courseware to see if you find anything of interest: http://ocw.mit.edu/OcwWeb/Global/all-courses.htm Also you seem to be focusing on math. A whole lot of other things come to mind. Computer history and pioneers, the basic structure of computers, high level languages and compliers, Turing machines, the evolution of hardware over time, the roles of govt., business, and individuals - e.g. ARPA NET, IBM, Apple, etc. === Subject: Differential equation with implicit solution My differential equations book asks the following: Show that -2x^2y + y^2 = 1 is a solution of the differential equation (in differential form) 2xy dx + (x^2 - y) dy = 0. Ok, that's trivial, one just implicitly differentiates the first equation, etc. They also ask to find at least one explicit solution. Now here's my problem: What explicit equation satisfies both the equations above? I've graphed the implicit solution (first equ above) and get what looks like almost y = 2x^2 + 1, and an inverted witch of agnesi with y = -1/((3/2)x^2 + 1). Unfortunately, these are not exact fits. I imagine that there is in fact such an explicit solution to both (on some interval of definition) and that perhaps there might be a method to derive it. Any help is greatly appreciated. R === Subject: Re: walking on a grate.. hello phil and Lynn -- thanks for help and explanation :) cheers, n. === Subject: Re: ATTENTION: Open dispute with my college about Procedures. Please read. > I am stupid You're not stupid, but your trolling skills need a *lot* of work. === Subject: Re: Ideas for course on great ideas in (theoretical) CS? Great ideas, mmmm....? - Reductionism - Existance of Complete problems - Construction with basic elements More specifically: - Coding (nearly all) problems using >1 symbols (say 0s and 1s) - Existance of a universal model of computation (Programability; transfering the computational power from structure of the machine to the programs and back) - Rewriting systems; generating a language - P vs. NP (as tractable vs. intractable) - Probabilistic problem solving; - Generating functions (really enjoyable discussion in Concrete Mathematics of Graham, Knuth, Patashnik) - Modeling with graphs - Object Oriented Programming - ... Siamak === Subject: Having integration trouble Hello. I am working on integrate sqrt ((4x^2) +9)/(x^4) using trig substitution. I have: u=2x a=3 2x/3 = tan theta dx=3/2 sec^2 theta d theta (sqrt (4x^2 + 9))/3 S 1/(sqrt (4x^2 +1)) dx= 3/2 S ((sec^2 theta d theta)/sec theta) =3/2 S (sec theta d theta) =3/2 ln abs (sec theta + tan theta) + C =3/2 ln [ abs ((sqrt (4x^2 +9) +2x/3)) +C Things that are throwing me: 1) the x^4 in the denominator of the problem and 2) the triangle setup I have looked in several books and all over the internet to find a problem like this to see if I was on the right track, but none that I found look like this. I don't know if the answer is right, but it is the best I could do with what I know. Stacy === Subject: Re: ATTENTION: Open dispute with my college about Procedures. Please read. > thanks for calling me names. i think it will help push me off the edge in the end. === Subject: Re: Axioms defining a finite field > No, it doesn't; your 2 is the 1 specified in the definition. It looks like a typo to me. === Subject: Re: No Set Contains Every Computable Natural <8JCdncK7SMPAiK7dRVn-vw@comcast.com <87hdxovh3h.fsf@phiwumbda.org> Assumption: There exists a 0 at the end of computation. >Of course there is. How can there not be? >My TM will never overwrite the last 0. > This is the source of your error (and just about all your errors.) There > is no last 0 on the *infinite* list, since there will always be a 0 > after it. I disagree. It is not the source of his error. It is a consequence of his error. The source of his error is an obsessive-compulsive disorder which manifests itself in commuting quantifiers willy-nilly, especially changing (A x)(E y) to (E y)(A x). Jesse Hughes How lucky we are to be able to hear how miserable Willie Nelson could imagine himself to be. -- Ken Tucker on Fresh Air === Subject: Re: No Set Contains Every Computable Natural <8JCdncK7SMPAiK7dRVn-vw@comcast.com <87hdxovh3h.fsf@phiwumbda.org> There will be such an N. N could be quite large (it might even equal 3). >Obviously, the original tape didn't contain every natural number. > Yes, it is obvious, and this is what everyone has been telling you all > along. (In case you forgot, you were the one claiming the existence of > such a tape) How many times do you need to be told that you will not have > a tape with all the natural numbers on it? Who says? Why not? He won't *if* we apply the condition that his input tape has finitely many ones, but there is nothing about that convention that is forced on us. Why not allow that he has a tape with all of N? After all, that assumption is not his problem. His problem is only in reasoning about what's on the tape after an infinite number of steps. If you simply say that infinite input tapes don't make sense and/or infinite computations are meaningless, then you avoid the real source of 's error. You rule out his thought experiment by fiat, without asking whether his assumptions are really sensible (they are) and trying to understand his reasoning (which is, of course, incorrect). But he himself was not to blame for his vices. They grew out of a personal defect in his mother. She did her best in the way of flogging him while an infant... but, poor woman! she had the misfortune to be left-handed, and a child flogged left-handedly had better be left unflogged. -- E.A. Poe === Subject: Re: Differential equation with implicit solution > My differential equations book asks the following: > Show that -2x^2y + y^2 = 1 is a solution of the differential equation > (in differential form) 2xy dx + (x^2 - y) dy = 0. Ok, that's trivial, > one just implicitly differentiates the first equation, etc. > They also ask to find at least one explicit solution. Now here's my > problem: What explicit equation satisfies both the equations above? Maybe that I've misunderstood your question, but it seems to me that all that you are supposed to do is to solve the equation -2x^2*y + y^2 = 1 in y. You'll get two solutions: y = x^2 + sqrt(x^4 + 1) and y = x^2 - sqrt(x^4 + 1). Jose Carlos Santos === Subject: Re: ATTENTION: Open dispute with my college about Procedures. Please read. > thanks for calling me names. i think it will help push me off the edge in > the end. And what names, pray tell, did I call you? === Subject: Re: Differential equation with implicit solution > My differential equations book asks the following: > Show that -2x^2y + y^2 = 1 is a solution of the differential equation > (in differential form) 2xy dx + (x^2 - y) dy = 0. Ok, that's trivial, > one just implicitly differentiates the first equation, etc. > They also ask to find at least one explicit solution. Now here's my > problem: What explicit equation satisfies both the equations above? > I've graphed the implicit solution (first equ above) and get what > looks like almost y = 2x^2 + 1, and an inverted witch of agnesi with y > = -1/((3/2)x^2 + 1). Unfortunately, these are not exact fits. > I imagine that there is in fact such an explicit solution to both (on > some interval of definition) and that perhaps there might be a method > to derive it. > Any help is greatly appreciated. > R Solving -2x^2y + y^2 = 1 gives two equations for x in terms of y both of which satisfy -2x^2y + y^2 = 1 and 2xy dx + (x^2 - y) dy = 0. Similarly solving -2x^2y + y^2 = 1 for y in terms of x, gives two equations, either of which is sufficiently explicit=== Subject: Re: JSH: Non-uniqueness of factorization <87ad3den4w.fsf@phiwumbda.org <25rZb.64909$KV5.49312@nwrdny01.gnilink.net <87wu6hd2zr.fsf@phiwumbda.org <5ruZb.63308$IF1.32769@nwrdny03.gnilink.net Discussion, linux) >I think MS is just erring on the side of caution by deleting a >string ending with :. One cannot fault them for not knowing who >JSH is because, after all, he's a fan of Java, not C#. >I guess we disagree on what constitutes caution. Mangling subject >lines is not cautious. The fact is that, JSH aside, there are plenty >of opportunities in which one wants to start a thread with a subject >of the form, Blah: blah blah blah>That the MS coders couldn't think of any isn't surprising. They >couldn't consider that (on rare occasions), lines might >begin... > Ouch, the mime parsing error is a really horrible thing. I can't fathom > that one. They must be working around some horrible kludge that requires > it. I can't imagine what kludge would prevent one from reading headers to determine whether the message is a mime message. I think what happened is this. Anyone may correct my mistakes, of course. Prior to the widespread use of mime, folks sent messages with uuencoded files in them. The folks on the other end knew to cut the uuenclosure out and decode it to recover the file. When MS designed OE, they remembered this old practice and realized that their users would not know what to do if they received a non-mime message with a uuenclosure. So, they (stupidly) decided to have their program protect users by looking for begin and parsing the rest as a uuenclosure. This is bad, but their implementation is particularly stupid. Why not check *first* for a matching end to the begin before parsing the contents as an enclosure? Any four-year-old knows that's better. As I said, it's not common for the bug to bite unintentionally, but it happens. I know a guy who uses a software package (an emacs mode, I think) that both right and left justifies his email and posts, by adding spacing on each line as needed. Now, we can debate whether he ought to use that and whether the result is pleasing, but he uses it. One of his posts is the first (maybe only) unintentional bite of the begin bug that I've seen. > I'll end this OT thread by saying that we do not disagree on the meaning of > 'caution'. I was commenting on MS and I *did* say that they 'erred'! > FWIW, I think user agents should leave the subject field alone. A sensible Re: convention seems perfectly reasonable to me. Too bad it's beyond the skills found at MS. Time and again, history has shown that people who think their beliefs trump reality lose, and lose badly. Luckily, I don't have to listen to you. -- James Harris on reality avoice === Subject: Re: Ideas for course on great ideas in (theoretical) CS? > Anyway, if anyone has any suggestions for material/topics > that I might cover, I would most appreciate it. Any pointers > would be accessible to students would be great (I have a couple) > would be great. Some excellent ideas in computer science are the ideas that lead to algorithms with computationally faster runtimes than an algorithm that a human would take in solving a problem. E.g. If humans are sorting a list of numbers, they usually use a kind of selection or insertion sort. I don't see why a human would manually perform quicksort, yet that is one of the more ideal ways of sorting. Another example is matrix multiplication... humans will likely apply the definition of matrix multiplication to find A*B while something simple like Strassen's algorithm (which people would probably not bother doing by hand) leads to a theoretically faster runtime. How about historical advances? Computing machines before electronics, for example? J === Subject: Re: No Set Contains Every Computable Natural <8765e4q2fe.fsf@phiwumbda.org <8JCdncK7SMPAiK7dRVn-vw@comcast.com> <87hdxovh3h.fsf@phiwumbda.org <87vfm4o85i.fsf@phiwumbda.org <87n07fo9qr.fsf@phiwumbda.org <87k72jnxtp.fsf@phiwumbda.org <87ptcau3xj.fsf@phiwumbda.org He won't *if* we apply the condition that his input tape has finitely > many ones, but there is nothing about that convention that is forced > on us. Why not allow that he has a tape with all of N? After all, > that assumption is not his problem. His problem is only in reasoning > about what's on the tape after an infinite number of steps. Yes, of course I agree that a TM can have anything written on its tape to start. The problem, of course, is when one talks of an output it implies that a TM has halted (in finite time) and thus only seen a finite amount of that input, so really having an infinite input is pretty useless. And, yes, we can take your earlier suggestions and define a tape cell to be, in a sense, finalised if the TM tape head is guaranteed to never visit that tape cell again... but then that still leads to his example with having none (instead of one) '0' symbols left in the end, which you and I have tried explaining to him. If you have the patience to handle him, I'll gladly leave this thread now. J === Subject: Re: Ideas for course on great ideas in (theoretical) CS? Distribution: inet > I have been coerced into teaching a Honors course the Fall > (mostly for non-CS freshman/sophomore Honors students). My > idea was to do some Great Ideas/Problems/Puzzles etc. from > computer science -- emphasis on theory/algorithms > and related areas like graph theory/combinatorics. Hey, great! I have one word for you: http://www.discretemath.com/ (One of the cool, yet vaguely disturbing, things about CMU is that our CS department gets these really crazy domain names.) That's the home page for 15-251 Great Theoretical Ideas in Computer Science I, which is a required course for CS majors at Carnegie Mellon. Now, I don't expect that you'd be able to pull off such a great job as Steven Rudich does at CMU, but I bet you can find some neat topics from 15-251's syllabus. I saw the course/web site at CMU Great Ideas in Theoretical > Computer Science and may use that as a guide for some of the > course. Well, I guess you knew that. Consider it seconded, then. :-D > Examples of some things I might discuss (besides a couple weeks on > basics/definitions/history) include Towers of Hanoi, Byzantine > Generals, I hadn't heard of Byzantine Generals before. But it made me think of the Firing Squad, and then the Dining Philosophers: two other interesting problems. > voting problems, You might even want to tie in the real-life theoretical problems involving voting secrecy and verifiability. What with the U.S. elections coming up, and a lot of states switching over to buggy voting software... ;-) But voting algorithms and Arrow's Paradox are fun too. > maybe a gentle discussion of interactive proofs, Yes, I seem to recall something about zero-knowledge proofs in 15-251, like the proof that I know a three-coloring of graph G. If that's what you mean. :) > prisoner's dilemma, I recently heard a talk by Christos Papadimitriou in which he brought up a prisoner's-dilemma style paradox involving routing. Suppose we have a graph, say a network of roads or Internet bandwidth, and we weight the time it takes to traverse an edge by a function of the percentage of traffic that's on it. For instance, a superhighway might take constant time to drive along: T(x)=1; while a back road might get jammed: T(x)=x. Then if we take a bunch of traffic and run it through this graph: o x /| 1 / | o |0 o | / 1 |/ x o obviously it's better for everyone involved if 50% of them go along the top and 50% along the bottom -- everyone finishes in 1.5 time units. But if we assume that the people are selfish, and will gravitate naturally toward the fastest route, we'll find that eventually everyone is mushing along the path x-0-x, and taking 2 time units to finish! So greedy routing ends up hurting, not helping, in this case. Discuss. ;-) > game of life, primality > testing, graph coloring ... anything that can be discussed in a > day or so to folks with no CS background, yet which has some > theory component to it ... stuff that is surprising or counter-intuitive > is all the better ;) Public-key cryptography. Wanna make 'em paranoid? Ken Thompson's On Trusting Trust. Lots of stuff from las Hofstadter might be appropriate. Ditto ditto Ray Smullyan ditto. (or a link to one) to comp.theory in a week or so. Sounds cool! Good luck! -Arthur === Subject: Re: No Set Contains Every Computable Natural <8765e4q2fe.fsf@phiwumbda.org> <8JCdncK7SMPAiK7dRVn-vw@comcast.com <87hdxovh3h.fsf@phiwumbda.org> He won't *if* we apply the condition that his input tape has finitely >many ones, but there is nothing about that convention that is forced >on us. Why not allow that he has a tape with all of N? After all, >that assumption is not his problem. His problem is only in reasoning >about what's on the tape after an infinite number of steps. > Yes, of course I agree that a TM can have anything written on its tape > to start. The problem, of course, is when one talks of an output it > implies that a TM has halted (in finite time) and thus only seen a finite > amount of that input, so really having an infinite input is pretty > useless. > And, yes, we can take your earlier suggestions and define a tape cell to > be, in a sense, finalised if the TM tape head is guaranteed to never visit > that tape cell again... but then that still leads to his example with > having none (instead of one) '0' symbols left in the end, which you and I > have tried explaining to him. Now, this is exactly right and the right way to explain it (not that is ever going to quit commuting quantifiers). I much prefer an explanation of why his reasoning is bad than a bald statement that infinite input and output is senseless. After all, working theorists *do* sometimes use infinite tapes. > If you have the patience to handle him, I'll gladly leave this thread > now. Well, I'm not taking any responsibility here, but you haven't any either. On occasion, I decide that responding to is diverting enough that I do so. Then, I get tired of the same old errors in new guises and wander away for a while. will always be there, making the same mistakes, if I want to return to the same well-tread ground. (Geez, putting it this way, I wonder *why* I ever do return.) Anyway, if you don't want to keep going in the thread, don't. 's confusion was here before you (I think) and it will be here after your gone. Yup, as far as I'm concerned, if you live out your lives smiling the entire time full of pride in your *believed* accomplishments, when you never had any, well that's ok with me. --James Harris, a man of remarkable accomplishments. === Subject: Re: Non-uniqueness of factorization Can someone please give me step by step details on how to ignore posts by > James Harris and replies to his posts? I just don't have the willpower to > not read his bull. > contains jstevh@msn.com in either header or body will be the best. > (Until he changes address of course.) Filtering on the body will require considerably longer times than filtering only on the standard headers returned by the news server (including from headers and also references, but since JSH posts from google, the reference header isn't so useful). Run mathematicians, RUN!!! I'm coming for you. It may take a few months, but I'll get [computer verification of my proof] and then your lives will be ended as you previously knew it. -- JSH meets PVS === Subject: Re: walking on a grate.. >hello everybody, >suppose I have a grate, N x M. how many paths are from node (0,0) to >(N-1,M-1) eg. from one corner to the diagonal-opposite one? >allowable paths are only with zero or positive change in node's coordinate. >how can I compute this? I would appreciate any suggestion and help.. >cheers, >n. > You should be able to calculate this. Take advantage of symmetry. > Start 1 position away from the end and work backwards. > Good luck > phil If you know basic combinatorics (which, of course, are proved using induction), you can get a much simpler solution than the one suggested. As a general hint, induction (or it's cool teenage brother the Zorn Lemma) is a good way to prove something, but not a good way to discover anything. There are exceptions, of course, but this isn't one of them; recurrence relations here are kinda silly. You're only allowed to move in a positive direction, and only along grid lines. So that means you move up n times and right n times, and the question is how many different orderings of these you can possibly have. That is, you're going to take exactly 2n steps, half of them north and half of them west. So simply pick which north steps you take. We are selecting n items of 2n, with order not mattering, so the answer is 2n choose n = (2n)! / (n!)^2. === Subject: First Cohomology group Hey I need to know the first cohomology group H^1(X,Z) of different sets X. Does any of you know a good website, where I can look these up? Some of the cases are, when X is a point, a circle, a sphere (S^n), the Cantor set and more. I don't need to know, how they are calculated, just the results. Mogens === Subject: Re: Axioms defining a finite field >No, it doesn't; your 2 is the 1 specified in the definition. >It looks like a typo to me. > Well, it wasn't. I carefully checked all the conditions, and forgot about this obvious possibility. Wim Benthem === Subject: Definition af quasi-unitary Hello If x is an element in a C*-algebra, what does it mean for x to be quasi-unitary? Mogens === Subject: Few problems about number. Here they are: 1. Draw 3 diagrams to show the relationship among numbers. 2. What are the different between number, numeral & digits? 3. What are the 3 functions & numbers? === Subject: Weierstrass Elliptic Function The addition theorem for Weierstrass elliptic function is rather complicated, isn't it? However after modifying Weierstrass elliptic function p(x) as p(x+a) where a is constant, the addition theorem is simplified as, p(x+y) = p(x)+p(y)+bp(x)p(y)+c{p(x)p(y)}^2 where b and c are constants. Can you believe it? Believe or not, please visit: http://139.134.5.123/tiddler2/cauchy/cauchyequation.htm there I explained how to calculate the constants b and c. === Subject: Re: ATTENTION: Open dispute with my college about Procedures. Please read. Did you take issue with the grading of your homework paper? It's absurd. If nobody has looked at it, he received a 25% deduction on a single problem for not writing (g o g) (x) = g(g(x)), (where the o is the composition symbol) and a comment that it was very hard to follow his work, even though it was quite easy to read. Indeed, it is the TA's writing that I find almost impossible to read. At one point, several points are taken off for use of notation in the work section of a problem which is not a proof (it is a numerical computation). In other places points are taken off for no apparent reason. At yet another place he shows that a function is not defined at a certain point and then asserts that it is noncontinuous at that point, at which the TA takes off points, apparently for not saying something like, Observe that if the function is not defined at this point, its limit at this point cannot equal its (nonexistant) value. As far as I can tell in the cursory glance I gave the test only one answer is actually incorrect. You have demonstrated your competence in the subject material of the class not only in the practical but in the theoretical. I don't really see what else the teacher could ask for. As for trials and lawyers, I'm not sure they'd do you much good. You'd have to dig up a law or something on the college's books against not letting you drop a class under certain conditions, and I'm pretty sure they didn't write such a thing in. I would argue for more points with the teacher, and if that doesn't suceed I'd retake the class. === Subject: Re: Ideas for course on great ideas in (theoretical) CS? Turing machines (or post production systems or lambda calculus) Church-Turing Thesis The Universal Turing Machine Halting Problem Algorithms and Big-O, P vs. NP conjecture. Information theory, coding theory Public-Key Encryption Pseudorandom number generation Quines etc. For theory applicable to the real world, use Knuth. If you're a mathematician like me and have your doubts that there is such a thing as a real world, an excellent intoductory text is Godel, Escher, Bach. === Subject: Re: Math notation question: '(' > I just ran across a math question in a review and I'm totally stumped > by the answer. Then I realized that I may have been misreading the > notation. > The statement was: > There exist x, y, and z that are nonzero. Such that 1 ( y>x and xy=z. > Does 1 ( y>x mean that both y & x are less than 1? Looking at the > answer that is the only thing that makes sense to me. I just don't > understand the notation. > Ben.. Never seen this sort of notation in my life. Looks like a typo in the book or something. Can we see the entire question and soluion? === Subject: Re: Having integration trouble integrate [I =] sqrt ((4x^2) + 9)/(x^4) using trig substitution. You can lose all the 4s and 9s by letting y = 2x/3 to give: I/2 = int (sqrt ((y^2 + 1)/y^4)) dy Then take 2y = (z^2 - 1) / z, so that 2.dy/dz = (z^2 + 1) / z^2 and: I/2 = int (1/z + 4z / (z^2 - 1)^2) dz = log(z) + 2 / (z^2 - 1) + C etc Cheers -------------------------------------------------------------- ------------- John R Ramsden (jr@adslate.com) -------------------------------------------------------------- ------------- Eternity is a long time, especially towards the end. Woody Allen === Subject: Re: proof of the dimension theorem of linear algebra all that --- is absolutely necessary, I mean I cannot see another way [using first order logic only? or is it still first order?]. But what confuses me is: shouldn't the same argument apply to the finite basis case? Why go for the other detailed (Gaussian) technique? We are actually interested in much less information that the algorithm (for elimination) provides, right? === Subject: Re: Simple idea, mathematics and common-sense > I've had a time explaining some *very* simple mathematical ideas that > lead to a few complexities, but it's been fruitful to explain, or try > to explain, as I work to figure out why these simple ideas either > excite derision, anger or confusion, and I think I have it figured > out. No, actually, you work to excite your own anger and confusion. Plus, this is all a psychological experiment to you, or maybe it's games theory, and you're gonna sic the generals on us, to kill everyone who doesn't buy your particular crock of sheeeit. > I'm sure many of you are put off by mathematics, so I assure you up > front that what I'll be talking about will mostly be *very* simple, > and there will only be a few slightly complicated things at the end. No, actually, it's *you* who are put off by mathematics, and who thinks that mathematicians are welfare leeches. > First of all, I'm going to talk about a case where mathematicians gave > up because they couldn't see something, and assumed that because they > couldn't see something it didn't exist! This is even more of a crock than what preceded. Everyone here knows that you've spent the better part of several years trying to show that mathematicians couldn't perform factorizations of some sorts, and each and every example you came up with turned out to be incorrect: The standard theory of algebraic integers proved to be able to produce factors that *you* said could not be found. Now, in a stunning display (well, stunning to those who haven't seen your continual use of this tactic) of revisionism, you have called this phenomenon (that is, JSH stating something could not be done within standard mathematics, but could be done by the miracle of Objects, with every case actually being readily handled by standard techniques): a case where mathematicians gave up because they couldn't see something, and assumed that because they couldn't see something it didn't exist! I'm wondering how your use of the Big Lie can be justified. Each case you said couldn't be handled, was in fact handled. Each time, you spent weeks, sometimes months, braying on about how the mathematicians were lying and cheating and engaging in fraud, and how you might just have to bring suit against one person or another, and how you had called some person's employer: I've deliberately involved an official at his college to take away plausible deniability for his school in a phone call I made months ago. Each time, you were shown to be incorrect, yet you call this whole episode a case where mathematicians gave up because they couldn't see something, and assumed that because they couldn't see something it didn't exist! Maybe your grasp of the English language isn't quite up to that we assume for a high-school graduate. Perhaps none of the above seems at all contradictory, that, hey, people who say the truth are really the liars, and frauds, and those who produce the results are those who ... gave up because they couldn't see something, and assumed that because they couldn't see something it didn't exist! You see, to my understanding, someone who produces results with standard techniques *isn't* giving up, but it's the person who claims that the techniques are inadequate who is giving up instead! > You know how with simple quadratics like x^2 + 3x + 2, it's easy > enough to see factors of 2 in the roots? > I mean, it's just (x^2 + 3x + 2) = (x+2)(x+1), and there they are. > However, if it's something like x^2 + 7x + 2, you can use the > quadratic formula and get the roots to find > x = (-7 +/- sqrt(41))/2 > and who can see factors of 2 in that thing? Man, are we back to your old confusion of terminology regarding factors? The values of x you show are divisors of 2, and 2 is a multiple of them. To refer to a factor of 2, one means that the number 2 divides the number under consideration. > There's something else important here which is the ambiguity of the > square root operator, which may sound complicated but it's easy to > demonstrate with another root of a quadratic: > (1+sqrt(9))/2 > and you may think, silly, why show sqrt(9) when sqrt(9) = 3, but yeah, > that's *one* of its solutions, as sqrt(9) = -3 as well, so you have > *two* numbers > (1+sqrt(9))/2 = 2 or -1 > as either solution will work. I've had people argue with me that by > definition (really by convention) you take the positive root. But > imagine the world of Contrary. On Contrary the mathematicians for > some odd reason *by definition* take the negative! Is mathematics > really changing depending on such decisions. Yes, everyone sees your idiosyncracy for what it is: an inability to deal with definitions. The square root function is no more ambiguous than the log function. One can well speak of sqrt(4) and -sqrt(4). The fact that you find it incomprehensible only fits in with your general level of confusion. > Imagine you've forgotten that you can resolve sqrt(9) to 3 or -3, so > you write these numbers like (1+sqrt(9))/2 and (1-sqrt(9))/2 and > mercifully discover that you can get rid of that square root sign by > adding them together, or multiplying them together. > Like adding them gives 1, and multiplying them together gives > (1 - 9)/4 = -2 > and your mathematicians scratched their heads and contemplated such > numbers, and decided that there was *no way* to understand factors of > 2 of numbers like > (1+sqrt(9))/2 and (1-sqrt(9))/2 > except as to consider them to be unique factors of 2, in some kind of > mysterious way. Yes, let's imagine that you have a clue about something. Imagination is a wonderful gift, right? > But wait, that's not a problem here, of course, because you can just > evaluate the square root, but look back now at x^2 + 7x + 2, where > x = (-7 +/- sqrt(41))/2 > and consider that you *cannot* resolve sqrt(41) in any way that will > help you here with this question. Why should I consider that? sqrt(41) is a fine number, somewhere between 6 and 7. On the other hand, I could produce a continued fraction to approximate the number, if that's what I had in mind. It's the elementary student who finds the square root to be so mysterious, not the professional. > Sure, you can write it out in decimal format with a lot of numbers > after the decimal place. My computer tells me that sqrt(41) > approximately equals 6.4031242374328486864882176746218, but of course, > you can keep going out to infinity trying just to see sqrt(41). > If you drop some of those numbers (to make it easier) and move those > decimal places, to get 64031242374, squaring gives > 4099999999957933155876, which looks VERY CLOSE to 41 * 10^20, and you > can see what our approximations actually are. > We approximate irrational numbers like square roots by finding some > REALLY BIG natural number, and moving the decimal place to the left. Why not take the continued fraction approximation? It converges faster than the decimal expansion. > However, you STILL don't have a simple idea of where factors of 2 go, > like you had with the easy and you now might think comforting example > of > (x^2 + 3x + 2) = (x+2)(x+1). You just can't get past those reducible polynomials, can you? > Now then numbers like (-7 + sqrt(41))/2 defy our ability to analyze > because we really, really like integers, but to get an integer you > have to use (-7 - sqrt(41))/2, as then you can add them together to > get -7 and multiply them to get 2, but mathematicians could not figure > out a way to handle such numbers in less than pairs! I sure don't know what you think you mean by that. Certainly a person can handle (-7 + sqrt(41))/2, all by itself. What on earth, aside from your own drunken stupor, would suggest otherwise? Why can't you multiply the number (-7 + sqrt(41))/2 by 2, add 7, and square the result? Doesn't that get you an integer? Does it require you to pair it with its conjugate? These numbers aren't paired for the reasons you think. > That's important. Mathematicians could never figure out a way to > handle such numbers in less than pairs. You have no clue about what's important or what's unimportant, let alone what's true or not. After all, you think it's important to report us to the FBI. You think it's important that you got a letter into Time magazine. You think that Z[1/2] is really the full set of real numbers. You think that Galois theory is false. > So some of them decided that what they couldn't see, wasn't > meaningful. Prove this remark. Mathematicians don't make such evaluations. > It'd be kind of like the weird mathematicians, who couldn't evaluate > sqrt(9), and were looking at (1+sqrt(9))/2 and (1-sqrt(9))/2 deciding > that where factors of 2 resided was a mystery to them. But we *can* > evaluate sqrt(9), but we *cannot* really evaluate sqrt(41) which > leaves a mystery with (-7 + sqrt(41))/2 and ((-7 - sqrt(41))/2, and > some mathematicians have decided that that's it. Oh, back to the Land of Make-Believe, are we? > For them, that's all you need to know. Here are these numbers where > we can't evaluate the square root. Sorry, no go there, they decided, > you're stuck, isn't it obvious? I really don't know what you're on about here. I think you were stuck with a four-function calculator, but no toilet paper as a youth, so you decided to work it out with a calculator when it came to matters of hygeine. How else to explain the bulk of excrement you smear across the newsgroups? > You see they decided that the limit on what they could *see* was a > limit on what could mathematically exist for those examples--irational > roots--where they could not see. As far as those mathematicians were > concerned, end of story. You have decided what motivated past generations of mathematicians, without so much as a smattering of knowledge about the field. In what way is that different from the settlers of the US having decided what motivated the Native American? Or, from whites in the Southern US having decided that their slaves really, really preferred slavery? What gives an outsider the right to decide why any group has developed the way it has? > But that's where my story begins. Well, here's something I agree with: your story begins at the end of a bogus revision of history, a tale full of trivialization, purposeful misstatement, and bigotry. That about sets the stage for your great and glorious research. > My mathematical research is about how the kind of patterns you see > with integers, like with > (x^2 + 3x + 2) = (x+2)(x+1) > where one root has all the meaningful factors of 2 *continues* into > realms where we can't see it directly because we can't get past those > ged irrationals coming in at least pairs! Again with the mis-characterization of algebraic irrationals. There is no truth to the rumor that your understanding of mathematics is deep enough to submerge the average dust mite. > I found an *indirect* way of looking, which involves putting > expressions like these polynomials I've shown here, but more > complicated into a special but VERY simple mathematical tool. > For example, > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > can be peered into, by figuring out that its roots can be considered > in the following mathematical structure: > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > where the a's are the roots of that complicated looking cubic that I > started with, and I've managed to place them in a structure opposite a > polynomial that has 49 as a factor. > In one sense that's easy. You can take just about any expression like > that, focus on some factor of its last term, and build something like > what I did. And it being so easy may explain some of the problems I'm > having with people taking it seriously! > What I figured out though is that what mathematicians couldn't see > before can now be logically seen, and in fact, the way you divide off > that 49 is to get something like > (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 a_3(x) + 7) = > 300125 x^3 - 18375 x^2 - 360 x + 22 Has anyone said that 49 *couldn't* be divided from the product in that fashion? No. In fact, it's *you* who has said that this is THE ONLY WAY to form the division. It's *you* who has said that there is no way to divide the product (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) by 49, and retain algebraic integers as factors. You stated that division as an impossibility, and also stated that THE ONLY WAY to do the division is to divide 7 into each of 2 of the factors. > but some vocal mathematicians don't like the idea as it contradicts > what they've been taught, and believed because by their thinking and > training, you can't get around the barrier. What barrier? You haven't taken the mathematics, you don't know what has been taught, and yet you continue to pontificate! Say, if I happen to have seen an episode of Sanford and Son, way back in the 70's (or was it the 80's?), does that qualify me to expound on the nature of African American culture? Do I get to say fo' shizzle and not be laughed out of the 'hood? > Notice that I picked two roots arbitrarily and in fact even if you > solve for the roots you can't look and tell *which* roots should be > divided by 7. All you know is that two of them should be, but can > never know *which* two. Such a crock. You can factor 49 arbitrarily in the complex numbers, into 3 factors, and distribute the factors at will. If you care to constrain the resulting factors, your factorization of 49 will be constrained, as will the freedom in placing the factors among the three parenthetical factors you started with. Where were you, over the past several months, while that exact fact was explored? Were you too busy working yourself up into a lather over the antics of the palindromic poster? Or, were you getting your panties into a pretzel over the algebraic ruminations of the cool Mister Winter? If you had followed your own stick to the math mantra, rather than your 7-year obsession stick it to the mathematicians I rather suspect you wouldn't have wasted your own several months whining and ranting and threatening and pleading. > Which makes my job of explaining harder, but it's just that ambiguity > thing coming back into place. We like integers. The math doesn't > care as it just handles numbers. What a dope. What a dope. YOU have to rescue mathematics because WE like integers. You fail to recognize that the mathematics that has been develops recognizes the special place occupied by the integers, and has a consistent view across more types of algebraic (and other) structures than you can imagine. You see a polynomial you can't see how to deal out the components of a factorization of 49, and you go ape about the Evil Empire of Mathematical Cabalists. > We can't look *directly* at the roots, but we can use logic to > consider them. What we can't see directly here *does* exist because > the mathematics says it does. Yeah, whatever. Your ability to tell what we can or can't do is not really credible, is it? > To some extent, my problems are with people and mathematicians who > trust logic less than what they've been taught and their own personal > sense of what makes since. Well, a full consideration of your problems is (as they say) outside the scope of this work. The people you are having the most difficulty with are quite simply *not* people who regurgitate what they've learned. These are people who are fully competent to develop completely original mathematics. > In a nutshell that's the basis for the arguments. Figures you'd put it into a nutshell, don't you think? > I say that just because we can't see the factors of irrational roots > like we can with rational ones our limitation is not a mathematical > constraint, while some people, including a lot of very obsessive > posters, refuse to acknowledge the possibility, fighting for their old > view. Please. You are the one who has claimed something to be impossible. Others have shown each and every one of your so-called impossible factorizations to be possible within the ring of algebraic integers. Who is it who is refusing to acknowledge possibility? Who is it who takes months on end, in the face of direct computations, to come to the correct conclusion? > Is it important? > Actually, it is important if mathematicians have an error in their > thinking. If? For seven years now, you've been taking every seeming discrepancy between standard mathematics and your own mutterings as a sign of the corruption of mathematics. It doesn't matter that you've ALWAYS been wrong, no, because in your BB-in-a-boxcar Brain, you've decided that you may be right at some point. Didn't you write this little attack: I hereby charge Rick Decker with academic fraud and note that his college is responsible for this rogue professor. I've deliberately involved an official at his college to take away plausible deniability for his school in a phone call I made months ago. His college has been made aware of his behavior. There may also be a civil matter involved at some point in the future. Why not stand up and do something, rather than haranguing the world with your interminable if they're wrong cant? Show someone wrong (and not yourself), or get off the can. > It may be the case that things thought to have been proven in > mathematics, really haven't. Oh, it's the world of maybe: It may be the case that monkeys fly out of my ass. You may already be a winner! Aliens may be living among us! Elvis may still be alive! You talk endlessly about evidence (which, BTW, is 100% against you), concern for the Truth (which concern of yours has been shown to be cynical in every sense but the one which connects it to Diogenes). > It's quite possible that any number of arguments claimed to be proofs > assumed that what they couldn't see, could not exist, as this issue > has been around for over a hundred years. You have never read a proof. You have no background to make this claim, and the evidence is that you will never produce a correct argument. Find the hole. I'll note that *all* of algebra is under *continual* examination by tens of thousands of students, *each one of whom* would be thrilled beyond comprehension to find a contradiction, ANY proof of a statement and its negation. None has been found, despite this intense scrutiny by wannabe experts and experts alike. > Yes, I can talk it all out rigorously and in a heavily mathematical > format, but I hope that giving you some idea what all the fighting is > really about, will help in understanding what's going on, as well as > why I don't just capitulate. Bull. You can make up pseudocrapological jargon, and repeat it until the cows come home, but nothing you have said is correct. You cannot fill in the details, because (1) you don't have the ability and (2) you don't have the patience. > Mathematicians may think that seeing is believing, but I think in > mathematics, logic is king. Crap. What you have seen with your own two eyes contradicts this: When faced with your claims that certain numbers couldn't have common divisors, mathematicians stepped forward to exhibit those divisors. You, on the other hand, put those meathooks of yours right over the eyes, and chanted loudly so you wouldn't have to face the truth. > James Harris Why not try to be a little more self-serving? Dale === Subject: Re: Ideas for course on great ideas in (theoretical) CS? > and may use that as a guide for some of the course. Examples > of some things I might discuss (besides a couple weeks on > basics/definitions/history) include Towers of Hanoi, Byzantine > Generals, voting problems, maybe a gentle discussion > of interactive proofs, prisoner's dilemma, game of life, primality > testing, graph coloring ... anything that can be discussed in a > day or so to folks with no CS background, yet which has some > theory component to it ... stuff that is surprising or counter-intuitive > is all the better ;) > Anyway, if anyone has any suggestions for material/topics > that I might cover, I would most appreciate it. Any pointers in FOCUS about 12 years or so ago that I recall that claimed the following, if I recall correctly: Given any theorem and a proof of that theorem, it is possible to construct a graph with a Hamiltonian circuit, and where given the graph and a Hamiltonian circuit, the proof can be reconstructed, and someone who does not know the proof can verify that the graph has this property, without gaining any information about the proof itself. (Someone jump in here, please, and correct any mistakes in the above. I defintely recall the parts below, but I'm fuzzy on the above. In particular, is the graph constructed from the theorem and the proof, or can the graph be constructed without knowing a proof of the theorem? I.e., for every theorem, does there exist a graph that contains a Hamiltonian circuit if and only if the theorem is true?) One of the common examples given to show zero-knowledge proofs is the problem of proving that you know a Hamiltonian circuit on a graph. Combine those, and use a non-interactive proof, and you get the surprising result that if you have a theorem and a proof of it, it is possible to prove to others that you indeed have a correct proof, without giving them any information about that proof! Imagine how frustrating that would be if it were actually practical and someone did that for an important problem, like one of the Millenium problems. (I wonder...would they get the prize?) > would be accessible to students would be great (I have a couple) > would be great. Take a look at The New Turing Omnibus by A. K. Dewdney. The book itself is probably too lightweight for your needs, but it will give you many topic ideas. For something different, how about sorting stacks of pancakes? Suppose you've got a stack of N pancakes, each a different size, and you'd like the stack sorted with bigger ones on the bottom, smaller on top (e.g., like a stack from the Towers of Hanoi). The only operation you have is to pick the topmost k pancakes (1 <= k <= n), and reverse them. So, if your stack is (from top to bottom) 1 5 3 2 4, and you flipped the top 4, you would have 2 3 5 1 4. All the usual questions can be asked about this: average number of flips required, worst case, etc. Besides being an interesting problem, there is an interesting bit of trivia associated with it. It is the subject of the one and only scientific paper by Gates: Gates, W. and Papadimitriou, C. Bounds for Sorting by Prefix Reversal. Discrete Math. 27, 47-57, 1979 This is why Gates has an Erdos number (4). === Subject: Re: Ideas for course on great ideas in (theoretical) CS? My CS teacher had a very good lecture on The set of all computable functions is countable. He told us: Think of the content of the memory of a computer, including executable code. It all boils down to a very big integer, doesn't it? The Knapsack problem and Travelling Salesman problem, IMHO, are worthy of paying attention. Or are you looking for something more dramatic? Of course, one thing you cannot forget to mention is the P=NP problem. === Subject: Re: Differential equation with implicit solution > My differential equations book asks the following: > Show that -2x^2y + y^2 = 1 is a solution of the differential equation > (in differential form) 2xy dx + (x^2 - y) dy = 0. Ok, that's trivial, > one just implicitly differentiates the first equation, etc. > They also ask to find at least one explicit solution. Now here's my > problem: What explicit equation satisfies both the equations above? If x = 0 then the DE becomes d/dy(y^2) = 0, which has a general solution y^2 = C. That could be what they're looking for. But this solution, and the one they gave, are both special cases of the general solution, which is: y^2 - 2.x^2.y = C If x != 0, take y = x^2.z in the DE, so dy/dx = x^2.dz/dx + 2xz and the DE becomes, after some manipulation: x.(z - 1)/2.dz/dx + (z - 1)^2 = 1 With t = (z - 1)^2 this becomes: dt/dx + (4/x).t = 4/x Multiplying by x^4 then gives: d/dx(t.x^4) = 4.x^3 i.e. t.x^4 = x^4 + C and the result follows. Cheers -------------------------------------------------------------- ------------- John R Ramsden (jr@adslate.com) -------------------------------------------------------------- ------------- Eternity is a long time, especially towards the end. Woody Allen === Subject: Re: 'erf' function in C ... > Is there someplace a table available of the true values of the > normal distribution up to 20 places? A 20 decimal place table of normal percentiles was published in JASA Percentiles by John S. White. It is downloadable from JSTOR if you have JSTOR access. Jerry === Subject: Re: integral >> can anyone help me in integrating from 0 to infinity e^(-(x^2))dx? > -Greg R. >I'd suggest trying a power series. >How could that possibly help? >I was trying to say take the power series for e^x, compose it with -x^2 and >then integrate that; since it's improper, you'd probably have to take a >limit. I've seen something like this done in my calculus book. As long as you integrate term-by-term from 0 to A and _then_ let A - infinity this is legal (from your first post I assumed you meant take the integral of each term from 0 to infinity - that's going to give nonsense.) But although it's legal it's not going to help, because there's no way you're going to be able to evaluate the limit of the resulting series. Hmm, I shouldn't jump to conclusions. Just because I can't doesn't mean you can't. What's the limit as A -> infinity of sum_0^infinity (-1)^n A^(2n+1) / ((2n+1)n!) ? >David Moran === Subject: Re: Having integration trouble > Hello. I am working on integrate sqrt ((4x^2) +9)/(x^4) using trig > substitution. > I have: > u=2x > a=3 > 2x/3 = tan theta > dx=3/2 sec^2 theta d theta > (sqrt (4x^2 + 9))/3 > S 1/(sqrt (4x^2 +1)) dx= 3/2 S ((sec^2 theta d theta)/sec theta) > =3/2 S (sec theta d theta) > =3/2 ln abs (sec theta + tan theta) + C > =3/2 ln [ abs ((sqrt (4x^2 +9) +2x/3)) +C > Things that are throwing me: 1) the x^4 in the denominator of the problem > and 2) the triangle setup > I have looked in several books and all over the internet to find a problem > like this to see if I was on the right track, but none that I found look > like this. I don't know if the answer is right, but it is the best I could > do with what I know. Substitute x = (3/2)*tan(u); dx = (3/2)*sec^2(u) du Then, sqrt(4x^2 + 9)/x^4 = 2*sqrt(x^2 + 9/4) / x^4 = 2*sqrt(9/4(tan^2(u) + 1) / ((81/16)*tan^4(u)) = (3*16/81)*sqrt(sec^2(u)) / tan^4(u) = (3 * 16/81) * sec(u) / tan^4(u) and f(u) du = (48/81)* sec^3(u) / tan^4(u) = cos(u) / sin^4(u) du Now set z = sin(u), du = -dz/cos(u) P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. === Subject: Re: Letter to Prof Ullrich and others > Try Google with > moron factorization bull > and guess who comes up. LOL - The first two hits, out of only seven in total. Cheers -------------------------------------------------------------- ------------- John R Ramsden (jr@adslate.com) -------------------------------------------------------------- ------------- Eternity is a long time, especially towards the end. Woody Allen === Subject: Re: Having integration trouble > Hello. I am working on integrate sqrt ((4x^2) +9)/(x^4) using trig > substitution. > I have: > u=2x > a=3 as you have never used this last bit (actually, you haven't yet introduced it properly!) I will ignore it. > 2x/3 = tan theta > dx=3/2 sec^2 theta d theta that's right. > (sqrt (4x^2 + 9))/3 > S 1/(sqrt (4x^2 +1)) dx= 3/2 S ((sec^2 theta d theta)/sec theta) > =3/2 S (sec theta d theta) > =3/2 ln abs (sec theta + tan theta) + C > =3/2 ln [ abs ((sqrt (4x^2 +9) +2x/3)) +C > Things that are throwing me: 1) the x^4 in the denominator of the problem > and 2) the triangle setup what did you do here? you cannot just chuck away bits of pieces *under* the integral sign! it is violating the law! remember the integral rules? i honestly think they should add this one: don't do anything just because you think you can do it. name the law you are using. if not found, call it quits. there is no rule of throwing away the denominator, by the way. > I have looked in several books and all over the internet to find a problem > like this to see if I was on the right track, but none that I found look > like this. I don't know if the answer is right, but it is the best I could > do with what I know. > Stacy i don't know if anyone is stupid enough to answer your question, because they should not, but i can help you with the triangle. > 2x/3 = tan theta that's right. draw the triangle. the adjacent is 3, opposite is 2x, that makes the hypotenuse sqrt{4x^2 + 9}. [i cannot resist a hint: keep the x^4 bit, convert it, then express it in terms of sines and cosines. always helps] === Subject: Re: Differential equation with implicit solution > My differential equations book asks the following: > Show that -2x^2y + y^2 = 1 is a solution of the differential equation > (in differential form) 2xy dx + (x^2 - y) dy = 0. Ok, that's trivial, > one just implicitly differentiates the first equation, etc. > They also ask to find at least one explicit solution. Now here's my > problem: What explicit equation satisfies both the equations above? both? solve the quadratic for y, i get y = x^2 (+/-) sqrt(x^4 + 1). check which branch is your solution (if any). i have found an even nicer ones -- y = 0, 2x^2! my trick was to note the right hand side 1 can be replaced by any constant (let's see -- zero!) and still the implicit differentiation works. of course, it is an explicit solution of the differential equation, not both. i don't think the question wanted both. > I've graphed the implicit solution (first equ above) and get what > looks like almost y = 2x^2 + 1, and an inverted witch of agnesi with y > = -1/((3/2)x^2 + 1). Unfortunately, these are not exact fits. > I imagine that there is in fact such an explicit solution to both (on > some interval of definition) and that perhaps there might be a method > to derive it. too simple a method perhaps. > Any help is greatly appreciated. > R === Subject: Shannon/Nyquist URL? I'm looking for a URL to give me an original (as far as possible) copy of the Shannon/Nyquist Sampling And Reconstruction Theorem. (I've obtained Shannon's, A Mathematical Theory Of Communication in the last couple of days, so I'm hoping that the former will also be available, but it hasn't appeared to me in any Google search) === Subject: Re: Differential equation with implicit solution >My differential equations book asks the following: >Show that -2x^2y + y^2 = 1 is a solution of the differential equation >(in differential form) 2xy dx + (x^2 - y) dy = 0. Ok, that's trivial, >one just implicitly differentiates the first equation, etc. >both? solve the quadratic for y, i get y = x^2 (+/-) sqrt(x^4 + 1). >check which branch is your solution (if any). >i have found an even nicer ones -- y = 0, 2x^2! This is hardly surprising given they are trivial linear combinations of the two general solutions. I'm not interested in mathematics that might have anything to do with reality. -- Easterly, in sci.math === Subject: Confused . . . expectation value of a random variable Definition: A *probability distribution* for a discrete random variable X with values { x_k | k in Z } where Z is the set of integers, is a set of numbers (I think they mean real) { p_k | k in Z } such that P(X = x_k) = p_k >= 0 and sum_{k = -infty}^infty p_k = 1 Definition: If X is a discrete random variable with values { x_k | k in Z } and probability distribution { p_k | l in Z } then the *mean* or *expected value* of X is E(X) = sum_{k = -infty}^infty x_k p_k Then comes the confusing bit: Proposition: Let Y = g(X) be a function on X. Then then *mean* or *expected value* of Y is E(Y) = sum{k = -infty}^infty g(x_k) p_k My question is, is that a definition? Because, in case of continuous random variable, they are clean in saying that it is. However, here, they comment: It is possible to give a general proof of this proposition, but we shall not do so here. Leaving me in eternal confusion that started the day I was born. When I was born I was so surprized that I couldn't talk for a year and a half --- I honestly don't remember who said that. === Subject: Re: Cavemen Hear! Hear! === Subject: Re: Help needed from a group theorist by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1LDK3j18310; Martin >An exponent E(G) of a group G is the lcm of all the orders o(g), >g in G.We have that G is cyclic if and only if E(G)=|G|. >Does anybody have an idea how can we find all n such that >the exponent of Z_n*,the group of units of Z/Zn is 2? >E(Z_n*)=2 n=? >This just means that x^2=1 for any invertible x in Z_n. Thus n cannot have >any prime factors other than 2 and 3. Write n=(2^s)(3^t). Splitting Z_n as a >product of two rings, the cases are >-- E(Z_(2^s)*)=2 and E(Z_(3^t)*)=1 i.e. s>=1 and t=0 (n=2,4,8,16,32,...) >-- E(Z_(2^s)*)=1 and E(Z_(3^t)*)=2 i.e. s=0 and t=1 (n=3 only) >-- E(Z_(2^s)*)=2 and E(Z_(3^t)*)=2 i.e. s>=1 and t=1 (n=6,12,24,48,...) >I hope I haven't blundered somewhere in here :) I still don't understand why is n=(2^s)(3^t),but I'll take that for granted. :) The order of Z_n* is phi(n) (Euler function),and since all the orders of the elements are 2, Z_n* is abelian. Can we determine the number of generators of Z_n* ? If there are m generators of Z_n*,then all we have to do is to solve phi(2^s*3^t)=2^m or 2^s*3^(t-1)=2^m This is all I can do and I hope someone can finish this. Martin === Subject: Re: errors in an argument by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1LDK2u18292; >My problem is more general. Too many times we get a very complicated and >fine tuned mechanism that would require a major jump in order to have any >effectiveness. Fish that use electricity to stun and kill opponents is a >remarkable one. The amount of electricity generated, and the coordination >involving it is very big. >The major jump is only in your mind. >It looks like it would take a big number of mutations to achieve it, with >any partial mutation or a small effect totally useless. A functional yet >small weapon of this kind is even punishing given the amount of energy >needed for it, and lack of effective results. >Two words: active sensor >http ://www.flmn h.ufl.edu/fish/tropical/JSA/gymno.htmNext time, do your own research. >http://w ww.google.com/search?q=evolution+%22electric+eel%22-- >iel W. Johnson >panoptes@iquest.net >http:// members.iquest.net/~pano ptes/039 53 36 N / 086 11 55 W Thank you very much for throwing phrases, pointing me to google, and telling me to do my own research in a discussion. My prime interest is math anyway and not biology, and I feel the same as No way about it being out of the scope of this board so I'll take it to a more relevant place when it bugs me again. === Subject: (A<-->B) as Set Derivative by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1LDJwU18174; Define: 1) (A-->B) = (AB')' = A' U B and therefore: 2) (A-->B)(B-->A) = (A' U B)(B' U A) = AB U A'B' where adjacent parentheses refer to set intersection. The symbol for (A-->B)(B-->A) should be familiar to people in Mathematical Logic, or to Logicians if a, b were propositions: 3) (a<-->b) = ~(a^~b)^~(b^~a) = (~aVb)(~bVa) = ab V ~a~b where ~ (tilde) represents negation (NOT), V disjunction (OR including AND), ^ conjunction (AND). The left-hand-side of (3) is written in Logic a iff b or a if and only if b, and the Set Theory analog is therefore for sets A, B: 4) (A<-->B) = (A-->B)(B-->A) = AB U A'B' Taking complements of both extremes in (4) yields: 5) (A<-->B)' = A'B U AB' which is the analog for Sets of: 6) (fg)' = f'g + fg' for differentiable functions f, g. Notice that although U in (5) is set Union (corresponds to logical AND/OR), it happens to be a disjoint or mutually exclusive Union since A'B does not intersect AB' (it has zero intersection). It is not unusual to write U as +, for example circled +, in such situations with regard to vector spaces and so on, but I will retain U to be technically correct. Readers can prove as an exercise that if u is a bounded Lebesgue measure (a Lebesgue measure on bounded sigma algebras, etc.), then: 7) u[(A<-->B)'] = u(A'B) + u(AB') which is even more similar to (6), where A, B are sets in the sigma algebra. To assert, then, that (A<-->B) (A if and only if B or A iff B) is the set analog of fg with regard to complementa- tion and ' as analogous to differentiation, seems quite intuitive to say the least, although we don't assert anything for higher primes like A or other contexts at this point. Obviously, A is (A')' = A but f is not f in general unless f is the exponential function exp(x). The exponential function, by the way, is not as trivial as it may seem, and I leave that as an exercise (hint: consider inverse functions). Osher Doctorow === Subject: Re: would you tell me exactely ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1LDJxB18185; >mathematics is something like Religion. >You can sell your talents if you're smart enough to have a bag of tricks >which impress employers or use the talent to get a Techie degree. But >the >spirit of math some transcends this. (something about the Magic feeling >you >get when you are learning something new, or finally see a connection, or >just being able to design an Excel sheet that will do stuff you would have >never had the imet ot do otherwise.) I am convinced that there is a great >number of people , in spirit, who are Mathematicians, but lack background >and are frustrated or feeL some loss, because of the madness that passes >for Education in grades 1-15. > Mathematicians don't have to worry about grades 1-15, > since they come out of the country's *education budget*. > Science and Mathematics has been placed in the country's Jesus Freak >Budget. thanks your message that it give me a new way of thinking what is the math. you said the math is like religion. i thought we had different opinion about it. in my opinion I thought math is like religion is the thing that you must stick everyday then you will find something special. I thought math is full of magic before but then I found the math is around us. that's one of the important reasons that's why I want to study it by my spare time. it can help you slove the problme , not just some math problme , it's a good thinking way that you can get a good decision when you fact to something. thank you sir , if you read my message I hope you can give your opinion and told about it details . I'm interested in your opinion. and also sorry about my bad english. 