mm-233 === Subject: Parallel lines questionOk maybe someone can help me. My math teacher wants us to write apaper on whether or not a line is parallel to itself and why. Thusfar, i'm pretty sure that a line is not parallel to itself becauseevery definition i have found specifically says two or more lines. Can anyone === Parallel Lines QuestionEuclid originally defined parallel lines as lines which, whenproduced (or extended) indefinitely, never intersect. So you can askyourself whether a line ever intersects itself or not.One of the nice things about words is that they allow you be creative,and sound like you're really saying something when actually whatyou've said is completely unclear. Of course the desire to be clear iswhat led the movement in mathematics to using other symbols whosemeaning can be made more precise; but as long as your teacher has theaudacity to give you a writing assignment in a math class, you mightas well have fun with it - in other words, you can probably chooseeither answer and find a way to verbally justify it.(Euclid: Euclid's Elements Volume One, tranla by Thomas Heath, goodchance its at your local library.) === points out of the space, A,B,C and D.We know that (A,B) pair forms a line and (C,D) pair forms another line.Somehow magically A and B are both on (C,D) line. (because we pick thepoints in the first place, we can make them so).Sometimes people define parallel lines as co-planer lines that never meet.You now have (A,B) line and (C,D) line meet more than once -- and maybeinfinitely times. Can you show that every point on (A,B) is also on (C,D)?Sometimes people define parallel lines as lines with constant distancebetween them. The distance between any point on (A,B) and line (C,D) iszero. (if you can show that every point on (A,B) is also on (C,D)?> Ok maybe someone can help me. My math teacher wants us to write a> paper on whether or not a line is parallel to itself and why. Thus> far, i'm pretty sure that a line is not parallel to itself because> every definition i have found specifically says two or more lines.> Can anyone tell me === lines questionI've always defined parallel lines (in part) as lines in the sameplane which share no common points, or never intersect. Using thisdefinition, I'd say that a line is not parallel to itself since all ofits points are === I'm modern geometry, and we had to define a line. This isthe 300 level in college definition which was a mind opening idea. Thinkabout the symmetries involved in a line, what makes lines parallel? How canyou prove this? Trying thinking in 3D, commonly known as 3 space. If youhave a line in space is it parallel to itself? Can you document that? If you can, then express what you have to say. Referto the documentation if you believe it to be necessary. It seems to me that iftwo lines in a plane have the same slope, then they are parallel. But a linebeing parallel to itself? Your teacher may be trying to provoke thoughtfulcritical reasoning from you to express this === lines question >But a line>being parallel to itself? Your teacher may be trying to provoke thoughtful>critical reasoning from you to express this in writing.Likely that is so, as there === Subject: Freeware Scientific CalculatorI have a freeware calculator that I have spent about 3 months updatingand now have up on my website for download. I need user feedback onusability, any bugs, sugges features, etc. It does expression evaluation, programable with VB Scriptinglanguage, does 2D (polar, cartesian and parametric) and 3D plotting,Matrix manipulation, complex number math, 1-variable equation solving,simultaneous equation solving (2x2 to 8x8), Integer math to 32000places, 2D statistics with graphing and regression curve fitting, unitconversions and geometric area/volume calculations. For documentingcalculations, it has a built in RTF editor, image editor, formulaillustrator and schematic capture module. The design star out as an aid for engineers but has evolved moreinto a learning tool for math students. The link is: www.dazyweblabs.com/dazycalc/index.html === TopologyThe following paper has been published:Algebraic and Geometric TopologyURL:http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3- 27.abs.htmlTitle:Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces Author(s):Hirotaka Tamanoi Abstract:Let G be a finite group and let M be a G-manifold. We introduce theconcept of generalized orbifold invariants of M/G associa to anarbitrary group Gamma, an arbitrary Gamma-set, and an arbitrarycovering space of a connec manifold Sigma whose fundamental groupis Gamma. Our orbifold invariants have a natural and simple geometricorigin in the context of locally constant G-equivariant maps fromG-principal bundles over covering spaces of Sigma to the G-manifoldM. We calculate generating functions of orbifold Euler characteristicof symmetric products of