008(it's my name). === Subject: advice on graduate studies by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1LDJxI18192; I am an undergrad who will be graduating this semester and will be going to grad school in fall((if they let me)) and i have a few questions. I have heard that you should not only think about what you like but also what is still a growing field. I really enjoy analysis.((classical,functional,Fourier..)) I really dislike (at least so far) algebra,number theory ,combinatorics. Now my question is : are the above named analysis's still a good field to get into or should i look more at say applied analysis.Someone once told me that unless you are a genious that harmonic analysis is not a good field to get into. Is this because the field is so old and developed? Would the same apply to classical and functional analysis? I have done some PDE theory and enjoyed working in the Sobolev spaces with the functional analysis techniques. If i was to go into PDE's would I expect more of the same (which i enjoyed) or does it turn into a grab bag of techniques ? I have never done any measure theoretic probability but considering my interests would this be another one to think about? Any replies would be greatly appreciated don === Subject: A slight generalization of Waring's problem 0 as n->infty. ...integrating from 0 to infinity e^(-(x^2))dx? > Let I be the number we want. > I^2 > = (integral from 0 to infinity e^(-(x^2))dx) > times (integral from 0 to infinity e^(-(y^2))dy) > = double integral (0 to infinity in x and y)[e^-(x^2+y^2)]dy*dx > (change to polar coordinates) > = double integral(0 <= theta <= pi/2)(0 <= r < infinity) > of the function: e^(-r^2) times r*dr*d(theta) > infinity > = (-1/2)e^(-r^2)| times pi/2 > 0 > = pi/4. > Since I^2 = pi/4, > I = (1/2)sqrt(pi). Holy smoke! That's neat! I have a question ... very silly ... regarding this business. How exactly this substitution thing work? SPECIFICALLY: What are the necessary and sufficient conditions for a successful substitution? Is there anything about multiple valued functions (probably inverse functions), open or closed domain and ranges, discontinuities, poles, whatever topological ... that I should be aware of while doing a substitution? I know for sure whatever can go wrong normally does not go wrong, or else I would have done very bad in the exams! Take this one for example ... coordinating (x,y) plane with (r,theta) has something special about it that scares me. x runs from -infinity to infinity, so does y, but r goes just half of that range, and theta is BOUNDED! I know 0 <= theta <= pi sounds fine, and that omitting the = in <= would do know harm ... et cetera ... but I just wanted a brief review of the whole story. Anyone care to go through the pain? Just to encourage you: remember how it felt when it all came together for you the first time? === Subject: Re: Having integration trouble ETAsAhQH37BTZh5i2Yb6PwwIFCN28K5rUwIULpT/vcZrvZRuTV+wlIE/ 9NAwy6I= If you have x = (3/2) tan (theta), then express x^4 in terms of theta along with the other factors. When you get your trigonometric integrand, it looks complicated; but express it in terms of sines and cosines and see what comes up. --OL === Subject: Re: x^2 + y^4 = z^4 ETAsAhRO71Sss8Pu0NTos/ w1OHq82cZgwwIUdz53qtcwP4RJywFO90nMnqpQ1us= If a^2+b^2 = 2c^2, then a and b have the same parity and thus for some integers m and n we have: a = m+n b = m-n Then a^2+b^2 = 2*(m^2+n^2) and therefore m^2+n^2 = c^2. The problem of three squares in arithmetic progression is thus reduced to the more familiar problem of Pythagorean triples. --OL === Subject: Re: there is no such thing as infinity Stop bragging guys ... please! I feel diminished ... I am only 22! :( Sometimes I get 'Ohhh!', 'Wow!', and 'Ooooh!'s for having a 386 [Who am I kidding! I *had* it, like, thousand years ago. The one I am 640K ought to be enough for anybody. - GATES, 1981 [I was born the following year :o] === Subject: Re: Confused . . . expectation value of a random variable > Definition: > A *probability distribution* for a discrete random variable X with > values > { x_k | k in Z } > where Z is the set of integers, is a set of numbers (I think they mean > real) > { p_k | k in Z } > such that > P(X = x_k) = p_k >= 0 > and > sum_{k = -infty}^infty p_k = 1 > Definition: > If X is a discrete random variable with values > { x_k | k in Z } > and probability distribution > { p_k | l in Z } > then the *mean* or *expected value* of X is > E(X) = sum_{k = -infty}^infty x_k p_k > Then comes the confusing bit: > Proposition: > Let Y = g(X) be a function on X. Then then *mean* or *expected value* > of Y is > E(Y) = sum{k = -infty}^infty g(x_k) p_k > My question is, is that a definition? Because, in case of continuous > random variable, they are clean in saying that it is. However, here, > they comment: > It is possible to give a general proof of this proposition, but we > shall not do so here. It is not a definition. If X is as described, then Y = g(X) is another discrete random variable, and it has value g(x_k) in the event that X has value x_k. The reason the proposition is not immediately obvious is that you could have g(x_k)=g(x_j) for different x_k, x_j. So the sets {Y = y_k} could be unions of several of the sets {X = x_k}, possibly even infinitely many such sets. G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: (A<-->B) as Set Derivative >Define: >1) (A-->B) = (AB')' = A' U B >and therefore: >2) (A-->B)(B-->A) = (A' U B)(B' U A) = AB U A'B' [...] >Taking complements of both extremes in (4) yields: >5) (A<-->B)' = A'B U AB' >which is the analog for Sets of: >6) (fg)' = f'g + fg' It also gives other interesting and useful differentiation formulas: (A u B)' = A'B' = A<-->B AB = A<-->B (A' U B')' which I suppose tells us that: (f + g)' = fg - (f' + g')' I'm not interested in mathematics that might have anything to do with reality. -- Easterly, in sci.math === Subject: Re: JSH: Non-uniqueness of factorization > I can't imagine what kludge would prevent one from reading headers to > determine whether the message is a mime message. I think what > happened is this. Anyone may correct my mistakes, of course. > Prior to the widespread use of mime, folks sent messages with > uuencoded files in them. Good call. Here's MS's page on the error. http://support.microsoft.com/?kbid=265230 === Subject: Re: Math notation question: '(' > The statement was: > There exist x, y, and z that are nonzero. Such that 1 ( y>x and xy=z. > The period after nonzero looks wrong. Is it a comma? Yeah, it shouldn't be there. Sorry. See whole whole transcription below. > Does 1 ( y>x mean that both y & x are less than 1? Looking at the > answer that is the only thing that makes sense to me. I just don't > understand the notation. > Neither do I. Have you accurately transcribed the _complete_ sentence? If x, y, and z are nonzero numbers such that 1(y>x and xy=z, which of the following CANNOT be true? 1) y>z 2) y=z 3) z=x 4) x>z 5) z>0 I think it's 4) x>z, but the answer in the review says it's 2) y=z. However, when x=1, y=2, and z=2 it meets all of the criteria and y=z, so #2 cannot be the answer. When looking at their explanation for why #2 is the answer they say because all of the other choises CAN be true #2 must be the correct answer. Which, to me, is not only a BS cop-out, but wrong. Also, in the #4 explanation it says According to the question, Y must be greater to or equal to one. Thus *my* question on the notation. Whatever happened to <=? I should also state that nowhere else in the review does it mention that the '(' to <= notation substitution exists. I think it's just a shoddily put-together review. === Subject: Re: JSH: Non-uniqueness of factorization <87ad3den4w.fsf@phiwumbda.org <25rZb.64909$KV5.49312@nwrdny01.gnilink.net <87wu6hd2zr.fsf@phiwumbda.org <5ruZb.63308$IF1.32769@nwrdny03.gnilink.net <87k72gdd8v.fsf@phiwumbda.org> I can't imagine what kludge would prevent one from reading headers to >determine whether the message is a mime message. I think what >happened is this. Anyone may correct my mistakes, of course. >Prior to the widespread use of mime, folks sent messages with >uuencoded files in them. > Good call. Here's MS's page on the error. > http://support.microsoft.com/?kbid=265230 It's a beautiful page. Here's the workaround for users of OE. To workaround this problem: * Do not start messages with the word begin followed by two spaces. * Use only one space between the word begin and the following data. * Capitalize the word begin so that it is reads Begin. * Use a different word such as start or commence. This is remarkably silly advice. The problem for the OE user who goes to this page is that he's received email (or read a posting) that he can't read. The only workaround that those big brains at MS have for him is advice on how to send email/posts that their broken software can read. They give no advice on what to do if you want to read something that OE can't display. Jesse Hughes [I]f gravel cannot make itself into an animal in a year, how could it do it in a million years? The animal would be dead before it got alive. --The Creation Evolution Encyclopedia === Subject: Re: ATTENTION: Open dispute with my college about Procedures. Please read. > Did you take issue with the grading of your homework paper? It's > absurd. If nobody has looked at it, he received a 25% deduction on a > single problem for not writing (g o g) (x) = g(g(x)), (where the o > is the composition symbol) and a comment that it was very hard to > follow his work, even though it was quite easy to read. Indeed, it > is the TA's writing that I find almost impossible to read. At one > point, several points are taken off for use of notation in the work > section of a problem which is not a proof (it is a numerical > computation). In other places points are taken off for no apparent > reason. At yet another place he shows that a function is not defined > at a certain point and then asserts that it is noncontinuous at that > point, at which the TA takes off points, apparently for not saying > something like, Observe that if the function is not defined at this > point, its limit at this point cannot equal its (nonexistant) value. > As far as I can tell in the cursory glance I gave the test only one > answer is actually incorrect. > You have demonstrated your competence in the subject material of > the class not only in the practical but in the theoretical. I don't > really see what else the teacher could ask for. > As for trials and lawyers, I'm not sure they'd do you much good. > You'd have to dig up a law or something on the college's books against > not letting you drop a class under certain conditions, and I'm pretty > sure they didn't write such a thing in. I would argue for more points > with the teacher, and if that doesn't suceed I'd retake the class. I looked over the test, and I think that the grading was fair. (For example, the 25% was taken off because he didn't do part of that question, that is, find the domain.) On the other hand, I do think that the test taker does show some competence - he definitely should not drop out of college. He could retake the course - indeed I didn't think it was bad enough to drop the course, unless he wants a sure A out of the course, which I don't think he will get at the end anyway. I don't think it is worth arguing the W, which is a fine grade to obtain for a course. Instead of going to lawyers and such like to argue for something which is not worth having, maybe the student should take his test and talk with the teacher about it - not to get the points back, but rather to see how he could do better next time. While I think that the lost points were in almost all cases fair, they were mostly lost over little things. The only question he got completely wrong was 6(b), and I bet most people in the class also got it wrong. I do agree with you that his handwriting was fine. Maybe it was late at night when the grader looked over his work. My advice would have been to continue with the course - do better in later tests, and be happy with the B or C that would come at the end. === Subject: Re: Cavemen > Where can I find pics of cavemens language e.g. A pic of a wall inprint This really isn't a math topic, but anyway, try: http://www.photovault.com/Link/Entertainment/Paintings/ Heiroglyphics/EPHVolu me01.html Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: Core error proof, simpler, shorter Discussion, linux) > It turns out you can prove that there's an error in core with rather > basic math, using a quadratic: > Let > P(x) = (x+8a)(x+b), and ab = 1, so P(x) = x^2 + (8a + b)x + 8. > Then > a = 1/b and b = 1/a I don't know why you write then here, but never mind. Let's stick with this as given. [...] > That means that 'a' *cannot* be an algebraic integer, as algebraic > integers cannot be roots of primitive non-monic polynomials > irreducible over Q! > But notice that 'b' IS an algebraic integer as it's the root of a > monic polynomial with integer coefficient. Did you just prove the deep and startling fact that sometimes, if b is an algebraic integer, 1/b is *not* an algebraic integer? Many argue that its programmers have turned out shoddy programs, but [their] objective is to make profit, not superlative programs per se. By the profit criterion, Microsoft has been one of the greatest companies in the history of this country. -- ADTI defends Microsoft === Subject: Re: Data analysis software > It plots and analyses any x-y data for peak location, peak height, > peak > width, semi-derivative, derivative, integral, semi-integral, > convolution, > deconvolution, curve fitting, and separating overlapped peaks and > background. > www.chemSoftware.com === Subject: Re: Letter to Prof Ullrich and others I am just curious: why are people like you > interested in even acknowleding James Harris and his ilk ? > This guy has been oopsing on sci.math for ages (at least several > years) I see. >come from the discussions. I have learned quite a bit myself. Also >interesting results come from it, like the Magidin-McKinnon theorem. >One of the things I learned was how to deal properly with quadratic >fields. Hear, hear! As a complete mathematics novice, I have learned enough maths from reading detailed and lengthy expositions to JSH that I am even dabbling with the idea of investigating some maths myself. If the estimable persons the OP addressed had not acknowledged JSH with their posts, even my (tiny) knowledge would have remained unadded to, and I could have made a complete and utter pillock of myself quoting him to my mathematical friends (including a teacher/graduate). As it is, I can have interesting conversations with them and, sticking with sci.math because of James' lunacy, I can even explain the 'waiters puzzle' to people (a minuscule victory, but mine own). Min So where are all the buffaloes? === Subject: Re: Hausdorff dimension of graph of Weierstrass function. >[...] >Indeed, if t is large enough a very crude argument >suffices to show Box = s. > Having dusted off a few cobwebs in the lacunary-series > section of my brain, it's clear that Box = s for t > 1 > and 1 < s < 2. > The proof depends on the fact that the supremum of > a lacunary series is larger than a constant times the > sum of the absolute value of the coefficients. This > is well-known, for example it can be proved using Riesz products (see Katznelson). We need a > slightly sophisticated local version, the proof of > which must be somewhere in the lacunary series > folder in the filing cabinet in my office: > Lemma. Suppose L > 1 and a > 0. There exists > C = C(L, a) > 0 such that if > f(x) = sum_1^N a_j cos(n_j x), > where n_{j+1}/n_j > L for all j, and I is an interval > of length a/n_1, then the sup of |f| on I is larger > than C sum |a_j|. > Assuming that, consider a square grid in the plane > with side length h. Take the sum and split into > the sum from 1 to N - 1 plus the tail, where N is > chosen so that the oscillation of the sum from > 1 to N - 1 on any interval of length h is less than > h (use obvious estimates on the derivatives). > If you choose N maximal with this property > it turns out that h > a/t^N for some constant > a > 0, so the lemma shows that the max of the > tail minus the min of the tail on any interval of > length h is large. Hence the same is true of f > itself, and this shows that the graph hits a large > number of squares in the grid, forcing Box > s. This seems to be similar to Falconer's proof. G.C. === Subject: Re: Pi gallons problem - any ideas? hrllo, >[...] >V2 = Ah/3 (r^(3/2) - 1) / (r - 1). >V2 is the volume of that subset of the frustum below a plane tangent to >the top and bottom circles at opposite points: i.e., the volume of water >that would remain in a frustum if it were tipped over until the water >just touched the bottom circle. >[...] > philippe 92 asked in message >I searched instead how to proove the >V2 = Ah/3 (r^(3/2) - 1) / (r - 1) >formula ... I didn't succeed.... >(V1 is easy, thanks) Jim Ferry replied: [...] > An easy way to prove this formula is to compute the volume of the shaded > region. The shaded region forms a cone (not a right circular cone, but > an oblique, elliptical cone), so its volume is (1/3) base x height. [...] Thank you for showing me this cone that I didn't see ... (I should). philippe (chephip at free dot fr) === Subject: Re: Shannon/Nyquist URL? been Airy R. Bean was cut from the mess that had once been his life, he managed to utter: >I'm looking for a URL to give me an original (as far as possible) >copy of the Shannon/Nyquist Sampling And Reconstruction Theorem. >(I've obtained Shannon's, A Mathematical Theory Of Communication >in the last couple of days, so I'm hoping that the former will also >be available, but it hasn't appeared to me in any Google search) Those intending to respond, should bear in mind that the 'technical' reading of this poster, is on a par with his wife Elisabeth's reading of Womans Weekly (or was it Peoples Friend?). Anytime you are ready for another go at The English Game, old fruit. You be sure and let me know. Old Nick. === Subject: JSH: Understand now? Frustration? Using P(x) = (x+8a)(x+b), where ab=1, and considering when (8a + b) is an integer can give you a perspective on what I've been saying, I hope. Basically the mathematical position of the mainstream--why it's a core error--is that you can never have a factor like 8 in an arena where -1 and 1 are the only integer units because you can't in the ring of algebraic integers, except for simple radical factors, if you don't have integer 'a' and 'b'. That's it. The arguments now should make sense to you as I talked about situation where 'a' *should* be a factor of 1, as it is in fact--in a more inclusive ring. The flaw with the understanding of the ring of algebraic integers is a fascinatingly frustrating one to explain!!! Posters found many ways to confuse the issue, and I still find it hard to believe that experienced professors like Arturo Magidin or Rick Decker never figured it out. But it's up to them to give a full accounting of why they've said what they've said. I'm just glad that I finally thought to put out something like P(x) = (x+8a)(x+b), where ab=1, so P(x) = x^2 + (8a + b)x + 8, with an emphasis on when (8a + b) is an integer. If you're STILL confused, I suggest you play with that quadratic for a while. First consider simple cases like I did, like a=b=1, and see what happens, and then move on to irrational 'a' and 'b', and see what happens. Oh yeah, I see nothing wrong with the label algebraic integers or their definition and wouldn't want either changed, but what *has* to be taught along with what has been previously taught are the interesting features and limitations of the ring. Universities and mathematicians who continue to refuse to tell the truth, shouldn't expect people to be on their side when at this point there's just no excuse. If you wish to be a rogue university, or a rogue math professor refusing to teach mathematics correctly, then I think you should be punished severely, and no one who respects society and progress should have pity on you. It is, after all, your choice. You have been repeatedly warned. James Harris === Subject: Re: Genetics and Math-Ability You avoid the question of why does it get inhereted, indeed why does it exist at all? The evolutionary pressures on our brains were forged in the Savanahas of Africa over a million year period. I can understand the evolutionary pressure on being able to count 1,2,many, many many Wildebeests. But calculate the surface area integral of the plain on which they run? You see some specilaised organ in Nature, and you look for the purpose which let it evolve. That doesn't exist for maths. There is a fashionable theory of intelligence, called the peacocks tail theory of human intelliegence. The analogy is to the peacock's tail, which has no functional use, except to attract mates. That alone allows the trait to develop. But do women really want to go to bed with guys who are good at math? And if that was so, could it have had enough evolutionary pressure to make something that can make http://mathworld.wolfram.com/? > Just curious: > What is the current scientific opinion regarding how math-ability is > inherited? > Is it, in general, a recessive trait or is it dominate? > (I say recessive.) > I am only half-serious, since math ability is most probably a result > of many traits and experiences and education. > But I was wondering today because, even though there are some in my > family who were math-teachers, almost everyone else had/has little > ability with mathematics...even very little ability as relative to > me... > ('relative to me'...snicker, snicker...) > Leroy Quet === Subject: Integration by parts help needed I'm having trouble integrating by parts. For example: I = Integral(x cox(2x) dx) dv = x, v = x^2/2 u = cos(2x), du = (1/2)sin(2x) I = ((x^2/2)cos(2x)) - ((1/4)Integral((x^2)sin(2x) dx)) Then I'm not sure what to do because the integral on the right seems just as complex as the one I started with...? BTW what do we use on here for the integration symbol? === Subject: Re: Understand now? Frustration? > Using P(x) = (x+8a)(x+b), where ab=1, and considering when (8a + b) is > an integer can give you a perspective on what I've been saying, I > hope. > Basically the mathematical position of the mainstream--why it's a core error--is that you can never have a factor like 8 in an arena > where -1 and 1 are the only integer units because you can't in the > ring of algebraic integers, except for simple radical factors, if you > don't have integer 'a' and 'b'. > That's it. The arguments now should make sense to you as I talked > about situation where 'a' *should* be a factor of 1, as it is in > fact--in a more inclusive ring. > The flaw with the understanding of the ring of algebraic integers is a > fascinatingly frustrating one to explain!!! > Posters found many ways to confuse the issue, and I still find it hard > to believe that experienced professors like Arturo Magidin or Rick > Decker never figured it out. > But it's up to them to give a full accounting of why they've said what > they've said. > I'm just glad that I finally thought to put out something like > P(x) = (x+8a)(x+b), where ab=1, > so P(x) = x^2 + (8a + b)x + 8, with an emphasis on when (8a + b) is an > integer. > If you're STILL confused, I suggest you play with that quadratic for a > while. First consider simple cases like I did, like a=b=1, and see > what happens, and then move on to irrational 'a' and 'b', and see what > happens. > Oh yeah, I see nothing wrong with the label algebraic integers or > their definition and wouldn't want either changed, but what *has* to > be taught along with what has been previously taught are the > interesting features and limitations of the ring. > Universities and mathematicians who continue to refuse to tell the > truth, shouldn't expect people to be on their side when at this point > there's just no excuse. > If you wish to be a rogue university, or a rogue math professor > refusing to teach mathematics correctly, then I think you should be > punished severely, and no one who respects society and progress should > have pity on you. > It is, after all, your choice. > You have been repeatedly warned. > James Harris Yeah, we've been warned by someone who doesn't even understand the arguments. Harris, you are a crank, and everyone but you knows it. Plus, how are you going to prove this math is wrong, say to a judge, if you take legal action? *giggle* Get a life, Harris. David Moran === Subject: Re: JSH: Understand now? Frustration? Discussion, linux) > If you wish to be a rogue university, or a rogue math professor > refusing to teach mathematics correctly, then I think you should be > punished severely, and no one who respects society and progress should > have pity on you. What if we suspect that our university is rogue? How can we tell for sure? Do rogue universities fly the Jolly Roger? Do they advertise for employment opportunities in rogue journals? I know a guy who wishes to be a rogue university. How should we punish him severely? Who do I call? So many questions... please help... I've ... contacted [some of the...] highest I.Q.'s in the country... I've even helped the FBI out a few times... I've met at least one governor..., a senator... and I've had some really good seats at sports games. My experiences are not your experiences. --JSH != you === Subject: Re: Probability of finding a Prime Number boundary=----=_NextPart_000_0008_01C3F868.B7918380 -------------------------------------------------------------- ------- charset=iso-8859-1 I have been traveling for work and didn't get a chance to respond sooner to Bob Silverman's note. As Bob points out there was a flaw in my reasoning. I should not have assumed that the probability of prime p divides N is independent of the probability that other primes divide N. I best understand this by mentally constructing a roulette wheel. The wheel has 6 positions on it numbered 1 to 6. I mark each position that is divisible by 2 in red. I randomly select a number by spining the wheel. The chances of landing on a red position are 1/2. Now, I construct a second wheel identical to the first and mark each position that is divisible by 3 in red. I randomly select a number by spining the wheel. The chances of landing on a red posibition are 2/3. For the two events to be truly independent the wheels would spin independently of each other and I could calculate the joint probability as 1/2 * 2/3. However, the requirement is that the two wheels be connected somehow so they stop at the same position. As a consequence the events are not independent. I can represent the two wheels as follows: (divisible by 2) bRbRbR (divisible by 3) bbRbbR where b is a black position and R is a red position. Notice that the last position for both wheels is R. Let's construct the two wheels so they have 12 positions: bRbRbRbRbRbR bbRbbRbbRbbR The pattern of black and red positions repeats. Whatever we can say about the first set of 6 positions holds for the second set of 6 positions. If we randomly choose a position (the same position for each wheel because they are connected), what is the probability that both wheels will be black (that is not divisible by 2 and not divisible by 3)? There are only 2 out of 6 ways for both wheels to be black. Interestingly 2/6 equals 1/2 * 2/3. I think this can be generalized ( and I emphaisize *think* not *prove* ) that for any number N less than p * q, the probability of N not being divisible by both p and q is ((p-1)/p) * ((q-1)/q). This appears to continue to hold if we go to 3 (say p, q, and s where N is less than p * q * s). The probability of N not being divisible by all three appears to equal ((p-1)/p) * ((q-1)/q) * ((s-1/s). This is easy to demonstrate if p = 2, q = 3 and s = 5 and p*q*s = 30. * * * * * * * * bRbRbRbRbRbRbRbRbRbRbRbRbRbRbR bbRbbRbbRbbRbbRbbRbbRbbRbbRbbR bbbbRbbbbRbbbbRbbbbRbbbbRbbbbR The asterisks indicate positions that are all black and there are 8 of them. Doing the above multiplication gives 8/30. This does not make those positions prime. It only means that they are not divisible by 2,3, and 5. Those positions could certainly be divisible by numbers greater than 5. The problem is that by considering numbers greater than 5 would require a roulette wheel with more positions. In the case of 2,3,5, and 7 the wheel would have to have 210 positions. I could find numbers not divisible by 2,3,5, and 7 using such a wheel but it would not work for numbers greater than 7. And, therein lies the error in my reasoning. David PS as for my statements about having a proof, I apologize. I was got a little too enthusiasitic. Furthermore, your argument is *almost*, but not mine, but part of that seems to have been left out. *Seemingly* the prob that N is prime should be prod(1-1/p) for p < sqrt(N). But there is a subtle problem that I discussed in the prior post, but which seems to have ben omitted. Let N be a large but unspecified integer. By the PNT, the probability that it is prime is ~1/log(N). Now consider the following. *IF* (and as we shall see this is a big if) the probability that a prime p divides N is independent of the probability that other primes divide N, then indeed the probability that N is prime is product of (1-1/p i) for p i < SQRT(N). However: Consider P = product(p i) for p i < x. As soon as x ~ log(N), the probabilities are no longer independent. Why? Because by the PNT we have psi(x) ~ x and thus P > N. i.e. once you get enough primes in the probability product, then the product of p will exceed N. Now, the probability of the next prime you add into your product is no longer independent. What you are attempting is a sieve method proof of the PNT and it can't be done. I now have to get technical. Something known as the fundamental lemma of the sieve gets in the way. You can not insert SQRT(N) primes into the product, because you lose independence. (also estimation error starts to creep in, but I don't discuss that here) See Halberstam & Richert Sieve Methods. Be careful; it is a difficult book. I have kicked it across the room more than once in frustration. Also look up Mertens' theorem in Hardy & Wright. The constant one gets in Mertens' theorem (e^gamma) differs from the constant in the PNT (which is 1) precisely because independence is lost as soon as one forces the number of primes in the product beyond log(N). Indeed. If one asks, what is the probability that a large integer N is not divisible by any primes less than x where x is SUFFICIENTLY SMALL WITH RESPECT TO N, then the answer is indeed product(1-1/p) for p < x. But you can't take x beyond log(N). And to determine the probability that N is prime you need to take x up to sqrt(N). Thus you can't really use this argument to estimate the probability that N is prime. === Subject: Re: Ideas for course on great ideas in (theoretical) CS? > Zero-knowledge non-interactive proofs could be good. I was surprised more people weren't mentioning this -- interactive proofs and zero-knowledge proofs are definitely cool. Something along (vaguely) similar lines would be probabilistically checkable proofs -- there were even high-level, about the concept section and in Discover magazine. This was probably in the 1990-1992 time range, but I can look up better references if you'd like. That's News To Me! newstome@comcast.net === Subject: Re: JSH: Understand now? Frustration? >If you wish to be a rogue university, or a rogue math professor >refusing to teach mathematics correctly, then I think you should be >punished severely, and no one who respects society and progress should >have pity on you. >What if we suspect that our university is rogue? How can we tell for >sure? George W. Bush has, with the help of JSH compiled a list of Rogue Universities... can't think of a punch line, well never mind. >Do rogue universities fly the Jolly Roger? Do they advertise for >employment opportunities in rogue journals? >I know a guy who wishes to be a rogue university. How should we >punish him severely? Who do I call? >So many questions... please help... === Subject: Re: x^2 + y^4 = z^4 > + y^4 = z^4 has no positive-integer solutions. Is the proof of this > result short enough for some kind soul to post it, or need I make a trip > to the library? (I have citations.) > An elementary proof (no elliptic curves) is based on descent. That > is, assume a solution exists in positive x, y, z, and show that > another one can be found with smaller z. The proof is not > particularly difficult but too long to type out here (one single-space > textbook page). It can be found in many basic number theory books, > e.g. Niven and Zuckerman in the chapter on Diophantine equations, p. > 107 in the 3rd edition. A proof of the Beal Conjecture was posted 2/18/04. Descent is not used. === Subject: Re: Integration by parts help needed > I'm having trouble integrating by parts. For example: > I = Integral(x cox(2x) dx) Hint: INT(u*cosu du) = u*sinu + INT(sinu du) = u*sin u + cosu. Do a differentiation to check and we get u*cosu - sin u - sin u = u*cosu When u dv does not work, do v du === Subject: Re: Integration by parts help needed >I'm having trouble integrating by parts. For example: >I = Integral(x cox(2x) dx) >dv = x, v = x^2/2 >u = cos(2x), du = (1/2)sin(2x) >I = ((x^2/2)cos(2x)) - ((1/4)Integral((x^2)sin(2x) dx)) >Then I'm not sure what to do because the integral on the right seems just as >complex as the one I started with...? A good idea when you are stuck with an integration by parts is to look for a different choice of u and dv. A better choice is u = x and dv = cos(2x) dx. Then v = sin(2x)/2 and du = dx. Integration by parts then yields | | x cos(2x) dx | | = x sin(2x)/2 - | sin(2x)/2 dx | = x sin(2x)/2 + cos(2x)/4 + C >BTW what do we use on here for the integration symbol? There is no single way that is used to represent the non-ASCII symbols. As long as others can understand what you meant them to, whichever way you choose will probably be fine. If people have a hard time figuring out what you have written, you might want to try something else. ASCII art symbols, such as I have used above, are okay, but the reader must be using a monospaced font in order for different rows to line up. I think that most people who read scientific newsgroups use such a font for just this reason. Rob Johnson I've had a time explaining some *very* simple mathematical ideas that > lead to a few complexities, but it's been fruitful to explain, or try > to explain, as I work to figure out why these simple ideas either > excite derision, anger or confusion, and I think I have it figured > out. > I'm sure many of you are put off by mathematics, so I assure you up > front that what I'll be talking about will mostly be *very* simple, > and there will only be a few slightly complicated things at the end. > First of all, I'm going to talk about a case where mathematicians gave > up because they couldn't see something, and assumed that because they > couldn't see something it didn't exist! You mean, like when you couldn't see Decker's or Ramsay's factorizations, you assumed they couldn't exist? > You know how with simple quadratics like x^2 + 3x + 2, it's easy > enough to see factors of 2 in the roots? > I mean, it's just (x^2 + 3x + 2) = (x+2)(x+1), and there they are. Just like in high school: Example 1 in the chapter on factoring by inspection ? > However, if it's something like x^2 + 7x + 2, you can use the > quadratic formula and get the roots to find > x = (-7 +/- sqrt(41))/2 > and who can see factors of 2 in that thing? Obviously if r1 and r2 are the two roots, r1*r2 = 2, so right there they are. Proving a point: you are really baffled by anything beyond factoring by inspection. That is all you seem capable of understanding. Even examples like this one that you create yourself seem beyond your grasp. > There's something else important here which is the ambiguity of the > square root operator, No, no, no, that's not it at all. The square root operator IS ambiguous in a sense, but not for positive numbers. The accepted convention is that if a > 0 , then sqrt(a) is the positive square root. No ambiguity at all. If a < 0, then sqrt(a) = i*sqrt(abs(a)). Here there is ambiguity, because putting i in there begs the question: how do I know whether it means +sqrt(-1) or -sqrt(-1) ? The answer is, there is no *algebraic* way to tell. There is an implicit agreement that, when two people write i, they both mean the same thing. That seems a bit crazy, but oddly it does not lead to problems. Everybody just agrees to it. > which may sound complicated but it's easy to > demonstrate with another root of a quadratic: > (1+sqrt(9))/2 > and you may think, silly, why show sqrt(9) when sqrt(9) = 3, but yeah, > that's *one* of its solutions, as sqrt(9) = -3 as well, No - if want -3, you write it as -sqrt(9). That's the accepted *definition* of the function. > so you have > *two* numbers > (1+sqrt(9))/2 = 2 or -1 No, the convention is that sqrt(9) = +3. Yes, 9 has two square roots. The other one is -sqrt(9) = -3. 9 has two square roots, but the square root *function* is well-defined by the convention described above. You are confusing the square root *function* with the fact that 9 has two square roots. The latter does not mean that it is impossible to create a well-defined *function*. > as either solution will work. I've had people argue with me that by > definition (really by convention) you take the positive root. But > imagine the world of Contrary. On Contrary the mathematicians for > some odd reason *by definition* take the negative! Is mathematics > really changing depending on such decisions. > Imagine you've forgotten that you can resolve sqrt(9) to 3 or -3, so > you write these numbers like (1+sqrt(9))/2 and (1-sqrt(9))/2 and > mercifully discover that you can get rid of that square root sign by > adding them together, or multiplying them together. > Like adding them gives 1, and multiplying them together gives > (1 - 9)/4 = -2 > and your mathematicians scratched their heads and contemplated such > numbers, and decided that there was *no way* to understand factors of > 2 of numbers like > (1+sqrt(9))/2 and (1-sqrt(9))/2 > except as to consider them to be unique factors of 2, in some kind of > mysterious way. This is pointless blathering. > But wait, that's not a problem here, of course, because you can just > evaluate the square root, but look back now at x^2 + 7x + 2, where > x = (-7 +/- sqrt(41))/2 > and consider that you *cannot* resolve sqrt(41) in any way that will > help you here with this question. Why do you need help? Sqrt(41) is perfectly well-defined. Your frustration is that when you look at it, you cannot see a 2 in it anywhere. You want to factor by inspection, and here it simply does not work. Trying to blame that on your imagined ambiguity of the square root function is just irrelevant. > Sure, you can write it out in decimal format with a lot of numbers > after the decimal place. My computer tells me that sqrt(41) > approximately equals 6.4031242374328486864882176746218, but of course, > you can keep going out to infinity trying just to see sqrt(41). > If you drop some of those numbers (to make it easier) and move those > decimal places, to get 64031242374, squaring gives > 4099999999957933155876, which looks VERY CLOSE to 41 * 10^20, and you > can see what our approximations actually are. > We approximate irrational numbers like square roots by finding some > REALLY BIG natural number, and moving the decimal place to the left. Your point being ??? > However, you STILL don't have a simple idea of where factors of 2 go, > like you had with the easy and you now might think comforting example > of > (x^2 + 3x + 2) = (x+2)(x+1). Yes! The point being, you're still frustrated that you cannot factor by inspection! > Now then numbers like (-7 + sqrt(41))/2 defy our ability to analyze *your* ability to analyze > because we really, really like integers, and you really, really like factoring by inspection, > but to get an integer you > have to use (-7 - sqrt(41))/2, as then you can add them together to > get -7 and multiply them to get 2, but mathematicians could not figure > out a way to handle such numbers in less than pairs! Ridiculous. Irrational square roots were invented to describe hypotenuses of triangles. In that case they represented something real and tangible and there was no question of pairs. I believe they were invented before negative numbers were invented. > That's important. Mathematicians could never figure out a way to > handle such numbers in less than pairs. > So some of them decided that what they couldn't see, wasn't > meaningful. Must you re-write all of history without even bothering to look it up? When negative numbers were invented, they were very quickly seen to be 'real' and useful, and MEANINGFUL. The same was true with irrationals, whether sqrt(2), pi, e, or sqrt(41). > It'd be kind of like the weird mathematicians, who couldn't evaluate > sqrt(9), and were looking at (1+sqrt(9))/2 and (1-sqrt(9))/2 deciding > that where factors of 2 resided was a mystery to them. But we *can* > evaluate sqrt(9), but we *cannot* really evaluate sqrt(41) which > leaves a mystery with (-7 + sqrt(41))/2 and ((-7 - sqrt(41))/2, and > some mathematicians have decided that that's it. Of course no one can write out all the digits of sqrt(41). However it is easily seen to be the hypotenuse of a right triangle with legs 5 and 4. That's tangible, real, representable, and useful. The fact that I cannot write out all the digits does not mean that I am in doubt about whether it is a bit bigger than 6.4 or a bit less that -6.4. I know that it is the former. > For them, that's all you need to know. Here are these numbers where > we can't evaluate the square root. Sorry, no go there, they decided, > you're stuck, isn't it obvious? Not so. You can evaluate it to your heart's content. It has a real and useful interpretation. If someone offers you sqrt(41) dollars, you expect to get $6.40 or possibly $6.41 if they are generous. No mathematician has trouble with it. You, however, clearly do. > You see they decided that the limit on what they could *see* was a > limit on what could mathematically exist for those examples--irational > roots--where they could not see. As far as those mathematicians were > concerned, end of story. Irrational roots CAN be seen as described above. They are real and useful. Far from having trouble with them, mathematicians INVENTED them. > But that's where my story begins. > My mathematical research is about how the kind of patterns you see > with integers, like with > (x^2 + 3x + 2) = (x+2)(x+1) > where one root has all the meaningful factors of 2 *continues* into > realms where we can't see it directly because we can't get past those > ged irrationals coming in at least pairs! > I found an *indirect* way of looking, which involves putting > expressions like these polynomials I've shown here, but more > complicated into a special but VERY simple mathematical tool. > For example, > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > can be peered into, by figuring out that its roots can be considered > in the following mathematical structure: > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > where the a's are the roots of that complicated looking cubic that I > started with, and I've managed to place them in a structure opposite a > polynomial that has 49 as a factor. > In one sense that's easy. You can take just about any expression like > that, focus on some factor of its last term, and build something like > what I did. And it being so easy may explain some of the problems I'm > having with people taking it seriously! > What I figured out though is that what mathematicians couldn't see > before can now be logically seen, and in fact, the way you divide off > that 49 is to get something like > (5 a_1(x)/7 + 1)(5 a_2(x)/7 + 1)(5 a_3(x) + 7) = > 300125 x^3 - 18375 x^2 - 360 x + 22 > but some vocal mathematicians don't like the idea as it contradicts > what they've been taught, and believed because by their thinking and > training, you can't get around the barrier. The opposite view is, you don't like it because it violates what *you've* been taught: factorization by inspection. Mathematicians take a broader view because they have learned more about factorization. *You're* the one wearing blinders here. You cannot see around the barrier because you don't know enough math. > Notice that I picked two roots arbitrarily and in fact even if you > solve for the roots you can't look and tell *which* roots should be > divided by 7. All you know is that two of them should be, but can > never know *which* two. None of them should be. All share factors in common with 7. None has a factor of 7. Just like with r1 = (-7 + sqrt(41))/2 and r2 = (-7 - sqrt(41))/2. Neither r1 nor r2 is divisible by 2. Nor is either coprime to 2. > Which makes my job of explaining harder, but it's just that ambiguity > thing coming back into place. We like integers. The math doesn't > care as it just handles numbers. It's not at all the supposed ambiguity of square roots or roots in general that is the issue here. The issue is, you cannot accept the obvious truth: even when it hits you square in the face. > We can't look *directly* at the roots, but we can use logic to > consider them. What we can't see directly here *does* exist because > the mathematics says it does. Ironically, what you say here is correct. Your logic however is faulty. You think the only possible factors are 7's and 1. But just as with r1 = (-7 + sqrt(41))/2 and (-7 - sqrt(41))/2, where NEITHER ONE is divisible by 2. so with the roots of your cubic: in general, NONE OF THEM IS DIVISIBLE BY 7. ALL of them are divisible by FACTORS of 7. > To some extent, my problems are with people and mathematicians who > trust logic less than what they've been taught and their own personal > sense of what makes since. You have it so totally backwards. Your problem is that you have not learned enough math to see that high-school methods that YOU have been taught, and your personal sense of what makes since, DO NOT APPLY. You need to learn some higher math to find your way here. > In a nutshell that's the basis for the arguments. > I say that just because we can't see the factors of irrational roots > like we can with rational ones our limitation is not a mathematical > constraint, while some people, including a lot of very obsessive > posters, refuse to acknowledge the possibility, fighting for their old > view. Again, you have it utterly backwards. You are fighting for a view which might have prevailed centuries before Gauss, Dedekind, and Kummer. You are fighting for good old factorization by inspection, because that is all you were taught to understand in high school. One would think by now, with your knowledge of Decker's example and Ramsay's example and others, that something might have started to penetrate. Evidently not. > Is it important? > Actually, it is important if mathematicians have an error in their > thinking. THEY DON'T. YOU DO. HAVEN'T YOU LEARNED *ANYTHING* IN THE LAST WEEK? > It may be the case that things thought to have been proven in > mathematics, really haven't. > It's quite possible that any number of arguments claimed to be proofs > assumed that what they couldn't see, could not exist, as this issue > has been around for over a hundred years. You are arguing for pre-19th century math, or even Dark Ages math. > Yes, I can talk it all out rigorously and in a heavily mathematical > format, There is not the slightest evidence that you can do that. All your attempts at rigorous math have been wretched failures. > but I hope that giving you some idea what all the fighting is > really about, will help in understanding what's going on, as well as > why I don't just capitulate. > Mathematicians may think that seeing is believing, but I think in > mathematics, logic is king. I could not invent a more perfect example of getting things backward. YOU think that if you cannot see a factor, it cannot be there: hence you think it MUST be true that 7 factors out of two of the linear terms of your cubic. You cannot conceive that there might be any other way to do it, a way that you cannot easily see explicitly. But there can. We have told you how repeatedly. You need some abstract ideas to understand it. 'Abstract' means: something BEYOND seeing is believing. You are essentially stuck with ancient fossil math, like bones in the LaBrea tarpits, obsolete. You yourself provide a perfectly good examply, with the roots of x^2 + 7*x + 2, and THEN YOU MISINTERPRET IT, and go off on ridiculous tangents. You want nothing but seeing-is-believing math, and then you accuse us of your own error! You saw Decker's original example. Evidently you *still* don't believe it. You saw Ramsay's example. You know that it showed something that was far from obvious. You saw it, but now you seem to be saying you still don't believe it. You say logic is king, but when your own logic defies hard evidence, you think the error must be in the evidence! It reminds me of the Rodney King tape. You apparently would have agreed with the first jury: seeing is not believing! Those policemen were not beating an unarmed helpless man on the ground with nightsticks! Like them, apparently, you prefer to believe your own preconceptions. Here's another example, perhaps more relevant. Consider x^2 - 6*x + 6. There are two roots: r1 = 3 + sqrt(-3) and r2 = 3 - sqrt(-3). Note that r1 * r2 = 6 = 2*3. It is easy to see that r1 has an algebraic integer factor in common with 3. You can see that *BY INSPECTION*, the only method that you actually trust. It is sqrt(3). Right? It's not so easy to see what factor r1 has in common with 2, is it? Can't get that one by inspection, right? So if we agree with your thinking, perhaps we conclude that it must be either 2 or 1. Clearly it is not. Nor do we need to invent an object ring, and declare that (3 +sqrt(-3))/2 is an object. 3 + sqrt(-3) DOES have an algebraic integer factor in common with 2. See if you can figure out what it is. Note that real mathematicians do not have trouble with this. They have ways of finding such things that go beyond your own seeing-is-believing method. You need to stretch your very limited imagination. Learn a little 19th century algebraic number theory and you might be able to see it also. Let us know if you can't get it. Nora B. > James Harris === Subject: Re: Simple idea, mathematics and common-sense > Yes, I can talk it all out rigorously and in a heavily mathematical > format, > James, > I have been following your threads for some time now and I have seen on > more occasions than I can count a request for you to do just this. Really? > I seem to recall that when you have deigned to acknowledge these requests > it wasn't ...[talked] out rigourously and in a heavily mathematical > format it was more of a diatribe against mathematicians and the evil > society that they control. Hey, there's this silly error, and I've explained it, but mathematicians seem either willing to ignore it, or engage in rather vicious personal attacks or arguments which I see as designed to hide the issue. > I have seen you claim that clearly stated definitions are wrong, that (as > you did in this post) there is some inherent ambiguity and that > professional mathematicians (as well as the very capable amatuers) are too > stupid to understand but I have *NEVER* seen you post anything that even I > would consider to be rigourous. Well I'm tired, so I'm pushing more on to people like yourself, so consider P(x) = (x + 8a)(x + b), where ab=1, and (8a + b) is an odd integer. You should have the expertise to play with that and figure out the error in core yourself! > Having now said that you can do this, when can I expect to see it? Play with P(x), and see if you're smart enough to see the mistake in reasoning with current teaching about the ring of algebraic integers. > I'm sure that if I have any difficulty then either you or somebody else > will be able to clarify. > Ivan. I certainly hope that a quadratic isn't too much for you! If so, then hey, it may bug you, but I may start talking about evil math society, and people who refuse to acknowledge the truth again!!! There's no more room for excuses, even for people like you Ivan. Either actually be a mathematician, and acknowledge mathematical truth, or just wait for the consequences of being rogue and against the best interests of society at large. Yup, if mathematicians refuse to teach correct mathematics, then they are fighting against humanity itself. And I think, will be treated accordingly, once that becomes common knowledge. Why fight against humanity, against progress and truth? James Harris === Subject: Give me that old time ontology: (was: the anticlassicalist }{ i: linguistic negation) > In message <1039dr5kpjloif1@corp.supernews.com>, galathaea >: >You should have realised that anything full of -ism and -ist which is > spewed >: >to sci.lang and sci.logic is not going to contain any physics. >: >: (Nor language; I can't speak for sci.logic ;-) >: >: I'm well aware of that. What I was wondering was whether the _OP_ >: realises that there's a difference between these isms and physics, and >: what it consists of. >Installment vi Before you lies the Void... details the properties of the closed linear subspace lattice >of a Hilbert space, detailing the relationship with observables and >prediction in quantum systems. This builds off of the mathematical theory >You do understand the physical content, do you not? > Haven't seen any. > Dirac compressed quantum mechanics into ~300 pages of terse elegance, > and didn't use ontology once. Well then, he was tersely codifying the mathematical structure as a given, which is the easier hard direction. Going in the less easy hard direction requires talking about something very much like ontology, whether or not we use the word ... like speaking prose, and all that. It's our old school chums deduction and induction. Basically, professor, it goes kind of like this (music please ...): Models as a Confidence Game In the beginning we are confronted with a blooming buzzing surface of sensory confusion. Being cerebral beings, we create a partially ordered heirarchy of model things, purported to be generative of this surface which we've been presented. The models are essentially, roughly and enoughly, our working ontology. There: I've used ontology in a sentence, and it wasn't so bad, was it? ;-) (Richard Herring vigorously spits and loudly calls for a pint). Some of these models; in a model itself received and conceived -- it's all heirarchical, you see -- are for us evolutionarily preconceived: in our lizard brains, ancient and tough, a world populated with persistent three-dimensional objects is enough. Onto these models we multiply and add; mathematical annexes, now more than a fad, allow us to predict with numeric decision the results of some experiments with admirable precision. This predictive concision increases our confidence that we're on to somethin', as we laboriously peel back the skins of the onion -- and well it should: a rigorous predictive structure is quite good. Confidence, confidence ... did I not add in my vision that each model is invested with Bayesian strength; and though we sometimes forget we should add at greater length they are all always subject to emprical revision? Hssst! The doggerels out. But I think you may get the idea: In the great chain of model being we may focus on some precise mathematical bits, the triumphs of a natural science known to be correct in a range of observation with a degree confidence only predicated on a few assumptions; like sanity and the persistence of order. Once given these bits, reasoning from their structure is primarily downward -- deductive -- which is something which can similarly be carried out with great rigor and confidence. But we may tend to forget that these models were not themselves handed to us on a platter -- or rather they were, if we adopt the stance that they are known givens -- but each the product of protracted groping by groups of the most talented people of their time, there being no mechanistic crank we can turn to integrate as readily as we differentiate. And in this groping, like it or not, something with a strong isometry to the ologies will out; a discussion of models and our confidence in models, hic nomen rosa. === Subject: Re: Hausdorff dimension of graph of Weierstrass function. >[...] >Indeed, if t is large enough a very crude argument >suffices to show Box = s. >Having dusted off a few cobwebs in the lacunary-series >section of my brain, it's clear that Box = s for t > 1 >and 1 < s < 2. >The proof depends on the fact that the supremum of >a lacunary series is larger than a constant times the >sum of the absolute value of the coefficients. This >is well-known, for example it can be proved using Riesz products (see Katznelson). We need a >slightly sophisticated local version, the proof of >which must be somewhere in the lacunary series >folder in the filing cabinet in my office: >Lemma. Suppose L > 1 and a > 0. There exists >C = C(L, a) > 0 such that if > f(x) = sum_1^N a_j cos(n_j x), >where n_{j+1}/n_j > L for all j, and I is an interval >of length a/n_1, then the sup of |f| on I is larger >than C sum |a_j|. >Assuming that, consider a square grid in the plane >with side length h. Take the sum and split into >the sum from 1 to N - 1 plus the tail, where N is >chosen so that the oscillation of the sum from >1 to N - 1 on any interval of length h is less than >h (use obvious estimates on the derivatives). >If you choose N maximal with this property >it turns out that h > a/t^N for some constant >a > 0, so the lemma shows that the max of the >tail minus the min of the tail on any interval of >length h is large. Hence the same is true of f >itself, and this shows that the graph hits a large >number of squares in the grid, forcing Box > s. >This seems to be similar to Falconer's proof. I'm certain it is - I finally got to the library a few days ago, and the proof of the upper bound that Hunt gives is _identical_ to the proof I came up with when you started this thread. That's because these are the easy bits. I was a little disappointed when I finally looked up Hunt's paper. Of course it's an extremely keen paper, but from the title I'd got the impression that he'd finally figured out the Hausdorff dimension of these suckers. I've been thinking a little bit about that - of course I haven't proved anything yet, but I also haven't yet convinced myself that it's all that hard... Taking the approach he takes, you just have to show that the set where f(t) - f(t+h) is small is small. The function's L^2 norm is right, and lacunary series sort of behave the same at every point, so it seems like it should be true. === Subject: Re: Help needed from a group theorist >An exponent E(G) of a group G is the lcm of all the orders o(g), >g in G.We have that G is cyclic if and only if E(G)=|G|. No we don't. Try the nonabelian group of order 6. >Does anybody have an idea how can we find all n such that >the exponent of Z_n*,the group of units of Z/Zn is 2? >E(Z_n*)=2 n=? >The solution is n=2,3,4,6,8,12,24 ,but I don't have a clue >where did this come from. Well Z_n* is the direct product of Z_{p^r}* for the prime power factors p^r of n, so this reduces to n = p^r. For that, Z_{p^r} is cyclic for p > 2 or for p^r = 2, 4, whereas Z_{2^r}* = C_2 x C_{2^{r-2}} for r >= 3. So Z_{p^r} has exponent 2 for p^r = 2, 4, 8, or 3, which means Z_n has exponent 2 for exactly those values of n that you list above. Derek Holt. === Subject: Re: JSH: Understand now? Frustration? >Using P(x) = (x+8a)(x+b), where ab=1, and considering when (8a + b) is >an integer can give you a perspective on what I've been saying, I >hope. >Basically the mathematical position of the mainstream--why it's a core error--is that you can never have a factor like 8 in an arena >where -1 and 1 are the only integer units because you can't in the >ring of algebraic integers, except for simple radical factors, if you >don't have integer 'a' and 'b'. Exactly what mathematician ever said that? (I mean, what mathematician ever said something about a factor like 8, for that matter?) You're projecting your own misconceptions and feeble understanding onto the mathematical mainstream>That's it. The arguments now should make sense to you as I talked >about situation where 'a' *should* be a factor of 1, as it is in >fact--in a more inclusive ring. >The flaw with the understanding of the ring of algebraic integers is a >fascinatingly frustrating one to explain!!! >Posters found many ways to confuse the issue, and I still find it hard >to believe that experienced professors like Arturo Magidin or Rick >Decker never figured it out. >But it's up to them to give a full accounting of why they've said what >they've said. >I'm just glad that I finally thought to put out something like >P(x) = (x+8a)(x+b), where ab=1, >so P(x) = x^2 + (8a + b)x + 8, with an emphasis on when (8a + b) is an >integer. >If you're STILL confused, I suggest you play with that quadratic for a >while. First consider simple cases like I did, like a=b=1, and see >what happens, and then move on to irrational 'a' and 'b', and see what >happens. Uh, we're _not_ confused. Well, I imagine I'm not the only one confused about the question of exactly what your point is here. But we're not confused about the math. >Oh yeah, I see nothing wrong with the label algebraic integers or >their definition and wouldn't want either changed, but what *has* to >be taught along with what has been previously taught are the >interesting features and limitations of the ring. >Universities and mathematicians who continue to refuse to tell the >truth, shouldn't expect people to be on their side when at this point >there's just no excuse. >If you wish to be a rogue university, or a rogue math professor >refusing to teach mathematics correctly, then I think you should be >punished severely, and no one who respects society and progress should >have pity on you. >It is, after all, your choice. >You have been repeatedly warned. And you're also sounding like a raving lunatic. >James Harris === Subject: Re: Probability of finding a Prime Number [Quoting Bob Silverman:] > See Halberstam & Richert Sieve Methods. > Be careful; it is a difficult book. I have kicked it > across the room more than once in frustration. It seems that the world is divided into the haves and the have nots, as this book seems to be no longer available at all. If anyone has a copy that they no longer care for, I'm certainly in the market for a copy. Amazon, however, does say: http://www.amazon.com/exec/obidos/tg/detail/-/0123182506/qid= 1077381040/sr=1 -4/ref=sr_1_4/104-5847946-5462337?v=glance&s=books <<< Customers interested in Sieve Methods may also be interested in: Horticultural Sieves (UK) Hand-made in Beech wood and steel UK based. In stock, quick delivery http://www.ctbaker.co.uk which is nice. Phil 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: how important is r^2 in statistics? I'm running a stepwise linear regression on a dependent variable. My statistics program created some models which in which the independent variables have t values of more than 3, which makes them significant on a 1% level. However, the r^2 of those models is never higher than .14. Of how much importance is this r^2? any comments are appreciated Nicolas === Subject: Re: branch of log z > What exactly is to choose a branch of log z for complex z? I'd > appreciate if someone help me picture this. There is an easier way than using a Riemann surface which is what the wolfram reference gives you. See Stewart's Advanced Calculus page 372. === Subject: Cyclotomic Equation Cyclotomic Equation is defined as x^p=1 The equation x^p=1 where solutions w = e^{2pi*k*i/p}are the roots of unity sometimes called de Moivre numbers. Now I wonder whether there is some math names for M^p=I M and I are nxn matrix, and I is indentity === Subject: Re: RFI:Wanless' Fourth Conjecture/Dirichlet's Geometric Theorem > STOP PRESS!! :-(( > I've been incredibly stupid! > The generalized form is clearly a load of dingo's kidneys: - > take a=5, b=3 for example - a^n+b will _always_ be even!! > Don't waste any more time on it... > J > The series a^n+b [hcf(a,b)=1] includes an infinite number of primes [over > all n] In the cold light of day, there _might_ be something of interest here - though I don't know how much of it has been covered already? If you're [still] interested check out: http://bearnol.is-a-geek.com/DGT.txt where comments/suggestions/further experimental testing is welcome... J === Subject: Re: Can something be undeterminable? |But is it logically possible for something, a mathematical statement, to be |absolutely undeterminable? Goedel once expressed an interest in developing a concept of absolute provability, but so far as I know nobody has done it, so the question is somewhat undefined. I think it's plausible that certain questions in mathematical logic are in fact undeterminable in a strong sense. Chaitin has defined an uncomputable real number called Omega. Omega is uncompressable in the sense that for each programming language there exists a constant C such that for any N, a program which correctly computes the first N bits of Omega is at least N-C bits long itself. One way to construct a program for computing bits of Omega is to have it search for proofs in a given formal system whose conclusion is that a given bit of Omega is 0 or that it is 1. The program can then output them in order. So any formal system capable of proving what the first N bits of Omega are has to be close to N bits long as well. I think this provides a pretty convincing argument that all but the first few bits of Omega are beyond any reasonable determination. There may be longish formal systems that we believe to be correct as far as their arithmetic consequences go, but presumably because of more fundamental grounds. It seems unlikely that we have a large amount of innate knowledge of mathematics, in the sense of knowing facts of arithmetic that absolutely cannot be justified except by assuming a lengthy set of axioms. I've seen ZF condensed to a relatively short (smaller) set of axioms. Higher axioms of infinity don't require much more. The theorem mentioned above leaves open the possibility, of course, of an axiom system being short and deciding the n-th bit of Omega for some large n, without deciding all the previous ones. Omega is defined as the probability of a Turing machine halting, given a certain way of constructing one at random. It seems implausible to me that a natural axiom system would decide bits very much beyond the first one it couldn't decide. This is just a plausibility argument, of course, and without an actual definition of absolute provability there's no way to make such an argument rigorous. Keith Ramsay === Subject: Re: Simple idea, mathematics and common-sense <3c65f87.0402210814.5111e751@posting.google.com Discussion, linux) >Yes, I can talk it all out rigorously and in a heavily mathematical >format, >I have seen you claim that clearly stated definitions are wrong, that (as >you did in this post) there is some inherent ambiguity and that >professional mathematicians (as well as the very capable amatuers) are too >stupid to understand but I have *NEVER* seen you post anything that even I >would consider to be rigourous. > Well I'm tired, so I'm pushing more on to people like yourself, so > consider > P(x) = (x + 8a)(x + b), where ab=1, and (8a + b) is an odd integer. > You should have the expertise to play with that and figure out the > error in core yourself! So you *can* talk it all out rigorously, but you're too tired. Boy, that's a shame. Maybe a good lie-down, and then you could tackle it tomorrow, right? > I certainly hope that a quadratic isn't too much for you! > If so, then hey, it may bug you, but I may start talking about evil > math society, and people who refuse to acknowledge the truth > again!!! Wait, what did the claim that you could add rigor mean? Did it mean that if posters on sci.math don't present your arguments in a rigorous format, then we get the evil mathematician harangue again? 'Cause I thought it meant that you would present a rigorous argument if asked[1]. Footnotes: [1] Well, I didn't really think that. I've been thinking about my problems with getting any kind of admission that my math arguments showing the core error in mathematics are correct, so I've gone to marketing books. -- James S. Harris, on when mathematics isn't enough === Subject: Re: I got low score on math test, please advise me and take a look > assuming that your report about progress feedback is accurate, a > rejection of the request above is a good glimpse into an incompetent > institution. what kind of justification were you offered? some > bureaucratic non-sense? > My guess is that justification offered was THE RULE THAT YOU CAN'T DROP A > COURSE AFTER A CERTAIN DATE. in other words, bureaucratic non-sense. don't you think that such rule should have a provision stateing that students need some performance feedback prior to the drop deadline? i now realise that i mis-read the guy's message. i thought he was denied to withdraw, when in fact he was denied to withdraw without a W. i take back my comment about incompetence. > === Subject: Re: Simple idea, mathematics and common-sense > I know I shouldn't try, but the light... it's so beautifu...zap! >There's something else important here which is the ambiguity of the >square root operator > It is not ambiguous. Sqrt(x) (or x^(1/2) or the notation with the > square root symbol) all define the principal square root (provided x > is a positive real number). So Sqrt(9) = 3. It is not -3 even though > -3 is a square root of 9. > Just read about it on the following link (You could also go to the > library and read a decent math-book there): > http://mathworld.wolfram.com/SquareRoot.html It doesn't matter how much mathematicians try to deny the obvious if people actually *check* what I'm saying. Like consider P(x) = (x + 8a)(x + b), with ab=1, and 8a + b = 17. If you work that out, then you have a = (17+/-sqrt(257))/16, now then can you tell me if (17+sqrt(257))/16 or (17-sqrt(257))/16 has 16 as a factor? Mathematicians arguing with me may promptly lash back with, yes, they can say that it is NOT a factor in the ring of algebraic integers! That's true, but so what? Now then, I dare any of you to try and see for yourselves if that 16 should divide back through, so that the number is actually more like an integer than a fraction. If that sounds silly consider (1+sqrt(-3))/2 as that's an algebraic integer and a factor of 1. It turns out that 1+sqrt(-3) has a factor that is 2. Think I'm crazy here? Well let 2x = 1 + sqrt(-3), then 2x - 1 = sqrt(-3), and squaring both sides 4x^2 - 4x + 1 = -3, so 4x^2 - 4x + 4 = 0, so x^2 - x + 1 = 0, so what I said IS correct. However, notice again that definition of algebraic integers as roots of monic polynomials with integer coefficients! If you believe that covers everything, then you're acting on faith, and not logic, as there's just no mathematical reasn for that belief. Any supposed proof depends on circular reasoning. I suggest those of you who don't believe me play with the expressions here. And remember, if mathematicians are teaching wrong mathematics, what good does that do anyone? The issue here may seem esoteric to some extent, but ultimately it's about the importance of truth in research. Professionals should teach things that are true. James Harris === Subject: Re: Integration by parts help needed En el mensaje:c17ttu$1fn13g$1@ID-83837.news.uni-berlin.de, Si escribi.97: > I'm having trouble integrating by parts. For example: > I = Integral(x cox(2x) dx) > dv = x, v = x^2/2 > u = cos(2x), du = (1/2)sin(2x) > I = ((x^2/2)cos(2x)) - ((1/4)Integral((x^2)sin(2x) dx)) > Then I'm not sure what to do because the integral on the right seems > just as complex as the one I started with...? > BTW what do we use on here for the integration symbol? I usr a funcional notation: Int(f(x), x) for indefinite integral and Int(f(x), x, a, b) for definite integral between a and b. In order to choose u and dv are two guidelines usually useful: 1) You must know how integrate dv (mandatory!) 2) du must be simpler that u And if you election leads to a more complicate integral, try a reverse one. Sometimes, you can get in the RHS the same integral, multiplied by a coefficient different from 1, and you can move it to LHS. === Subject: Simple vector spaces theorem I am trying to prove the following theorem regarding vector spaces: If u_1, u_2, ..., u_n span the vector space V, then some subset of {u_1, u_2, ..., u_n} is a basis of V. The following is my attempt at a proof and I'd be very grateful if someone could verify its correctness. I also use a well-known lemma which I will refer to as lemma 1. This lemma is: ** Lemma 1: Suppose f_1, f_2, ..., f_n span a vector space Q, then one of the following holds: i). f_1, f_2, ..., f_n is a basis of Q. ii). There exists a j (0 < j <= n), such that removing f_j from f_1, f_2, ..., f_n still leaves a spanning set of Q. ** I argue by cases. Either V = {0} or V != {0} (where {0} is the set consisting of the zero-vector, not a scalar 0, and '!=' means 'not equal to'). If V = {0}, then by definition, its basis is the empty set. Clearly the empty set is a subset of {u_1, u_2, ..., u_n}. For the case V != {0}, we aim for a contradiction by assuming that there doesn't exist a subset of {u_1, u_2, ..., u_n} which is a basis of V. Now since u_1, u_2, ..., u_n span V, one of the following is true by lemma 1: i). u_1, u_2, ..., u_n is a basis of V. ii). There exists a j (0 < j <= n), such that removing u_j from u_1, u_2, ..., u_n still leaves a spanning set of V. By our assumption, i) is not true, hence ii) must be. We now consider the subset of {u_1, u_2, ..., u_n}, which still spans V, formed by removing the said element. Therefore this subset must also satisfy either i) or ii) of lemma 1, and again since it cannot satisfy i), it must satisfy ii). Clearly we can repeat this argument until we conclude that the empty set spans V, which is clearly a contradiction since V != {0}. Hence our original assumption is false. Q.E.D. James Radcliffe. === Subject: Re: JSH: My fear, consider this >Previously I noted that I could use >P(x) = (x+8a)(x+b), and ab = 1, so P(x) = x^2 + (8a + b)x + 8. >And I considered 8a + b = 17, as then you have >8/b + b = 17, so b^2 - 17b + 8 = 0. assuming and b are reciprocal units in the ring of algebraic integers otherwise you are talking nonsense >Now imagine *any* non-unit factor f in the ring of algebraic integers, >like 1+i that might be a factor of b, and let b = fz, and substitute >and you get >f^2 z^2 - 17fz + 8 = 0, so >f z^2 - 17z + 8/f = 0 >and notice you STILL have that f on the front. >I don't want to hear that it isn't applicable because f isn't an >integer, as if you will have to get a polynomial reducible over Q if >you pick the right f, as that's just bogus. >The problem is that 17. For it to work, you need to have something >even! The problem is you insistence that 17 can be written as 8a+b where a and b are algebriac integers and a=1/b, all you you do is prove you cannot do tihs. It's a result, not the one you are claiming though. Although you do not explicitly state a and b are algebraic integers, if they aren't then this is vacuous (even more so than before, I mean). >I'm so damned tired. I can't be sure if anyone will listen to me. >Arturo Magidin or Dik Winter or Nora Baron or Rick Decker will come >back like they have before, now won't they? In proving you don't know what you're talking about? Yes. Once more your manupulations are correct but your conclusions are wrong. >They'll come back and post something stupid, and wrong, and just plain >evil, and you'll go along like you've done before. >It's so evil, so frustrating. Nothing matters in mathematics, >mathematicians don't give a damn about the truth. >NOBODY ING CARES ABOUT THE >TRUTH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!! !!!!!!!!!!!!!!! >JSH === Subject: Re: I got low score on math test, please advise me and take a look > assuming that your report about progress feedback is accurate, a > rejection of the request above is a good glimpse into an incompetent > institution. what kind of justification were you offered? some > bureaucratic non-sense? > My guess is that justification offered was THE RULE THAT YOU CAN'T DROP A > COURSE AFTER A CERTAIN DATE. > in other words, bureaucratic non-sense. don't you think that such rule > should have a provision stateing that students need some performance > feedback prior to the drop deadline? How do you know that he didn't get any performance feedback before the deadline? He only said that he didn't get THIS TEST back before the deadline. How do you know if this is the first test? How do you know if he's had other assignments, or how many? This troll has only told us what he wants us to know. And whatever respect I had for him, I lost when he started sending me unsolicited e-mail. === Subject: Re: Simple idea, mathematics and common-sense -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 >It doesn't matter how much mathematicians try to deny the obvious if >people actually *check* what I'm saying. >Like consider P(x) = (x + 8a)(x + b), with ab=1, and 8a + b = 17. >If you work that out, then you have a = (17+/-sqrt(257))/16, now then >can you tell me if (17+sqrt(257))/16 or (17-sqrt(257))/16 has 16 as a >factor? The question seems meaningless, or at least pointless. In order too talk about factor, you need to identify the ring in which the factorization occurs. You have not done so. If your ring is the ring of reals, then your point is trivial and uninteresting. If your ring is that of the algebraic integers, you would first need to show that a is an algebraic integer. If your ring is something else, you need to say what it is. >Mathematicians arguing with me may promptly lash back with, yes, they >can say that it is NOT a factor in the ring of algebraic integers! >That's true, but so what? Now then, I dare any of you to try and see >for yourselves if that 16 should divide back through, so that the >number is actually more like an integer than a fraction. What does more like an integer than a fraction mean? Or are you just making things up as you go along? -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.2.4 (SunOS) iD8DBQFAN58gvmGe70vHPUMRAq2tAKCiQncwU0RPhaucggEco0T7Frb/ iACcC1mu /e5+DUjmmwvqf/Jepyj8RBU= =MOSF -----END PGP SIGNATURE----- === Subject: anybody can read out eigenvalues/vectors by eye-inspection? For simple matrices, such as [6 4; 4 6] [a b; b c] [a b; -b c] [3 2; 2 3] etc... Any tricks? === Subject: Re: Simple vector spaces theorem > I am trying to prove the following theorem regarding vector spaces: > If u_1, u_2, ..., u_n span the vector space V, then some subset of {u_1, > u_2, ..., u_n} is a basis of V. > The following is my attempt at a proof and I'd be very grateful if someone > could verify its correctness. It's correct. Jose Carlos Santos === Subject: Re: HELL :-) (Was: Re: the anticlassicalist }{ ii: the spectre continues) > [...] > , > I don't snip here as a skip. I just wanted to say that I see clearly > you are one who has already learned that the only secret behind any of > this is the hunt for patterns. Going out and grabbing disparate works to > look for connections, then hunting down connections to those, and building > them into pictures. When the connections become rigorously symbolised, we > have math (or logic, or numerology, or symbology -- whatever you like). > When a model connects ontology to epistemology, a theory is born. Or lack thereof. Paraconsistency does not admit theory. I want to say thanks. The graphical representation in it expresses the basis behind a lot of my confusion. There is a tendency for everyone to decompose problems. When they set up a foundation for reasoning, they always seem to do this in the same way. Well, that is just not a sensible global outlook. The equilibrium state of cross-purposeful reasoning will end up with a reflective structure in which two alignments will exchange 0's and 1's across two consistent frameworks. The classical truth table semantics reflect idempotence in the sense of de literature, de Morgan idempotence can account for game-theoretic communication (tit-for-tat) whereas Boolean idempotence can merely represent coherent, incomplete, static theories. Just some big mathematics to account for the role of essence in Aristotle's delineation between homonymous and synonymous usage. Linguists must have a lot of fun. So, here is another post that is uninteresting or irrelevant or uniformative or... whatever other criticisms have been thrown my way for lack of a theory. :-) > Its nice watching people have fun with their patterns, play with them, > fit the pieces together. Its almost as fun as doing it myself. > I wanted to say this because I see many on usenet who frown on such > behavior. They seem to look only for some kind of stale acceptance of a > never surprising universe, and look to attack any excitement or creativity > they see. > So I sometimes want to bce the powers, give praise at the > interesting twists and turns I see people playing with. I have a child like > wonder for child like wonder. > You even picked up Conway and Sloan... That was the first book I ever > checked out of a university library (I didn't understand it then, but I kept > coming back to try again). > ===-=-=-=-=- > === Subject: Re: x^2 + y^4 = z^4 > A proof of the Beal Conjecture was posted 2/18/04. Descent is not used. The second of what? Have you won the dosh yet? Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Model theory puzzle > Find a finite system of axioms (feel free to introduce operations, > relations, constants) so that all of its finite models would have a > prime number of elements, and for every prime p there's a model to > that system of size p. Start with the axioms for a field and introduce a third binary operation ^ for exponentiation, satisfying the following axioms: For all y, 0^y=0. For all nonzero x, x^0=1. For all x and for all y not equal to -1, x^(y+1)=x*(x^y) For all nonzero x, we have x^(-1) = 1 In the field with p elements, for nonzero x, one defines x^y to be the product of z copies of x, where z is the integer between 0 and p-1 representing y, with the convention that x^y is 1 for y=0. If a finite field F of characteristic p satisfies these axioms, the last axiom implies that the product of p-1 copies of a nonzero element x equals 1, which implies that the field has only p elements. Allan Adler ara@zurich.ai.mit.edu **** * * * Disclaimer: I am a guest and *not* a member of the MIT Artificial * * Intelligence Lab. My actions and comments do not reflect * * in any way on MIT. Moreover, I am nowhere near the Boston * * metropolitan area. * * * **** === Subject: (A<-->B) as Non-Derivative Re: (A<-->B) as Set Derivative > Define: > 1) (A-->B) = (AB')' = A' U B I presume that ' denotes the complement of a set (in some presumed universe of discourse X) and AB denotes the intersection of A and B. Then this is just X - (A - B) or X(AB) where - or denotes set difference. > and therefore: > 2) (A-->B)(B-->A) = (A' U B)(B' U A) = AB U A'B' The complement of the symmetric difference. > where adjacent parentheses refer to set intersection. > The symbol for (A-->B)(B-->A) should be familiar to people in > Mathematical Logic, or to Logicians if a, b were propositions: > 3) (a<-->b) = ~(a^~b)^~(b^~a) = (~aVb)(~bVa) = ab V ~a~b > where ~ (tilde) represents negation (NOT), V disjunction (OR > including AND), ^ conjunction (AND). The left-hand-side of (3) > is written in Logic a iff b or a if and only if b, and > the Set Theory analog is therefore for sets A, B: > 4) (A<-->B) = (A-->B)(B-->A) = AB U A'B' > Taking complements of both extremes in (4) yields: > 5) (A<-->B)' = A'B U AB' the symmetric difference of A and B. > which is the analog for Sets of: > 6) (fg)' = f'g + fg' No, it isn't. A possible analogue for sets of this would be (adopting your notation) (A<-->B)' = (A'<-->B) U (A<-->B') or (AB)' = (A'B) U (AB'), Slight problem .... both of these are false in general. > for differentiable functions f, g. Notice that although U in (5) is > set Union (corresponds to logical AND/OR), it happens to be a > disjoint or mutually exclusive Union since A'B does not intersect > AB' (it has zero intersection). It is not unusual to write U as > +, for example circled +, in such situations with regard to vector > spaces and so on, but I will retain U to be technically correct. > Readers can prove as an exercise that if u is a bounded Lebesgue > measure (a Lebesgue measure on bounded sigma algebras, etc.), then: Seems you don't know what Lebesgue measure is (I'll tell you: it's a measure (not bounded) on a specific sigma-algebra on R^n). > 7) u[(A<-->B)'] = u(A'B) + u(AB') > which is even more similar to (6), No, it isn't. > where A, B are sets in the > sigma algebra. To assert, then, that (A<-->B) (A if and only if > B or A iff B) is the set analog of fg with regard to complementa- > tion and ' as analogous to differentiation, seems quite intuitive seems quite stupid, to say the least > to say the least, although we don't assert anything for higher > primes like A or other contexts at this point. Very sensible. > Obviously, A > is (A')' = A but f is not f in general unless f is the exponential > function exp(x). or a multiple of the exponential. Of course this shows the daftness of this bogus analogy> The exponential function, by the way, is not > as trivial as it may seem, and I leave that as an exercise (hint: > consider inverse functions). Now what would it mean for the exponential function to be as trivial as it may seem? What's the point of this posting anyway, apart from a compulsive desire to see analogies where none exist? Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: (A<-->B) as Non-Derivative Re: (A<-->B) as Set Derivative > Define: > 1) (A-->B) = (AB')' = A' U B I presume that ' denotes the complement of a set (in some presumed universe of discourse X) and AB denotes the intersection of A and B. Then this is just X - (A - B) or X(AB) where - or denotes set difference. > and therefore: > 2) (A-->B)(B-->A) = (A' U B)(B' U A) = AB U A'B' The complement of the symmetric difference. > where adjacent parentheses refer to set intersection. > The symbol for (A-->B)(B-->A) should be familiar to people in > Mathematical Logic, or to Logicians if a, b were propositions: > 3) (a<-->b) = ~(a^~b)^~(b^~a) = (~aVb)(~bVa) = ab V ~a~b > where ~ (tilde) represents negation (NOT), V disjunction (OR > including AND), ^ conjunction (AND). The left-hand-side of (3) > is written in Logic a iff b or a if and only if b, and > the Set Theory analog is therefore for sets A, B: > 4) (A<-->B) = (A-->B)(B-->A) = AB U A'B' > Taking complements of both extremes in (4) yields: > 5) (A<-->B)' = A'B U AB' the symmetric difference of A and B. > which is the analog for Sets of: > 6) (fg)' = f'g + fg' No, it isn't. A possible analogue for sets of this would be (adopting your notation) (A<-->B)' = (A'<-->B) U (A<-->B') or (AB)' = (A'B) U (AB'), Slight problem .... both of these are false in general. > for differentiable functions f, g. Notice that although U in (5) is > set Union (corresponds to logical AND/OR), it happens to be a > disjoint or mutually exclusive Union since A'B does not intersect > AB' (it has zero intersection). It is not unusual to write U as > +, for example circled +, in such situations with regard to vector > spaces and so on, but I will retain U to be technically correct. > Readers can prove as an exercise that if u is a bounded Lebesgue > measure (a Lebesgue measure on bounded sigma algebras, etc.), then: Seems you don't know what Lebesgue measure is (I'll tell you: it's a measure (not bounded) on a specific sigma-algebra on R^n). > 7) u[(A<-->B)'] = u(A'B) + u(AB') > which is even more similar to (6), No, it isn't. > where A, B are sets in the > sigma algebra. To assert, then, that (A<-->B) (A if and only if > B or A iff B) is the set analog of fg with regard to complementa- > tion and ' as analogous to differentiation, seems quite intuitive seems quite stupid, to say the least > to say the least, although we don't assert anything for higher > primes like A or other contexts at this point. Very sensible. > Obviously, A > is (A')' = A but f is not f in general unless f is the exponential > function exp(x). or a multiple of the exponential. Of course this shows the daftness of this bogus analogy> The exponential function, by the way, is not > as trivial as it may seem, and I leave that as an exercise (hint: > consider inverse functions). Now what would it mean for the exponential function to be as trivial as it may seem? What's the point of this posting anyway, apart from a compulsive desire to see analogies where none exist? Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: 'erf' function in C > A possible improvement is to note that the even part of R(x), > (R(x)+R(-x))/2, is equal to 1/phi(x), thus only the odd part of R(x), > (R(x)-R(-x))/2, needs to be computed: > double Phi(double x) > {long double s,t=0,b=1,pwr=x; > int i; > s=x; > for(i=2;s!=t;i+=2) > { b/=(i+1); > pwr=pwr*x*x; > t=s; > s+=pwr*b; > } > return .5+s*exp(-.5*x*x-.91893853320467274178L); > } > Very nice, and a definite improvement*! It speeds up the convergence > considerably which, quite possibly, accounts for more accurate results. > i.e. w/CVF on Wintel, > x phi(x) > 0.123 0.5489464510164368 > 1.200 0.8849303297782917 > 2.400 0.9918024640754040 > 6.100 0.9999999994696567 > -6.100 0.0000000005303433 > -1.100 0.1356660609463827 > 7.200 0.9999999999996979 > * Actually, it's an understatement considering the Syziphian effort at > LANL some odd thirty years ago. See, netlibfn lib. I translated the C code to Fortran 95, as shown below. With CVF, the results agree with those of B. Voh. Using the Lahey/Fujitsu Fortran 95 compiler with the option -quad, which extends DOUBLE PRECISION to quadruple precision, the results agree with the true values posted by George Marsaglia to at least 18 decimal places, and often the full 20. Translating both the double and long double C variables to DOUBLE PRECISION in Fortran seem wrong to me, although the results are good. What is the proper translation? x phi(x) 0.123 0.54894645101643675908 1.200 0.88493032977829173198 2.400 0.99180246407540387056 6.100 0.99999999946965767371 -6.100 0.00000000053034232629 -1.100 0.13566606094638267517 7.200 0.99999999999969893720 elemental function phi(x) result(y) implicit none double precision, intent(in) :: x double precision :: y double precision :: s,t,b,pwr integer :: i t = 0.0d0 b = 1.0d0 pwr = x i = 2 s = x do if (s == t) exit b = b/(i+1) pwr = pwr*x**2 t = s s = s + pwr*b i = i + 2 end do y = 0.5d0 + s*exp(-0.5*x**2 - 0.91893853320467274178d0) end function phi program xphi implicit none integer, parameter :: n = 7 double precision, parameter :: x(n) = (/0.123d0,1.200d0,2.400d0,6.100d0, & -6.100d0,-1.100d0,7.200d0/) integer :: i double precision :: phi external phi do i=1,n write (*,(f8.3,f24.20)) x(i),phi(x(i)) end do end program xphi === Subject: Great mistakes of the physicists http://hamidvansari.topcities.com/ === Subject: Re: Genetics and Math-Ability > You avoid the question of why does it get inhereted, indeed why does it > exist at all? > The evolutionary pressures on our brains were forged in the Savanahas of > Africa over a million year period. I can understand the evolutionary > pressure on being able to count 1,2,many, many many Wildebeests. But > calculate the surface area integral of the plain on which they run? You see > some specilaised organ in Nature, and you look for the purpose which let it > evolve. That doesn't exist for maths. The ability to recognise (and perhaps invent) patterns, to observe similarities, to argue with neighbours are important too; all of these contribute to mathematical ability. Moreover, it is not the inheritance of special abilities, that is really needed; inheritance of a suitable general ability that can be adapted towards a special ability (depending on enviromental pressures) is important. === Subject: Re: Ideas for course on great ideas in (theoretical) CS? > I have been coerced into teaching a Honors course the Fall > (mostly for non-CS freshman/sophomore Honors students). My > idea was to do some Great Ideas/Problems/Puzzles etc. from > computer science -- emphasis on theory/algorithms > and related areas like graph theory/combinatorics. > Of course, the honors college wants a syllabus in one week! > I looked at the book, Great Ideas in CS and though a nice > book, seems a bit light on the theory side ... given that > I want to focus on theory to keep me interested. I saw the > course/web site at CMU Great Ideas in Theoretical Computer Science > and may use that as a guide for some of the course. Examples > of some things I might discuss (besides a couple weeks on > basics/definitions/history) include Towers of Hanoi, Byzantine > Generals, voting problems, maybe a gentle discussion > of interactive proofs, prisoner's dilemma, game of life, primality > testing, graph coloring ... anything that can be discussed in a > day or so to folks with no CS background, yet which has some > theory component to it ... stuff that is surprising or counter-intuitive > is all the better ;) > Anyway, if anyone has any suggestions for material/topics > that I might cover, I would most appreciate it. Any pointers > would be accessible to students would be great (I have a couple) > would be great. > (or a link to one) to comp.theory in a week or so. > Chip Klostermeyer One of my favorite topics is Turing machines and the Halting Problem. It has a very immediate application, accessible to most undergraduate students, to wit the difference between syntax and semantics. Compilers, programs themselves, routinely check all programs in a language for syntactical correctness. But by the Halting Problem result, no program can check all programs for infinite loops. === Subject: Re: Simple idea, mathematics and common-sense >I know I shouldn't try, but the light... it's so beautifu...zap! >>There's something else important here which is the ambiguity of the >>square root operator >It is not ambiguous. Sqrt(x) (or x^(1/2) or the notation with the >square root symbol) all define the principal square root (provided x >is a positive real number). So Sqrt(9) = 3. It is not -3 even though >-3 is a square root of 9. >Just read about it on the following link (You could also go to the >library and read a decent math-book there): >http://mathworld.wolfram.com/SquareRoot.html > It doesn't matter how much mathematicians try to deny the obvious if > people actually *check* what I'm saying. Mathematicians try to deny the obviousAnd what here is obvious? That it is impossible to define the square root as being the value on the principal branch of z^(1/2)? Wow. I guess you really got us there. My calculator's sqrt button doesn't work, all my software that calculates sqrt will have to be returned, it's such a waste. But, there you go, it's so *obvious* that this is impossible, I suppose you gotta be right, there, ace. > Like consider P(x) = (x + 8a)(x + b), with ab=1, and 8a + b = 17. > If you work that out, then you have a = (17+/-sqrt(257))/16, now then > can you tell me if (17+sqrt(257))/16 or (17-sqrt(257))/16 has 16 as a > factor? So, what does this P(x) have to do with anything? You've set up a quadratic equation: 8a + 1/a = 17 8a^2 - 17a + 1 = 0. you thought having an unusual factorization of P(x) = x^2 + 17x + 8 would impress everyone. Think again. Your question about whether a is divisible by 16 is meaningless unless you specify which ring should contain the result. Surely that ring isn't the ring of algebraic integers. > Mathematicians arguing with me may promptly lash back with, yes, they > can say that it is NOT a factor in the ring of algebraic integers! I see. Anyone who responds to deny Your Eminence is now lashing back! Is it appropriate to refer to your alleged mathematics as asswipes? You may promptly asswipe back that I am wrong. > That's true, but so what? Now then, I dare any of you to try and see > for yourselves if that 16 should divide back through, so that the > number is actually more like an integer than a fraction. True? The number a isn't even an algebraic integer. If it were an algebraic integer (meaning, if you were to take another value), there is an easy test to establish divisibility: find the minimal polynomial of the quotient. There is also a norm function (defined on algebraic number fields) that allows one to discriminate in many cases whether an algebraic integer is or is not divisible by some specific factor. > If that sounds silly consider (1+sqrt(-3))/2 as that's an algebraic > integer and a factor of 1. It turns out that 1+sqrt(-3) has a factor > that is 2. > Think I'm crazy here? Well let 2x = 1 + sqrt(-3), then > 2x - 1 = sqrt(-3), and squaring both sides > 4x^2 - 4x + 1 = -3, so > 4x^2 - 4x + 4 = 0, so > x^2 - x + 1 = 0, > so what I said IS correct. However, notice again that definition of > algebraic integers as roots of monic polynomials with integer > coefficients! Is this JSH holding his Masters' Class on Algebraic Number Theory? Exactly who is the teacher here? Certainly not Mister James Z[1/2] is the entire set of real numbers Harris? > If you believe that covers everything, then you're acting on faith, > and not logic, as there's just no mathematical reasn for that belief. > Any supposed proof depends on circular reasoning. What do you mean to say here? That it's a mistake to think that the whole world of algebraic numbers comes down to whether the number (1 + sqrt(-3))/2 is an algebraic integer? Besides yourself, who on earth would think such a thing? Or, are you saying that there's just no mathematical reasn for the belief that algebraic integers are *defined* by the condition you cited: roots of monic polynomials over Z? Here you are back to your oft-repeated claim of circular reasoningYou seem to think that any use of the definition constitutes circular reasoning. A definition, to the extent that it specifies a unique object, is simply a matter of fixing terminology, and its use in an argument doesn't amount to any sort of flaw. This is not a matter of what you want to call mathematical reasn or of its absence. that of direct algebraic manipulation. You don't seem to recognize the use of induction, or of any of the standard arguments of algebra. If you care to exhibit the argument you're referring to, with your commentary as to what steps are OK, and what ones constitute circular reasoning, I'm sure we would all be amused by your flailing at that one. > I suggest those of you who don't believe me play with the expressions > here. What expressions? I showed that neither of the roots for a is an algebraic integer. Big deal. As usual, you haven't proven a thing. > And remember, if mathematicians are teaching wrong mathematics, what > good does that do anyone? The issue here may seem esoteric to some > extent, but ultimately it's about the importance of truth in research. What about those people who make careers out of false claims about the correctness of mathematics? What about people who continually cast aspersions on the credibility, competence, and honesty of those whose arguments refute those false claims? > Professionals should teach things that are true. Oh, I get it. ONLY professionals are supposed to tell the truth. Amateurs get a free ride, not only with respect to the level of rigor that is required, but also with regard to the need for honesty. > James Harris Dale. === Subject: Re: The Lost Proof of Fermat A+B = C A^2 + B^2 = C + A^2 - A + B^2 - B A^2 + B^2 = [A+B] + A[A-1] + B[B-1] A^p + B^p = [A+B] + A*[A^(p-1)-1] + B*[B^(p-1)-1] 3^2 + 4^2 = 5^2 3^3 + 4^3 = 5^3 - 34 34 is a Fibonacci number. I wonder if the Fibonacci series can lead to a proof of Fermat's Last Theorem? 1,2,3,5,8,13,21,34,55,89,144,233,377,610,... 5^2 - 3^2 = 2^4 [2^2 - 1^2]^2 + [2*2*1]^2 = [2^2 + 1^2]^2 5^2 + [2*2*3]^2 = 13^2 Squares of fibonacci numbers give fibonacci numbers or multiples of them: 1^2 + 2^2 = 5 2^2 + 3^2 = 13 3^2 + 5^2 = 34 5^2 + 8^2 = 89 8^2 + 13^2 = 233 But Fibonacci cubes aren't so predictable? 1^3 + 2^3 = 9 = 8 + 1 2^3 + 3^3 = 35 = 34 + 1 3^3 + 5^3 = 152 = 144 + 8 5^3 + 8^3 = 637 = 1 + 5 + 21 + 610 Interesting... === Subject: Re: Simple idea, mathematics and common-sense >I know I shouldn't try, but the light... it's so beautifu...zap! >There's something else important here which is the ambiguity of the >square root operator >It is not ambiguous. Sqrt(x) (or x^(1/2) or the notation with the >square root symbol) all define the principal square root (provided x >is a positive real number). So Sqrt(9) = 3. It is not -3 even though >-3 is a square root of 9. >Just read about it on the following link (You could also go to the >library and read a decent math-book there): >http://mathworld.wolfram.com/SquareRoot.html >It doesn't matter how much mathematicians try to deny the obvious if >people actually *check* what I'm saying. >Like consider P(x) = (x + 8a)(x + b), with ab=1, and 8a + b = 17. >If you work that out, then you have a = (17+/-sqrt(257))/16, now then >can you tell me if (17+sqrt(257))/16 or (17-sqrt(257))/16 has 16 as a >factor? A factor in what sense? >Mathematicians arguing with me may promptly lash back with, yes, they >can say that it is NOT a factor in the ring of algebraic integers! No, it's not a factor, in the algebraic integers. How is that lashing back? Are you claiming that it _is_ a factor in the algebraic integers? >That's true, but so what? Now then, I dare any of you to try and see >for yourselves if that 16 should divide back through, so that the >number is actually more like an integer than a fraction. Huh? Exactly what does more like an integer than a fraction mean? >If that sounds silly consider (1+sqrt(-3))/2 as that's an algebraic >integer and a factor of 1. It turns out that 1+sqrt(-3) has a factor >that is 2. >Think I'm crazy here? Could be. You seem to be insisting that you're right and everyone else is wrong, but you're not even giving a _hint_ what it is that you're right about. Exactly what is it that you're asserting, that you claim is correct even though the evil establishment denies it? >Well let 2x = 1 + sqrt(-3), then >2x - 1 = sqrt(-3), and squaring both sides >4x^2 - 4x + 1 = -3, so >4x^2 - 4x + 4 = 0, so >x^2 - x + 1 = 0, >so what I said IS correct. _What_ is correct? I can't figure out what it is you're saying. > However, notice again that definition of >algebraic integers as roots of monic polynomials with integer >coefficients! >If you believe that covers everything, then you're acting on faith, >and not logic, as there's just no mathematical reasn for that belief. >Any supposed proof depends on circular reasoning. >I suggest those of you who don't believe me play with the expressions >here. >And remember, if mathematicians are teaching wrong mathematics, what >good does that do anyone? The issue here may seem esoteric to some >extent, but ultimately it's about the importance of truth in research. >Professionals should teach things that are true. >James Harris === Subject: Re: RFI:Wanless' Fourth Conjecture/Dirichlet's Geometric Theorem Gerry Myerson > The sequence 2^n + 78557 includes no primes. Perhaps you meant 78557*2^n + 1 is always composite? See e.g. http://mathworld.wolfram.com/ SierpinskiNumberoftheSecondKind.html Rainer Rosenthal r.rosenthal@web.de === Subject: Re: Simple idea, mathematics and common-sense >[...] >If so, then hey, it may bug you, but I may start talking about evil >math society, and people who refuse to acknowledge the truth again!!! My god you're a ing idiot. It doesn't _bug_ people when you talk about evil math society, it leads to gales of laughter. You really haven't figured that out yet? Wow. >There's no more room for excuses, even for people like you Ivan. >Either actually be a mathematician, and acknowledge mathematical >truth, or just wait for the consequences of being rogue and against >the best interests of society at large. >Yup, if mathematicians refuse to teach correct mathematics, then >they are fighting against humanity itself. >And I think, will be treated accordingly, once that becomes common >knowledge. >Why fight against humanity, against progress and truth? Yup, every single mathematician on the planet - the people here, the famous ones you harass via email, the professors you visit, the editors of those journals, _every_ _single_ _one_ of them is fighting against humanity, against progress and truth. You really must have no idea how totally wacky that sounds. >James Harris === Subject: Re: Infinite Galois Groups Boruch asks: >Let F be the field of lengths constuctible by compass and >straighthedge. What is G(F/Q)? This is off the top of my head, so I might be wrong about some or all of this, but here goes. I'm not sure G(F/Q) has to be normal over Q. If one insists that the lengths have to be real, it isn't (e.g. sqrt(sqrt(2)+sqrt(3)) has sqrt(sqrt(2)-sqrt(3)) for a conjugate). That can make things a lot more complicated, since one has to worry about reality at every stage of the construction of a given number. For the sake of simplicity, it might be better to first ask for the properties of the smallest field E containing Q and closed under taking square roots. The field E is normal over Q and one can try to look at G(E/Q). Then G(E/Q) is a pro-2-group. Therefore, E is contained in the maximal normal 2-extension of Q. Conversely, every normal 2-extension of Q is obtained as a tower of quadratic extensions, so it lies in E. Therefore, E is the maximal pro-2-extension of Q. I haven't read much about this, but I think that a certain amount is known about the group G(E/Q). In fact it might be known completely. I think Iwasawa and Shafarevich did work on this and I think there is also something by Serre, but I don't know the relevant literature. Actually, if b is a complex number constructible by straight edge and compass and c is its complex conjugate, then b+c and bc are real and constructible. So, maybe the answer to the original question about constructible lengths is that one gets the maximal real subfield of E. Ignorantly, Allan Adler ara@zurich.ai.mit.edu **** * * * Disclaimer: I am a guest and *not* a member of the MIT Artificial * * Intelligence Lab. My actions and comments do not reflect * * in any way on MIT. Moreover, I am nowhere near the Boston * * metropolitan area. * * * **** === Subject: Re: The Lost Proof of Fermat > Squares of fibonacci numbers give fibonacci numbers or multiples of them: Since one is a Fibonacci number, this isn't too interesting. === Subject: Fundamental Theorems of Calculus I am a high school Calculus student and we've just covered the Fundamental Theorems in class. While I understand how to use the equations and don't have a problem with doing the assigned problems, I'm still having trouble logically working out exactly how these Theorems work. I have a few specific questions, and I'd appreciate it if someone could tell me where my thought process is wrong or advise how I might better understand how the Theorems work. In this post, FT1 means The Fundamental Theorem, Part I, and FT2 means The Fundamental Theorem, Part II or The Integral Evalulation Theorem. 1. In FT1, can 'a' (the lower limit of integration) be chosen to be any value at all (even if it winds up being greater than or equal to this question is yes, but that seems to cause problems. For example, what if a=4 and we were evaluating the equation when x=4 as well. Wouldn't the answer then be zero, no matter what f(t) or f(x) is? Consider the same situation again, when we are evaluating when x=4. Then if 'a' was chosen to be less than 4, let's assume the answer was positive. Then if 'a' was changed to 4, it would be zero and if 'a' was changed to greater than four, the answer would be negative. If I continue this train of thought, I realize that if 'a' is changed at all the answer would be different, since you're measuring a different area under the curve. It seems that the f(x) produced by FT1 would completely depend on that 'a' value, and any change in a would result in a different function f(x). I don't understand how 'a' can be any value at all and FT1 still yields the same result: f(x). 2. When studying FT2, the book defined F(x) as the signed area function: F(x) = INT[from a to x](f(t)dt) When evaluating definite integrals, the formula F(b) - F(a) is used, and the subtraction works in a way that it cancels out or negatives whatever the 'a' value was chosen to be in the original definition. However, I was wondering if F(x) can be used by itself, for example would a question ever give you a function f(x) and ask you to find F(2). Since there's no subtraction, just as in my first question, it seems here that the 'a' value would affect the answer. 3. I've gone through the derivation in my textbook of FT1 and I basically understand how they find that theorem, but I'm still having trouble with actually grasping how that equation works. What is wrong with my thought process in the following: Let's say we're trying to find f(2), and we know from FT1 that f(x) = d/dx INT[from a to x](f(t)dt). Therefore, let's start by evaluating the integral part when x=2. In this case, the definite integral from a (whatever that is) to x (which is 2) is simply a number, since it's just giving you the signed area under the curve between those two points. You then differentiate that number (the d/dx part), and, since the derivative of a constant is zero, the answer for f(2) must be zero. Under this reasoning, any function comes out to be y=0 at all points. I don't see how you can take a definite integral (which yields a constant), then differentiate and get anything but zero. However, maybe my reasoning is wrong because we're dealing with x so the answer is really a function, not a constant (but I can't understand how this can be the case). Even if we do get a function, it still doesn't make sense to me. First, you're integrating f(t)dt, so you get some function in terms of t. You then differentiate with respect to x, so you'd have to use implicit differentiation and you'd wind up with both a t and an x in the answer. If the result of that integral is a function in terms of the variable t, where does the t go in the answer? 4. I was trying to visualize what the signed area function F(x) actually looks like, so I considered the two functions 2x and x^2. If I understand the inverse nature of differentiation and integration correctly, x^2 should be an integral of 2x or the F(x) that we're dealing with when 2x is our f(x). In this situation, I see how the right half of the graph makes sense and the quadratic line shows how the area under the straight line is increasing exponentially, but it doesn't seem to work on the left side of the graph. Here, 2x goes negative and the x*2 line, it seems, should also be negative since we're dealing with signed area. The only way I could see this working is if the 'a' value is zero. Then the area left of the graph would be taken going from right to left, and this going backwards would cancel out the other negative value finishing with a net positive value. However, this makes me wonder about that 'a' even more. What if a=1? Then the area would be correct for everything to the right of one but if you tried to find the area between zero and one you'd be going backwards and the answer would be negative even though that portion of the graph is above the x-axis. The value chosen for 'a' seems to be causing problems in my reasoning no matter how I look at it. I can proceed with my work in the class, but I don't like to go ahead on assumptions without really understanding the theorems that I'm using, and I'm pretty confused as to how FT1 and FT2 work with an 'a' that can be chosen arbitrarily. I'd appreciate any input that would help lead me in the right direction. Johnathan === Subject: Re: My fear, consider this James, I finally decided to take a look at something that you've said, rather than just reading the commands from others and assuming that you're wrong. I completely followed your math up to the b^2-17b+8 = 0, your algebra seems correct (assuming that ab = 1) ... then you assume that b is not prime ... you say that fz = b and you substitute fz for b in your equation: f^2z^2 - 17fz + 8 = 0 Then you divided by f: fz^2 - 17z + 8/f = 0 So far, all is well. But then came this: > and notice you STILL have that f on the front. what is the 'front'? do you mean the fact that there is an f that appears on the left hand side of the equation? > I don't want to hear that it isn't applicable because f isn't an > integer, as if you will have to get a polynomial reducible over Q if > you pick the right f, as that's just bogus. What? Why isn't f an integer? What are you talking about here? What you do mean by if you will have to get a polynomial reducible over Q if you pick the right f? What is Q? Why is this bogus? > The problem is that 17. For it to work, you need to have something > even! What is 'it'? Why do you need to have something even? In short, I don't have a clue what this is all about. I really tried to take you seriously here, but it is just impossible because you are not being very precise about the terms that you use. For instance, you say that something is 'bogus' ... does that mean that it leads to a contradiction? If so, can you show me the contradiction that it leads to? If I may be so bold as to suggest a style change to you: please try to fight the temptation to describe things in english ... you tend to be imprecise with your words (i.e. 'bogus', 'the problem is that 17' what problem? 'For it to work' what's 'it'?) ... I followed your brush with algebra at the beginning ... you could continue to use symbolic arguments through to the end and I'll bet the average non-mind-reader could follow. > It's so sad that I've tried to come up with simple examples for so > long, and posters like Nora Baron, Dik Winter and Arturo Magidin > have *successfully* come back with their own posts and kept winning at > convincing you all that I was wrong! > It's been so frustrating that I'm terrified that they will just get > away with it again, so here's another angle as I desperately try yet > again to get someone to care about what's mathematically correct > knowing the kind of people who are out there to come right back and > push incorrect math. > Previously I noted that I could use > P(x) = (x+8a)(x+b), and ab = 1, so P(x) = x^2 + (8a + b)x + 8. > And I considered 8a + b = 17, as then you have > 8/b + b = 17, so b^2 - 17b + 8 = 0. > Now imagine *any* non-unit factor f in the ring of algebraic integers, > like 1+i that might be a factor of b, and let b = fz, and substitute > and you get > f^2 z^2 - 17fz + 8 = 0, so > f z^2 - 17z + 8/f = 0 > and notice you STILL have that f on the front. > I don't want to hear that it isn't applicable because f isn't an > integer, as if you will have to get a polynomial reducible over Q if > you pick the right f, as that's just bogus. > The problem is that 17. For it to work, you need to have something > even! > I'm so damned tired. I can't be sure if anyone will listen to me. > Arturo Magidin or Dik Winter or Nora Baron or Rick Decker will come > back like they have before, now won't they? > They'll come back and post something stupid, and wrong, and just plain > evil, and you'll go along like you've done before. > It's so evil, so frustrating. Nothing matters in mathematics, > mathematicians don't give a damn about the truth. > NOBODY ING CARES ABOUT THE TRUTH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!! !!!!!!!!!!!!!!!! > JSH === Subject: Re: My fear, consider this James, One more point (and with all due respect) ... Imagine that I knew the answer to the famous Computer Science problem that asks about the relationship between the sets P and NP, imagine, also, that I only know how to speak the word 'blah' ... so, in my brain I had this wonderful answer all worked out, but when I tried to communicate to others all they heard was 'blah blah blah blah ...' ... do you see that it is not THEIR fault ... nor, really, is it mine, but it would behoove me to learn how to communicate with the other people (as then I could claim the $1M prize that goes along with a solution) ... just an analogy that may help. > James, > I finally decided to take a look at something that you've said, rather than > just reading the commands from others and assuming that you're wrong. I > completely followed your math up to the b^2-17b+8 = 0, your algebra seems > correct (assuming that ab = 1) ... then you assume that b is not prime ... > you say that fz = b and you substitute fz for b in your equation: > f^2z^2 - 17fz + 8 = 0 > Then you divided by f: > fz^2 - 17z + 8/f = 0 > So far, all is well. But then came this: > and notice you STILL have that f on the front. > what is the 'front'? do you mean the fact that there is an f that appears > on the left hand side of the equation? > I don't want to hear that it isn't applicable because f isn't an > integer, as if you will have to get a polynomial reducible over Q if > you pick the right f, as that's just bogus. > What? Why isn't f an integer? What are you talking about here? What you > do mean by if you will have to get a polynomial reducible over Q if you > pick the right f? What is Q? Why is this bogus? > The problem is that 17. For it to work, you need to have something > even! > What is 'it'? Why do you need to have something even? > In short, I don't have a clue what this is all about. I really tried to > take you seriously here, but it is just impossible because you are not being > very precise about the terms that you use. For instance, you say that > something is 'bogus' ... does that mean that it leads to a contradiction? > If so, can you show me the contradiction that it leads to? If I may be so > bold as to suggest a style change to you: please try to fight the > temptation to describe things in english ... you tend to be imprecise with > your words (i.e. 'bogus', 'the problem is that 17' what problem? 'For it to > work' what's 'it'?) ... I followed your brush with algebra at the beginning > ... you could continue to use symbolic arguments through to the end and I'll > bet the average non-mind-reader could follow. > It's so sad that I've tried to come up with simple examples for so > long, and posters like Nora Baron, Dik Winter and Arturo Magidin > have *successfully* come back with their own posts and kept winning at > convincing you all that I was wrong! > It's been so frustrating that I'm terrified that they will just get > away with it again, so here's another angle as I desperately try yet > again to get someone to care about what's mathematically correct > knowing the kind of people who are out there to come right back and > push incorrect math. > Previously I noted that I could use > P(x) = (x+8a)(x+b), and ab = 1, so P(x) = x^2 + (8a + b)x + 8. > And I considered 8a + b = 17, as then you have > 8/b + b = 17, so b^2 - 17b + 8 = 0. > Now imagine *any* non-unit factor f in the ring of algebraic integers, > like 1+i that might be a factor of b, and let b = fz, and substitute > and you get > f^2 z^2 - 17fz + 8 = 0, so > f z^2 - 17z + 8/f = 0 > and notice you STILL have that f on the front. > I don't want to hear that it isn't applicable because f isn't an > integer, as if you will have to get a polynomial reducible over Q if > you pick the right f, as that's just bogus. > The problem is that 17. For it to work, you need to have something > even! > I'm so damned tired. I can't be sure if anyone will listen to me. > Arturo Magidin or Dik Winter or Nora Baron or Rick Decker will come > back like they have before, now won't they? > They'll come back and post something stupid, and wrong, and just plain > evil, and you'll go along like you've done before. > It's so evil, so frustrating. Nothing matters in mathematics, > mathematicians don't give a damn about the truth. > NOBODY ING CARES ABOUT THE TRUTH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!! > !!!!!!!!!!!!!!!! > JSH === Subject: Re: 'erf' function in C [...] > I translated the C code to Fortran 95, as shown below. With CVF, the > results agree with those of B. Voh. Using the Lahey/Fujitsu Fortran 95 > compiler with the option -quad, which extends DOUBLE PRECISION to > quadruple precision, the results agree with the true values posted by > George Marsaglia to at least 18 decimal places, and often the full 20. > Translating both the double and long double C variables to DOUBLE > PRECISION in Fortran seem wrong to me, although the results are good. > What is the proper translation? [...] Double in C and Fortran 95 should be the same, ie, 64 bit. Lahey's quad is 128 bit, and C's long double is 80 bit. The only Fortran 95 compilers for Win32 with support for a long double type are the two British compilers, Salford and NAS. Ciao, Gerry T. === Subject: Re: the anticlassicalist }{ ii: the spectre continues Excuse the delayed reply. I dropped alt.philosophy, sci.lang, and sci.physics. | > 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 simplifiedBoolean algebras are a special case of Heyting algebras, and there are plenty of finite Heyting algebras even excluding finite boolean algebras. | > And since Heyting algebras have a potential universality hinted by the | > Curry-Howard isomorphism, why is it so necessary to fall back on the | > classical approach. I still do not see from where this desire arises at | > forcing the ontology of a model... | | Perhaps the following may help (and perhaps not :-) ). | | A Heyting algebra is a mathematical structure of some kind. | It's defined as an infinite set, together with some operators and | relations, that satisfies certain (first-order logic) conditions. | So even to understand the notion of a Heyting algebra, most | people require a good intuitive picture of the Tarski semantics | of first order (classical) predicate logic. | | Constructivism has so many variants that, in order to study | them well, most systems are defined and studied using classical | means and classical thinking habits. In more vague terms: | 'reasoning on the meta level is still classical'. I don't think 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 which 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. 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 things as Martin-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. If you were to count as schools of classical mathematics all the different nonconstructive formal systems in which 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 which _some_ mathematics has actually been done, you can find set theorists who've taken as their starting points initial assumptions of varying strengths. The kicker here is that my statement about Bishop constructivism applies to classical mathematics as well. Most classical mathematics also starts from Bishop's constructivism plus a few extra axioms. I would argue that this is evidence that it would be a good choice of metatheory. There's a basic asymmetry between a theory having extra axioms and one which simply leaves them out. Having a metatheory which makes more assumptions than the theory being considered sometimes leads to confusion. There's an odd little theorem in topos theory whose proof starts something like this: either every null object is an initial object, or there exists a null object that isn't an initial object.... This is an application of the law of excluded middle. It's sort of like allowing the classicality of the metatheory to bleed into the object theory. And then this is overcome by considering topos objects in toposes! Many of the problems people believe they have with constructivism also are less liable to apply while doing metamathematics. The theorems people try to prove tend to be of low logical complexity, things like If X is a theorem of S, then Y is a theorem of T. Theorems of that form are guaranteed to be constructively valid if they are classically valid. There's a tradition in logic of trying to pare down the principles needed to prove metamathematical results to the most elementary kind. Making them also constructive is just a natural next step in this direction. | You may find that awful, but that's the way it is. One reason | for this is that there is no reason to prefer one form of | constructivism over another. But there is. All else being equal, one should be sparing with one's assumptions. If I'm a Bishop constructivist, and you are a fan of the Markov school's work, I can accomodate your results in my scheme of things by considering them proofs of consequences of Markov's rule and the recursiveness of all functions from N to N. One should only regard assumptions as permanent axioms once it's clear that they've become thoroughly intertwined in your work, and that your work really doesn't go through without them. | You like Heyting algebras, | someone else likes some other system. And unlike | the several formalizations of classical logic, these variants | are far from being mutually equivalent or translatable. Heyting algebras are quite standard. Heyting algebras bear the same relationship to constructive reasoning as boolean algebras bear to classical reasoning. They reflect the propositional calculus. One *could* consider (as logicians have, being very thorough) logics intermediate between the usual propositional calculus of constructive and classical mathematics, but I would be very surprised if anybody were actually doing mathematics that way. Even on the level of predicate calculus, is anybody seriously using something in between intuitionist logic and classical logic? | Using classical definitions of the lot allows us at least | to understand all variants at the same time, and to make | mutual comparisons. Having to use a classical metatheory for constructive reasoning creates certain confusions. People wind up using techniques for circumventing the assumption of excluded middle, such as topos theory, Kripke or Beth models. Crudely speaking, the problem is that one has to keep going around thinking, The law of excluded middle is actually true, but we're going to pretend like it isn't. I don't know what problems the reverse is supposed to create; one is just considering some extra assumption | There is plenty of room for more | research and other points of view, but as something that | is supposed to give students a good basis for further | research, classical logic is still essential, constructivism | an extra. Classical logic is overwhelmingly the popular choice, whether it deserves it or not. Sort of like Microsoft Windows. It may make pragmatic sense to defer to it, given the place it has in the world, but it's not a really persuasive argument for its being inherently better. | That may serve as an explanation why not many | have responded very enthousiastic to your pamflet. :-) I'm sorry to say that there are other reasons why some of us haven't responded enthusiastically to Galathaea's pamphletIt's probably rather rare for someone to be more sympathetic to the constructivists than I am. I still feel like I don't know enough about it to make a really fair evaluation of it, but as you can see I'm more optimistic about it than people usually are. It seems to me that we need both to know more constructive mathematics, and to have better reasons for what we think it's pros and cons are. The lack of experience leaves our general idea of the relationship between constructive and classical mathematics resting on too little. On the other hand, weaknesses in people's general ideas and justifications leads them to chalk certain things up as advantages of classical reasoning, that certain specific theorems can be proven, for instance, not taking other subtleties properly into account, like the fact that they are taken to mean something different constructively, and that alternative paths to the same practical end result exist. 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. I would generally advise against being a self-proclaimed liar, even if this is meant in a humorous way (which I don't know). 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 things in less depth; a diver deals in fewer at greater depth. There's nothing 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 something in your hands! It seemed to me that a lot of the examples, and maybe all of them, of things whose logic is constructive (whatever that is supposed to mean, specifically) that we've seen here, are just special cases of the topological interpretation. This 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 things a bit, one could say that the complementary perception, that something 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. But that's a poor argument for dropping the law of excluded middle. It's more realistic to say that the concept of being a W has a grey area on its borders. So really one has an argument in favor of fuzzy logic. As far as I can tell, the jury is still out on fuzzy logic. Such an argument in favor of fuzzy logic is surely only a motivational argument. We can't say so easily whether your logic is the right place to introduce awareness of fuzziness. 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. If someone wants to chip in on the mathematical side of constructivism, try helping me satisfy some of my curiosity. I've had the question of the degree to which the Jor-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 isomorphisms 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. I'm interested in a strong version of the concept of simple group (with apartness). I'm willing to assume that simplicity holds for the quotient groups in the following form. If x and y are elements of the group, and x<>1, then y can be expressed as a product of conjugates of x and x^{-1}. I think 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. Keith Ramsay === Subject: Re: the anticlassicalist }{ ii: the spectre continues > Excuse the delayed reply. I dropped alt.philosophy, sci.lang, > and sci.physics. Not quite... :-) I dropped them here (I think). [...] > [Keith Ramsay] > Even on the level of predicate > calculus, is anybody seriously using something in between intuitionist > logic and classical logic? I want to say yes to this... > | Using classical definitions of the lot allows us at least > | to understand all variants at the same time, and to make > | mutual comparisons. > Having to use a classical metatheory for constructive reasoning creates > certain confusions. People wind up using techniques for circumventing > the assumption of excluded middle, such as topos theory, Kripke or Beth > models. Crudely speaking, the problem is that one has to keep going around > thinking, The law of excluded middle is actually true, but we're going to > pretend like it isn't. I don't know what problems the reverse is > supposed to create; one is just considering some extra assumption There are good reasons for this being a default behavior or attitude. But, understanding the reason is onerous... > | There is plenty of room for more > | research and other points of view, but as something that > | is supposed to give students a good basis for further > | research, classical logic is still essential, constructivism > | an extra. > Classical logic is overwhelmingly the popular choice, whether it deserves > it or not. Sort of like Microsoft Windows. It may make pragmatic sense to > defer to it, given the place it has in the world, but it's not a really > persuasive argument for its being inherently better. Right. > | That may serve as an explanation why not many > | have responded very enthousiastic to your pamflet. :-) > I'm sorry to say that there are other reasons why some of us haven't > responded enthusiastically to Galathaea's pamphlet> It's probably rather rare for someone to be more sympathetic to the > constructivists than I am. I still feel like I don't know enough about it > to make a really fair evaluation of it, but as you can see I'm more > optimistic about it than people usually are. It seems to me that we need > both to know more constructive mathematics, and to have better reasons for > what we think it's pros and cons are. The lack of experience leaves our > general idea of the relationship between constructive and classical > mathematics resting on too little. And a lack of historical perspective. There are entire lines of inquiry that are ignored. The general sense of mathematics as some fixed collection of truths does not reflect the various debates. I just ran across a reference on John Baez' home page to a philosopher of mathematics calling for a new foundational approach that reflects what mathematicians do. There is no need for that. > On the other hand, weaknesses in > people's general ideas and justifications leads them to chalk certain > things up as advantages of classical reasoning, that certain specific > theorems can be proven, for instance, not taking other subtleties properly > into account, like the fact that they are taken to mean something > different constructively, and that alternative paths to the same practical > end result exist. [...] Well, no one takes my posts seriously, but here goes... Where you ask about someone using something between intuitionist and classical logic, I need to redirect to semantics. Tarski initiated algebraic semantics with cylindrical algebras. When you pursue this line of inquiry, you get to Halmos' polyadic algebras and finally to the representation theorem I recently found in Balbes and Dwinger, If L is a closure algebra, then L^o is a Heyting algebra under the partial ordering of L and is also a D01-subalgebra of L More remarkable is the fact that every Heyting algebra can be represented this way The closure algebras are Boolean algebras, A Boolean algebra L with an additive closure operator ^c in which 0^c=0 is called a closure algebra. An element (a e L) is called open if (a-bar e L^c). The set of open elements is denoted by L^o So, there is a close relationship between Heyting algebras and Boolean algebras that seems to be ignored. There is additional structure involved that relates the two. There are semantics that do not seem to interest many in questions as they apply in foundations. Yet, if you are going to embed a Heyting algebra in such a way that its specification induces a D01-subalgebra, one might ask if the representation deserves to first be interpreted as a system rather than in the pieces that can be syntactically represented in isolation. In other words, is one man's D01-subalgebra another man's bilattice? This is what I mean when I ask why anyone should consider a new attempt at foundations. During my argument with George Greene this year I learned a lot by looking for sources to understand his objections. In large part, it seems as if bibliographies are partitioned. It is not at all clear to me that there is any debate left. I really believe there is enough information for a coherent integration of these questions. You talk about pretending that the law of excluded middle is not true while implicitly depending upon it. I think I need to see a formal specification where the natural language used is not depending on AND and OR in the sense of classical logic. The best paper I have found thus far is from Jacek Malinowski analyzing Strawson presupposition. Getting to classical logic from something that is not classical logic simply does not seem to make sense. But, this is so far removed from constructive mathematics, that someone who learns Bishop or Markov with an interest in *mathematics* certainly has no interest in the regress to presupposition via negation. And, then there is someone like me who stumbled on de Morgan fields and de Morgan isomorphisms without any clue and without being able to explain the skepticism in any coherent fashion. The questions are so disjointed. I mentioned the historical perspective just because of my own experience. I have been all over the place. I do not have any particular disagreement with Frege's ontology of number. But, it is not at all clear that that is the concept of number applicable under algebraic semantics. I intuitively understood that term in the same sense that others refer to descriptive set theory. And, I have finally found a modern philosopher who has discussed the distinction between Frege's resolution of sematnical applicability and the problem of descriptive applicability. In my own researches, I can find particularly similar statements concerning two fixed values in Dedekind's writings, Abraham Robinson's note on threshold logic, and the papers I recently found from the Steklov Institute of Mathematics in Russia. All of this work seems to relate to the algebraic semantics. And, just for kicks... I would love to be able to even talk to you about the Jor-Holder theorem. But,... :-) I started out thinking classically and ended up defending intuitionism with which I did not even agree from the standpoint of a Heyting algebra. I am not even close to thinking about this in terms of a chain of groups. But, I have done a lot of diving even if it seems otherwise. So, I started this part by saying that no one took my posts seriously. I cannot say that I blame anyone for that. Nevertheless, the issues are so spread out over so many topics, it is hard to be coherent until you figure out what someone else knows or does not know. :-) === Subject: Re: the anticlassicalist }{ ii: the spectre continues sci.lang: > Excuse the delayed reply. I dropped alt.philosophy, sci.lang, > and sci.physics. Actually, you didn't, but the thought was appreciated. [...] Brian === Subject: Re: 'erf' function in C And what about the inverse cumulative normal distribution? I used to implement an algorithm you published, but do not like it because it is machine-dependent. What in your opinion is the best algorithm? === Subject: Bayesian Class and Math/Stat Teaching Techniques A few weeks ago I posted a message asking about books on Bayesian Unfortunately I have since dropped the class and am wondering about whether I should continue the degree (Masters in Applied Statistics) and would like some thoughts from the thoughtful people here. I have a BS in Electrical Engineering and an MBA, both from the University of Michigan. What I really liked about the MBA program is that it was almost 100% applied. Probably 50-70%+ of the classes was case study classes, a trend mostly propagated and refined by Harvard Business School where supposedly 100% of the classes are case study classes. Apparently more and more law programs have more and more case study classes as well. Case study classes are really as applied as you get because concepts and theories are learned in the context of real world situations and circumstances. I am also the type of person who is a very intuitive learner and has a much easier time learning when I see how what I am learning relates to challenges in real life (eg: business, which is what I do). So given that the degree I am pursuing is called Masters in *APPLIED* Statistics, I thought the the courses would be heavily applied and taught in the context of solving real world problems. No dice. Both courses I took in the first semester (part-time evening program) had heavy theory. The Bayesian class was not even as bad as the other one (Mathematical Statistics). There was essentially no attempt on the part of the professor to relate the theory to real world programs or to even give real world examples to illustrate the concepts. It was formula, theory, formula, theory, theory, formula, etc. I asked him about that and he said there's no way around the theory. I'm not trying to get around the theory but theories and formulas mean nothing to me without real world context. I'm not stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE and 98+ percentile overall. My thinking right now is that my expectations were just off and disciplines like Math/Statistics are just not as, ummm, progressive as Business/Law when it comes to teaching (please -- no hate mail). Those teaching Math/Stat may also be too smart and are not interested in mune day-to-day business/industry problems (hopefully that will stop the hate mail!). So what's up with that? Why is a degree called Masters in Applied Statistics so heavy in theory? 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. and not to learn esoteric statistical theory in the absence of application that I will surely forget an hour after the final exam. I am not trying to slam the field. I am interested in some opinions from those in the field, especially those teaching it, to help me determine if I === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > My thinking right now is that my expectations were just off and > disciplines like Math/Statistics are just not as, ummm, progressive as > Business/Law when it comes to teaching (please -- no hate mail). Those > teaching Math/Stat may also be too smart and are not interested in > mune day-to-day business/industry problems (hopefully that will > stop the hate mail!). So what's up with that? Why is a degree called Masters in Applied Statistics so heavy in theory? 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. and not to learn esoteric statistical theory > in the absence of application that I will surely forget an hour after > the final exam. Personally, for me, and I am an applied mathematician, I think the applied people actually have to learn more theory than the theoretical mathematicians. The reason being that the theoretical mathematicians are all happy to be doing math just for the sake of math. Me, however, I need to why someone would ever do it, then once I know why someone would, I need to why it actually works. (or sometimes why something works, and then what its purpose is) For example, why does linear regression give the best straight line fit to model data? I can do it no problem, but if it's all for naught, why bother learning it. Those are just my thoughts. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > There was essentially no attempt on the part of the professor > to relate the theory to real world programs or to even give real world > examples to illustrate the concepts. It was formula, theory, formula, > theory, theory, formula, etc. I asked him about that and he said there's no > way around the theory. To me, math is like a Swiss army knife: It's a tool. A neat and handy tool, and it can do all sorts of smart things. Some parts I use everyday, other parts I hardly know what do. You could also compare math to a car. People have different opinions about a car. To some, it is merely a means of transportation. They don't care how it works, they just want to be able to drive to work every day. For others, a car is a toy. They can spend hours every day fiddling with it, turning a knob, drilling a hole, adjusting a valve, just to see how it works. Tools for me include: Swiss army knife, a car, and math. On the other hand, a computer is like a toy to me, but that is O.T. BUT, that is not all. When using a tool, whether it is math or a car, you need to learn how to use it. You also need to know the basics of how it works. For a car, you need to know the traffic laws, basic maintenance, and you need to know not to drive too fast in adverse weather and road conditions. In short, you need some theory, before you can start to drive a car. The same applies to math. I've taken lots of so-called applied math courses, without seeing a single application. But then again, there are different opinions. I believe some parts of math are more deep than others. For instance, the proof of Fermats Last Theorem involves concepts that I have never heard (e.g. modular forms), even though I have taken several courses in discrete math. So, I consider such concepts deep. On the other hand, stuff like calculus and linear algebra are more likely to find everyday use. One of my prof's expressed the opinion that all undergraduate classes are applied in the above sense. I agree with that sentiment. Despite that, there were no real-worls applications in any of them. That's too bad. I've been a T.A. in undergraduate math for several years, and it was very common to be asked about potential every-day applications. I found it very difficult to give such answers, but I attribute that to lack of training/experience on my part. I've certainly learned that many students - like the O.P. - find it much easier to grasp the concepts when they get related to something less abstract. It is the teachers job to explain both the theory *and* give some reasonable applications. The latter will be the most challenging for me, should I ever get a job teaching math. -Michael. P.S. The O.P. has noticed a difference between learning Math and Business. Perhaps there is - generally - a different teaching culture between the two. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > My thinking right now is that my expectations were just off and > disciplines like Math/Statistics are just not as, ummm, progressive as > Business/Law when it comes to teaching (please -- no hate mail). Those > teaching Math/Stat may also be too smart and are not interested in > mune day-to-day business/industry problems (hopefully that will > stop the hate mail!). So what's up with that? Why is a degree called Masters in Applied Statistics so heavy in theory? 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. and not to learn esoteric statistical theory > in the absence of application that I will surely forget an hour after > the final exam. > Personally, for me, and I am an applied mathematician, I think the applied > people actually have to learn more theory than the theoretical > mathematicians. The reason being that the theoretical mathematicians are > all happy to be doing math just for the sake of math. Me, however, I need > to why someone would ever do it, then once I know why someone would, I need > to why it actually works. (or sometimes why something works, and then what > its purpose is) For example, why does linear regression give the best > straight line fit to model data? I can do it no problem, but if it's all > for naught, why bother learning it. > Those are just my thoughts. The way the business schools and law schools would teach that concept (why does linear regression give the best straight line fit to model data) would be to have a case where, say, someone using another method got less than optimal results that ended in disaster and then show/teach what the calculations might be with linear regression. The way the Applied Statistics classes I was in would have taught that would have been to produce a bunch of incomprehensible formulas without any real world examples illustrating the concept and the pluses/minuses of each method. In fact, I specifically remember a class in business school illustrating a statistical problem and how/why the scientists at Morton Thiokol did not catch the potential problem with the o-ring on the space shuttle Challenger that caused the tragedy -- they didn't do tests nor have data points for the o-ring at low temperatures, and the temperature at launch was very cold for central Florida. I am certainly not saying don't teach theory. I am just questioning the point of incessantly teaching theory in the complete *absence* of application for a degree called Masters of Applied Statistics=== Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > To me, math is like a Swiss army knife: It's a tool. A neat and handy tool, > and it can do all sorts of smart things. Some parts I use everyday, other > parts I hardly know what do. That is *exactly* why I enrolled in Masters of Applied Statistics -- I want to use it like a Swiss Army Knife. I just happen to be an outdoorsman so let's take your analogy and run with it. If I showed a person who has never seen a Swiss Army Knife (or Gerber, Leatherman, etc.) a Swiss Army Knife and I cannot explain how/when to use the various tools on the Swiss Army Knife without hours and hours and hours of theory without actually referring *to* the knife or demonstrating with the knife, what's the point? I should just throw the knife away and talk about metallurgy, the history of man and tools, how to manufacture knives, or something equally theoretical. An *applied* class in Swiss Army Knives should involve lots of actual usage of Swiss Army Knives in real world situations with very short blurbs about the genesis of Swiss Army Knives, metallurgy, etc. > You could also compare math to a car. People have different opinions about a > car. To some, it is merely a means of transportation. They don't care how it > works, they just want to be able to drive to work every day. For others, a > car is a toy. They can spend hours every day fiddling with it, turning a > knob, drilling a hole, adjusting a valve, just to see how it works. Sure. I also happen to be an amateur mechanic. If I enroll in a class called, say, Applied Automotive Repair, the theory necessary for the repairs should be taught in the context of the actual repairs. If the classes of Applied Automotive Repair have almost not hands on work on cars and are instead lots and lots of formulas and theory (again, without actualy hands on work), it's not really applied -- it's Theory of Automotive Design and Repairs> One of my prof's expressed the opinion that all undergraduate classes are applied in the above sense. I agree with that sentiment. Despite that, > there were no real-worls applications in any of them. That's too bad. That is unfortunate but it's not nearly as unfortunate as a part-time graduate program called Masters of Applied Statistics that's got lots of theory without applications. Why? Almost all undergrads have no career-oriented work experience and wouldn't even know how/where to apply what they learn. Grads in part-time graduate programs called Masters of Applied Statistics have years and years of professional work experience and can all conceptualize where and how they can use what they just learned. It's a pity that those people go back to school and get essentially treated programs do not fail their students. > I've been a T.A. in undergraduate math for several years, and it was very > common to be asked about potential every-day applications. I found it very > difficult to give such answers, but I attribute that to lack of > training/experience on my part. I've certainly learned that many students - > like the O.P. - find it much easier to grasp the concepts when they get > related to something less abstract. Don't take this as an insult -- because it's not meant as one -- but I suspect you have been in academia most/all of your life because people who have been in industry would probably have a much easier job of relating what they are teaching to the undergrads and how it might be used in the real world> It is the teachers job to explain both the theory *and* give some reasonable > applications. The latter will be the most challenging for me, should I ever > get a job teaching math. > -Michael. Hopefully you succeed. If you have not been in industry you might find it very useful to be in industry for a few years before you teach math. > P.S. The O.P. has noticed a difference between learning Math and Business. > Perhaps there is - generally - a different teaching culture between the two. Almost certainly. Business is focused on making money, period. Everything else is a means to that end; It's the ultimate in applied discipline. All the professors in the business school I went to still worked/consulted in industry so they are in touch with the real world and still very much have their applied caps on (and almost none had PhDs). Everyone is focused on how to *solve problems* such that profits are maximized. department was just different. They liked theory and they liked formulas. They liked elegant solutions and proofs, even if they were irrelevant to application. I sensed a certain disdain for word problems and real world analogies and explanations to help the students conceptualize the theory because real math students don't need those crutches. They're very smart people who would probably look contemptuously at the description I gave above for business schools (No PhDs?!?! Only interested in profit?!?! Theory only useful if taught in conjuction with application?!?! How grotesque, how low-brow, how coarse!!!). === Subject: Re: the anticlassicalist }{ ii: the spectre continues > Excuse the delayed reply. I dropped alt.philosophy, sci.lang, > and sci.physics. Good idea. > | > 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. You're right. Finite topological spaces, right? Cheers, Herman Jurjus PS Thank you for the rest of your contribution as well. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > Don't take this as an insult -- because it's not meant as one -- but I > suspect you have been in academia most/all of your life because people who > have been in industry would probably have a much easier job of relating what > they are teaching to the undergrads and how it might be used in the real > worldYep. I had a career in academia, but then I switched to industry. The change was coincidental at the time, more or less forced upon me by the job situation at the time, but I've come to like it out here in the real world! Nevertheless, I appreciate the theory and its importance. It's not a matter of either/or. I believe one should attempt to combine both theory and application, possibly 50% time and effort spent on each. Even in a course specifically labelled applied> It is the teachers job to explain both the theory *and* give some > reasonable > applications. The latter will be the most challenging for me, should I > ever > get a job teaching math. > -Michael. > Hopefully you succeed. If you have not been in industry you might find it > very useful to be in industry for a few years before you teach math. Absolutely. For the very same reason, I actually would prefer teaching physics, simply because it is easier to relate to real-world situations. Anyway, most applied mathematics rests on some kind of model of the real world, so teaching applied math actually requires the students are well versed in creating mathematical models describing a specific problem. Converting an every-day problem into a mathematical description is really an art, something that takes years of training. I suppose such training should be a part of the applied courses. > P.S. The O.P. has noticed a difference between learning Math and Business. > Perhaps there is - generally - a different teaching culture between the > two. > Almost certainly. Business is focused on making money, period. Everything > else is a means to that end; It's the ultimate in applied discipline. All > the professors in the business school I went to still worked/consulted in > industry so they are in touch with the real world and still very much have > their applied caps on (and almost none had PhDs). Everyone is focused on > how to *solve problems* such that profits are maximized. > department was just different. They liked theory and they liked formulas. > They liked elegant solutions and proofs, even if they were irrelevant to > application. I sensed a certain disdain for word problems and real world > analogies and explanations to help the students conceptualize the theory > because real math students don't need those crutches. They're very smart > people who would probably look contemptuously at the description I gave > above for business schools (No PhDs?!?! Only interested in profit?!?! > Theory only useful if taught in conjuction with application?!?! How > grotesque, how low-brow, how coarse!!!). I'm detecting some contempt on your part towards mathematicians in general, or perhaps I am mistaken. Anyway, mathematics (and mathematicians) have a justification on its own. We (read: civilization) need people who are able to develop and expand on the mathematical tools available. But there is a gap between mathematics and business, in what you have described. I suppose it's up to people like you and I to close that gap... -Michael. === Subject: Re: Minimally simple finite groups? >Which of the finite simple groups are minimally simple, i.e., >have all of their proper subgroups solvable? Obviously the only >alternating group that qualifies is A_5 =~ L(2,4) =~ L(2,5), and >I know the list also includes L(2,7) =~ L(3,2), L(2,8), L(2,13), >... On the other hand, I also know it *doesn't* include L(2,9) =~ >A_6 or L(2,11), both of which contain subgroups isomorphic to >A_5. > The classification of minimal simple groups was a consequence of > John Thomspon's classification of nonsolvable groups in which nontrivial > solvable subgroups have solvable normlaizers, which was one of the big > classification theorems that came before CFSG. > This is in Bull. Amer. Math. Soc. 74, 1968, 383-437. > The simple groups coming out of Thompson's Theorem are L_2(q), Sz(q), > L_3(3), M_11, A_7, U_3(3). Of course, these are not all minimal simple. > L_3(3) is, but M_11, A_7, U_3(3) are not. > Sz(2^e) is minimal simple whenever it does not contain a smaller > Sz(2^f), which I guess is equivalent to e being prime. Is it easy to see why Sz(2^e) can't contain (as sections) any non-abelian simple groups other than the obvious Sz(2^f) for all f dividing e? Or do you need substantial machinery from Thompson's Theorem? > For L_2(p^e) to be minimal simple, its order must not be divisible by 60 > (otherwise it contains A_5). Ah. Why must it contain A_5 if its order is divisible by 60? > Also it must not contain any smaller > simple L_2(p^f), so if p>=3, then we must have e=1. But L_2(2^e) and > L_2(3^e) will be minimal simple for e prime provided their order is > no divisible by 60. You must mean so if p>3, of course. Again, is it easy to see why other non-obvious simple groups are excluded from being sections of L_2(p^e)? (Or are they? Can there be larger groups, but containing A_5?) > Something like that, anyway! As always, thanks for another of your wonderfully informative posts! === Subject: Re: the anticlassicalist }{ : ism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.com >No, it means that at some point you have to hook into what physicists >*observe*. Will it tell us the energy levels of a hydrogen atom and >predict the Lamb shift? If you're interested in developing a better >mathematical formalism, that's fine, go ahead. Just don't tell us it's >physics. >It is not about the formalism. It is about whether assumptions in the >formalism >that make it coherent are even being respected. [...] >Your claim I'm not making a claim, just a request. >translates into a definition for physics as nothing more than a >chaotically organized aggregate of partial results. You make that sound like a bad thing ;-) It's based on inductive reasoning from observation, so it can't help but be partial. As for chaotic, I don't know. > If that is the case, you should >not be relying on mathematical formalism for justifications--as in the >many times I >have seen the statement it works. If mathematics is a symbolic tool with no >inherent connection to the material and the natural language descriptions are >nonsensical, then you are just idiots generating random information. But what I'm requesting *is* your inherent connection to the material. If this stuff is important for physics, you need to explain why, and why it's better than what is presently in use. You need to *sell* it, but so far all I have seen is thousands of words of unnecessary detail that apparently say no more than look how smart I am Richard Herring === Subject: Re: the anticlassicalist }{ : ism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.com <6IvurhFpHeNAFwU6@baesystems.com> <103cn32aodr3m3a@corp.supernews.com In message <103cn32aodr3m3a@corp.supernews.com>, galathaea >: No, it means that at some point you have to hook into what physicists >: *observe*. Will it tell us the energy levels of a hydrogen atom and >: predict the Lamb shift? If you're interested in developing a better >: mathematical formalism, that's fine, go ahead. Just don't tell us it's >: physics. >Actually, all I did was point to the connection between the foundational >objects of the theory and the observational objects it predicts. I called >them the ontology and the epistemology of the model in a rigorous way to >accord somewhat with common usage, but any other names would work. But the >formalism is all about observational propositions (what is the likelihood >that A and B happen?, if C happens, what does that imply for D?, where >the letters stand for quantum events like spontaneous decay). Anywhere you >have time series of quantum events or concurrent systems, you implicitly or >explicitly use the logic I mention, often in an algebraic setting. OK. So if I'm already unknowingly using this logic, what is the point you're trying to make? Believe me, it hasn't yet emerged from all the verbiage. >It _is_ physics. I _am_ interested in developing a better mathematical >formalism, but alas cannot take credit for this one. This one has been >studied by physicists now for 3 quarters of a century. Richard Herring === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques >One of my prof's expressed the opinion that all undergraduate classes are applied in the above sense. I agree with that sentiment. Despite that, >there were no real-worls applications in any of them. That's too bad. >That is unfortunate but it's not nearly as unfortunate as a part-time >graduate program called Masters of Applied Statistics that's got lots of >theory without applications. Why? Because any real-life applications would almost certainly be too complex and time-consuming to be covered? Besides, you dropped the class, how do you know there weren't any practical applications introduced later on after the theory was covered? >Almost all undergrads have no >career-oriented work experience and wouldn't even know how/where to apply >what they learn. Grads in part-time graduate programs called Masters of >Applied Statistics have years and years of professional work experience and >can all conceptualize where and how they can use what they just learned. >It's a pity that those people go back to school and get essentially treated >programs do not fail their students. That's because in business and law, how you handle real-life situations is what it's all about. In math, while you can certainly cram yourself with formulas and examples for three years, what happens the first time you run into a problem which was not covered in the Applied Applied Statistics course? >Don't take this as an insult -- because it's not meant as one -- but I >suspect you have been in academia most/all of your life because people who >have been in industry would probably have a much easier job of relating what >they are teaching to the undergrads and how it might be used in the real >worldThat's a stretch. Most applied mathematicians will do their masters or doctorate work on problems directly relating to some real-life problem. I doubt a statistician working in the industry would be able to give any more practical applications for abstract probability theory than someone with years of experience in the academia. Odds are the industry statistician would just blather on about his own work, never mind if it has any relevance to the topic being covered. >Almost certainly. Business is focused on making money, period. Everything >else is a means to that end; It's the ultimate in applied discipline. All >the professors in the business school I went to still worked/consulted in >industry so they are in touch with the real world and still very much have >their applied caps on (and almost none had PhDs). Everyone is focused on >how to *solve problems* such that profits are maximized. If they were really interested in maximizing profits, they would have just given up with the math and outsourced it to some outfit in China. >department was just different. They liked theory and they liked formulas. >They liked elegant solutions and proofs, even if they were irrelevant to >application. That's why it's called the math department! Would you appreciate a business school that dwelled on non-business related topics like philosophy, physics and women studies? >I sensed a certain disdain for word problems and real world >analogies and explanations to help the students conceptualize the theory >because real math students don't need those crutches. Any math department that discourages against methods that help the students learn doesn't sound very promising, I agree, but then again I'm personally frustrated with the eternal what practical applications does theory X have? inquiries. Abstraction is an important method for reducing the solution of one problem to such a general form that it can be applied to a multitude of different problems. By teaching abstract theories instead of rigid one-problem-solving methods, you actually enable the students to tackle a larger problem set (provided they have the ability to make the leap from abstract definitions to real-life parameters). >They're very smart >people who would probably look contemptuously at the description I gave >above for business schools (No PhDs?!?! Only interested in profit?!?! >Theory only useful if taught in conjuction with application?!?! How >grotesque, how low-brow, how coarse!!!). This would be an interesting argument if it was an actual quote from someone instead of just a strawman erected to bolster your point of view. I'm not interested in mathematics that might have anything to do with reality. -- Easterly, in sci.math === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > Why is a degree called Masters in Applied Statistics so heavy in > theory? 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. and > not to learn esoteric statistical theory in the absence of > application that I will surely forget an hour after the final exam. I my understanding, Applied Math is solid, well-funded math which has interesting applications and whose development is mostly driven by real-world needs (as opposed to inner mathematical needs). Applied Math does not mean that there are less theorems, or that it's just know how to apply a formula. The problem with just teaching how to apply concept X is that in some situations X cannot be applied as usual. If you don't understand the exact limitations of your methods the bridge you build will collapse or the insurance company you advise will go bankrupt. One of the best ways to make sure you understand a theorem is to prove it. In my experience, proof by trying (doing a case study) is, unfortunately, not an option. If you are lucky, you might find a teacher who gives you both, a translucent introduction into the theory as well as fascinating case studies. But these people are rare. And you must be able to adopt to people will expect that she be able to do both, understand applications _and_ develop new theory where necessary. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques >> One of my prof's expressed the opinion that all undergraduate classes are >applied in the above sense. I agree with that sentiment. Despite that, >> there were no real-worls applications in any of them. That's too bad. >That is unfortunate but it's not nearly as unfortunate as a part-time >graduate program called Masters of Applied Statistics that's got lots of >theory without applications. Why? > Because any real-life applications would almost certainly be too > complex and time-consuming to be covered? How then are the students going to learn how to use the theory? Only a very few of them will be able to work that out by themselves. >Almost all undergrads have no >career-oriented work experience and wouldn't even know how/where to apply >what they learn. Grads in part-time graduate programs called Masters of >Applied Statistics have years and years of professional work experience and >can all conceptualize where and how they can use what they just learned. >It's a pity that those people go back to school and get essentially treated >programs do not fail their students. > That's because in business and law, how you handle real-life > situations is what it's all about. In math, while you can certainly > cram yourself with formulas and examples for three years, what happens > the first time you run into a problem which was not covered in the Applied Applied Statistics course? That's going to the other extreme, completely neglecting the theory and instead just memorizing a bunch of formulas without knowing how or why they work. I'm advocating for combining theory and practical applications, teaching the students why the tool theory works *and* how to use it. >Don't take this as an insult -- because it's not meant as one -- but I >suspect you have been in academia most/all of your life because people who >have been in industry would probably have a much easier job of relating what >they are teaching to the undergrads and how it might be used in the real >world> That's a stretch. Most applied mathematicians will do their masters or > doctorate work on problems directly relating to some real-life > problem. I doubt a statistician working in the industry would be > able to give any more practical applications for abstract probability > theory than someone with years of experience in the academia. I disagree, but I have no data to back me up. I'm quite curious and I would like it if someone could give a few data points. Perhaps reality is mixed: You could probably find both good and bad applicationers in both academia and the industry. > Odds are the industry statistician would just blather on about his > own work, never mind if it has any relevance to the topic being > covered. I would think that risc to be equally applicable to a math prof :-) >I sensed a certain disdain for word problems and real world >analogies and explanations to help the students conceptualize the theory >because real math students don't need those crutches. > Any math department that discourages against methods that help the > students learn doesn't sound very promising, I agree, but then again > I'm personally frustrated with the eternal what practical > applications does theory X have? inquiries. > Abstraction is an important method for reducing the solution of one > problem to such a general form that it can be applied to a multitude > of different problems. By teaching abstract theories instead of rigid > one-problem-solving methods, you actually enable the students to > tackle a larger problem set (provided they have the ability to make > the leap from abstract definitions to real-life parameters). Actually, the real difficulty lies in the reverse direction, going from real-life descriptions to an abstract model, at least when you trying to solve a concrete problem. This modelling step is something one really only learns through experience and lots of practice. Now, how do the students get that experience? This is also the reason behind the question what practical applications does theory X have? The question really boils down to hands-on training with modelling. i.e. teaching the students how to *use* the theory. Some people are able to work out the practical applications without needing any help. They would probably end up doing a PhD a few years down the road. For all the rest of the students, they need training and training and more training. -Michael. === Subject: Re: the anticlassicalist } <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.com > But what I'm requesting *is* your inherent connection to the material> If this stuff is important for physics, you need to explain why, and why > it's better than what is presently in use. You need to *sell* it, but > so far all I have seen is thousands of words of unnecessary detail that > apparently say no more than look how smart I amFair enough. For my part, what you have are academic disciplines making contradictory claims on the nature of truth. But, the experimental aspect of physics is directly outside of that debate--thankfully. Indirectly, there is some impact on how decisions are made. In contrast to experimental physics, the theoretical physics is pushing mathematical methods to the point of inconsistency. While I want to respect the physical intuition developed in the study of physics, intepreting mathematics without regard for its historical development is unsound. I do respect the fact that mathematical physics is not physics in the experimental sense. And, I have already removed sci.physics from parts of this thread. Also, I thank you for your responses which have not been entirely hostile and extend an apology for the parts of my posts that were. Now, you ask if anything I talked about could be better than anything in use. I would argue that my interpretation of the mathematics might give a correct explanation for how information-theoretic interpretations of quantum mechanics should be understood. Nevertheless, there are no numeric predictions. Unfortunately, I get a little pissed off when logicians don't think they have any obligation to justify their explanation for mathematics and when physicists don't perspective I have to read nonsense like http://plato.stanford.edu/entries/mathematics-inconsistent/ because the bureaucratic structure of universities permits people making contrary claims on mathematical thought to operate independently. corrective. :-) === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > A few weeks ago I posted a message asking about books on Bayesian > Unfortunately I have since dropped the class and am wondering about whether > I should continue the degree (Masters in Applied Statistics) and would like > some thoughts from the thoughtful people here. > I have a BS in Electrical Engineering and an MBA, both from the University > of Michigan. What I really liked about the MBA program is that it was almost > 100% applied. Probably 50-70%+ of the classes was case study classes, a > trend mostly propagated and refined by Harvard Business School where > supposedly 100% of the classes are case study classes. Apparently more and > more law programs have more and more case study classes as well. Case study > classes are really as applied as you get because concepts and theories are > learned in the context of real world situations and circumstances. I am also > the type of person who is a very intuitive learner and has a much easier > time learning when I see how what I am learning relates to challenges in > real life (eg: business, which is what I do). > So given that the degree I am pursuing is called Masters in *APPLIED* > Statistics, I thought the the courses would be heavily applied and taught in > the context of solving real world problems. No dice. Both courses I took in > the first semester (part-time evening program) had heavy theory. The > Bayesian class was not even as bad as the other one (Mathematical > Statistics). There was essentially no attempt on the part of the professor > to relate the theory to real world programs or to even give real world > examples to illustrate the concepts. It was formula, theory, formula, > theory, theory, formula, etc. I asked him about that and he said there's no > way around the theory. I'm not trying to get around the theory but > theories and formulas mean nothing to me without real world context. I'm not > stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE > and 98+ percentile overall. That makes your comments even less acceptable. > My thinking right now is that my expectations were just off and disciplines > like Math/Statistics are just not as, ummm, progressive as Business/Law when > it comes to teaching (please -- no hate mail). Those teaching Math/Stat may > also be too smart and are not interested in mune day-to-day > business/industry problems (hopefully that will stop the hate mail!). So > what's up with that? Why is a degree called Masters in Applied Statistics > so heavy in theory? 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. and not to learn > esoteric statistical theory in the absence of application that I will surely > forget an hour after the final exam. Can you think back to your EE course? Differentiation, integration, Coulomb's law, Newton's law, ... -- what would have happened had your high-school teachers decided to skip all that 'theory'? Have you ever attempted to apply theory without having covered the basis of that theory? Why are statistics books full of mathematics? So that professors can show off? Or because they like to keep sales down (Assuming that this is not a troll.) Could you elaborate on what you think you mean by: ... how to use statistical software to solve real world problems, etc...? Loading a set numbers into a calculating program and getting back some numbers? Doing experiments with dice and balls and urns? And maybe we could hear something of the law and business studies 'case studies', so that we could attempt to suggest why that approach is not applicable here? > I am not trying to slam the field. I am interested in some opinions from > those in the field, especially those teaching it, to help me determine if I If I was hiring an MSc in Applied Statistics (or doing such a degree myself), I'd be most disappointed if there wasn't a strong theoretical content. At least, I'd expect graduates to be prepared for writing (new) programs (or devising algorithms) to solve (unsolved) problems. What did the rest of the class think? Jon C. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques Just a comment on a detail here - [ delete, some other stuff] > In fact, I specifically remember a class in business school illustrating a > statistical problem and how/why the scientists at Morton Thiokol did not > catch the potential problem with the o-ring on the space shuttle Challenger > that caused the tragedy -- they didn't do tests nor have data points for the > o-ring at low temperatures, and the temperature at launch was very cold for > central Florida. [ ... ] As I recall reading about it, it eventually came out that there were scientists/ engineers around who had wanted to scrub the cold-weather launch. *Since* they did not have data, they were wise enough to have serious doubts -- but they were unable to convince the administrators, who were not 'technical people.' I wonder what the point was, in a business school class? - Keep channels open to your technical people? - or, S**t happens? Rich Ulrich, wpilib@pitt.edu http://www.pitt.edu/~wpilib/index.html Taxes are the price we pay for civilization. === Subject: Re: Minimally simple finite groups? >Which of the finite simple groups are minimally simple, i.e., >have all of their proper subgroups solvable? Obviously the only >alternating group that qualifies is A_5 =~ L(2,4) =~ L(2,5), and >I know the list also includes L(2,7) =~ L(3,2), L(2,8), L(2,13), >... On the other hand, I also know it *doesn't* include L(2,9) =~ >A_6 or L(2,11), both of which contain subgroups isomorphic to >A_5. >The classification of minimal simple groups was a consequence of >John Thomspon's classification of nonsolvable groups in which nontrivial >solvable subgroups have solvable normlaizers, which was one of the big >classification theorems that came before CFSG. >This is in Bull. Amer. Math. Soc. 74, 1968, 383-437. >The simple groups coming out of Thompson's Theorem are L_2(q), Sz(q), >L_3(3), M_11, A_7, U_3(3). Of course, these are not all minimal simple. >L_3(3) is, but M_11, A_7, U_3(3) are not. >Sz(2^e) is minimal simple whenever it does not contain a smaller >Sz(2^f), which I guess is equivalent to e being prime. >Is it easy to see why Sz(2^e) can't contain (as sections) any >non-abelian simple groups other than the obvious Sz(2^f) for all >f dividing e? Or do you need substantial machinery from >Thompson's Theorem? It is easy to see it, because Sz(2^e) are the only finite nonabelian simple groups with order not divisible by 3. That was actually proved before the classification, again by Thompson, but it is was a highly nontrivial result, probably harder than the odd order theorem. I am sure there are much easier ways to analyse the subgroups of Sz(2^e) though! >For L_2(p^e) to be minimal simple, its order must not be divisible by 60 >(otherwise it contains A_5). >Ah. Why must it contain A_5 if its order is divisible by 60? >Also it must not contain any smaller >simple L_2(p^f), so if p>=3, then we must have e=1. But L_2(2^e) and >L_2(3^e) will be minimal simple for e prime provided their order is >no divisible by 60. >You must mean so if p>3, of course. Again, is it easy to see >why other non-obvious simple groups are excluded from being >sections of L_2(p^e)? (Or are they? Can there be larger groups, >but containing A_5?) The subgroups of PSL(2,q) were all classified by L.E. Dickson in about 1900. I am not sure what the best reference for that is. It is in Huppert's book Endliche Gruppen but that is in German of course! Anyway, the subgroups are roughly cyclic groups, dihedral groups of order dividing q-1 or q+1 (q odd) or 2(q-1) or 2(q+1) (q even), semidirect products PD for a p-group P of order dividing q and cyclic group D of order dividing q-1, A_4, S_4 whenever 24 divides order, A_5 whenever 60 divides order, and PSL(2,r) and sometimes PGL(2,r) where q is a power of r. In other words, A_5 is the only `sporadic' simple subgroup of PSL(2,q). The fact that it occurs whenever 60 divides the order probably follows from the fact that SL(2,5) has a 2-dimensional complex representation, but don't push me for details there! Derek Holt. === Subject: Re: 'erf' function in C > And what about the inverse cumulative normal distribution? > I used to implement an algorithm you published, > but do not like it because it is machine-dependent. > What in your opinion is the best algorithm? With the improvement on my method for evaluating cPhi(x) as R(x)*phi(x), when the Taylor series for R(x) is about zero, (suggested by Daly), the simple C function double Phi(double x) { long double t=0,b=1,s=x,pwr=x; int i; for(i=2;s!=t;i+=2) { b/=(i+1); pwr=pwr*x*x; t=s; s+=pwr*b;} return .5+s*exp(-.5*x*x-.91893853320467274178L); } should serve very well for solving the equation Phi(X)=U for positive X, given U>1/2, by Newton's method: Start with an initial estimate x=sqrt(-1.6*ln(1-U^2)); then repeat x=x-(Phi(x)-U)/phi(x); until you get no further changes in x. The paper that you refer to must be where I want to generate a normal X by means of solving Phi(X)=U, given a random U, uniform in [0,1), by getting an integer j, formed from certain bits of the exponent part of the floating point representation of U. Then use j to access tabled values A[j] and B[j] to initalize the Taylor series for Phi inverse, using the fractional part of U as the argument in the series. It is designed for speed in generating a normal variate directly as Phi^(-1)(U), but only provides accuracy to 6/7 digits, (As I recall, the Taylor series was easy and fast only for the first few terms.) It is described in ``Normal (Gaussian) random variables in supercomputers'', (1991) Journal of Supercomputing}, v 5, 49--55, where the method is particularly suited for parallel computation. George Marsaglia === Subject: Trying to unify axioms. Hi Gregory L. Hansen, You said, I think a lot of people invoke Godel's theorem as a vague analogy for a philosophical principle that's been known for a long time and that is more easily understood without the analogy. Any logical system has a set of postulates that cannot be explained, or derived from more fundamental assumptions. If they could be derived, they would be conclusions, not postulates, and not the axiomatic basis of a theory. Lately, I've been failing to mention that Godel's incompleteness theorem is a merely strong argument against trying to unify axioms. Number one, it can't be done. Number two, it gets too convoluted. === Subject: Re: Trying to unify axioms. Nothing. You forever spewing ing imbecile, an axiom by definition is irreducible and unprovable. Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Trying to unify axioms. Hi Uncle Al, You mentioned, an axiom by definition is irreducible and unprovable. . I was talking about Godel's incompleteness theorem, which is about unifying axioms into a consistent set. Notice the word Set . === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques >department was just different. They liked theory and they liked formulas. >They liked elegant solutions and proofs, even if they were irrelevant to >application. I sensed a certain disdain for word problems and real world >analogies and explanations to help the students conceptualize the theory >because real math students don't need those crutches. My experience, and that of other math/stat instructors whom I've talked to, is quite the opposite. It's the STUDENTS who don't like word problems, and resist applications (eg, to physics), because to do them they have to actually understand the mathematical material (and even some physics!), rather than just applying formulas without really knowing what they're doing. This may not be true of real math students, however, who ought to be able to do the word problems (but who may find the standard ones to be too easy to be interesting). Radford Neal -------------------------------------------------------------- -------------- Radford M. Neal radford@cs.utoronto.ca Dept. of Statistics and Dept. of Computer Science radford@utstat.utoronto.ca University of Toronto http://www.cs.utoronto.ca/~radford -------------------------------------------------------------- -------------- === Subject: Re: Trying to unify axioms. > Hi Uncle Al, You mentioned, an axiom by definition is irreducible and unprovable. . > I was talking about Godel's incompleteness theorem, > which is about unifying axioms into a consistent set. You forever spewing ing imbecile, Godel's incompleteness theorem is the Liar's Paradox writ large. If you have nothing to say, don't. BTW, the barber shaves herself. Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques I once took a two semester class on Foundations of Applied MathematicsThe textbooks were: So given that the degree I am pursuing is called Masters in *APPLIED* > Statistics, I thought the the courses would be heavily applied and taught in > the context of solving real world problems. No dice. Both courses I took in > the first semester (part-time evening program) had heavy theory. The > Bayesian class was not even as bad as the other one (Mathematical > Statistics). There was essentially no attempt on the part of the professor > to relate the theory to real world programs or to even give real world > examples to illustrate the concepts. It was formula, theory, formula, > theory, theory, formula, etc. I asked him about that and he said there's no > way around the theory. I'm not trying to get around the theory but > theories and formulas mean nothing to me without real world context. I'm not > stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE > and 98+ percentile overall. There's just a certain amount of theory you'll have to get through. It is building the basis for the practical applications. And, unless you got into the measure theory foundations of probability, the theory they were teaching you was likely a lot of simple calculus applied to statistics. One small example of learning theory versus practical applications: There was an undergrad operations research course slanted towards engineering students. The weekly assignment had a problem that had to be solved with Newton's method -- fine, no problem, there's a formula. But this particular problem was mistated by the prof, and Newton's method basically hiked over an asymptote and headed off to infinity, for just about any guessable starting position. It suddenly became a more interesting problem. So, one solution was to slightly modify Newton's method, and basically give it a bit of drag. By adjusting the drag, and with a bit of luck, the asymptotes can be avoided, and a solution pops out. But most of the students didn't do that -- they didn't know enough about the basic workings of Newton's method to modify it. Theory is important. Real world problems often don't have standard solutions. === Subject: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1NNFGi03865 equation except factoring, quadratic formula, completing the square,graphing or the square root method. I went to http://www.geocities.com/dirkie6/page4.html but those methods are too complex. === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1NNk3i06416 X-Orig-Trace: 1077579954 4259 211.30.65.105 That page just provides three different proofs that the solutions are (-b +- sqrt(b2 - 4ac)) / (2a) That's the formula; it doesn't get any simpler than that. If the proofs themselves are too difficult, which line of which proof don't you understand? > equation except factoring, quadratic formula, completing the > square,graphing or the square root method. > I went to http://www.geocities.com/dirkie6/page4.html but those > methods are too complex. === Subject: Re: Trying to unify axioms. > Hi Uncle Al, You mentioned, an axiom by definition is irreducible and unprovable. . > I was talking about Godel's incompleteness theorem, > which is about unifying axioms into a consistent set. > You forever spewing ing imbecile, Godel's incompleteness theorem > is the Liar's Paradox writ large. If you have nothing to say, don't. > BTW, the barber shaves herself. Sounds as if someone has been applying Occam's razor a bit too liberally. An axiom is a self-evident truth. An example would be Where there is life, there is death. This example is irreducible and it is provable. Without the use of axioms, we would be unable to make decisions. Sine qua non. As applies to Godel, an axiom need not be transitive or commutative, only reflexive. === Subject: Re: Trying to unify axioms. > Nothing. > You forever spewing ing imbecile, an axiom by definition is > irreducible and unprovable. False. First off, an axiom isn't necessarily irreducable, unless you want to claim that reducing it makes the axiom not reducable and therefore not an axiom. Sometimes we have an axiom that we find out can be reduced into other axioms. Does that therefore disprove the axiom, or destroy the usefulness of the axiom, or of taking it as such? No it doesn't. Second, axioms are not unprovable. They CAN be proven. For an example of this, consider the law of identity. Can you prove it? If not, then how do we even know it's true? We do know it's true, and it IS an axiom, so that just proves that axioms are not unprovable. (...Starblade Riven Darksquall...) === Subject: Re: Trying to unify axioms. > Hi Uncle Al, You mentioned, an axiom by definition is irreducible and unprovable. . > I was talking about Godel's incompleteness theorem, > which is about unifying axioms into a consistent set. > You forever spewing ing imbecile, Godel's incompleteness theorem > is the Liar's Paradox writ large. If you have nothing to say, don't. > BTW, the barber shaves herself. > Sounds as if someone has been applying Occam's razor a bit too > liberally. An axiom is a self-evident truth. An example would be Where there is life, there is death. This example is irreducible > and it is provable. Without the use of axioms, we would be unable to > make decisions. Sine qua non. > As applies to Godel, an axiom need not be transitive or commutative, > only reflexive. Euclid's Fifth Postulate (i.e., Playfair's Axiom) is so obvious it is plain wrong right here on Earth. A gravitation theorem can be constructed with (metric, Einstein) or without (affine,Weitzenboek) the Equivalence Principle. All derived predictions are identical. Do a pair of local test masses in vacuum fall along parallel and non-parallel trajectories simultaneously? Philosophers don't need waste crocks; scientists do. Where there is life, there is death. Spermatogonia are immortal. Hitler's showers opened their doors to show death without life. Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Trying to unify axioms. > Nothing. > You forever spewing ing imbecile, an axiom by definition is > irreducible and unprovable. > False. > Second, axioms are not unprovable. They CAN be proven. For an example > of this, consider the law of identity. Can you prove it? If not, then > how do we even know it's true? We do know it's true, and it IS an > axiom, so that just proves that axioms are not unprovable. Idiot. > (...Starblade Riven Darksquall...) (...Bull Spewing Horse...) Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > A few weeks ago I posted a message asking about books on Bayesian > Unfortunately I have since dropped the class and am wondering about whether > I should continue the degree (Masters in Applied Statistics) and would like > some thoughts from the thoughtful people here. > I have a BS in Electrical Engineering and an MBA, both from the University > of Michigan. What I really liked about the MBA program is that it was almost > 100% applied. Probably 50-70%+ of the classes was case study classes, a > trend mostly propagated and refined by Harvard Business School where > supposedly 100% of the classes are case study classes. Apparently more and > more law programs have more and more case study classes as well. Case study > classes are really as applied as you get because concepts and theories are > learned in the context of real world situations and circumstances. I am also > the type of person who is a very intuitive learner and has a much easier > time learning when I see how what I am learning relates to challenges in > real life (eg: business, which is what I do). > So given that the degree I am pursuing is called Masters in *APPLIED* > Statistics, I thought the the courses would be heavily applied and taught in > the context of solving real world problems. No dice. Both courses I took in > the first semester (part-time evening program) had heavy theory. The > Bayesian class was not even as bad as the other one (Mathematical > Statistics). There was essentially no attempt on the part of the professor > to relate the theory to real world programs or to even give real world > examples to illustrate the concepts. It was formula, theory, formula, > theory, theory, formula, etc. I asked him about that and he said there's no > way around the theory. I'm not trying to get around the theory but > theories and formulas mean nothing to me without real world context. I'm not > stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE > and 98+ percentile overall. > My thinking right now is that my expectations were just off and disciplines > like Math/Statistics are just not as, ummm, progressive as Business/Law when > it comes to teaching (please -- no hate mail). Those teaching Math/Stat may > also be too smart and are not interested in mune day-to-day > business/industry problems (hopefully that will stop the hate mail!). So > what's up with that? Why is a degree called Masters in Applied Statistics > so heavy in theory? 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. and not to learn > esoteric statistical theory in the absence of application that I will surely > forget an hour after the final exam. > I am not trying to slam the field. I am interested in some opinions from > those in the field, especially those teaching it, to help me determine if I I can relate to some extent. While in theory I love theory, in practice I often wonder how what I'm studying relates to what I want The best math course I ever had was diff eq using _Differential Equations with Applications and Historical Notes_ by George Simmons as the text, in part because the applications gave me something concrete with which to relate. I'm not surprised that the Mathematical Statistics course was heavily theory, and in fact many of the stat books I've looked at appeared tilted that way. I've probably learned more statistics from books on signal processing than stat texts. Someone might argue that's why I know so little about statistics. :-) Perhaps it is just the particular school/program you're in, but I don't know. Maybe you should consider economics and econometrics, which can be heavy with stats and purport, at least, to deal with real world problems. Good luck. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques boundary=------------070201070602030005080405 -------------------------------------------------------------- ------- I am a geologist whose career has been based on statistics. I suffered through the theory classes and the theorems. It did not help that many of my classmates were physics majors who were actually fairly narrow probabalists. The background math / theory has turned out to be critical. Far too often the applied instructors (in the business college, in chemometrics, etc.) don't remember (or care) that all procedures have limited validity (robustness). A prime advantage is to understand where procedures make no sense because the system of interest does not cooperate. Hence the mean has limited practical usefulness, as does the standard deviation, regression statistics, factors and the like. Using the procedures without understanding ( and investigating) the underlying assumptions can be professionally suicidal for an applied statistician. One of my avocations is evaluating the work of bozos like you in lawsuits. It's like hitting herring in a barrel. A few weeks ago I posted a message asking about books on Bayesian >Unfortunately I have since dropped the class and am wondering about whether >I should continue the degree (Masters in Applied Statistics) and would like >some thoughts from the thoughtful people here. >I have a BS in Electrical Engineering and an MBA, both from the University >of Michigan. What I really liked about the MBA program is that it was almost >100% applied. Probably 50-70%+ of the classes was case study classes, a >trend mostly propagated and refined by Harvard Business School where >supposedly 100% of the classes are case study classes. Apparently more and >more law programs have more and more case study classes as well. Case study >classes are really as applied as you get because concepts and theories are >learned in the context of real world situations and circumstances. I am also >the type of person who is a very intuitive learner and has a much easier >time learning when I see how what I am learning relates to challenges in >real life (eg: business, which is what I do). >So given that the degree I am pursuing is called Masters in *APPLIED* >Statistics, I thought the the courses would be heavily applied and taught in >the context of solving real world problems. No dice. Both courses I took in >the first semester (part-time evening program) had heavy theory. The >Bayesian class was not even as bad as the other one (Mathematical >Statistics). There was essentially no attempt on the part of the professor >to relate the theory to real world programs or to even give real world >examples to illustrate the concepts. It was formula, theory, formula, >theory, theory, formula, etc. I asked him about that and he said there's no >way around the theory. I'm not trying to get around the theory but >theories and formulas mean nothing to me without real world context. I'm not >stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE >and 98+ percentile overall. >My thinking right now is that my expectations were just off and disciplines >like Math/Statistics are just not as, ummm, progressive as Business/Law when >it comes to teaching (please -- no hate mail). Those teaching Math/Stat may >also be too smart and are not interested in mune day-to-day >business/industry problems (hopefully that will stop the hate mail!). So >what's up with that? Why is a degree called Masters in Applied Statistics >so heavy in theory? 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. and not to learn >esoteric statistical theory in the absence of application that I will surely >forget an hour after the final exam. >I am not trying to slam the field. I am interested in some opinions from >those in the field, especially those teaching it, to help me determine if I I can relate to some extent. While in theory I love theory, in >practice I often wonder how what I'm studying relates to what I want >The best math course I ever had was diff eq using _Differential >Equations with Applications and Historical Notes_ by George Simmons >as the text, in part because the applications gave me something >concrete with which to relate. I'm not surprised that the Mathematical >Statistics course was heavily theory, and in fact many of the stat >books I've looked at appeared tilted that way. I've probably learned >more statistics from books on signal processing than stat texts. >Someone might argue that's why I know so little about statistics. :-) >Perhaps it is just the particular school/program you're in, but I don't >know. Maybe you should consider economics and econometrics, which >can be heavy with stats and purport, at least, to deal with real world >problems. Good luck. === Subject: Re: the anticlassicalist }{ iv: from maps to logic : You mean it provides a natural framework for conjectures ? Model theory as a whole does provide a natural framework for theories if you distinguish or attach an interpretive layer for observation (which in and of itself is not distinguished in classical model theory expositions). The maps themselves and the notion of distinguishing derived and defined existence is somewhat commonly developed in the more constructive schools and is also well established in the foundational analyses, where such a distinction is important for forcing theory and the general study of axiomatic consequences. So this distinction is important to axiomatisation of theories (and I don't make much a distinction between theories and conjectures excepth that conjectures often refer to added axioms inside a preexisting theoretical context), but I would call the entire model theoretic approach with epistemology added as : > duality : : Doesn't tell much about that. Duality is negation in boolean algebras, isn't it ? The definitions of T and _|_ are dual in this regard, so there is some sense where boolean negation is represented in dualities. Paticularly with / and / sharing a dual nature, in a boolean framework with excluded middle much of the structural theory of boolean algebras can be understood through the study of dualities. However in the general case, duality can be used in many circumstances to extend theorems on semilattices up to a complete lattice, which saves some work and repetition, but does not confine us yet to any boolean model. ===-=-=-=-=- === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > Just a comment on a detail here - > [ delete, some other stuff] > In fact, I specifically remember a class in business school illustrating a > statistical problem and how/why the scientists at Morton Thiokol did not > catch the potential problem with the o-ring on the space shuttle Challenger > that caused the tragedy -- they didn't do tests nor have data points for the > o-ring at low temperatures, and the temperature at launch was very cold for > central Florida. > [ ... ] > As I recall reading about it, it eventually came out that there > were scientists/ engineers around who had wanted to scrub > the cold-weather launch. *Since* they did not have data, > they were wise enough to have serious doubts -- but they > were unable to convince the administrators, who were not > 'technical people.' > I wonder what the point was, in a business school class? > - Keep channels open to your technical people? > - or, S**t happens? > Rich Ulrich, wpilib@pitt.edu > http://www.pitt.edu/~wpilib/index.html Taxes are the price we pay for civilization. just because there are many data points, one may still not have enough information with which to proceed. All the data points they had indicated that the o-ring was fine but the had no data points for how the o-ring might behave in cold weather. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques I wonder if the course becomes more applied in future semesters? The two subjects you've mentioned I don't think anyone would expect to be light on the theory - especially anything called 'Mathematical Statistics'. Based on my experience (which is based on the exposure to statistics teaching of only one university mind you) I would expect a Masters of Applied Statistics to perhaps start off with some statistical theory courses and progress into what I would call 'applied statistics' topics like generalised linear models, survival analysis techniques, time series techniques etc. Now whether these would be what /you/ would call applied (case-study driven) I don't know - probably depends on the faculty culture, the teaching staff preferences etc. But a course that covers theory of GLMs, theory of survival analysis, theory of time series analysis I would still look at as a valid Applied Statistics course. Of course it would be /nice/ if each of those topics included real-life case-studies as well, but I don't think it would be false advertising if it didn't. Basically, I think the applied in the degree title probably refers to the topics covered - not the teaching style. Getting back to the Swiss Army Knife analogy, I think it would be valid to have the theory of Swiss Army Knives topic in an Applied Knifework course, along with theory of cleavers, theory of bayonets etc. In my experience it is these 'applied' topics where the examples and case-studies come into it more. There isn't an awful lot of 'real-world' examples that would be applicable to a Mathematical Statistics topic, but when dealing with survival analysis (as an example) there are lots of data sets, the theory and application. It might be worth checking out future topics and their syllabus before deciding whether to withdraw. Kylie. > A few weeks ago I posted a message asking about books on Bayesian > Unfortunately I have since dropped the class and am wondering about whether > I should continue the degree (Masters in Applied Statistics) and would like > some thoughts from the thoughtful people here. > I have a BS in Electrical Engineering and an MBA, both from the University > of Michigan. What I really liked about the MBA program is that it was almost > 100% applied. Probably 50-70%+ of the classes was case study classes, a > trend mostly propagated and refined by Harvard Business School where > supposedly 100% of the classes are case study classes. Apparently more and > more law programs have more and more case study classes as well. Case study > classes are really as applied as you get because concepts and theories are > learned in the context of real world situations and circumstances. I am also > the type of person who is a very intuitive learner and has a much easier > time learning when I see how what I am learning relates to challenges in > real life (eg: business, which is what I do). > So given that the degree I am pursuing is called Masters in *APPLIED* > Statistics, I thought the the courses would be heavily applied and taught in > the context of solving real world problems. No dice. Both courses I took in > the first semester (part-time evening program) had heavy theory. The > Bayesian class was not even as bad as the other one (Mathematical > Statistics). There was essentially no attempt on the part of the professor > to relate the theory to real world programs or to even give real world > examples to illustrate the concepts. It was formula, theory, formula, > theory, theory, formula, etc. I asked him about that and he said there's no > way around the theory. I'm not trying to get around the theory but > theories and formulas mean nothing to me without real world context. I'm not > stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE > and 98+ percentile overall. > My thinking right now is that my expectations were just off and disciplines > like Math/Statistics are just not as, ummm, progressive as Business/Law when > it comes to teaching (please -- no hate mail). Those teaching Math/Stat may > also be too smart and are not interested in mune day-to-day > business/industry problems (hopefully that will stop the hate mail!). So > what's up with that? Why is a degree called Masters in Applied Statistics > so heavy in theory? 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. and not to learn > esoteric statistical theory in the absence of application that I will surely > forget an hour after the final exam. > I am not trying to slam the field. I am interested in some opinions from > those in the field, especially those teaching it, to help me determine if I === Subject: Re: the anticlassicalist }{ v: universal truths : : > And on these lattices we can make more explicit our theories of negation. : : > On sci.logic, recently gave a very interesting post on Stone algebras. : > When we want to model uses of various definitions, we need to show them as : > an axiom set. Stone algebras study the axiom set: : : > ~(a) / ~(~(a)) <--> T : : : Btw, can complex numbers model it ? Let's see : : ~ : r exp(i s) --> r/2 exp(i s/2) : : a = 1 : : ~(a) = -1/2 : : ~(~(a)) = i/4 : : let's say T is 0... : : Well, then I am sure a Lorentz-Moebius transform can save the situation, I : mean, provide a natural interpretation to the same formula graphics. The : Moebius group is strictly 3-transitive, and : : z / w <---> u : : can pass as denoting -the- moebius transform that swaps z and w while keeping : u invariant. Now idea if the thing can mimick a lattice any further, though. I think I would need a little more exposition. In particular, would such a lattice define T and _|_. Also, there would be two fixed points in any moebius transformation, so I'm wondering if an identification is assumed to be made. Or maybe my confusion lies at what exactly the objects of the lattice are. In other words, are they transforms (the lattice being all Moebius transforms) or points under a given transform. Although the negation definition does seem to be on points inside the transformation given, I have difficulty reading the / on points unambiguously. I can see how the symbols z / w <-> u can be used to single out a transformation, but wouldn't z / w <-> v (where v was the other fixed point of the transformation) denote the same transformation, implying that either / was not unique or that some kind of collapse was in order? There is, however, a natural way to identify certain lattice substructures to operators obeying Hecke algebras, if that is what you are looking at. Certainly, the study of modular forms and lattice structures is intricately connected. : Isn't the symmetric group attached to the boolean algebra in such a manner : that extremely transitive (but non-symmetric) groups shine as interesting : analogues ? I think these kinds of questions are best analysed in the context of topoi, where we can generalise the relationships. At least, when you mention symmetric group attached to the boolean algebra, I think of the natural power objects in a topos defined in terms of a subobject classifier. Symmetric groups can be found in extensions of power objects in the category of sets, and we can lift the natural boolean structure of the subobject classifier in a standard way. In a general topos, however, all we have is a Heyting algebra defined on the subobject classifier, and the lifting implies a certain, more general structure. Now I am aware of relationships that identify the permutation groups to certain commutative dynamical flows (not quite in relation to group commutativity and k-commutativity), with other groups denoting flows of deeper structure (braidings and knots embedded, for example, with no crossing type theorems), and I am also aware of the deep relationship between such groups and Hopf algebras. But I have not gone so far as to make the full connection except in the study of quantum groups and braiding categories. So, although I cannot answer your questions, maybe what I have provided is a start for discussion? ===-=-=-=-=- === Subject: Re: Trying to unify axioms. > Hi Gregory L. Hansen, You said, I think a lot of people invoke Godel's theorem > as a vague analogy for a philosophical principle > that's been known for a long time > and that is more easily understood without the analogy. > Any logical system has a set of postulates > that cannot be explained, or derived > from more fundamental assumptions. > If they could be derived, > they would be conclusions, not postulates, > and not the axiomatic basis of a theory. > Lately, I've been failing to mention that > Godel's incompleteness theorem > is a merely strong argument against > trying to unify axioms. > Number one, it can't be done. > Number two, it gets too convoluted. You're right, but Goedel's Theorem has nothing to do with it. Since Goedel's Theorem concerns set theory and mathematical axioms, not postulates. Postulates are like propositions, which are known to fit tidly into first order logic. Even more tidly than Goedel's Completeness Theorem concern Logic. Since even Goedel claimed they didn't. === Subject: Re: the 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. I wanted to stress that both algebras are finitely generated, however, and it is only through the potential application of axioms over an infinity that any infinities arise. This, 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 this realm, the finite sharing most properties with the boolean. : | > And since Heyting algebras have a potential universality hinted by the : | > Curry-Howard isomorphism, why is it so necessary to fall back on the : | > classical approach. I still do not see from where this desire arises at : | > forcing the ontology of a model... : | : | Perhaps the following may help (and perhaps not :-) ). : | : | A Heyting algebra is a mathematical structure of some kind. : | It's defined as an infinite set, together with some operators and : | relations, that satisfies certain (first-order logic) conditions. : | So even to understand the notion of a Heyting algebra, most : | people require a good intuitive picture of the Tarski semantics : | of first order (classical) predicate logic. : | : | Constructivism has so many variants that, in order to study : | them well, most systems are defined and studied using classical : | means and classical thinking habits. In more vague terms: : | 'reasoning on the meta level is still classical'. : : I don't think 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 which 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. Heyting algebras are much more general than merely models for constructive mathematics. They also underly a huge number of models in the sciences, which 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. : 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 : things as Martin-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 this 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. : If you were to count as schools of classical mathematics all the : different nonconstructive formal systems in which 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 which _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 this narrows the applicability of the logical structure far too much. 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 this 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 highly influenced by its Heyting structure, and I am looking for why this is not so in more fields that implicitly have the structure hiding away in their analyses. : The kicker here is that my statement about Bishop constructivism applies : to classical mathematics as well. Most classical mathematics also starts : from Bishop's constructivism plus a few extra axioms. I would argue that : this is evidence that it would be a good choice of metatheory. There's a : basic asymmetry between a theory having extra axioms and one which simply : leaves them out. Having a metatheory which makes more assumptions than : the theory being considered sometimes leads to confusion. There's an odd : little theorem in topos theory whose proof starts something like this: : either every null object is an initial object, or there exists a null : object that isn't an initial object.... This is an application of the : law of excluded middle. It's sort of like allowing the classicality of : the metatheory to bleed into the object theory. And then this is overcome : by considering topos objects in toposes! There is a lot of research in topoi that uses the metatheory of classical first-order predication, just as in many areas of math. But again, that is a choice for metatheory which does not in any way determine the model logic determined by the language inside the topoi. For example, numerous theories work inside the topos of directed graphs (automata, phylogenetics, etc.), and although a particular analyses may use classical logic to analyses the model, propositions inside the model still must be evaluated in the logic of the topos (which is Heyting). I would like to hear more about your mentioned theorem, though, as it certainly does sound like an odd beginning. : Many of the problems people believe they have with constructivism also : are less liable to apply while doing metamathematics. The theorems people : try to prove tend to be of low logical complexity, things like If X is : a theorem of S, then Y is a theorem of T. Theorems of that form are : guaranteed to be constructively valid if they are classically valid. : There's a tradition in logic of trying to pare down the principles needed : to prove metamathematical results to the most elementary kind. Making them : also constructive is just a natural next step in this direction. Yes, and of course you have Kleene truth and Goedel decidability and the structure of metamathematics requiring evaluations of certain kinds of propositions in alternate logics for the sake of consistency... : | You may find that awful, but that's the way it is. One reason : | for this is that there is no reason to prefer one form of : | constructivism over another. : : But there is. All else being equal, one should be sparing with one's : assumptions. If I'm a Bishop constructivist, and you are a fan of the : Markov school's work, I can accomodate your results in my scheme of : things by considering them proofs of consequences of Markov's rule and : the recursiveness of all functions from N to N. One should only regard : assumptions as permanent axioms once it's clear that they've become : thoroughly intertwined in your work, and that your work really doesn't : go through without them. This kind of minimality is also why I believe that mathematicians often add value to constructive proofs, even when they may be less committal to the entire constructivist ansatz. : | You like Heyting algebras, : | someone else likes some other system. And unlike : | the several formalizations of classical logic, these variants : | are far from being mutually equivalent or translatable. : : Heyting algebras are quite standard. Heyting algebras bear the same : relationship to constructive reasoning as boolean algebras bear to : classical reasoning. They reflect the propositional calculus. : : One *could* consider (as logicians have, being very thorough) logics : intermediate between the usual propositional calculus of constructive : and classical mathematics, but I would be very surprised if anybody : were actually doing mathematics that way. Even on the level of predicate : calculus, is anybody seriously using something in between intuitionist : logic and classical logic? I want to point out that many regularly used models have a model logic with more structure than just the definition of a Heyting algebra. For example, a topologist will work in a given topology, and this defines a logic with Heyting structure plus the additional structure defining the particular topology. : | Using classical definitions of the lot allows us at least : | to understand all variants at the same time, and to make : | mutual comparisons. : : Having to use a classical metatheory for constructive reasoning creates : certain confusions. People wind up using techniques for circumventing : the assumption of excluded middle, such as topos theory, Kripke or Beth : models. Crudely speaking, the problem is that one has to keep going around : thinking, The law of excluded middle is actually true, but we're going to : pretend like it isn't. I don't know what problems the reverse is : supposed to create; one is just considering some extra assumption I think this highlights the differnce between thinking _inside_ the model's logic, and thinking _about_ the model's logic. : | There is plenty of room for more : | research and other points of view, but as something that : | is supposed to give students a good basis for further : | research, classical logic is still essential, constructivism : | an extra. : : Classical logic is overwhelmingly the popular choice, whether it deserves : it or not. Sort of like Microsoft Windows. It may make pragmatic sense to : defer to it, given the place it has in the world, but it's not a really : persuasive argument for its being inherently better. As a metatheory, yes. But models abound that do not have classical logic inherent to them. Often, like quantum mechanics, this is avoided by not reasoning inside the models logic directly but instead working with the algebraic translation of the logic. But I believe this is merely because people are not used to thinking of their model's structure in a logical way and feel much more comfortable in algebraic or other formulations. The Heyting algebraic structure, though, is itself quite common (though often unacknowledged). : | That may serve as an explanation why not many : | have responded very enthousiastic to your pamflet. :-) : : I'm sorry to say that there are other reasons why some of us haven't : responded enthusiastically to Galathaea's pamphlet: : It's probably rather rare for someone to be more sympathetic to the : constructivists than I am. I still feel like I don't know enough about it : to make a really fair evaluation of it, but as you can see I'm more : optimistic about it than people usually are. It seems to me that we need : both to know more constructive mathematics, and to have better reasons for : what we think it's pros and cons are. The lack of experience leaves our : general idea of the relationship between constructive and classical : mathematics resting on too little. On the other hand, weaknesses in : people's general ideas and justifications leads them to chalk certain : things up as advantages of classical reasoning, that certain specific : theorems can be proven, for instance, not taking other subtleties properly : into account, like the fact that they are taken to mean something : different constructively, and that alternative paths to the same practical : end result exist. What I have tried to show is that this is only one of many, many reasons for studying the underlying logic. The bibliography I gave was only a small fraction of the work I have seen, but it touches on foundational models in numerous fields. : 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? What I have tried to find is people in all of these groups from several directions (which 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 which I felt might share a common goal. 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. 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 this absolutely clear. : I would generally advise against being a self-proclaimed liar, even if : this is meant in a humorous way (which 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. 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, which never knows itself to be true but seeks justification. 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. : 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 things in less depth; a diver deals in fewer at greater depth. : There's nothing 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 something in your hands! Be careful here. This 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. A lot of my research has focused on structural analyses of topics and foundationalist approaches, 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 isomorphism 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 Hilbert 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. : It seemed to me that a lot of the examples, and maybe all of them, of : things whose logic is constructive (whatever that is supposed to mean, : specifically) that we've seen here, are just special cases of the : topological interpretation. This 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 things a bit, one could say that the complementary : perception, that something 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 that's a poor argument for dropping the law of excluded middle. It's : more realistic to say that the concept of being a W has a grey area on : its borders. So really one has an argument in favor of fuzzy logic. As far : as I can tell, the jury is still out on fuzzy logic. Such an argument in : favor of fuzzy logic is surely only a motivational argument. We can't say : so easily whether your logic is the right place to introduce awareness of : fuzziness. It is funny then that I've also pointed to the natural Heyting structure of fuzzy logic in the literature! =) : 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 the lack of content has been from those spamming their own newsgroups, not asking for intelligent discussion, just spamming with insults and the like. 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 chip in on the mathematical side of constructivism, : try helping me satisfy some of my curiosity. I've had the question of the : degree to which the Jor-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 isomorphisms 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 isomorphism theorem of groups as not being constructive? Although I haven't explored this before, I never noticed anything hiding 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 this? : I'm interested in a strong version of the concept of simple group (with : apartness). I'm willing to assume that simplicity holds for the quotient : groups in the following form. If x and y are elements of the group, and : x<>1, then y can be expressed as a product of conjugates of x and x^{-1}. : I think 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 this in the opposite direction when looking for constructive deductions. 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. This class of groups is quite large. Then look at the structure required to constructively prove elements apart from x that have this conjugation construction equal to the entire group as well. For finite groups, of course, this can be carried out to completion constructively. For infinite groups, of course, its much more difficult, although the relationship being implied is finite between all elements. Constructing the elements to conjugate through would get you there, though. I think this approach is very basic to the logic that I would desire being taught more. This 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, which does oppose the infinite axiomatics underlying certain classical constructions. It is very much the difference between bottom-up and top-down approaches. I know this 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 thinking of Jor-Holder and similar theorems in a constructive light yet, so I will need to revisit some of my materials. Thank you for your interest, by the way! ===-=-=-=-=- === Subject: Re: Minimally simple finite groups? [...] > The subgroups of PSL(2,q) were all classified by L.E. Dickson in about > 1900. I am not sure what the best reference for that is. It is in > Huppert's book Endliche Gruppen but that is in German of course! Is it covered in Carter's Simple Groups of Lie Type? How about the Atlas? (Not that a copy of that can be had for love nor money, as far as I've been able to discover. :-( > Anyway, the subgroups are roughly cyclic groups, dihedral groups > of order dividing q-1 or q+1 (q odd) or 2(q-1) or 2(q+1) (q even), > semidirect products PD for a p-group P of order dividing q and cyclic > group D of order dividing q-1, Isn't P always a Sylow p-subgroup, and elementary abelian? And |D| = q-1 for q even, |D| = (q-1)/2 for q odd, D a Cartan subgroup and so PD a Borel subgroup? Are the Borel groups the maximal parabolics? Is the normalizer N(D) of D always dihedral? > A_4, S_4 whenever 24 divides order, ? PSL(2,8) has order 504 = 2^3 * 3^2 * 7, but no subgroup isomorphic to S_4 or A_4. > A_5 whenever > 60 divides order, and PSL(2,r) and sometimes PGL(2,r) where q is a power > of r. Under what circumstances PGL(2,r)? Guessing from PSL(2,9) =~ A_6, maybe when the order-r subgroups split into 2 conjugacy classes? (Which seems like it should be easy to see by looking at the Borel groups?) > In other words, A_5 is the only `sporadic' simple subgroup of > PSL(2,q). The fact that it occurs whenever 60 divides the order probably > follows from the fact that SL(2,5) has a 2-dimensional complex > representation, but don't push me for details there! === Subject: Re: Minimally simple finite groups? >[...] >The subgroups of PSL(2,q) were all classified by L.E. Dickson in about >1900. I am not sure what the best reference for that is. It is in >Huppert's book Endliche Gruppen but that is in German of course! >Is it covered in Carter's Simple Groups of Lie Type? How about >the Atlas? (Not that a copy of that can be had for love nor >money, as far as I've been able to discover. :-( I don't believe that the full description of the subgroups is in either of those. You will find maximal subgroups of individual groups in the Atlas. >Anyway, the subgroups are roughly cyclic groups, dihedral groups >of order dividing q-1 or q+1 (q odd) or 2(q-1) or 2(q+1) (q even), >semidirect products PD for a p-group P of order dividing q and cyclic >group D of order dividing q-1, >Isn't P always a Sylow p-subgroup, and elementary abelian? And >|D| = q-1 for q even, |D| = (q-1)/2 for q odd, D a Cartan >subgroup and so PD a Borel subgroup? Are the Borel groups the >maximal parabolics? Is the normalizer N(D) of D always dihedral? The largest such P is a Sylow p-subgroup and is elementary abelian. But I was describing an arbitrary subgroup of PSL(2,q). Again, the maximal D has the order you say, and the maximal PD is a Borel subgroup. Yes, the normalizer of D is dihedral provided D is nontrivial. >A_4, S_4 whenever 24 divides order, >? PSL(2,8) has order 504 = 2^3 * 3^2 * 7, but no subgroup >isomorphic to S_4 or A_4. Yes you are right. The S_4 occur only for odd q. >A_5 whenever >60 divides order, and PSL(2,r) and sometimes PGL(2,r) where q is a power >of r. >Under what circumstances PGL(2,r)? Guessing from PSL(2,9) =~ >A_6, maybe when the order-r subgroups split into 2 conjugacy >classes? (Which seems like it should be easy to see by looking >at the Borel groups?) I think, for odd q, PSL(2,q) contains PGL(2,r) iff q is an even power of r. For example PSL(2,p^2) contains PGL(2,p), but PSL(2,p^3) does not. Of course, for even q, PSL(2,q) = PGL(2,q). Derek Holt. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > I am a geologist whose career has been based on statistics. I've worked in lots of fields myself. Right now my work is in meteorology and climatatology. Lots of applied statistics in all the things I've done. I enjoy theory, but it is difficult to find teachers or books that relate it well to practice, but they are out there and can be found with some effort. > I suffered through the theory classes and the theorems. It did not help > that many of my classmates were physics majors who were actually fairly > narrow probabalists. The background math / theory has turned out to be > critical. Far too often the applied instructors (in the business > college, in chemometrics, etc.) don't remember (or care) that all > procedures have limited validity (robustness). A prime advantage is to > understand where procedures make no sense because the system of interest > does not cooperate. Hence the mean has limited practical usefulness, as > does the standard deviation, regression statistics, factors and the > like. Using the procedures without understanding ( and investigating) > the underlying assumptions can be professionally suicidal for an applied statistician. One of my avocations is evaluating the work of > bozos like you in lawsuits. Why do think I'm a bozo? > It's like hitting herring in a barrel. First, you have to get a barrel of herring. ;-) >A few weeks ago I posted a message asking about books on Bayesian >Unfortunately I have since dropped the class and am wondering about whether >>I should continue the degree (Masters in Applied Statistics) and would like >>some thoughts from the thoughtful people here. >I have a BS in Electrical Engineering and an MBA, both from the University >>of Michigan. What I really liked about the MBA program is that it was almost >>100% applied. Probably 50-70%+ of the classes was case study classes, a >>trend mostly propagated and refined by Harvard Business School where >>supposedly 100% of the classes are case study classes. Apparently more and >>more law programs have more and more case study classes as well. Case study >>classes are really as applied as you get because concepts and theories are >>learned in the context of real world situations and circumstances. I am also >>the type of person who is a very intuitive learner and has a much easier >>time learning when I see how what I am learning relates to challenges in >>real life (eg: business, which is what I do). >So given that the degree I am pursuing is called Masters in *APPLIED* >>Statistics, I thought the the courses would be heavily applied and taught in >>the context of solving real world problems. No dice. Both courses I took in >>the first semester (part-time evening program) had heavy theory. The >>Bayesian class was not even as bad as the other one (Mathematical >>Statistics). There was essentially no attempt on the part of the professor >>to relate the theory to real world programs or to even give real world >>examples to illustrate the concepts. It was formula, theory, formula, >>theory, theory, formula, etc. I asked him about that and he said there's no >>way around the theory. I'm not trying to get around the theory but >>theories and formulas mean nothing to me without real world context. I'm not >>stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE >>and 98+ percentile overall. >My thinking right now is that my expectations were just off and disciplines >>like Math/Statistics are just not as, ummm, progressive as Business/Law when >>it comes to teaching (please -- no hate mail). Those teaching Math/Stat may >>also be too smart and are not interested in mune day-to-day >>business/industry problems (hopefully that will stop the hate mail!). So >>what's up with that? Why is a degree called Masters in Applied Statistics >>so heavy in theory? 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. and not to learn >>esoteric statistical theory in the absence of application that I will surely >>forget an hour after the final exam. >I am not trying to slam the field. I am interested in some opinions from >>those in the field, especially those teaching it, to help me determine if I > >I can relate to some extent. While in theory I love theory, in >practice I often wonder how what I'm studying relates to what I want >The best math course I ever had was diff eq using _Differential >Equations with Applications and Historical Notes_ by George Simmons >as the text, in part because the applications gave me something >concrete with which to relate. I'm not surprised that the Mathematical >Statistics course was heavily theory, and in fact many of the stat >books I've looked at appeared tilted that way. I've probably learned >more statistics from books on signal processing than stat texts. >Someone might argue that's why I know so little about statistics. :-) >Perhaps it is just the particular school/program you're in, but I don't >know. Maybe you should consider economics and econometrics, which >can be heavy with stats and purport, at least, to deal with real world >problems. Good luck. === Subject: Re: Minimally simple finite groups? >[...] >The subgroups of PSL(2,q) were all classified by L.E. Dickson in about >1900. I am not sure what the best reference for that is. It is in >Huppert's book Endliche Gruppen but that is in German of course! >Is it covered in Carter's Simple Groups of Lie Type? How about >the Atlas? (Not that a copy of that can be had for love nor >money, as far as I've been able to discover. :-( John Roberts-Jones === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1OCNHi01492 > equation except factoring, quadratic formula, completing the > square,graphing or the square root method. > I went to http://www.geocities.com/dirkie6/page4.html but those > methods are too complex. You can do the following: ax^2 + bx + c = 0 is your equation. suppose m and n are roots of this equation. a(x-m)(x-n) = 0 ax^2 - (m + n)x + mn = 0 then you have the system : (A) m + n = -b (B) mn = c (A)^2-4(B) gives m-n, and you can solve the system. Actually if you go through symbolically with this, you'll end up with the quadratic formula. === Subject: Re: Trying to unify axioms. >Nothing. >You forever spewing ing imbecile, an axiom by definition is >irreducible and unprovable. >False. >First off, an axiom isn't necessarily irreducable, unless you want to >claim that reducing it makes the axiom not reducable and therefore not >an axiom. Sometimes we have an axiom that we find out can be reduced >into other axioms. Does that therefore disprove the axiom, or destroy >the usefulness of the axiom, or of taking it as such? No it doesn't. >Second, axioms are not unprovable. They CAN be proven. For an example >of this, consider the law of identity. Can you prove it? If not, then >how do we even know it's true? We do know it's true, and it IS an >axiom, so that just proves that axioms are not unprovable. >(...Starblade Riven Darksquall...) If an axiom were reducible or proveable, it would be a conclusion, not an axiom. The axioms would become the axioms used to reach that conclusion. Usenet is like a herd of performing elephants with diarrhea -- massive, difficult to redirect, awe-inspiring, entertaining, and a source of mind-boggling amounts of excrement when you least expect it. -- Gene Spafford, 1992 === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques >department was just different. They liked theory and they liked formulas. >They liked elegant solutions and proofs, even if they were irrelevant to >application. I sensed a certain disdain for word problems and real world >analogies and explanations to help the students conceptualize the theory >because real math students don't need those crutches. >My experience, and that of other math/stat instructors whom I've >talked to, is quite the opposite. It's the STUDENTS who don't like >word problems, and resist applications (eg, to physics), because to do >them they have to actually understand the mathematical material (and >even some physics!), rather than just applying formulas without really >knowing what they're doing. This may not be true of real math >students, however, who ought to be able to do the word problems (but >who may find the standard ones to be too easy to be interesting). > Radford Neal The students want to know how to plug things into formulas and get answers. This is the LAST step in applying statistics, mathematics, or whatever. One can study proofs from axiomatic approaches by themselves. But when one has a real-world problem, the most important, and often hardest, part is to translate that problem into a pure mathematics or statistics problem, so what is known from those fields can be applied. This requires knowing the CONCEPTS, and being able to formulate the word problems, with the solution often having to be done by computers or often by those who can use the power of the subject, and possibly even extend it. The physicist applying mathematics, or the economist or biologist applying statistics, have to state their formal assumptions, after which the full power can be used, sometimes showing that the assumptions are not what the user thought they were. One needs a little more care with word problems than is often the case. The economist may not be able to formulate physics word problems, and vice versa. But when a word problem is properly formulated, solving it does not require knowing physics or economics. If this is not the case, at least the formulation is incomplete. Teaching statistical methods without concepts only gets them used as religion. In engineering, errors usually show themselves quickly, but in statistics, this is not the case, and I know much harm which has been done. This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: the anticlassicalist }{ ii: the spectre continues [...] >: One *could* consider (as logicians have, being very thorough) logics >: intermediate between the usual propositional calculus of constructive >: and classical mathematics, but I would be very surprised if anybody >: were actually doing mathematics that way. Even on the level of predicate >: calculus, is anybody seriously using something in between intuitionist >: logic and classical logic? > I want to point out that many regularly used models have a model logic with > more structure than just the definition of a Heyting algebra. For example, > a topologist will work in a given topology, and this defines a logic with > Heyting structure plus the additional structure defining the particular > topology. Which is generally of no interest to this topologist. [...] >: 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? You obviously have in respect of sci.lang. You also obviously either violated one of the first principles of netiquette, namely, that one should become familiar with the actual content and customary practices of a newsgroup before posting to it, or deliberately posted to sci.lang something that you should have known was inappropriate. Quite clearly you *do* feel entitled to grab attention at others' expense. [...] > I have seen many pleas against the cross posting. Which is prima facie evidence that it was inappropriate. It is ultimately the members of a newsgroup who determine what is appropriate, not some document. If I want to read physics, I'll go to sci.physics; I don't want it cluttering up sci.lang. If I want to read mathematics, I'll go to sci.math. If I want to read logic, I'll go to sci.logic. If I want to read philosophy, I'm ill. > None of them have been > very convincing in my opinion, Which is largely irrelevant. [...] >: 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 the lack of content has been from those spamming their own > newsgroups, not asking for intelligent discussion, just spamming with > insults and the like. Which again is a very good indication that your content was widely considered inappropriate. Like it or not, many newsgroups are communities. Outsiders are not necessarily unwelcome, but outsiders who barge in and presume to lecture from a pedestal are likely to get the rough reception that they've earned. Bluntly, you're a rude, arrogant bastard with the social intelligence of a pet rock. On top of that you write some of the flabbiest, most turgid prose that it's been my misfortune to read anywhere, let alone on Usenet, and exhibit several of the familiar stigmata of the Usenet crank or monomaniac. If you don't like your reception, mend your manners. [...] === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques boundary=------------080505090508040604090003 -------------------------------------------------------------- ------- Sorry . you are not a bozo. I was referring to JW. Maybe we should establish a business selling standard barrels of herring for the national shoot-the-herring-in-the-barrel tournament? Can we come up with a test to see whether or not the differences due to Matjes vs Schmaltz herring are signficant w/r to herring marknanship? . I am a geologist whose career has been based on statistics. I've worked in lots of fields myself. Right now my work is in meteorology >and climatatology. Lots of applied statistics in all the things I've done. >I enjoy theory, but it is difficult to find teachers or books that relate >it well to practice, but they are out there and can be found with some >effort. I suffered through the theory classes and the theorems. It did not help >that many of my classmates were physics majors who were actually fairly >narrow probabalists. The background math / theory has turned out to be >critical. Far too often the applied instructors (in the business >college, in chemometrics, etc.) don't remember (or care) that all >procedures have limited validity (robustness). A prime advantage is to >understand where procedures make no sense because the system of interest >does not cooperate. Hence the mean has limited practical usefulness, as >does the standard deviation, regression statistics, factors and the >like. Using the procedures without understanding ( and investigating) >the underlying assumptions can be professionally suicidal for an applied statistician. One of my avocations is evaluating the work of >bozos like you in lawsuits. Why do think I'm a bozo? It's like hitting herring in a barrel. First, you have to get a barrel of herring. ;-) >A few weeks ago I posted a message asking about books on Bayesian >Unfortunately I have since dropped the class and am wondering about whether >I should continue the degree (Masters in Applied Statistics) and would like >some thoughts from the thoughtful people here. >I have a BS in Electrical Engineering and an MBA, both from the University >of Michigan. What I really liked about the MBA program is that it was almost >100% applied. Probably 50-70%+ of the classes was case study classes, a >trend mostly propagated and refined by Harvard Business School where >supposedly 100% of the classes are case study classes. Apparently more and >more law programs have more and more case study classes as well. Case study >classes are really as applied as you get because concepts and theories are >learned in the context of real world situations and circumstances. I am also >the type of person who is a very intuitive learner and has a much easier >time learning when I see how what I am learning relates to challenges in >real life (eg: business, which is what I do). >So given that the degree I am pursuing is called Masters in *APPLIED* >Statistics, I thought the the courses would be heavily applied and taught in >the context of solving real world problems. No dice. Both courses I took in >the first semester (part-time evening program) had heavy theory. The >Bayesian class was not even as bad as the other one (Mathematical >Statistics). There was essentially no attempt on the part of the professor >to relate the theory to real world programs or to even give real world >examples to illustrate the concepts. It was formula, theory, formula, >theory, theory, formula, etc. I asked him about that and he said there's no >way around the theory. I'm not trying to get around the theory but >theories and formulas mean nothing to me without real world context. I'm not >stupid either -- I score in the 99+ percentile on quantitative SAT/GMAT/GRE >and 98+ percentile overall. >My thinking right now is that my expectations were just off and disciplines >like Math/Statistics are just not as, ummm, progressive as Business/Law when >it comes to teaching (please -- no hate mail). Those teaching Math/Stat may >also be too smart and are not interested in mune day-to-day >business/industry problems (hopefully that will stop the hate mail!). So >what's up with that? Why is a degree called Masters in Applied Statistics >so heavy in theory? 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. and not to learn >esoteric statistical theory in the absence of application that I will surely >forget an hour after the final exam. >I am not trying to slam the field. I am interested in some opinions from >those in the field, especially those teaching it, to help me determine if I >>I can relate to some extent. While in theory I love theory, in >>practice I often wonder how what I'm studying relates to what I want >>The best math course I ever had was diff eq using _Differential >>Equations with Applications and Historical Notes_ by George Simmons >>as the text, in part because the applications gave me something >>concrete with which to relate. I'm not surprised that the Mathematical >>Statistics course was heavily theory, and in fact many of the stat >>books I've looked at appeared tilted that way. I've probably learned >>more statistics from books on signal processing than stat texts. >>Someone might argue that's why I know so little about statistics. :-) >>Perhaps it is just the particular school/program you're in, but I don't >>know. Maybe you should consider economics and econometrics, which >>can be heavy with stats and purport, at least, to deal with real world >>problems. Good luck. -- > === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1OH9Mi28590 > equation except factoring, quadratic formula, completing the > square,graphing or the square root method. > I went to http://www.geocities.com/dirkie6/page4.html but those > methods are too complex. Well, any methods more advanced than the ones you mentioned will be more complex. It will be hard to find a simpler method than the ones you mentioned. If such a method existed, it would be presented more widely. But other methods do exist. I will give a method based on Galois theory. I wish to find the roots of the equation x^2 + bx + c = 0, expressed in terms of b and c, using radicals. Let r be a root. Note r^2 + b*r + c = 0. Using long division, I find that x^2 + bx + c = (x - r) * (x + (b + r)). Thus, the other root is -(b + r). I will assume that the polynomial x^2 + bx + c is irreducible. You can get around this assumption by assuming that b and c are variable parameters rather than complex numbers. It is easy to see that by adjoining r to the base field that we get a splitting field and that the Galois group is cyclic of order two. As prescribed by Galois theory, I find the two Lagrange resolvents v_1 and v_2 given by: v_1 = r - (-b-r) = 2r + b v_2 = r + (-b-r) = -b Since v_1 + v_2 = 2r and r is not in the base field, then either v_1 or v_2 is not in the base field. Here, v_1 must not be in the base field, since -b is. Note that v_1 * v_1 = 4*r^2 + 4*b*r + b^2 = 4*(r^2 + b*r) + b^2 = 4*(-c) + b^2 = b^2 - 4c Thus, v_1 * v_1 is in the base field since b and c are. It is easy to see that 1 and r is a basis for the splitting field. It is easy to see that 1 and v_1 is a basis also. But, 1 = 1 v_1 = b + 2r Inverting, I get 1 = 1 r = (-b + v_1) / 2 Since v_1 = sqrt(b^2 - 4c), I have r = (-b + sqrt(b^2 - 4c)) / 2 This is the quadratic formula for one of the roots. The formula for the other root -(b + r) can be obtained by eliminating the r. -- Hale === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1OIiVi05040 > equation except factoring, quadratic formula, completing the > square,graphing or the square root method. > I went to http://www.geocities.com/dirkie6/page4.html but those > methods are too complex. Factoring, quadratic formula, completing the square are all one method: the quadratic formula is proved by completing the square; if you can't see how to factor a quadratic, complete the square. Since you mention graphing, I take it that approximate methods are acceptable to you, so consider Newton's method (aka the Newton-Raphson method): http://mathworld.wolfram.com/NewtonsMethod.html. What's the square root method? G.C. === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1OIuii06204 X-Orig-Trace: DXC=kIXg>bnAYaPlmRcVd::IAPc[3Bf`1S5QPF0iV`L?jh__>nYnb3]1j]P< CUP2SXDnOFkB5 QgBnAWKiXW<6[I>JenabflWQj]HL2J4S7LQ > equation except factoring, quadratic formula, completing the > square,graphing or the square root method. > I went to http://www.geocities.com/dirkie6/page4.html but those > methods are too complex. > Well, any methods more advanced than the ones you mentioned > will be more complex. It will be hard to find a simpler > method than the ones you mentioned. If such a method existed, > it would be presented more widely. > But other methods do exist. > I will give a method based on Galois theory. > I wish to find the roots of the equation x^2 + bx + c = 0, > expressed in terms of b and c, using radicals. > Let r be a root. > Note r^2 + b*r + c = 0. > Using long division, I find that > x^2 + bx + c = (x - r) * (x + (b + r)). > Thus, the other root is -(b + r). > I will assume that the polynomial x^2 + bx + c is irreducible. > You can get around this assumption by assuming that b and > c are variable parameters rather than complex numbers. > It is easy to see that by adjoining r to the base field > that we get a splitting field and that the Galois group > is cyclic of order two. > As prescribed by Galois theory, I find the two Lagrange > resolvents v_1 and v_2 given by: > v_1 = r - (-b-r) = 2r + b > v_2 = r + (-b-r) = -b > Since v_1 + v_2 = 2r and r is not in the base field, > then either v_1 or v_2 is not in the base field. > Here, v_1 must not be in the base field, since -b is. > Note that > v_1 * v_1 = 4*r^2 + 4*b*r + b^2 > = 4*(r^2 + b*r) + b^2 > = 4*(-c) + b^2 > = b^2 - 4c > Thus, v_1 * v_1 is in the base field since b and c are. > It is easy to see that 1 and r is a basis for the splitting field. > It is easy to see that 1 and v_1 is a basis also. > But, > 1 = 1 > v_1 = b + 2r > Inverting, I get > 1 = 1 > r = (-b + v_1) / 2 > Since v_1 = sqrt(b^2 - 4c), I have > r = (-b + sqrt(b^2 - 4c)) / 2 > This is the quadratic formula for one of the roots. > The formula for the other root -(b + r) can be obtained > by eliminating the r. > -- Hale Are you sure you couldn't find a heavier hammer to crack that egg? === Subject: Re: the anticlassicalist }{ : ism for Daleks : OK. So if I'm already unknowingly using this logic, what is the point : you're trying to make? Believe me, it hasn't yet emerged from all the : verbiage. Well, from a practical point of view, understanding the logic of orthomodular lattices that arises in propositions on a Hilbert space can prevent a lot of early conceptual difficulties in the use of quantum mechanics. Students first approaching the topic can understand that much of the strangeness of quantum mechanics, many quotes by various famous physicists from early quantum foundations about the difficulty (or even impossibility) to understand the subject, can be dismissed as merely problems in understanding the logic of quantum mechanics. Because quantum mechanics really is a well established model in physics which I believe can be understood completely by mere humans, and I think that some people have fallen into certain antisymbolic modes of thought by a myriad of pop-physics book which agonise over this strangenessAs I keep trying to point out, my focus is on education with these series of posts. The theory of models and their logics is crucial to all science. Indeed, it rigorously formalises the notion of a theory. Many models across disciplines have a Heyting structure, and these are often unknown to practitioners in the various fields, so I use that as my archetype for education reform. There are many cases where the unfamiliarity with the logical structure of the models used presents confusion to the students early on. But understanding the logic of the models can also have important theoretical components as well. For quantum logic, as the case in question, understanding the structural differences in the logics of classical and quantum mechanics gives us information (of a cohomological nature) concerning quantisation programs. Although canonical quantisation is fine to mislead first year students about our understanding of quantisation, it ignores the actual difficulties and ambiguities that arise. Instead one needs to go from prequantisations to metaplectic corrections and all sorts of fun stuff along the way. Avoiding obstructions to quantisation and (somewhat oppositely) understanding the ambiguities that arise in the process requires one be comfortable with logical structures of classical and quantum models. Extend to quantum field theories, strings, and such, and maybe it gets clearer why I see the necessity to alter education of logic in physics curricula. And this does not even touch upon the theory of causal sets and other directions in physics where such an education would be beneficial. ===-=-=-=-=- === Subject: Re: 'erf' function in C > A possible improvement is to note that the even part of R(x), > (R(x)+R(-x))/2, is equal to 1/phi(x), thus only the odd part of R(x), > (R(x)-R(-x))/2, needs to be computed: > double Phi(double x) > {long double s,t=0,b=1,pwr=x; > int i; > s=x; > for(i=2;s!=t;i+=2) > { b/=(i+1); > pwr=pwr*x*x; > t=s; > s+=pwr*b; > } > return .5+s*exp(-.5*x*x-.91893853320467274178L); > } > Very nice, and a definite improvement*! It speeds up the convergence > considerably which, quite possibly, accounts for more accurate results. > i.e. w/CVF on Wintel, > x phi(x) > 0.123 0.5489464510164368 > 1.200 0.8849303297782917 > 2.400 0.9918024640754040 > 6.100 0.9999999994696567 > -6.100 0.0000000005303433 > -1.100 0.1356660609463827 > 7.200 0.9999999999996979 > * Actually, it's an understatement considering the Syziphian effort at > LANL some odd thirty years ago. See, netlibfn lib. It is possible to compress the code further at the expense of clarity (warning: this is not for the faint of heart): double Phi(double x) {long double s=x,t=0,b=x,x2=x*x,i=1; while(s!=t) s=(t=s)+(b*=x2/(i+=2)); return .5+s*exp(-.5*x2-.91893853320467274178L); } This avoids the unnecessary computation of i+1 followed by the unnecessary conversion of the result from int to long double, eliminates the unnecessary variable pwr, and precomputes x*x. === Subject: Re: Alternative ways to solve a quadratic equation >equation except factoring, quadratic formula, completing the >square,graphing or the square root method. >I went to http:// www.geocities.co m/d >irkie6/page4.html but those >methods are too complex. > How about Newton's method? Guess, iterate to a better guess, > etc. Do you consider this a solution? > phil One problem with it is that when there is a rational root and you do not guess it exactly right at first, you never get it exactly right. === Subject: Re: Trying to unify axioms. > Nothing. > You forever spewing ing imbecile, an axiom by definition is > irreducible and unprovable. > False. > First off, an axiom isn't necessarily irreducable, unless you want to > claim that reducing it makes the axiom not reducable and therefore not > an axiom. Sometimes we have an axiom that we find out can be reduced > into other axioms. Does that therefore disprove the axiom, or destroy > the usefulness of the axiom, or of taking it as such? No it doesn't. > Second, axioms are not unprovable. They CAN be proven. For an example > of this, consider the law of identity. Can you prove it? If not, then > how do we even know it's true? We do know it's true, and it IS an > axiom, so that just proves that axioms are not unprovable. > (...Starblade Riven Darksquall...) An axiom is unprovable by definition. Usually, if an axiom can be shown to be provable based on other axioms or postulates, it no longer remains an axiom and is technically a theorem, which by definition is provable. And of course, nothing is ever provable in and of itself, which is why a single isolated statement can only be true by definition. === Subject: Documents about multiplicative order in general and mersenne numbers could anybody of you tell me where to find material about the following topics: - multiplicative order in general - divisors of mersenne numbers Unfortunately I couldn't find anything except for some smaller Because divisors of Mersenne-numbers with an exponent which is prime can only be numbers with the same multiplicative order 2. Any and all help is appreciated :o) Thank you in advance. Juergen Bullinger === Subject: Re: the anticlassicalist }{ ii: the spectre continues : > I want to point out that many regularly used models have a model logic with : > more structure than just the definition of a Heyting algebra. For example, : > a topologist will work in a given topology, and this defines a logic with : > Heyting structure plus the additional structure defining the particular : > topology. : : Which is generally of no interest to this topologist. Whereas this topologist finds that much of the work by other topologists, much like work of the quantum physicists, merely re-expresses the logical structure in other constructs. Often, I find that it is not a question of interest, but more of just being unaware that the manipulations being done fit any kind of logical formalism. My findings are, of course, not final! : > 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? : : You obviously have in respect of sci.lang. You also : obviously either violated one of the first principles of : netiquette, namely, that one should become familiar with the : actual content and customary practices of a newsgroup before : posting to it, or deliberately posted to sci.lang something : that you should have known was inappropriate. Quite clearly : you *do* feel entitled to grab attention at others' expense. Expense? I took the time to write several long posts concerning a topic relevant to all newsgroups posted to. I am earnestly interested in an educational deficiency which has been quite well demonstrated by many posts of others in this thread. I came out attacking no one. I was upfront about my agenda. And I quarantined my ramblings to 2 intended (and 2 incidental due to newsserver problems) threads, all of which are not forced upon any newsgroup reader and may easily be avoided. The netiquette problems are from those who come attacking, giving no constructive material in which to discuss. That now includes you, mister Scott. As for sci.lang, obviously there are problems in understanding reasoning on topological trees (and linguistic phylogeny) that have been evidenced several times recently in that forum. Plus the whole cognitive origins of language, semiotics and natural language models, etc. features of my exposition that I've been willing to discuss in depth make your statement patently false. Oh, did you miss the thread on modality in language as well? Yeah, I'm reading... Are you? : [...] : : > I have seen many pleas against the cross posting. : : Which is prima facie evidence that it was inappropriate. It : is ultimately the members of a newsgroup who determine what : is appropriate, not some document. If I want to read : physics, I'll go to sci.physics; I don't want it cluttering : up sci.lang. If I want to read mathematics, I'll go to : sci.math. If I want to read logic, I'll go to sci.logic. : If I want to read philosophy, I'm ill. No, it's evidence of nothing of the kind! None of those pleas has ever described in any way how I have violated the topicality of their newsgroup. Even you didn't make any such description above. You just accuse and fight your alpha games like you are the arbiter of truth and justice. Your desire to want to avoid certain discussions of an interdisciplinary nature is easily avoided by ignoring threads you find distasteful. I can walk you through that procedure if you are having any difficulties. : > None of them have been : > very convincing in my opinion, : : Which is largely irrelevant. Unfortunately, that is quite relevant. Convincing me is the only way someone is going to get me to stop posting. : > Most of the lack of content has been from those spamming their own : > newsgroups, not asking for intelligent discussion, just spamming with : > insults and the like. : : Which again is a very good indication that your content was : widely considered inappropriate. Like it or not, many : newsgroups are communities. Outsiders are not necessarily : unwelcome, but outsiders who barge in and presume to lecture : from a pedestal are likely to get the rough reception that : they've earned. No, its an indication that there are quite a lot of jerks out there who, when faced with a topic they do not understand and do not want to understand, find solace in insults. I do like the fact that newsgroups are communities, particularly that they are communities of wide ranges of views about the topics they discuss. There are certainly members, such as yourself mister Scott, who dislike the fact that others may begin a discussion confident of the knowledge that they have such a right, but unfortunately you are in the wrong and I am in the right. And playing your power games, with their complete absence of any rational points, just illustrates to me that you recognise your complete lack of power in this circumstance. : Bluntly, you're a rude, arrogant bastard with the social : intelligence of a pet rock. On top of that you write some : of the flabbiest, most turgid prose that it's been my : misfortune to read anywhere, let alone on Usenet, and : exhibit several of the familiar stigmata of the Usenet crank : or monomaniac. If you don't like your reception, mend your : manners. I have never been rude to anyone who was not first rude to me, and then only enough to play the alpha game they initiated to its proper closure. I don't seek contentless arguments; it is others who feel inclined to provide me with such. I am a bastard; that is true. I was born with an unmarried mother. I can be quite humble when speaking to others who engage in rational critique or otherwise educate me of my errors. I see you borrowed the use of turgidity from the other newsgroup spammer. Feeling uncreative today? the writing process. I have tried to be exciting yet remain technically correct. Sure, I've failed in places, but that is part of the process of my finding out how to express these ideas. What I see is that there is a group of sterile, uncreative, alpha posturing jerks who enjoy attempting to crush any form of expression that does not conform to their galatea of elegance, and they joyfully make no contribution to the technical discussion to keep their spamming focused. Only part of my reception I do not like. Mend your manners, mister Scott. ===-=-=-=-=- === Subject: Interesting problem I have the following 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? Thank u for your attention. {V} === Subject: Re: Documents about multiplicative order in general and mersenne numbers > could anybody of you tell me where to find material about the > following topics: > - multiplicative order in general > - divisors of mersenne numbers http://www.cerias.purdue.edu/homes/ssw/cun/ is the homepage for the Cunningham project with is concerned with factoring numbers of the form b^n + 1 and b^n - 1. Material on multiplicative order can be found in pretty much any book with a title like Introduction to Number Theory, Elementary Number Theory, etc. === Subject: Re: Minimally simple finite groups? >For L_2(p^e) to be minimal simple, its order must not be divisible by 60 >(otherwise it contains A_5). > Ah. Why must it contain A_5 if its order is divisible by 60? Hi Jim, Maybe the following idea can help to answer your question. If in a field F the equations a^2 + a - 1 = 0 b^2 + 1 = 0 have a solution, then the matrices 0 1 A = -1 a -1 b B = b 0 ( which are in SL(2,F) ) satisfy the relations A^5 = B^3 = (AB)^4 = [A, (AB)^2] = 1. However these are defining relations of the group SL(2,5) (with generators A,B). Hence, SL(2,F) contains a homomorphic image of SL(2,5). -1 0 Since (AB)^2 = 0 -1 we have a proper embedding unless F has characteristic 2, in which case there is an embedding A_5 -> SL(2,F) = PSL(2,F). It remains to see for which finite F=GF(q) the above two equations can be solved. Maybe the condition that 60 divides |L_2(q)| is sufficient for this. Anvita === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1OLdki21021 required us to find a totally new method to solve a quadratic equation with. It can involve the quadratic formula but has to be different. === Subject: Re: 'erf' function in C > It is possible to compress the code further at the expense of > clarity (warning: this is not for the faint of heart): > double Phi(double x) > {long double s=x,t=0,b=x,x2=x*x,i=1; > while(s!=t) s=(t=s)+(b*=x2/(i+=2)); > return .5+s*exp(-.5*x2-.91893853320467274178L); > } > This avoids the unnecessary computation of i+1 followed by the > unnecessary conversion of the result from int to long double, > eliminates the unnecessary variable pwr, and precomputes x*x. Have you checked this algorithm in the tail? For, say, x = -50, I either get inf/nan or an incorrect answer (0.5), depending on which compiler I use. The compiler in my tests is gcc, specifically DJGPP or MingW. The problem is overflow in s. Furthermore, Phi(x) is not increasing in x for values between -13 and -7, see below (DJGPP using long double). x Phi(x) -13 -2.94464e-15 -12 -2.89807e-15 -11 6.27672e-16 -10 6.49085e-16 -9 6.65294e-16 -8 1.28093e-15 -7 1.28042e-12 === Subject: Re: the anticlassicalist }{ ii: the spectre continues > I came out attacking no one. I was upfront about > my agenda. And I quarantined my ramblings to 2 intended (and 2 incidental > due to newsserver problems) threads, all of which are not forced upon any > newsgroup reader and may easily be avoided. The netiquette problems are > from those who come attacking, giving no constructive material in which to > discuss. That now includes you, mister Scott. All right, gaga-la-t.8et.8ee, here, fished out of google === Subject: Re: New human migration map based upon mtDNAs : : > Groups L1, L2, M & N can be likened to language groups. : : : Since languages have been demonstrated to evolve at wildly : different rates, that can only be: : : (1) a chance coincidence (helped by cherry picking the : language groups) : : or : : (2) a self-delusion, like Ruhlen and Cavalli-Sforza's two : trees, linguistic vs genetic : : or : : (3) evidence that the mtDNA of speakers of any given language : evolves at the same rate as that language. Jacques, you are using these wildly different rates as your sword to cut and slash what you do not understand. ^^^^^^^^^^^^^^^^^^^^^^^^^^ [1] They in no way imply these three choices of yours, and I _do_ accept that rates are quite far from constant across linguistic evolution. Different rates can easily point to the same or similar tree topologies. This is phylogenetic bioinformatics 101. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ [2] And there is a hypothesis underlying this. That languages used by communities as humans migrate tend to maintain similarities from previous languages used by the groups, and that genetic isolation can often imply some amount of linguistic isolation. It's not pseudoscience (like I would ^^^^^^^^^^^^ characterise your theory on the three possibilities), it is the same sort of ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ [3] hypothesis that working archaeologists use to understand a language used in a find and hypothesise on where the people came from. : > Therefore, Jacques, there can be such a map. If sides 1 - 12 contain : > too much theory for you, then ignore them. : : If the three points above contain too much logic for : you, then ignore them and hop onto the bandwagon, which : looks about to join a nice gravy train. I can offer some books on phylogenetic reconstruction if the ideas are so ^^^^^^^^^^^^^^^^^^^ difficult you need to resort to name calling. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ [4] : I myself favour (3) above, as it is sufficiently ridiculous : to get published in Nature. And it goes a long way : towards proving the innateness of grammar and whatnot. : Can't lose with such a hand. I figure you'd like 3. Its the culmination of your pseudoscience. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ [5] The five segments highlighted above are five mealy-mouthed repetitions of the same plain English words you are full of . None of them follows a demonstration, not even an argument, nor a piece of evidence, and none leads to what follows. And you, miserable mealy-mouthed two-faced drivel-dribbling lying snake, have the cheek of complaining that > netiquette problems are > from those who come attacking, giving no constructive material in which to > discuss. That now includes you, mister Scott. === Subject: Re: the anticlassicalist }{ ii: the spectre continues Jacques, your first response to my first post (which was purely a scientific description of phylogenetic techniques with no attacks to anyone) in that thread was: ===-=-=-=-=- Yes, I know that. Like the drunk on all fours. Comes a policeman. Hello there, Sir, in a bit of trouble are we? I'm jusht looking for me housh keys. Oh dear. And whereabouts have you lost them? On t'other shide of the shtreet Er... and shouldn't you be looking there? It'sh dark there. There'sh light here. ===-=-=-=-=- (...and you _were_ wrong in the places I pointed out.) Your response to my post in Out of Anatolia contained the lovely little: ===-=-=-=-=- What does Heyting algebra have to contribute to unmasking such hocus pocus merchants? Let me guess: that is not what it is for. Perhaps it even helps camouflage the snake oil as pure olive oil? ===-=-=-=-=- Again you demonstrate my point about who is picking the fights, in brilliant clarity, for the small little section of usenet unfortunate enough to have to partake in your ugliness. Littering the world with your hatred does not make you look strong. ===-=-=-=-=- === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > Sorry . you are not a bozo. I was referring to JW. Glad to hear that. I thought that might be the case. > Maybe we should establish a business selling standard barrels of > herring for the national shoot-the-herring-in-the-barrel tournament? > Can we come up with a test to see whether or not the differences due > to Matjes vs Schmaltz herring are signficant w/r to herring > marknanship? . :-) I think to make it a viable business we have to have the contestants use a small enough caliber weapon that there is enough of the herring left to process for food. > I am a geologist whose career has been based on > statistics > I've worked in lots of fields myself. Right now my work is > in meteorology > and climatatology. Lots of applied statistics in all the > things I've done. > I enjoy theory, but it is difficult to find teachers or > books that relate > it well to practice, but they are out there and can be found > with some > effort. > I suffered through the theory classes and the > theorems. It did not help > that many of my classmates were physics majors who > were actually fairly > narrow probabalists. The background math / theory > has turned out to be > critical. Far too often the applied instructors > (in the business > college, in chemometrics, etc.) don't remember > (or care) that all > procedures have limited validity (robustness). A > prime advantage is to > understand where procedures make no sense because > the system of interest > does not cooperate. Hence the mean has limited > practical usefulness, as > does the standard deviation, regression > statistics, factors and the > like. Using the procedures without understanding > ( and investigating) > the underlying assumptions can be professionally > suicidal for an > applied statistician. One of my avocations is > evaluating the work of > bozos like you in lawsuits. > Why do think I'm a bozo? > It's like hitting herring in a barrel. > First, you have to get a barrel of herring. ;-) A few weeks ago I posted a > message asking about books on > Bayesian > Statistics. I got some nice > Unfortunately I have since > dropped the class and am > wondering about whether > I should continue the degree > (Masters in Applied > Statistics) and would like > some thoughts from the > thoughtful people here. > I have a BS in Electrical > Engineering and an MBA, both > from the University > of Michigan. What I really > liked about the MBA program is > that it was almost > 100% applied. Probably 50-70%+ > of the classes was case > study classes, a > trend mostly propagated and > refined by Harvard Business > School where > supposedly 100% of the classes > are case study classes. > Apparently more and > more law programs have more > and more case study classes as > well. Case study > classes are really as applied > as you get because concepts > and theories are > learned in the context of real > world situations and > circumstances. I am also > the type of person who is a > very intuitive learner and has > a much easier > time learning when I see how > what I am learning relates to > challenges in > real life (eg: business, which > is what I do). > So given that the degree I am > pursuing is called Masters in > *APPLIED* > Statistics, I thought the the > courses would be heavily > applied and taught in > the context of solving real > world problems. No dice. Both > courses I took in > the first semester (part-time > evening program) had heavy > theory. The > Bayesian class was not even as > bad as the other one > (Mathematical > Statistics). There was > essentially no attempt on the > part of the professor > to relate the theory to real > world programs or to even give > real world > examples to illustrate the > concepts. It was formula, > theory, formula, > theory, theory, formula, etc. > I asked him about that and he > said there's no > way around the theory. I'm not > trying to get around the > theory but > theories and formulas mean > nothing to me without real > world context. I'm not > stupid either -- I score in > the 99+ percentile on > quantitative SAT/GMAT/GRE > and 98+ percentile overall. > My thinking right now is that > my expectations were just off > and disciplines > like Math/Statistics are just > not as, ummm, progressive as > Business/Law when > it comes to teaching (please > -- no hate mail). Those > teaching Math/Stat may > also be too smart and are > not interested in mune > day-to-day > business/industry problems > (hopefully that will stop the > hate mail!). So > what's up with that? Why is a > degree called Masters in > Applied Statistics > so heavy in theory? 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. and > not to learn > esoteric statistical theory in > the absence of application > that I will surely > forget an hour after the final > exam. > I am not trying to slam the > field. I am interested in some > opinions from > those in the field, especially > those teaching it, to help me > determine if I > should continue further. > I can relate to some extent. While in > theory I love theory, in > practice I often wonder how what I'm > studying relates to what I want > to understand, whether the field is pure > The best math course I ever had was diff > eq using _Differential > Equations with Applications and > Historical Notes_ by George Simmons > as the text, in part because the > applications gave me something > concrete with which to relate. I'm not > surprised that the Mathematical > Statistics course was heavily theory, > and in fact many of the stat > books I've looked at appeared tilted > that way. I've probably learned > more statistics from books on signal > processing than stat texts. > Someone might argue that's why I know so > little about statistics. :-) > Perhaps it is just the particular > school/program you're in, but I don't > know. Maybe you should consider > economics and econometrics, which > can be heavy with stats and purport, at > least, to deal with real world > problems. Good luck. -- Martin R. L. Martin and Associates, Consultants in .martin@wdn.com Science and Technology http://www.rmartin.com All too often the study of data requires care. === Subject: Re: Interesting problem Trivially not ! take the element ab where a in A and b in B, then it is neither in A nor in B on account of the mult. closure > I have the following 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? > Thank u for your attention. > {V} === Subject: Re: Interesting problem >Trivially not ! >take the element ab where a in A and b in B, >then it is neither in A nor in B on account >of the mult. closure perhaps that isn't what is meant by multiplicative closure - that would seem to be ideal closure, but I believe it can be extended to show the answer. If ab is in B, and 1/b is in B, then so is a# so 1/b is in A. Now wlog 1 is in B, and hence so is 2, thus 1/2 is in A, hence 1/2+1/2=1 is in A# >I have the following 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? >Thank u for your attention. >{V} === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1ONkTi00513 > required us to find a totally new method to solve a quadratic equation > with. It can involve the quadratic formula but has to be different. I assume you mean totally new to you as opposed to totally new to mathematics. You might look for some techniques for solving higher order equations. Or here is one very simple and inefficient one called the half interval search - once you have established that a root lies between point A and B, because of a sign change. Evaluate the function at (A + B)/2. If it is the same sign as A this becomes the new A otherwise the new B. Keep going until you are tired or happy. Or you might try to find some kind of analog computer. That is, instead of using mathematics to represent a physical phenomenon (the usual way of doing things) find some sort of physical representation of the math. and go measure it. === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P00Ki02172 Originator: fishbowl@conservatory.com (james) >required us to find a totally new method to solve a quadratic equation >with. Totally new? That would be quite a feat. Oh, not totally new as in, hasn't been discovered before. Your chances of finding a method that the Greeks didn't know are pretty slim :-) >It can involve the quadratic formula but has to be different. I'd look for a quadratic identity using trig substitution, or maybe find roots with the Newton Method. Sounds like your assignment is supposed to be an exercise in finding identities for the quadratic formula. If you come up with something totally new here *DON'T* give it to your professor, submit a paper to James watched Good Will Hunting one time too many M. === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P02Ri02705 Originator: fishbowl@conservatory.com (james) >But other methods do exist. >I will give a method based on Galois theory. William, you've been folderized. === Subject: Re: Interesting problem >perhaps that isn't what is meant by multiplicative closure - that >would seem to be ideal closure, but I believe it can be extended to >show the answer. >If ab is in B, and 1/b is in B, then so is a# contradiction assuming a in A and ab in B, that is... >so 1/b is in A. Now wlog 1 is in B, and hence so is 2, thus 1/2 is in >A, hence 1/2+1/2=1 is in A# No, you could only say thus 1/2 is in A if there was some a in A such that 2a is in B. Of course that can't happen. Why can't 1/2, 1, 2 all be in B? In fact if 1 is in B, then all positive rationals are in B. Moreover, all rational powers of rationals are in B. But there are lots of other numbers, e.g. perhaps some transcendentals, that could be in A. Department of Mathematics http://www.math.ubc.ca/~israel === Subject: Re: 'erf' function in C Yes, the paper cited below is the one I used to implement the inverse cumulative normal function. In that paper you recommended that such a function be used to generate random normally distributed numbers, instead of the Box-Muller method. 1) Would you still recommend this? 2) The approach suggested below seems to be aiming more for accuracy, rather than speed. On the other hand, the published algorithm is machine-dependent and therefore a bit of a headache to maintain. Is there another fast but portable algorithm with sufficient accuracy available? >And what about the inverse cumulative normal distribution? >I used to implement an algorithm you published, >but do not like it because it is machine-dependent. >What in your opinion is the best algorithm? > With the improvement on my method for > evaluating cPhi(x) as R(x)*phi(x), > when the Taylor series for R(x) is about zero, > (suggested by Daly), the simple C function > double Phi(double x) > { long double t=0,b=1,s=x,pwr=x; > int i; > for(i=2;s!=t;i+=2) > { b/=(i+1); pwr=pwr*x*x; t=s; s+=pwr*b;} > return .5+s*exp(-.5*x*x-.91893853320467274178L); > } > should serve very well for solving the equation > Phi(X)=U > for positive X, given U>1/2, by Newton's method: > Start with an initial estimate > x=sqrt(-1.6*ln(1-U^2)); > then repeat > x=x-(Phi(x)-U)/phi(x); > until you get no further changes in x. > The paper that you refer to must be where I want > to generate a normal X by means of solving > Phi(X)=U, given a random U, uniform in [0,1), > by getting an integer j, formed from certain > bits of the exponent part of the floating point > representation of U. Then use j to access tabled > values A[j] and B[j] to initalize the Taylor series > for Phi inverse, using the fractional part of U > as the argument in the series. It is designed for > speed in generating a normal variate directly > as Phi^(-1)(U), but only provides accuracy to 6/7 digits, > (As I recall, the Taylor series was easy and fast only for the first few > terms.) > It is described in > ``Normal (Gaussian) random variables in supercomputers'', (1991) > Journal of Supercomputing}, v 5, 49--55, > where the method is particularly suited for parallel computation. > George Marsaglia === Subject: Re: the anticlassicalist }{ ii: the spectre continues sci.lang,sci.logic,sci.math: [...] >:> 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? >: You obviously have in respect of sci.lang. You also >: obviously either violated one of the first principles of >: netiquette, namely, that one should become familiar with the >: actual content and customary practices of a newsgroup before >: posting to it, or deliberately posted to sci.lang something >: that you should have known was inappropriate. Quite clearly >: you *do* feel entitled to grab attention at others' expense. > Expense? Yes, expense. You are, for example, directly responsible for cluttering sci.lang with off-topic mathematics and complaints about the lack of physical content in your posts from sci.physics. > I took the time to write several long posts concerning a topic > relevant to all newsgroups posted to. In your opinion. The relevance is not apparent to many of those better qualified than you to judge. [...] > As for sci.lang, obviously there are problems in understanding reasoning on > topological trees (and linguistic phylogeny) that have been evidenced > several times recently in that forum. Jacques erred in calling his three trees topologically equivalent (though in fact two of them are -- as unlabelled trees), but I doubt that he was trying very hard: you've offered nothing to suggest that you should be taken any more seriously on the subject of linguistic phylogeny than, say, Cavalli-Sforza, who is a linguistic ignoramus. > Plus the whole cognitive origins of > language, semiotics and natural language models, etc. features of my > exposition that I've been willing to discuss in depth make your statement > patently false. Bollocks. I've not seen you offer anything substantive on any of those topics, which with the possible exception of the first are in any case not topics of great interest to most of the regulars here, judging by the eight or so years that I've been reading the group. > Oh, did you miss the thread on modality in language as well? What, Andrew Patterson's nonsense? That was (a) independent of your posturing and (b) obviously received largely with indifference. > Yeah, I'm reading... > Are you? Everything? Of course not. I read the threads in which the professional linguists and knowledgeable sci.lang regulars participate. I've also read some of the stuff dragged in from sci.physics. I read this bit because from the days when I used to read sci.math I recognized Keith Ramsay as someone likely to have something sensible to say. >: [...] >:> I have seen many pleas against the cross posting. >: Which is prima facie evidence that it was inappropriate. It >: is ultimately the members of a newsgroup who determine what >: is appropriate, not some document. If I want to read >: physics, I'll go to sci.physics; I don't want it cluttering >: up sci.lang. If I want to read mathematics, I'll go to >: sci.math. If I want to read logic, I'll go to sci.logic. >: If I want to read philosophy, I'm ill. > No, it's evidence of nothing of the kind! None of those pleas has ever > described in any way how I have violated the topicality of their newsgroup. You still don't get it, do you? What's on topic is determined by the group itself, especially when it has a core of regulars. If you'd paid the slightest attention, you'd have recognized that sci.lang runs to historical linguistics (about which you've demonstrated considerable ignorance) and sociolinguistics, with occasional forays into syntax. (Oh, and food.) > Even you didn't make any such description above. You just accuse and fight > your alpha games like you are the arbiter of truth and justice. Not at all, though I fear that you are far too self-centred and convinced of your own virtue to recognize that what you call an accusation was in fact (1) a partial explanation of the reception you've had, (2) implicit advice on how to get a better reception, and (3) a fairly mild expression of exasperation. In point of fact I have for the most part treated you as you deserve: apart from a brief observation and followup when you first appeared, I have ignored you. But your question to Keith was too inviting an opportunity to be ignored. > Your desire to want to avoid certain discussions of an interdisciplinary > nature is easily avoided by ignoring threads you find distasteful. I can > walk you through that procedure if you are having any difficulties. The interdisciplinary nature of these 'discussions' is mostly in your own mind. What's been posted here has mostly been mathematics and handwaving. >:> None of them have been >:> very convincing in my opinion, >: Which is largely irrelevant. > Unfortunately, that is quite relevant. Convincing me is the only way > someone is going to get me to stop posting. I quite believe it. I was discussing the facts of the matter, to which your opinion is largely irrelevant, and not your future behavior. >:> Most of the lack of content has been from those spamming their own >:> newsgroups, not asking for intelligent discussion, just spamming with >:> insults and the like. >: Which again is a very good indication that your content was >: widely considered inappropriate. Like it or not, many >: newsgroups are communities. Outsiders are not necessarily >: unwelcome, but outsiders who barge in and presume to lecture >: from a pedestal are likely to get the rough reception that >: they've earned. > No, its an indication that there are quite a lot of jerks out there who, > when faced with a topic they do not understand and do not want to > understand, find solace in insults. I do like the fact that newsgroups are > communities, particularly that they are communities of wide ranges of views > about the topics they discuss. There are certainly members, such as > yourself mister Scott, who dislike the fact that others may begin a > discussion confident of the knowledge that they have such a right, but > unfortunately you are in the wrong and I am in the right. And playing your > power games, with their complete absence of any rational points, just > illustrates to me that you recognise your complete lack of power in this > circumstance. I'm afraid that it's you who are playing power games. You are the one thumbing his nose at the rest of us and going 'Nyaa, you can't make me leave'. In this you are quite correct: Usenet is an open forum, and I wouldn't have it any other way. But just as you are free to shove your id.8ee fixe in front of people's faces at preposterous (and singularly ineffective) length, so am I free to point out that you are doing so. >: Bluntly, you're a rude, arrogant bastard with the social >: intelligence of a pet rock. On top of that you write some >: of the flabbiest, most turgid prose that it's been my >: misfortune to read anywhere, let alone on Usenet, and >: exhibit several of the familiar stigmata of the Usenet crank >: or monomaniac. If you don't like your reception, mend your >: manners. > I have never been rude to anyone who was not first rude to me, and then only > enough to play the alpha game they initiated to its proper closure. You are mistaken, owing to your inability to recognize the rudeness of your behavior thus far. > I don't > seek contentless arguments; it is others who feel inclined to provide me > with such. I am a bastard; that is true. I was born with an unmarried > mother. Irrelevant, since I use the term in its figurative sense, and certainly no concern of mine in any case. > I can be quite humble when speaking to others who engage in > rational critique or otherwise educate me of my errors. I see you borrowed > the use of turgidity from the other newsgroup spammer. No, I did not. I have no idea even to whom you refer. But I am hardly surprised that someone else used so obviously apt a description. [...] If you are serious about getting together those who are genuinely interested in your views, I suggest that you set up a web-based bulletin board; I'm given to understand that this is very easy to do these days. You can then announce it in the newsgroups in which you think it might be of interest. Reply or not, as you wish; I'll not be responding again outside of threads with genuine linguistic content, if I even bother to read. === Subject: Re: the anticlassicalist }{ ii: the spectre continues sci.lang,sci.logic,sci.math: [...] >:> 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? >: You obviously have in respect of sci.lang. You also >: obviously either violated one of the first principles of >: netiquette, namely, that one should become familiar with the >: actual content and customary practices of a newsgroup before >: posting to it, or deliberately posted to sci.lang something >: that you should have known was inappropriate. Quite clearly >: you *do* feel entitled to grab attention at others' expense. > Expense? Yes, expense. You are, for example, directly responsible for cluttering sci.lang with off-topic mathematics and complaints about the lack of physical content in your posts from sci.physics. > I took the time to write several long posts concerning a topic > relevant to all newsgroups posted to. In your opinion. The relevance is not apparent to many of those better qualified than you to judge. [...] > As for sci.lang, obviously there are problems in understanding reasoning on > topological trees (and linguistic phylogeny) that have been evidenced > several times recently in that forum. Jacques erred in calling his three trees topologically equivalent (though in fact two of them are -- as unlabelled trees), but I doubt that he was trying very hard: you've offered nothing to suggest that you should be taken any more seriously on the subject of linguistic phylogeny than, say, Cavalli-Sforza, who is a linguistic ignoramus. > Plus the whole cognitive origins of > language, semiotics and natural language models, etc. features of my > exposition that I've been willing to discuss in depth make your statement > patently false. Bollocks. I've not seen you offer anything substantive on any of those topics, which with the possible exception of the first are in any case not topics of great interest to most of the regulars here, judging by the eight or so years that I've been reading the group. > Oh, did you miss the thread on modality in language as well? What, Andrew Patterson's nonsense? That was (a) independent of your posturing and (b) obviously received largely with indifference. > Yeah, I'm reading... > Are you? Everything? Of course not. I read the threads in which the professional linguists and knowledgeable sci.lang regulars participate. I've also read some of the stuff dragged in from sci.physics. I read this bit because from the days when I used to read sci.math I recognized Keith Ramsay as someone likely to have something sensible to say. >: [...] >:> I have seen many pleas against the cross posting. >: Which is prima facie evidence that it was inappropriate. It >: is ultimately the members of a newsgroup who determine what >: is appropriate, not some document. If I want to read >: physics, I'll go to sci.physics; I don't want it cluttering >: up sci.lang. If I want to read mathematics, I'll go to >: sci.math. If I want to read logic, I'll go to sci.logic. >: If I want to read philosophy, I'm ill. > No, it's evidence of nothing of the kind! None of those pleas has ever > described in any way how I have violated the topicality of their newsgroup. You still don't get it, do you? What's on topic is determined by the group itself, especially when it has a core of regulars. If you'd paid the slightest attention, you'd have recognized that sci.lang runs to historical linguistics (about which you've demonstrated considerable ignorance) and sociolinguistics, with occasional forays into syntax. (Oh, and food.) > Even you didn't make any such description above. You just accuse and fight > your alpha games like you are the arbiter of truth and justice. Not at all, though I fear that you are far too self-centred and convinced of your own virtue to recognize that what you call an accusation was in fact (1) a partial explanation of the reception you've had, (2) implicit advice on how to get a better reception, and (3) a fairly mild expression of exasperation. In point of fact I have for the most part treated you as you deserve: apart from a brief observation and followup when you first appeared, I have ignored you. But your question to Keith was too inviting an opportunity to be ignored. > Your desire to want to avoid certain discussions of an interdisciplinary > nature is easily avoided by ignoring threads you find distasteful. I can > walk you through that procedure if you are having any difficulties. The interdisciplinary nature of these 'discussions' is mostly in your own mind. What's been posted here has mostly been mathematics and handwaving. >:> None of them have been >:> very convincing in my opinion, >: Which is largely irrelevant. > Unfortunately, that is quite relevant. Convincing me is the only way > someone is going to get me to stop posting. I quite believe it. I was discussing the facts of the matter, to which your opinion is largely irrelevant, and not your future behavior. >:> Most of the lack of content has been from those spamming their own >:> newsgroups, not asking for intelligent discussion, just spamming with >:> insults and the like. >: Which again is a very good indication that your content was >: widely considered inappropriate. Like it or not, many >: newsgroups are communities. Outsiders are not necessarily >: unwelcome, but outsiders who barge in and presume to lecture >: from a pedestal are likely to get the rough reception that >: they've earned. > No, its an indication that there are quite a lot of jerks out there who, > when faced with a topic they do not understand and do not want to > understand, find solace in insults. I do like the fact that newsgroups are > communities, particularly that they are communities of wide ranges of views > about the topics they discuss. There are certainly members, such as > yourself mister Scott, who dislike the fact that others may begin a > discussion confident of the knowledge that they have such a right, but > unfortunately you are in the wrong and I am in the right. And playing your > power games, with their complete absence of any rational points, just > illustrates to me that you recognise your complete lack of power in this > circumstance. I'm afraid that it's you who are playing power games. You are the one thumbing his nose at the rest of us and going 'Nyaa, you can't make me leave'. In this you are quite correct: Usenet is an open forum, and I wouldn't have it any other way. But just as you are free to shove your id.8ee fixe in front of people's faces at preposterous (and singularly ineffective) length, so am I free to point out that you are doing so. >: Bluntly, you're a rude, arrogant bastard with the social >: intelligence of a pet rock. On top of that you write some >: of the flabbiest, most turgid prose that it's been my >: misfortune to read anywhere, let alone on Usenet, and >: exhibit several of the familiar stigmata of the Usenet crank >: or monomaniac. If you don't like your reception, mend your >: manners. > I have never been rude to anyone who was not first rude to me, and then only > enough to play the alpha game they initiated to its proper closure. You are mistaken, owing to your inability to recognize the rudeness of your behavior thus far. > I don't > seek contentless arguments; it is others who feel inclined to provide me > with such. I am a bastard; that is true. I was born with an unmarried > mother. Irrelevant, since I use the term in its figurative sense, and certainly no concern of mine in any case. > I can be quite humble when speaking to others who engage in > rational critique or otherwise educate me of my errors. I see you borrowed > the use of turgidity from the other newsgroup spammer. No, I did not. I have no idea even to whom you refer. But I am hardly surprised that someone else used so obviously apt a description. [...] If you are serious about getting together those who are genuinely interested in your views, I suggest that you set up a web-based bulletin board; I'm given to understand that this is very easy to do these days. You can then announce it in the newsgroups in which you think it might be of interest. Reply or not, as you wish; I'll not be responding again outside of threads with genuine linguistic content, if I even bother to read. === Subject: Re: Alternative ways to solve a quadratic equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1P5KUi29951 === Subject: Re: the anticlassicalist }{ ii: the spectre continues > Jacques erred in calling his three trees topologically > equivalent (though in fact two of them are -- as unlabelled > trees) Well, in fact, I would not mind arguing that they are. But since I left the vital bit out, arguing so would smack of a post-hoc argument. > but I doubt that he was trying very hard: you've > offered nothing to suggest that you should be taken any more > seriously on the subject of linguistic phylogeny than, say, > Cavalli-Sforza, who is a linguistic ignoramus. Just right. I'll just give a hint of what I left out, as to me it went without saying for any comparative linguist with a smattering of topology and graph theory. The arcs are not one-dimensional lines, they are (sorry, don't know the proper jargon) they are ribbons, paths, roads with a _width_. > Unfortunately, that is quite relevant. Convincing me is the only way > someone is going to get me to stop posting. Post, post, post, post. And post. On s'en bat l'oeil, Ducon. (Ack, argh, help! Ad hominem, ad hominem! --No, ad mentulam, dickhead--, where's the editor of Maledicta when we need him?) >:> Most of the lack of content has been from those spamming their own >:> newsgroups, not asking for intelligent discussion, just spamming with >:> insults and the like. At least, insults, in these circumstances, have the merit of being honest. J'appelle un chat un chat et Rollet un fripon. As for you, gaga-lact.8ee, it's all mealy-mouthed innuendos and the arguments statements ex cathedra perc.8ee. > No, its an indication that there are quite a lot of jerks out there who, > when faced with a topic they do not understand and do not want to > understand, find solace in insults. Well, well, well... like people who, faced with matters of comparative linguistics, find solace in calling comparative linguists jerks eh? === Subject: Re: Trying to unify axioms. > Nothing. > You forever spewing ing imbecile, an axiom by definition is > irreducible and unprovable. > False. > First off, an axiom isn't necessarily irreducable, unless you want to > claim that reducing it makes the axiom not reducable and therefore not > an axiom. Sometimes we have an axiom that we find out can be reduced > into other axioms. Does that therefore disprove the axiom, or destroy > the usefulness of the axiom, or of taking it as such? No it doesn't. Uh, actually, axioms are irreducible. > Second, axioms are not unprovable. They CAN be proven. For an example > of this, consider the law of identity. Can you prove it? If not, then > how do we even know it's true? We do know it's true, and it IS an > axiom, so that just proves that axioms are not unprovable. The Law of Identity cannot be proven. It is taken to be an axiom of the first-order language with equality, and is only supported by our intuition. Try proving the axiom of commutativity. Oh wait, here's a counter-example: Matrix multiplication isn't commutative! All axioms do is place constraints on what sorts of objects are being studied. 'cid 'ooh === Subject: Zorn's Lemma Question Let S be the real numbers (0,1) Since it is a set of real numbers it is partially ordered and every chain is obviously bounded by 1 Yet (0,1) does not have a maximal element. What am I missing here? === Subject: Re: Zorn's Lemma Question >Let S be the real numbers (0,1) >Since it is a set of real numbers it is partially ordered and every chain is >obviously bounded by 1 >Yet (0,1) does not have a maximal element. >What am I missing here? The chain (0,1), for example, does *not* have a least upper bound *in S.* Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: re:Alternative ways to solve a quadratic equation Here's a simple method which I came up with before I learnt quadratic equations: Write the equation in the form (x + b)x = c Find A and B so that A + B = x + b and A - B = x Obviously A = x + b/2, B = b/2 The equation becomes A^2 - B^2 = c, or A^2 = c + b^2 / 4 === Subject: Re: Interesting problem Robert Israel ha scritto nel messaggio > No, you could only say thus 1/2 is in A if there was some a in A such > that 2a is in B. Of course that can't happen. Why can't 1/2, 1, 2 all > be in B? In fact if 1 is in B, then all positive rationals are in B. > Moreover, all rational powers of rationals are in B. > But there are lots of other numbers, e.g. perhaps some transcendentals, > that could be in A. I can show that both A and B have to be dense in |R+. The problem i found in this problem are trascendental numbers.... i have no idea about a solution... === Subject: Re: Trying to unify axioms. Hi Gregory L. Hansen, You mentioned, If an axiom were reducible or provable, it would be a conclusion, not an axiom. . But if you kept reducing axioms ad infinitum, all the while keeping all axioms everywhere consistent ... then you'd be a masochist, and your result would be convoluted. In short, you'd be a string theorist. If an axiom works it works ... that's good enough. No need to further reduce it. Physicalism is the only theory of everything, and it contains not one equation. Matter is the only reality. Time is perfectly spatial. The future is perfectly fixed but imperfectly known. === Subject: Re: Trying to unify axioms. >> Nothing. > You forever spewing ing imbecile, an axiom by definition is >> irreducible and unprovable. >False. >First off, an axiom isn't necessarily irreducable, unless you want to >claim that reducing it makes the axiom not reducable and therefore not >an axiom. Sometimes we have an axiom that we find out can be reduced >into other axioms. Does that therefore disprove the axiom, or destroy >the usefulness of the axiom, or of taking it as such? No it doesn't. >Second, axioms are not unprovable. They CAN be proven. For an example >of this, consider the law of identity. Can you prove it? If not, then >how do we even know it's true? We do know it's true, and it IS an >axiom, so that just proves that axioms are not unprovable. >(...Starblade Riven Darksquall...) > If an axiom were reducible or proveable, it would be a conclusion, not an > axiom. The axioms would become the axioms used to reach that conclusion. So, then, the axioms in the system of the human mind are human perception and automatic thought processes, and the ability to learn? The fact is, the very idea that we need axioms is false. What we need are principles. Those can be proven but that does not mean they are therefore reducable to other facts. (...Starblade Riven Darksquall...) === Subject: Re: Trying to unify axioms. > Nothing. > You forever spewing ing imbecile, an axiom by definition is > irreducible and unprovable. > False. > First off, an axiom isn't necessarily irreducable, unless you want to > claim that reducing it makes the axiom not reducable and therefore not > an axiom. Sometimes we have an axiom that we find out can be reduced > into other axioms. Does that therefore disprove the axiom, or destroy > the usefulness of the axiom, or of taking it as such? No it doesn't. > Second, axioms are not unprovable. They CAN be proven. For an example > of this, consider the law of identity. Can you prove it? If not, then > how do we even know it's true? We do know it's true, and it IS an > axiom, so that just proves that axioms are not unprovable. > (...Starblade Riven Darksquall...) > An axiom is unprovable by definition. Usually, if an axiom can be > shown to be provable based on other axioms or postulates, it no longer > remains an axiom and is technically a theorem, which by definition is > provable. And of course, nothing is ever provable in and of itself, > which is why a single isolated statement can only be true by > definition. Then explain the human mind. Which axioms did we start with? If I can explain it to you using reason, then does it no longer become an axiom? (...Starblade Riven Darksquall...) === Subject: Re: the anticlassicalist }{ : ism for Daleks <40325A2C.21936CE9@hate.spam.net> <103513r3gafocff@corp.supernews.com <40328829.1E5C341D@hate.spam.net> <1035esdt30h4d38@corp.supernews.com <1039dr5kpjloif1@corp.supernews.com <6IvurhFpHeNAFwU6@baesystems.com> <103cn32aodr3m3a@corp.supernews.com <103n8ds9mudn455@corp.supernews.com In message <103n8ds9mudn455@corp.supernews.com>, galathaea >: OK. So if I'm already unknowingly using this logic, what is the point >: you're trying to make? Believe me, it hasn't yet emerged from all the >: verbiage. >Well, from a practical point of view, understanding the logic of >orthomodular lattices that arises in propositions on a Hilbert space can >prevent a lot of early conceptual difficulties in the use of quantum >mechanics. Really? My experience is that the difficulties arise because quantum objects don't behave intuitively, i.e. like the objects of everyday experience. No amount of bafflegab about orthomodular lattices will change that. [snip another few hundred words...] >Extend to quantum field theories, strings, and such, and maybe it gets >clearer why I see the necessity to alter education of logic in physics >curricula. No. No clearer. >And this does not even touch upon the theory of causal sets and other >directions in physics where such an education would be beneficial. Richard Herring === Subject: Re: the anticlassicalist }{ : ism for Daleks [...] > As I keep trying to point out, my focus is on education with these > series of posts. The theory of models and their logics is crucial > to all science. Oh, no it isn't! > Indeed, it rigorously formalises the notion of a theory. A philosopher of science may have a professional duty to give a toss about such a thing, but a hardworking biochemist or oceanographer by no means does. > Many models across disciplines have a Heyting structure, and these > are often unknown to practitioners in the various fields, so I use > that as my archetype for education reform. There are many cases > where the unfamiliarity with the logical structure of the models > used presents confusion to the students early on. Hello childrens! Today we're going to learn about counting! But first we will need a few set-theoretical preliminaries so as to avoid any confusion about the structure of the natural number system. Des has no Bourbaki today [T]he structural trend in linguistics which took root with the International Congresses of the twenties and early thirties [...] had close and effective connections with phenomenology in its Husserlian and Hegelian versions. -- Roman Jakobson === Subject: Re: Zorn's Lemma Question >Let S be the real numbers (0,1) >Since it is a set of real numbers it is partially ordered and every chain is >obviously bounded by 1 >Yet (0,1) does not have a maximal element. >What am I missing here? >The chain (0,1), for example, does *not* have a least upper bound *in S.* More relevant to a question about Zorn's lemma is the fact that (0,1) does not have an upper bound in S. === Subject: Re: Trying to unify axioms. >> Nothing. > You forever spewing ing imbecile, an axiom by definition is >> irreducible and unprovable. >False. >First off, an axiom isn't necessarily irreducable, unless you want to >claim that reducing it makes the axiom not reducable and therefore not >an axiom. Sometimes we have an axiom that we find out can be reduced >into other axioms. Does that therefore disprove the axiom, or destroy >the usefulness of the axiom, or of taking it as such? No it doesn't. >Second, axioms are not unprovable. They CAN be proven. For an example >of this, consider the law of identity. Can you prove it? If not, then >how do we even know it's true? We do know it's true, and it IS an >axiom, so that just proves that axioms are not unprovable. >(...Starblade Riven Darksquall...) >If an axiom were reducible or proveable, it would be a conclusion, not an >axiom. The axioms would become the axioms used to reach that conclusion. >So, then, the axioms in the system of the human mind are human >perception and automatic thought processes, and the ability to learn? >The fact is, the very idea that we need axioms is false. What we need >are principles. Those can be proven but that does not mean they are >therefore reducable to other facts. The neat thing about building a math with axioms, is that you can first build one, then go back and change just one axioms slightly. Go through the exercise of building the math again, and see the differences between the first and the second build. It's fun to do. The neat thing about math is one doesn't have to include a reality check. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Interesting 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? >In fact if 1 is in B, then all positive rationals are in B. >Moreover, all rational powers of rationals are in B. >But there are lots of other numbers, e.g. perhaps some >transcendentals, that could be in A. Generalization: b in B ==> bQ+ = { bq | q in Q, q > 0 } subset B b in B ==> b^Q+ = { b^q | q in Q, q > 0 } subset B b in B ==> [b^Q+ * Q+]^Q+ = b^Q+ * (Q+)^Q+ subset B and similar for A. Thus as Q+ subset B, B is dense subset R+ and if a in A, aQ+ subset A shows A is dense subset R+ ---- === Subject: Re: Trying to unify axioms. Then explain the human mind. Not all things can be explained with logic, especially men. Hi Gregory L. Hansen, You mentioned, If an axiom were reducible or provable, > it would be a conclusion, not an axiom. . >But if you kept reducing axioms ad infinitum, > all the while keeping all axioms everywhere consistent ... > then you'd be a masochist, > and your result would be convoluted. That's true, but some stopping points are more useful than others. For instance, you might want to stop at something related to physically meaningful measurements, say the invariance of the speed of light, for example. If your postulates go beyond the measurable, you start to just make things up. > In short, you'd be a string theorist. >If an axiom works it works ... that's good enough. > No need to further reduce it. Well, system of axioms, really. >Physicalism is the only theory of everything, > and it contains not one equation. > Matter is the only reality. > Time is perfectly spatial. > The future is perfectly fixed but imperfectly known. Are those morons getting dumber or just louder? -- Mayor Quimby === Subject: Re: Trying to unify axioms. >An axiom is unprovable by definition. Usually, if an axiom can be >shown to be provable based on other axioms or postulates, it no longer >remains an axiom and is technically a theorem, which by definition is >provable. And of course, nothing is ever provable in and of itself, >which is why a single isolated statement can only be true by >definition. >Then explain the human mind. Which axioms did we start with? If I can >explain it to you using reason, then does it no longer become an >axiom? Poincare has a lot to say about that in his book Science and Hypothesis, $4 at Amazon. Basically, you can't explain it using reason. You can't explain things like width to someone that hasn't shared some of your experiences. Very well, he replied, I allow you cow's dung in place of human excrement; bake your bread on that. -- Ezekiel 4:15 === Subject: Re: Trying to unify axioms. >> An axiom is unprovable by definition. In logic, an axiom is provable by definition. === Subject: Re: Trying to unify axioms. >> An axiom is unprovable by definition. > In logic, an axiom is provable by definition. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate. If an axiom was provable, what would you call the statements from which the axiom is deduced? Don't try to teach a pig how to sing. You'll waste your time and annoy the pig. === Subject: Re: Trying to unify axioms. > If an axiom was provable, what would you call the statements from which > the axiom is deduced? In logic, an axiom is provable because it is deducible from itself. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > The way the business schools and law schools would teach that concept > (why does linear regression give the best straight line fit to model > data) would be to have a case where, say, someone using another method > got less than optimal results that ended in disaster and then show/teach > what the calculations might be with linear regression. That wouldn't teach *why* linear regression gives the best straight line. In law school, the case method of teaching is used for a couple of reasons. First, in many areas of law, cases are the *source* of the law. For example, consider copyright law. The statute is in many areas, such as fair use, very vague. It is simply not possible to decide from the statute in most situations whether or not something is fair use. You *must* read with the Supreme Court and other courts have said in actual cases, and from that you can develop a notion of what constitutes fair use. Second, because of the way the precedence system works, citations to previous cases are an important part of any legal argument. === Subject: Re: Trying to unify axioms. >If an axiom was provable, what would you call the statements from which >the axiom is deduced? > In logic, an axiom is provable because it is deducible from itself. That doesn't even make sense. If we assume this is a refrigerator, it follows that this is a refrigerator. That's not a deduction, it's a restatement of the assumption. For every problem there is a solution which is simple, clean and wrong. -- Henry Louis Mencken === Subject: Re: Trying to unify axioms. >That doesn't even make sense. It makes perfect sense. It's a simple observation about how provable is used in logic. === Subject: Re: Trying to unify axioms. > If an axiom was provable, what would you call the statements from which > the axiom is deduced? > In logic, an axiom is provable because it is deducible from itself. What a slick way to do away with critical thinking. Tommy Aquinas would be proud. Isn't this the basis of religious belief? === Subject: Re: Trying to unify axioms. > If an axiom was provable, what would you call the statements from which > the axiom is deduced? > In logic, an axiom is provable because it is deducible from itself. Only one line can be drawn parallel to a given line through an exterior point. Everybody knows that from high school geometry. It is obviously true as Euclid's Fifth Postulate (here restated as Playfair's Axiom). There is only one glitch: It is empirically wrong. There are no lines parallel to a given line on the Earth' surface. You cannot accurately navigate or survey with Euclid. A mile square on the Earth's surface bounds more than a square mile. All triangles on Earth's surface have their interior angles sum to more than 180 degrees (as much as 540 degrees!). Given a circle drawn on the surface of the Earth, the ratio of the circumference to the diameter is always less than pi. By trivial demonstration, you are full of . A axiom is a stated unprovable assumption, It is indefensible for being an axiom - and can be falsified by a single reproducible counterdemonstration. A refrigerator is not a refrigerator if its motor is run backwards to create an oven. All acceptible theories of gravitation must give the same predictions to the extreme limits of experimental error because they all describe the same unique reality. There are only two exceptions: 1) They can disagree about observations that have not been made (e.g., Planck energy regimes), and 2) They can disagree about the Equivalence Principle - that all local bodies fall identically in vacuum. An unmade observation cannot be a constraint. The Equivalence Principle has not been exhaustively tested for violation. Like Euclid, the most elegant and comprehensive internally self-consistent and exhaustively empirically confirmed axiomatic system is only as strong as its weakest axiom. Given just one reproducible counterdemonstration, it all comes crashing down. Philosophy is crap. Humanity languished in pestilence, disease, poverty, famine, filth, and ignorance given 5000 years of continuously refined philosophies. If you want to flush away the crap, you need an engineer. If you wish to wash your hands of it, you need a chemist. Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Trying to unify axioms. > What a slick way to do away with critical thinking. You might as well say that 1+1=2 does away with critical thinking. === Subject: Re: Trying to unify axioms. > A axiom is a stated > unprovable assumption, It is indefensible for being an axiom - and > can be falsified by a single reproducible counterdemonstration. Falsifiability, indefensible - this has nothing to do with it. It's simply a fact that in logic, every axiom of a theory T is provable in T. === Subject: Re: Trying to unify axioms. > A axiom is a stated > unprovable assumption, It is indefensible for being an axiom - and > can be falsified by a single reproducible counterdemonstration. > Falsifiability, indefensible - this has nothing to do with it. It's > simply a fact that in logic, every axiom of a theory T is provable in T. Only one line can be drawn parallel to a given line through an exterior point. Everybody knows that from high school geometry. It is obviously true as Euclid's Fifth Postulate (here restated as Playfair's Axiom). There is only one glitch: It is empirically wrong. There are no lines parallel to a given line on the Earth' surface. Pookie pookie. The are an infinite number of lines that can be drawn parallel to a given line through an exterior point on a hyperbolic surface. Euclid is incomplete for his fifth postulate. Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Trying to unify axioms. Only one line can be drawn parallel to a given line through an > exterior point. Everybody knows that from high school geometry. It > is obviously true as Euclid's Fifth Postulate (here restated as > Playfair's Axiom). There is only one glitch: It is empirically > wrong. There are no lines parallel to a given line on the Earth' > surface. True or false is irrelevant. Every axiom of a theory T is provable in T. This is just a matter of terminology. === Subject: Re: Trying to unify axioms. > What a slick way to do away with critical thinking. > You might as well say that 1+1=2 does away with critical thinking. It does if you proclaim it and fail to define the meaning of addition, the concept of number, etc., and then show how 1+1=2 follows from the definitions. Hey, by your way of definition, I could proclaim 1+1=4 as an axiom and get away with it. === Subject: Re: Trying to unify axioms. >What a slick way to do away with critical thinking. > You might as well say that 1+1=2 does away with critical thinking. Revisit my example. If we assume that this is a refrigerator, it follows that this is a refrigerator. That is, R->R. Or (R ^ R) v (-R ^ -R). Classic tautology, it's true whether R is true or false. So which is it, true or false? R can't resolve that on its own. So choose a postulate: 1a) R 1b) -R Once you've chosen one we can say until the cows come home that R->R or (-R)->(-R), but nothing can be decided until you pick. 1+1=2, on the other hand, is not just a restatment of a postulate. There's actually a chain of reasoning involved that comes to a conclusion that's different from any of the postulates. Irony: Small businesses want relief from the flood of spam clogging their in-boxes, but they fear a proposed national 'Do Not Spam' registry will make it impossible to use e-mail as a marketing tool. === Subject: Re: Trying to unify axioms. > Hey, by your way of definition, I could proclaim 1+1=4 as > an axiom and get away with it. In a theory in which 1+1=4 is an axiom it is, trivially, provable that 1+1=4. === Subject: Re: Documents about multiplicative order in general and mersenne numbers > http://www.cerias.purdue.edu/homes/ssw/cun/ is the homepage for > the Cunningham project with is concerned with factoring numbers > of the form b^n + 1 and b^n - 1. > Material on multiplicative order can be found in pretty much > any book with a title like Introduction to Number Theory, > Elementary Number Theory, etc. Hi Gerry, thank you for this reply. I'll look at the homepage of this project. Is it really true that everything which is known on the multiplicative order is written in introductory books to number theory? I already browsed some. Is this really all which is known about multiplicative order: Ord a (mod n) | Phi(n) Ord a (mod pq) = lcm(Ord a (mod p),Ord a (mod q)) And that for each p which is prime there exist Phi(p-1) primitive roots a where Ord a (mod p) = p-1 (if you take p and q as two distinct primes and n as an arbitrary integer greater than 1) Is this really all? Maybe I forgot some point, but this is what I found in two good introductory books about number theory. I can hardly believe that this is everything which was ever found out about multiplicative order. Could somebody shed a light, please? Thank you in advance kind regards Juergen Bullinger === Subject: Re: Trying to unify axioms. Hey, by your way of definition, I could proclaim 1+1=4 as > an axiom and get away with it. Of course you could. Why not? Radicals are interesting because they were considered 'radical' by modern mathematics depends on. --Another JSH history lesson === Subject: Re: Trying to unify axioms. <0q4%b.127219$FO1.2443626@weber.videotron.net > What a slick way to do away with critical thinking. > You might as well say that 1+1=2 does away with critical thinking. > Revisit my example. If we assume that this is a refrigerator, it follows > that this is a refrigerator. > That is, R->R. > Or (R ^ R) v (-R ^ -R). > Classic tautology, it's true whether R is true or false. So which is it, > true or false? R can't resolve that on its own. Huh? Which is what? R -> R is clearly true, regardless of the truth value of R. What's the issue? > So choose a postulate: > 1a) R > 1b) -R > Once you've chosen one we can say until the cows come home that R->R or > (-R)->(-R), but nothing can be decided until you pick. Absolute nonsense. No matter whether R is true or false, *both* R -> R and ~R -> ~R are true. > 1+1=2, on the other hand, is not just a restatment of a postulate. > There's actually a chain of reasoning involved that comes to a > conclusion that's different from any of the postulates. Oh? What are the postulates from which you derived 1+1=2? Jesse Hughes Besides, discoverers are too proud to kiss butt. Indiana Jones would never kiss some academic's ass to get published, and neither will I. --James Harris === Subject: Re: Trying to unify axioms. > Hi Gregory L. Hansen, You mentioned, If an axiom were reducible or provable, > it would be a conclusion, not an axiom. . > But if you kept reducing axioms ad infinitum, > all the while keeping all axioms everywhere consistent ... > then you'd be a masochist, > and your result would be convoluted. > In short, you'd be a string theorist. *Laughs* That's a good one. > If an axiom works it works ... that's good enough. > No need to further reduce it. Good point. > Physicalism is the only theory of everything, > and it contains not one equation. > Matter is the only reality. > Time is perfectly spatial. > The future is perfectly fixed but imperfectly known. Actually that's not true. Nature is fixxed, but nature is always changing. In that way we can say nature IS progress. The future is not set, but the means by which we must get to the future is essentially set. However, it also posesses degrees of freedom. (...Starblade Riven Darksquall...) === Subject: Re: 'erf' function in C > It is possible to compress the code further at the expense of > clarity (warning: this is not for the faint of heart): > double Phi(double x) > {long double s=x,t=0,b=x,x2=x*x,i=1; > while(s!=t) s=(t=s)+(b*=x2/(i+=2)); > return .5+s*exp(-.5*x2-.91893853320467274178L); > } > This avoids the unnecessary computation of i+1 followed by the > unnecessary conversion of the result from int to long double, > eliminates the unnecessary variable pwr, and precomputes x*x. > Have you checked this algorithm in the tail? For, say, x = -50, I either > get inf/nan or an incorrect answer (0.5), depending on which compiler I > use. The compiler in my tests is gcc, specifically DJGPP or MingW. The > problem is overflow in s. For large arguments one might want to switch to asymptotics. I think, that is what Genz makes in his approx for cdf normal. === Subject: This Week's Finds in Mathematical Physics (Week 202) Also available at http://math.ucr.edu/home/baez/week202.html This Week's Finds in Mathematical Physics - Week 202 John Baez This week I'll deviate from my plan of discussing number theory, and instead say a bit about something else that's been on my mind lately: structure types. But, you'll see my fascination with Galois theory lurking beneath the surface. Andre Joyal invented these in 1981 - he called them especes de structure. Basically, a structure type is just any sort of structure we can put on finite sets: an ordering, a coloring, a partition, or whatever. In combinatorics people count such structures using generating functions. A generating function is a power series where the coefficient of x^n keeps track of how many structures of the given kind you can put on an n-element set. By playing around with these functions, you can often figure out the coefficients and get explicit formulas - or at least asymptotic formulas - that count the structures in question. The reason this works is that operations on generating functions come from operations on structure types. For example, in week190, I described how addition, multiplication and composition of generating functions correspond to different ways to get new structure types from old. Joyal's great contribution was to give structure types a rigorous definition, and use this to show that many calculations involving generating functions can be done directly with structure types. It turns out that just as generating functions form a *set* equipped with various operations, structure types form a *category* with a bunch of completely analogous operations. This means that instead of merely proving *equations* between generating functions, we can construct *isomorphisms* between their underlying structure types - which imply such equations, but are worth much more. It's like the difference between knowing two things are equal and knowing a specific reason WHY they're equal! Of course, this business of replacing equations by isomorphisms is called categorification. In this lingo, structure types are categorified power series, just as finite sets are categorified natural numbers. A while back, James Dolan and I noticed that since you can use power series to describe states of the quantum harmonic oscillator, you can think of structure types as states of a categorified version of this physical system! This gives new insights into the combinatorial underpinnings of quantum physics. For example, the discrete spectrum of the harmonic oscillator Hamiltonian can be traced back to the discreteness of finite sets! The commutation relations between annihilation and creation operators boil down to a very simple fact: there's one more way to put a ball in a box and then take one out, than to take one out and then put one in. Even better, the whole theory of Feynman diagrams gets a simple combinatorial interpretation. But for this, one really needs to go beyond structure types and work with a generalization called stuff typesI've been thinking about this business for a while now, so last fall I decided to start giving a year-long course on categorification and quantization. The idea is to explain bunches of quantum theory, quantum field theory and combinatorics all from this new point of view. It's fun! Derek Wise has been scanning in his notes, and a bunch of people have been putting their homework online. So, you can follow along if you want: 1) John Baez and Derek Wise, Categorification and Quantization. I'd like to give you a little taste of this subject now. But, instead of explaining it in detail, I'll just give some examples of how structure types yield some far-out generalizations of the concept of cardinality. This stuff is a continuation of some themes developed in week144, week185, week190, so I'll start with a review. Suppose F is a structure type. Let F_n be the *set* of ways we can put this structure on a n-element set, and let |F_n| be the *number* of ways to do it. In combinatorics, people take all these numbers |F_n| and pack them into a single power series. It's called the generating function of F, and it's defined like this: |F_n| |F|(x) = sum ----- x^n n! It may not converge, so in general it's just a formal power series - but for interesting structure types it often converges to an interesting function. What's good about generating functions is that simple operations on them correspond to simple operations on structure types. We can use this to count structures on finite sets. Let me remind you how it works for binary trees! There's a structure type T where a T-structure on a set is a way of making it into the leaves of a binary tree drawn in the plane. For example, here's one T-structure on the set {a,b,c,d}: b d a c / / / / / / / / / / on an n-element set is n! times the number of binary trees with n leaves. Annoyingly, the latter number is traditionally called the (n-1)st Cat number, C_{n-1}. So, we have: |T|(x) = sum C_{n-1} x^n where the sum starts with n = 1. There's a nice recursive definition of T: To put a T-structure on a set, either note that it has one element, in which case there's just one T-structure on it, or chop it into two subsets and put a T-structure on each one. In other words, any binary tree is either a degenerate tree with just one leaf: X or a pair of binary trees stuck together at the root: ----- ----- | | | | | T | | T | | | | | ----- ----- / / / / We can write this symbolically as T = X + T^2 Here's why: X is a structure type called being the one-element set, + means exclusive or, and squaring a structure type means you chop your set in two parts and put that structure on each part. (I explained these rules more carefully in week190.) I should emphasize that the equals sign here is really an *isomorphism* between structure types - I'm only using equals because the isomorphism key on my keyboard is stuck. But if we take the generating function of both sides we get an actual equation, and the notation is set up to make this really easy: |T| = x + |T|^2 In week144 I showed how you can solve this using the quadratic equation: |T| = (1 - sqrt(1 - 4x))/2. and then do a Taylor expansion to get |T| = x + x^2 + 2x^3 + 5x^4 + 14x^5 + 42x^6 + ... Lo and behold! The coefficient of x^n is the number of binary trees with n leaves! There's also another approach where we work directly with the structure types themselves, instead of taking generating functions. This is harder because we can't subtract structure types, or divide them by 2, or take square roots of them - at least, not without stretching the rules of this game. All we can do is use the isomorphism T = X + T^2 and the basic rules of category theory. It's not as efficient, but it's illuminating. It's also incredibly simple: we just keep sticking in X + T^2 wherever we see T on the right-hand side, over and over again. Like this: T = X + T^2 T = X + (X + T^2)^2 T = X + (X + (X + T^2)^2)^2 and so on. You might not think we're getting anywhere, but if you stop at the nth stage and expand out what we've got, you'll get the first n terms of the Taylor expansion we had before! At least, you will if you count stages and terms correctly. I won't actually do this, because it's better if you do it yourself. When you do, you'll see it captures the recursive process of building a binary tree from lots of smaller binary trees. Each time you see a T and replace it with an X + T^2, you're really taking a little binary tree: ----- | | | T | | | ----- and replacing it with either a degenerate tree with just a single leaf: X or a pair of binary trees: ----- ----- | | | | | T | | T | | | | | ----- ----- / / / / So, each term in the final result actually corresponds to a specific tree! This is a good example of categorification: when we calculate the coefficient of x^n this way, we're not just getting the *number* of binary planar trees with n leaves - we're getting an actual explicit description of the *set* of such trees. Now, what happens if we take the generating function |T|(x) and evaluate it at x = 1? On the one hand, we get a divergent series: |T|(1) = 1 + 1 + 2 + 5 + 14 + 42 + ... This is the sum of all Cat numbers - or in other words, the number of binary planar trees. On the other hand, we can use the formula |T| = (1 - sqrt(1 - 4x))/2 to get |T|(1) = (1 - sqrt(-3))/2 It may seem insane to conclude 1 + 1 + 2 + 5 + 14 + 42 + ... = (1 - sqrt(-3))/2 but Lawvere noticed that there's a kind of strange sense to it. The trick is to work not with generating function |T| but with the structure type T itself. Since |T|(1) is equal to the *number* of planar binary trees, T(1) should be naturally isomorphic to the *set* of planar binary trees. And it is - it's obvious, once you think about what it really means. The number of binary planar trees is not very interesting, but the set of them is. In particular, if we take the isomorphism T = X + T^2 and set X = 1, we get an isomorphism T(1) = 1 + T(1)^2 which says a planar binary tree is either the tree with one leaf or a pair of planar binary trees. Starting from this, we can derive lots of other isomorphisms involving the set T(1), which turn out to be categorified versions of equations satisfied by the number |T|(1) = (1 - sqrt(-3))/2 For example, this number is a sixth root of unity. While there's no one-to-one correspondence between 6-tuples of trees and the 1 element set, which would categorify the formula |T|(1)^6 = 1 there *is* a very nice one-to-correspondence between 7-tuples of trees and trees, which categorifies the formula |T|(1)^7 = |T|(1) Of course the set of binary trees is countably infinite, and so is the set of 7-tuples of binary trees, so they can be placed in one-to-one correspondence - but that's boring. When I say very nice, I mean something more interesting: starting with the isomorphism T = x + T^2 we get a one-to-one correspondence T(1) = 1 + T(1)^2 which says that any binary planar tree is either degenerate or a pair of binary planar trees... and using this we can *construct* a one-to-one correspondence T(1)^7 = T(1) The construction is remarkably complicated. Even if you do it as efficiently as possible, I think it takes 18 steps, like this: T(1)^7 = T(1)^6 + T(1)^8 = T(1)^5 + T(1)^7 + T(1)^8 . . . = 1 + T(1) + T(1)^2 + T(1)^4 = 1 + T(1) + T(1)^3 = 1 + T(1)^2 = T(1) I'll let you fill in the missing steps - it's actually quite fun if you like puzzles. If you get stuck, you can look up the answer in a couple of different places. While Lawvere was the first to figure this out, the first to write it up was Andreas Blass: 2) Andreas Blass, Seven trees in one, Jour. Pure Appl. Alg. 103 (1995), 1-21. Also available at http://www.math.lsa.umich.edu/~ablass/cat.html There's also a nice treatment based on more general results here: 3) Marcelo Fiore, Isomorphisms of generic recursive polynomial types, to appear in 31st Symposium on Principles of Programming Languages (POPL04). Also available at http://www.cl.cam.ac.uk/~mpf23/papers/Types/recisos.ps.gz In fact, Fiore and Leinster have a nice general theory that explains why the set T(1) acts so much like a sixth root of unity: 4) Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers, available as math.CT/0212377. The idea is that whenever we have an object Z in a distributive category (a category with finite products and coproducts, the former distributing over the latter), and it's equipped with an isomorphism Z = P(Z) where P is a polynomial with natural number coefficients, we can associate to it a cardinality |Z|, namely any complex solution of the equation |Z| = P(|Z|) Which solution should we use? Well, for simplicity let's consider the case where P has degree at least 2 and the relevant Galois group acts transitively on the roots of this equation, so all roots are created equal. Then we can pick *any* solution as the cardinality |Z|. Any polynomial equation with natural number coefficients satisfied by one solution will be satisfied by all, so it won't matter which one we choose. Now suppose the cardinality |Z| satisfies such an equation: Q(|Z|) = R(|Z|) where neither Q nor R is constant. Then the results of Fiore and Leinster say we can construct an isomorphism Q(Z) = R(Z) in our distributive category! In other words, a bunch of equations satisfied by the object's cardinality automatically come from isomorphisms involving the object itself. This explains why the set T(1) of binary trees acts like it has cardinality |T|(1) = (1 - sqrt(-3))/2 or equally well, |T|(1) = (1 + sqrt(-3))/2 (Since the relevant Galois group interchanges these two numbers, we can use either one.) More generally, the set T(n) consisting of binary trees with n-colored leaves acts a lot like the number |T|(n). This has gotten me interested in trying to find a nice model of what I call the Golden Object: an object G in some distributive category that's equipped with an isomorphism G^2 = G + 1 The Golden Object doesn't fit into Fiore and Leinster's formalism, since this isomorphism is not of the form G = P(G) where P has natural number coefficients. But, it still seems that such an object deserves to have a cardinality equal to the golden ratio. James Propp came up with an interesting idea related to the Golden Object: consider what happens when we evaluate the generating function for binary trees at -1. On the one hand we get an alternating sum of Cat numbers: |T|(-1) = -1 + 1 - 2 + 5 - 14 + 42 + ... On the other hand, we can use the formula |T| = (1 - sqrt(1 - 4x))/2 to get |T|(1) = (1 - sqrt(5))/2 which is -1 divided by the Golden Ratio. Of course, it's possible we should use the other sign of the square root, and get |T|(1) = (1 + sqrt(5))/2 which is just the Golden Ratio! Galois theory says these two roots are created equal. Either way, we get a bizarre and fascinating formula: - 1 + 1 - 2 + 5 - 14 + 42 + ... = (1 +- sqrt(5))/2 Can we fit this into some clear and rigorous framework, or is it just nuts? We'd like some generalization of cardinality for which the set of binary trees with -1-colored leaves has cardinality equal to the Golden Ratio. James Propp suggested one avenue. Following Schanuel's ideas on Euler characteristic as a generalization of cardinality, it makes sense to treat the real line as a space of cardinality -1. This will sound crazy unless you go back and read week147, so please do that! Anyway, using this idea it seems reasonable to consider the space of binary trees with leaves labelled by real numbers as a rigorous version of the set of binary trees with -1-colored leaves. So, we just need to figure out what generalization of Euler characteristic gives this space an Euler characteristic equal to the Golden Ratio. It would be great if we could make this space into a Golden Object in some distributive category, but that may be asking too much. Whew! There's obviously a lot of work left to be done here. Here's something easier: a riddle. What's this sequence? un, dos, tres, quatre, cinc, seis, set, vuit, nou, deu,... Now I'd like to mention some important papers on n-categories. You may think I'd lost interest in this topic, because I've been talking about other things. But it's not true! Most importantly, Tom Leinster has come out with a big book on n-categories and operads: 5) Tom Leinster, Higher Operads, Higher Categories, Cambridge U. Press, As you'll note, he managed to talk the press into letting him keep his book freely available online! We should all do this. Nobody will ever make much cash writing esoteric scientific tomes - it takes so long, you could earn more per hour digging ditches. The only *financial* benefit of writing such a book is that people will read it, think you're smart, and want to hire you, promote you, or invite you to give talks in cool places. So, maximize your chances of having people read your books by keeping them free online! People will still buy the paper version if it's any good.... And indeed, Leinster's book has many virtues besides being free. He gracefully leads the reader from the very basics of category theory straight to the current battle front of weak n-categories, emphasizing throughout how operads automatically take care of the otherwise mind-numbing thicket of coherence laws that inevitably infest the subject. He doesn't take well-established notions like monoidal category and bicategory for granted - instead, he dives in, takes their definitions apart, and compares alternatives to see what makes these concepts tick. It's this sort of careful thinking that we desperately need if we're ever going to reach the dream of a clear and powerful theory of higher-dimensional algebra. He does a similar careful analysis of operads and multicategories before presenting a generalized theory of operads that's powerful enough to support various different approaches to weak n-categories. And then he describes and compares some of these different approaches! In short: if you want to learn more about operads and n-categories, this is *the* book to read. Leinster doesn't say too much about what n-categories are good for, except for a nice clear introduction entitled Motivation for Topologists, where he sketches their relevance to homology theory, homotopy theory, and cobordism theory. But this is understandable, since a thorough treatment of their applications would vastly expand an already hefty 380-page book, and diffuse its focus. It would also steal sales from *my* forthcoming book on higher- dimensional algebra - which would be really bad, since I plan to retire on the fortune I'll make from this. Secondly, Michael Batanin has worked out a beautiful extension of his ideas on n-categories which sheds new light on their applications to homotopy theory: 6) Michael A. Batanin, The Eckmann-Hilton argument, higher operads and E_n spaces, available as math.CT/0207281. Michael A. Batanin, the combinatorics of iterated loop spaces, available as math.CT/0301221. Getting a manageable combinatorial understanding of the space of loops in the spaces of loops in the space of loops... in some space has always been part of the dream of higher-dimensional algebra. These k-fold loop spaces or have been important in homotopy theory since the 1970s - see the end of week199 for a little bit about them. People know that k-fold loop spaces have k different products that commute up to homotopy in a certain way that can be summarized by saying they are algebras of the E_k operad, also called the little k-cubes operad. However, their wealth of structure is still a bit mind-boggling. James Dolan and I made some conjectures about their relation to k-tuply monoidal categories in our paper Categorification (see week121), and now Batanin is making this more precise using his approach to n-categories - which is one of the ones described in Leinster's book. There's also been a lot of work applying higher-dimensional algebra to topological quantum field theory - that's what got me interested in n-categories in the first place, but a lot has happened since then. For a highly readable introduction to the subject, with tons of great pictures, try: 7) Joachim Kock, Frobenius Algebras and 2D Topological Quantum Field This is mainly about 2d TQFTs, where the concept of Frobenius algebra reigns supreme, and everything is very easy to visualize. When we go up to 3-dimensional spacetime life gets harder, but also more interesting. This book isn't so easy, but it's packed with beautiful math and wonderfully drawn pictures: 8) Kerler and Volodymyr L. Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners, Lecture Notes in Mathematics 1765, Springer, Berlin, 2001. The idea is that if we can extend the definition of a quantum field theory to spacetimes that have not just boundaries but *corners*, we can try to build up the theory for arbitrary spacetimes from its behavior on simple building blocks - since it's easier to chop manifolds up into a few basic shapes if we let those shapes have corners. However, it takes higher-dimensional algebra to describe all the ways we can stick together manifolds with corners! Here Kerler and Lyubashenko make 3-dimensional manifolds going between 2-manifolds with boundary into a double category... and make a bunch of famous 3d TQFTs into double functorsClosely related is this paper by Kerler: 9) Kerler, Towards an algebraic characterization of 3-dimensional math.GT/0008204. It relates the category whose objects are 2-manifolds with a circle as boundary, and whose morphisms are 3-manifolds with corners going between these, to a braided monoidal category freely generated by a quasitriangular Hopf algebra object. (I'm leaving out some fine print here, but probably putting in more than most people want!) It comes close to showing these categories are the same, but suggests that they're not quite - so the perfect connection between topology and higher categories remains elusive in this important example. Answer to the riddle: these are the Cat numbers - i.e., the natural numbers as written in Cat. This riddle was taken from the second volume of Stanley's book on enumerative combinatorics (see week144). -------------------------------------------------------------- --------- mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html === Subject: Re: Documents about multiplicative order in general and mersenne numbers > Is it really true that everything which is known on the multiplicative > order is written in introductory books to number theory? I'm sorry. I didn't realize that you wanted to know everything that is known about multiplicative order. Since you didn't indicate that you knew anything about multiplicative order, I thought you wanted to know where to begin to learn about multiplicative order. Maybe it would work better if you let us know what it is that you actually want to know about multiplicative order. Then perhaps someone could tell you whether or not what you want to know is known, and, if it is, where to find it. === Subject: Difficulty of calculus vs. discrete math I have taken college courses both in calculus and in discrete mathematics. What surprised me was the difficulties that other students were having with discrete mathematics. For some reason, they found calculus much easier. The discrete math was baby stuff: formal logic, divisibility, combinatorics, and the like. Why would one find calculus easy and discrete math difficult? === Subject: Pigeonhole Speed-Up Consider the following inference, based on the Pigeonhole Principle: (i) There are at least m objects (ii) There are exactly n A's (iii) The f of each object is an A Therefore, (iv) There are two distinct objects a, b such that f(a) = f(b). For any particular choices of m and n, we can express (i)-(iv) as first-order formulas (note that m and n are not variables. When we express (i) as a first-order formula, it just contains a sequence of m existential quantifiers, followed by conjunction of inequalities between the variables). If m>n, this argument is valid in FOL. (When n = 0, premises (ii) and (iii) are inconsistent, so the argument is still valid.) There is a deductive proof of (iv) from {(i), (ii), (iii)} in your favourite proof system for FOL. In a sense, this is a mathematics-free proof. At no point does one refer to numbers, sets or functions. What interests me is the unfeasible length of this mathematics-free proof, when m, n are not tiny. Consider the formula [(i) & (ii) & (iii)] -> (iv) Call this PHP(m,n). If m>n, then PHP(m,n) is true. The informal set theoretical proof of the fact that if m>n, then PHP(m, n) is true is not too long. Problem 1 (Technical): Given m and n such that m>n, how long is a FOL proof of PHP(m,n)? I don't know too much about complexity theory, where such matters are studied. I do know that people have studied the complexity of proofs of propositional formulations of PHP, giving detailed bounds. As a piece of experimental mathematics, I set up my SPASS theorem prover on my PC to prove PHP(m,n), for some small m,n. PHP(2,1), PHP(3,1), PHP(4,1) were very quick (less than a second). For various small m, PHP(m, 2) was very quick also, only a few seconds for PHP(7,2). Curiously, PHP(4,2) took less than PHP(3,2), but then it got longer again. Letting n=3, PHP(5,3) and PHP(6,3) were both several minutes. However PHP(4,3) took 1 hour and 16 minutes! I suspect that PHP(5,4) would take a rather large amount of computing time. (The real-world example below concerns PHP(10^9, 180).) Problem 2 (Philosophical): We all know that PHP(m,n) is true, if m>n. The problem is *how* we know that it's always true. In particular, consider an application of this. E.g., (i) There are at least one ion people (ii) There are exactly 180 countries (iii) Each person is in exactly one country, So, (iv) There are at least two distinct people in the same country. Presumably everyone agrees that (iv) follows logically from (i),(ii),(iii). But in order to justify this belief, one must appeal to genuine mathematics, as one cannot actually perform the maths-free computation feasibly. If the speed-up of such proofs goes as fast as I suspect it does, then it's possible that a first-order proof of (iv) from (i),(ii),(iii) may have more However, informal mathematical reasoning, about a *set* of people, and a *set* of countries, and their cardinal numbers, and a *function* from people to countries, provides a quick proof of (iv) from (i),(ii),(iii). In other words, there seems to be a problem for a very strict formalist about mathematics: they cannot accept the above reasoning, since it directly appeals to numbers, sets and functions. --- Jeff === Subject: Re: Zorn's Lemma Question >>Let S be the real numbers (0,1) >>Since it is a set of real numbers it is partially ordered and every chain is >>obviously bounded by 1 >>Yet (0,1) does not have a maximal element. >What am I missing here? The chain (0,1), for example, does *not* have a least upper bound *in S.* > More relevant to a question about Zorn's lemma is the fact that (0,1) > does not have an upper bound in S. Historical aside... 1. Kuratowski published Zorn's lemma in 1922, Zorn in 1935. Why didn't K's discovery catch on? Maybe because his paper is 34 pages long, and Zorn's only four pages :) 2. I'm not certain, but I think that both of those authors spoke of _least_ upper bounds, and someone later noticed that any upper bound will do. 3. For an ordered set to have a maximal element, it is sufficient that any _well-ordered_ subset have an upper bound. Again I'm not sure who was first, but it might have been someone in the Bourbaki group. === Subject: Re: Difficulty of calculus vs. discrete math > I have taken college courses both in calculus and in discrete > mathematics. What surprised me was the difficulties that other > students were having with discrete mathematics. For some reason, they > found calculus much easier. > The discrete math was baby stuff: formal logic, divisibility, > combinatorics, and the like. > Why would one find calculus easy and discrete math difficult? Well, calculus is - in some weird sense - just the continuum limit of discrete maths. It's like - again in some very warped analogy - the difference between classical mechanics and quantum mechanics. The latter has discrete energy levels, the difference between them being proportional to Planck's constant. Just to give an example of what I mean. You could pretty quickly figure out the definite integral of 1/x from x=4 to x=1000, say. But say you needed the sum of 1/n from n=4 to n=1000. It's certainly possible, but would probably take more time to figure out. The same happens in probability. The Binomial distribution can - in some cases - be approximated with the Normal distribution, and the latter is often much easier to work with. So my conjecture is that calculus is easier to work with, because of the lact of discreteness. Of course, the continuum limit possesses its own quirks, including (but not limited to) the fact that integration and differentiation is not always interchangeable. Another reason might be that calculus is more intuitive. It seems pretty easy understanding integration as a means of calculating area, and differentiation as a means of calculating slope. But combinatorics is often quite surprising, even though in principle it is just a matter of counting. -Michael. === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > So given that the degree I am pursuing is called Masters in *APPLIED* > Statistics, I thought the the courses would be heavily applied and taught in > the context of solving real world problems. No dice. Both courses I took in > the first semester (part-time evening program) had heavy theory. The > Bayesian class was not even as bad as the other one (Mathematical > Statistics). There was essentially no attempt on the part of the professor > to relate the theory to real world programs or to even give real world > examples to illustrate the concepts. It was formula, theory, formula, > theory, theory, formula, etc. I asked him about that and he said there's no > way around the theory. I too recently took a Math Statistics and introductory Bayesian class from a department that calls itself Applied Math and Statistics. But before I took Math Statistics--an introductory class, I tried to take another more advanced class because that was all that was on offer at the time. I found that I would have benefitted more from the advanced class where applied problems were discussed, had I had some exposure to the concepts covered in the Math Statistics class. So I can see how there is no getting around the theory. Perhaps your professor could not think of any examples, perhaps he didn't know of any (Math Stat professors are often theoreticians), perhaps he was just trying to cover the syllabus and examples had not been designed in. Some of these introductory classes try to do too much. The class I took had students from economics, psychology, astronomy, physics, math, bioinformatics, computer science and engineering as well as MS and PhD Applied Statistics students. I personally don't think even one of these groups was well served and this was reflected in the grade distribution which was multi-modal. The course itself tried to fit a years worth of material in a ten-week quarter. The professor barely had time to catch his breath, let alone give relevant examples. Mine was a bad class but I suspect your class too had to accomodate different kinds of students, not just those who were getting a Master's degree in Applied Statistics. One of the challenges of being a graduate student is to cope with and find a way to learn and get something out of a process and system that is not very efficient. -neel === Subject: Re: the anticlassicalist }{ ii: the spectre continues [...] |> Even on the level of predicate |> calculus, is anybody seriously using something in between intuitionist |> logic and classical logic? | |I want to say yes to this... Can you name names? Remember that I was referring to actual doing of mathematics *in* such a logic, not just doing metamathematics of it. [...] |> It seems to me that we need |> both to know more constructive mathematics, and to have better reasons for |> what we think it's pros and cons are. The lack of experience leaves our |> general idea of the relationship between constructive and classical |> mathematics resting on too little. | |And a lack of historical perspective. There are entire lines of inquiry |that are ignored. Excuse me if I word-wrap your text a bit. What's an example of an entire line of enquiry being ignored? I don't doubt that they exist, but I don't know what kind of thing you have in mind. |The general sense of |mathematics as some fixed collection of truths does not reflect the |various debates. It's not clear to me that thinking of it as a fixed collection of truths is a problem per se. I can see that ignoring all the difficulties of interpretation or identification of truths can cause problems. |I just ran across a reference on John Baez' home page |to a philosopher of mathematics calling for a new foundational approach |that reflects what mathematicians do. There is no need for that. I don't see the connection between this remark and the previous one. [...] |Where you ask about someone using something between intuitionist and |classical logic, I need to redirect to semantics. Tarski initiated |algebraic semantics with cylindrical algebras. When you pursue this line |of inquiry, you get to Halmos' polyadic algebras and finally to the |representation theorem I recently found in |Balbes and Dwinger, | |If L is a closure algebra, then L^o is a Heyting |algebra under the partial ordering of L and is also |a D01-subalgebra of L | |More remarkable is the fact that every Heyting |algebra can be represented this way I doubt this is constructively valid, is it? | The closure algebras are Boolean algebras, With more structure. |A Boolean algebra L with an additive closure |operator ^c in which 0^c=0 is called a closure |algebra. An element (a e L) is called open if |(a-bar e L^c). The set of open elements is |denoted by L^o | |So, there is a close relationship between Heyting algebras and Boolean |algebras that seems to be ignored. Are you sure you're correctly gauging what's being ignored? If the closure operator is the familiar operation of topological closure, then the L^o are just the examples I mentioned already, the open sets of topological spaces ordered by inclusion. This is familiar and also covers many of the cases most often used by logicians. |There |is additional structure involved that relates the two. There are semantics |that do not seem to interest many in |questions as they apply in foundations. Well, you certainly can find Heyting algebras that aren't Boolean, and that satisfy additional logical laws beyond intuitionist logic. That gives you a way of defining an intermediate logic. But then what do you plan to do with it once you have it? |Yet, if you are going to embed a Heyting algebra in such a way that |its specification induces a D01-subalgebra, one might ask if the |representation deserves to first be interpreted as a system rather than |in the pieces that can be syntactically represented in isolation. In |other words, is one man's D01-subalgebra another man's bilattice? Well, this kind of mixed structure is a lot like one of the logics in which there are both constructive and classical operators or quantifiers. It's all fine and all, but I think you need some idea of what you intend to do with it. |This is what I mean when I ask why anyone should consider a new attempt |at foundations. During my argument with George Greene this year I |learned a lot by looking for sources to understand his objections. In |large part, it seems as if bibliographies are partitioned. It is not at |all clear to me that there is any debate left. I really believe there |is enough information for a coherent integration of these questions. What questions? Really, you need to pose more definite questions. |You talk about pretending that the law of excluded middle is not true |while implicitly depending upon it. Usually it's pretty overt dependence on it, in the sense that no attempt is made to make the theory constructive. |I think I need to see a formal specification where the natural language |used is not depending on AND and OR in the sense of classical logic. The |best paper I have found thus far is from Jacek Malinowski analyzing |Strawson presupposition. Getting to classical logic from something that |is not classical logic simply does not seem to make sense. But, this is |so far removed from constructive mathematics, that someone who learns |Bishop or Markov with an interest in *mathematics* certainly has no |interest in the regress to presupposition via negation. I don't see how you think such a formal specification will help you. Formal specification of what, by the way? Why doesn't a formal specification done constructively count, or would it? |And, then there is someone like me who stumbled on de Morgan fields and |de Morgan isomorphisms without any clue and without being able to explain |the skepticism in any coherent fashion. Skepticism about what? |The questions are so disjointed. Then you have to address them separately. It seemed for a moment like your thesis was going to be that there were, contrary to my impression, people doing mathematics in intermediate logics (i.e., between intuitionist and classical), but now I can't tell. Give sticking to one thesis a try. |I mentioned the historical perspective just because of my own experience. |I have been all over the place. I do not have any particular |disagreement with Frege's ontology of number. I don't think Frege's ontology of number is a phrase that would be immediately understood by many people. |But, it is not at all clear that that is |the concept of number applicable under algebraic semantics. I don't know what concept you mean. |I intuitively understood that term in the same |sense that others refer to descriptive set theory. You've lost me. I have some idea of what descriptive set theory is, an inkling of algebraic semantics, and have read a few introductory descriptions of Frege's work. I don't see that it makes any sense to say that you understand algebraic semantics to mean descriptive set theory|And, I have finally found a modern philosopher who has |discussed the distinction between Frege's resolution of sematnical |applicability and the problem of descriptive applicability. | |In my own researches, I can find particularly similar statements |concerning two fixed values in Dedekind's writings, Abraham Robinson's |note on threshold logic, and the papers I recently found from the Steklov |Institute of Mathematics in Russia. All of this work seems to relate to |the algebraic semantics. | |And, just for kicks... I would love to be able to even talk to you about |the Jor-Holder theorem. But,... :-) I started out thinking |classically and ended up defending intuitionism with which I did not even |agree from the standpoint of a Heyting algebra. What does it mean to say you agree with intuitionism from the standpoint of a Heyting algebra? |I am not even close to thinking about this in terms of a chain of groups. |But, I have done a lot of diving even if it seems otherwise. | |So, I started this part by saying that no one took my posts seriously. |I cannot say that I blame anyone for that. Nevertheless, the issues are |so spread out over so many topics, it is hard to be coherent until you |figure out what someone else knows or does not know. | |:-) I don't really buy this complaint. It's one thing to be hoping that your audience (some of them) have heard of algebraic semantics, Frege, descriptive set theory, representation theorems for distributive lattices, and Strawsonian presupposition. It's quite a different thing to pretend you're in doubt as to whether the thread you think you see running through all that stuff can be properly conveyed to your audience by this kind of loose wandering around between them. A rule of thumb is that if you can't express your issue as a thesis of some kind, it probably won't come off as a coherent thesis to your audience. Keith Ramsay === Subject: Re: Zorn's Lemma Question >>Let S be the real numbers (0,1) >>Since it is a set of real numbers it is partially ordered and every >chain is >>obviously bounded by 1 >>Yet (0,1) does not have a maximal element. >>What am I missing here? >The chain (0,1), for example, does *not* have a least upper bound *in S.* >More relevant to a question about Zorn's lemma is the fact that (0,1) >does not have an upper bound in S. >Historical aside... >1. Kuratowski published Zorn's lemma in 1922, Zorn in 1935. Why didn't K's >discovery catch on? Maybe because his paper is 34 pages long, and Zorn's >only four pages :) >2. I'm not certain, but I think that both of those authors spoke of _least_ >upper bounds, and someone later noticed that any upper bound will do. Well that's curious, if true. Raises an obvious question... hmm. Ok, let's call that LZL, and the currently standard Zorn's lemma ZL. In fact LZL is as strong as ZL, because LZL allows you to well-order any set by showing that there is a maximal well-ordered subset, and you can use that to prove ZL. Otoh there are situations where ZL applies but LZL cannot be applied directly, for example if we were using ZL to show that (0,1) union (1,2] has a maximal element. Of course that's an absurdly phony example, since there's no way to verify the hypothesis of ZL before you notice that there's a maximal element. Hence the question: Are there any actual (or natural) applications of ZL where not every chain has a least upper bound? >3. For an ordered set to have a maximal element, it is sufficient that any >_well-ordered_ subset have an upper bound. Again I'm not sure who was first, >but it might have been someone in the Bourbaki group. >LH === Subject: Re: Difficulty of calculus vs. discrete math > I have taken college courses both in calculus and in discrete > mathematics. What surprised me was the difficulties that other > students were having with discrete mathematics. For some reason, they > found calculus much easier. > The discrete math was baby stuff: formal logic, divisibility, > combinatorics, and the like. > Why would one find calculus easy and discrete math difficult? I find that discrete objects often have less structure than continuous objects. Hence one will often be stuck with problems with no analytical solution more often in discrete math which makes calculations and proofs more difficult and time consuming. Also, one will find a lot less 'tricks' to solve huge classes of problems in discrete math, while such mechanical tricks abound in continuum math. Hence students often find discrete math a subject where almost every problem requires one to essentially 'start from scratch'. I remember reading just such a post from a student learning graph theory. Note, this is from my own personal experience. === Subject: Re: Difficulty of calculus vs. discrete math |> Why would one find calculus easy and discrete math difficult? i think that there's some potential for talking at cross-purposes in a discussion about this question, because there are both some good reasons and some bad ones, the bad ones relating to the fraudulent corrupt nature of the educational systems in many places around the world. i don't feel like getting into a discussion of the bad reasons at the moment; instead i just want to make a minor comment about some of the good ones, which are mostly very different from the bad ones. |Well, calculus is - in some weird sense - just the continuum limit of |discrete maths. It's like - again in some very warped analogy - the |difference between classical mechanics and quantum mechanics. The latter has |discrete energy levels, the difference between them being proportional to |Planck's constant. | |Just to give an example of what I mean. You could pretty quickly figure out |the definite integral of 1/x from x=4 to x=1000, say. But say you needed the |sum of 1/n from n=4 to n=1000. It's certainly possible, but would probably |take more time to figure out. | |The same happens in probability. The Binomial distribution can - in some |cases - be approximated with the Normal distribution, and the latter is |often much easier to work with. | |So my conjecture is that calculus is easier to work with, because of the |lact of discreteness. Of course, the continuum limit possesses its own |quirks, including (but not limited to) the fact that integration and |differentiation is not always interchangeable. | |Another reason might be that calculus is more intuitive. It seems pretty |easy understanding integration as a means of calculating area, and |differentiation as a means of calculating slope. But combinatorics is often |quite surprising, even though in principle it is just a matter of counting. it's true that sometimes formulas look simpler in the continuum limit because annoyingly complicated high-frequency terms may vanish there. however, as for this business about calculus allegedly being more intuitive, that's highly arguable (both for and against, but especially against), and it's probably better to describe the situation rather differently: in calculus, you pat yourself on the back whenever you manage to solve _any_ problem (for example calculating the area under a curve), because from an intuitive point of view it's actually surprising that you can accomplish _anything_ in the continuum limit. solving the discrete analog of the same problem is often so ridiculously simple (in the area-under-the-curve case it's simply a matter of counting discrete boxes) from a conceptual viewpoint that it doesn't give you much of a sense of accomplishment. thus generally in discrete mathematics all the easy problems have already been done practically even before you get started and what's left is the hard problems, whereas in continuum mathematics it's possible to impress people to a certain extent while working only on problems that are secretly actually surprisingly easy. another phenomenon is this: discrete means zero-dimensional and continuous means higher-dimensional, and naively it seems like higher-dimensional problems should be more difficult than lower-dimensional ones. but sometimes dimension enters into a problem in a negative way, so that the higher-dimensional version of a problem is actually easier than the lower-dimensional one. for example according to what i've heard, the study of dynamical systems has some tendency to get more difficult as the dimension of the orbit space gets higher, but the dimension of the orbit space gets lower as the dimension of the time semigroup gets higher. i said i wouldn't comment on the bad reasons, but i changed my mind: another phenomenon is that in discrete mathematics formulas are often rather complicated though easy to logically derive, while in the continuum limit the analogous formulas are often simpler though more difficult to logically derive. thus at a certain stage of learning, formula-derivers may enjoy discrete mathematics more while formula-memorizers enjoy continuum mathematics more. [e-mail address jdolan@math.ucr.edu] === Subject: Re: Zorn's Lemma Question > 3. For an ordered set to have a maximal element, it is sufficient that any > _well-ordered_ subset have an upper bound. Can we prove that is sufficient without using the Axiom of Choice? In a discussion of Zorn's Lemma, one might like to avoid using AC. === Subject: Re: Zorn's Lemma Question I don't know if this is relevant to the original poster's reason for > 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 which 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 his remarks on the lemma that the upper bound > must be in L. Did your teacher explain why he asked you to deface your books? As there is no nonempty before chain, it's not needed before partially 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 think the in which takes care of that; the pronoun which refers to L. Putting in which at the front tells us that the chain and the upper bound are both in L. === Subject: Re: ellipse 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 are consistent with an ellipse, there will be an infinity of ellipses going through those points. Consider the four points (+/- 1, +/- 1). iel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: Interesting problem >I'm still not 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 everything 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 sets neither of which is void. Maybe I didn't make it clear I want subsets of |R+ so their union is |R+. To rephrase my question, why isn't that an answer of yes to the question? I suppose the answer may be trivial but I don't know that much about the algebra of transcendentals. --Lynn === Subject: Re: ellipse from 4 points In sci.math, : > I 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, this 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. 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 something degrees-of-freedom wise. #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: ellipse from 4 points > I 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, this 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. > You 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 which 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.) > JoeS The test condition, on equations in the general form of form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, is the value of B^2-4*A*C. This 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 degenerate circle with no real points. === Subject: Re: Interesting problem charset=Windows-1252 >>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 everything else? > To rephrase my question, why isn't that an answer of yes to the question? Consider 5 + pi and 5 - pi. === Subject: Re: the 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 than the Boolean algebras they embed in? I would guess that a finitely generated Heyting algebra need not embed in any finitely generated boolean algebra. |and it |is only through the potential application of axioms over an infinity that |any infinities arise. I don't think the phrase application of axioms over an infinity carries any specific meaning. |This, 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 this realm, the finite sharing most |properties with the boolean. The notion of potential infinity is not often stressed by constructivists, in my experience. It's stressed by some people talking *about* constructivism, in an attempt to characterize the difference between constructivism and classical philosophy of mathematics. It sounds like you mean to say that finite Heyting algebras share some natural properties with Boolean algebras. Perhaps you'd like to state some? [...] |: | Constructivism has so many variants that, in order to study |: | them well, most systems are defined and studied using classical |: | means and classical thinking habits. In more vague terms: |: | 'reasoning on the meta level is still classical'. |: |: I don't think 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 which 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 think this project will get very far unless they see some kind of 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, which 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 add up to much of a motivation to think 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 |: things as Martin-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 this 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 in which he presents it in terms of axioms about questions, which amount to the closed subspaces of a Hilbert space. I'm afraid using orthomodular lattices doesn't really keep people from one slit, or through both at the same time?, nor should it. As Feynman once said, all the weirdness boils down to just that one simple case. If you can make coherent sense of that situation, then the rest falls into place. I don't doubt that orthomodular lattices are a useful concept, but I don't think it has much to do with the conceptual difficulties people tend to have with quantum mechanics. |: If you were to count as schools of classical mathematics all the |: different nonconstructive formal systems in which 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 which _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 this narrows the applicability of the logical structure far too |much. Please remember that this was all in response to the remark that if you use a constructive foundation, you suffer from a wealth of alternative constructivisms. Obviously applying a logic foundationally is not the only 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 have a 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 this 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 highly influenced by its Heyting |structure, and I am looking for why this is not so in more fields that |implicitly have the structure hiding away in their analyses. Best of luck with your investigation. But it seems to me that you're being a little too glib with the attribution of deep identificationI once saw a book that argued for the existence of God on the basis that the golden ratio (sqrt(5)+1)/2 occurred in a supernatural profusion. I think they failed to appreciate the extent to which this occurred simply because 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 obvious partial ordering in the background, and the Heyting algebra structure comes from it. Sure, this is commonplace, but where does that really leave us? For a concept to be really fruitful or deep, it has to be more than just frequent in occurrence. If you err too far on the side of generality, you wind up with many examples, but not being able to add much content to any of them. On the other hand, if you managed to uncover a good constructive theorem or a few, you might really have something. [...] |: 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 to which you posted them. I think perhaps you are rather generous in general in attributing mutual relevance to ideas you like, for one thing. Remember that the intended purpose of these rules is to make it easier for people to read what they want to and not be bothered with what they don't want to. I think you've approached this in a manner much like the typical excessive crossposter does. They usually hugely overestimate the interest that their particular message will have for the readership of the groups they're posting to (or don't care). People post stuff to sci.logic about arguments for or against the existence of God, with the explanation that they're interested in discussing the logic of the argument or seeing experts in logic expose the fallacies of their opponents. People will likewise start crossposting threads to sci.math in which 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 this the kind of thing that a person who would read that group would be especially likely to be interested in reading?. It's not some abstract conceptual problem; it's a practical matter of trying to keep some kind of loose order in order to make it easier for people to read what they want and avoid what they don't want. Whether you realize it or not, the fact that a person reads sci.lang is really poor evidence that they'd find this kind of thread interesting. |What I have tried to find is people in all of these groups from several |directions (which 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 which I felt might share a common goal. Often the best approach is to make a brief mention in various groups that you intend to start a discussion somewhere else, and then leave those groups alone. If anybody actually is keenly interested in reading it, they will find it easy to subscribe to the group in which the discussion is now taking 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, they get to keep deleting it as postings generated from other groups keep showing 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 this absolutely |clear. | Massive crossposting has been dubbed velveeta to distinguish it from spam proper. Anything that spreads a message around more than ten times the number of places where it really belongs, though, falls into a common category, however you want to term it. |: I would generally advise against being a self-proclaimed liar, even if |: this is meant in a humorous way (which 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 more context is a little bit like calling someone a tennis player because they have, at some point, played tennis. I am not a badminton playerI am someone who's at some time played badminton. If there's nothing more to it than this, then I'd have to say it's a bit of not especially amusing whimsy. | 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, |which 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 an annoying 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 things in less depth; a diver deals in fewer at greater depth. |: There's nothing 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 something in your hands! | |Be careful here. This 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 this 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 isomorphism 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 Hilbert 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 as necessarily derogatory. Her point was to help people realize that they might be more one way or the other, and that they should quit trying to apply a style that doesn't really suit them, and instead develop whatever their own appropriate style is. On the axis between believing in one right way to do everything, and believing we each should do our own thing, this author certainly would fall well toward the liberal end. I mention it as a way of indicating that I also accept such differences as naturally present, and don't intend to count it against someone if they merely have a different style from mine. On the other hand, your postings I've seen so far have been far from dense in content. I suspect you think they are, because you count as contentual remarks that are only suggestive. |: It seemed to me that a lot of the examples, and maybe all of them, of |: things whose logic is constructive (whatever that is supposed to mean, |: specifically) that we've seen here, are just special cases of the |: topological interpretation. This 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 things a bit, one could say that the complementary |: perception, that something 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 simple algebraic structure like a commutative ring. Whether they actually serve as examples in which the fact of its being a ring is interesting is a different 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 the lack of content has been from those spamming their own |newsgroups, not asking for intelligent discussion, just spamming with |insults and the like. I was talking about your postings, not theirs. Please don't stretch the term spamming beyond all sense. Spamming implies wide distribution. It's often a convenient excuse by massive crossposters that they're not to blame for the resulting flood 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 chip in on the mathematical side of constructivism, |: try helping me satisfy some of my curiosity. I've had the question of the |: degree to which the Jor-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 isomorphisms 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 isomorphism theorem of groups as not being |constructive? The second isomorphism theorem says that if H and K are subgroups of a group, with K normal, then HK/K is isomorphic to H/(H^K). That's the one used in the theorem that any two decompositions (whether with simple quotients 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 the factor 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 the class in HK/K that maps to it. There's no problem there. |Although I haven't explored this before, I never noticed anything hiding 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 this? Take the special case of a group with two decomposition series of length 2: suppose I have a group G with two simple normal subgroups N1 and N2 with simple quotients G/N1 and G/N2. I can write a common refinement of the two decompositions either as 11, then y can be expressed as a product of conjugates of x and x^{-1}. |: I think 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 this in the |opposite direction when looking for constructive deductions. After reading the following, I wonder in what sense you think this is opposite to my approach. A sort of generators-and-relations approach to constructing examples is often good. I think it makes sense also to try 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. This class of |groups is quite large. Then look at the structure required to |constructively prove elements apart from x that have this conjugation |construction equal to the entire group as well. For finite groups, of |course, this can be carried out to completion constructively. For infinite |groups, of course, its much more difficult, although the relationship being |implied is finite between all elements. Constructing the elements to |conjugate through would get you there, though. This is fine; I just don't know as much about infinite simple groups as I'd like. One really wants something like the possibility of making slight deformations in the subgroups, so that they're not too easily distinguished up to isomorphism. |I think this approach is very basic to the logic that I would desire being |taught more. This 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, |which 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 think serious researchers in these areas do combinations of what you would call top down and bottom up. |I know this 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 |thinking of Jor-Holder and similar theorems in a constructive light yet, |so I will need to revisit some of my materials. | |Thank you for your interest, by the way! Keith Ramsay === Subject: Re: Bayesian Class and Math/Stat Teaching Techniques > One of my avocations is evaluating the work of bozos > like you in lawsuits. > Why do think 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 === Subject: Re: ellipse from 4 points > In sci.math, > : > I 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, this 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. > 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 something degrees-of-freedom wise. Pascal's theorem (about an inscribed hexagon) can be rephrased to give a parametric expression for one point on the conic, in terms of five others. Say A,B,C,X,Y are five points. The general point Z is (A(w(CX)))(B(w(CY)) where w is a variable line through (AY)(BX). In the improvised notation here, the product of two lines is their point of intersection, and the product of two points is the line connecting them. === Subject: Re: Bayesian Class and Math/Stat Teaching 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 practices as 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 is the 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 maybe some software you're comfortable with, and start working through some problems on your own. Working on problems will help motivate later study -- working on a problem that you can't solve is 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 -- textbook problems are phrased in a way that supposes one will apply the right technique, i.e., whatever was covered in the chapter. You'll learn more by finding & stating problems in the wildFor what it's worth, Robert Dodier === Subject: Re: Interesting problem >>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 everything else? > To rephrase my question, why isn't that an answer of yes to the question? > Consider 5 + pi and 5 - pi. That's for closure uder addition. Consider pi times 1/pi for closure under multiplication. === Subject: Re: 'erf' function in C > It is possible to compress the code further at the expense of > clarity (warning: this is not for the faint of heart): > double Phi(double x) > {long double s=x,t=0,b=x,x2=x*x,i=1; > while(s!=t) s=(t=s)+(b*=x2/(i+=2)); > return .5+s*exp(-.5*x2-.91893853320467274178L); > } > This avoids the unnecessary computation of i+1 followed by the > unnecessary conversion of the result from int to long double, > eliminates the unnecessary variable pwr, and precomputes x*x. > Have you checked this algorithm in the tail? For, say, x = -50, I either > get inf/nan or an incorrect answer (0.5), depending on which compiler I > use. The compiler in my tests is gcc, specifically DJGPP or MingW. The > problem is overflow in s. > Furthermore, Phi(x) is not increasing in x for values between -13 and -7, > see below (DJGPP using long double). > x Phi(x) > -13 -2.94464e-15 > -12 -2.89807e-15 > -11 6.27672e-16 > -10 6.49085e-16 > -9 6.65294e-16 > -8 1.28093e-15 > -7 1.28042e-12 The attraction of the method for evaluating cPhi(x)=1-Phi(x) through the intermediary R(x)=cPhi(x)/phi(x) is that R(x) has an easily developed Taylor series. Thus an easy way to evaluate R(z+h), given a known value a=R(z), is through b=z*a-1; pwr=1; s=a+h*b; for(i=2;s!=t;i+=2) { a=(a+z*b)/i; b=(b+z*a)/(i+1); pwr=pwr*h*h; t=s; s=s+pwr*(a+h*b); } It happens that even using the Taylor series for z=0 provides better accuracy for Phi(x) than most users are likely to need for applications in probability and statistics. But at the expense of speed and restriction to practical values of x , say, -7[...] >> The subgroups of PSL(2,q) were all classified by L.E. Dickson in about >> 1900. I am not sure what the best reference for that is. It is in >> Huppert's book Endliche Gruppen but that is in German of course! [...] >> Anyway, the subgroups are roughly cyclic groups, dihedral groups >> of order dividing q-1 or q+1 (q odd) or 2(q-1) or 2(q+1) (q even), >> semidirect products PD for a p-group P of order dividing q and cyclic >> group D of order dividing q-1, >Isn't P always a Sylow p-subgroup, and elementary abelian? And >|D| = q-1 for q even, |D| = (q-1)/2 for q odd, D a Cartan >subgroup and so PD a Borel subgroup? Are the Borel groups the >maximal parabolics? Is the normalizer N(D) of D always dihedral? > The largest such P is a Sylow p-subgroup and is elementary abelian. > But I was describing an arbitrary subgroup of PSL(2,q). Yes, I see that now. Clearly there will be such non-abelian subgroups PD for some non-Sylow p-subgroups P whenever PSL(2,p^e) has e composite, and other times. (For an example of the latter, PSL(2,9) =~ A_6 has subgroups isomorphic to S_3.) I think I've got a handle on the e-composite case, but I'll have to ponder the other times case some more. Maybe it's equivalent to the embedded PGL(2,r) situation below? > Again, the maximal D has the order you say, and the maximal PD is a > Borel subgroup. Yes, the normalizer of D is dihedral provided D is > nontrivial. OK. I was thinking |N(D)| = 2|D|, and it probably is for the Cartan subgroups (right?), but I see now that it's not true in general. (The S_3 subgroups of PSL(2,9), as well as its subgroups of order 18, are counterexamples.) Yet Another Thing to Think About (YATtTA). And the Borel subgroups are the maximal parabolics? (This would seem to be strongly implied by what you've said above.) [...] >> A_5 whenever >> 60 divides order, and PSL(2,r) and sometimes PGL(2,r) where q is a >> power of r. >Under what circumstances PGL(2,r)? Guessing from PSL(2,9) =~ >A_6, maybe when the order-r subgroups split into 2 conjugacy >classes? (Which seems like it should be easy to see by looking >at the Borel groups?) > I think, for odd q, PSL(2,q) contains PGL(2,r) iff q is an even power of > r. For example PSL(2,p^2) contains PGL(2,p), but PSL(2,p^3) does not. OK. YATtTA. > Of course, for even q, PSL(2,q) = PGL(2,q). === Subject: Re: Minimally simple finite groups? >> For L_2(p^e) to be minimal simple, its order must not be divisible by >> 60 (otherwise it contains A_5). > Ah. Why must it contain A_5 if its order is divisible by 60? > Hi Jim, > Maybe the following idea can help to answer your question. > If in a field F the equations [...] === Subject: Re: Zorn's Lemma Question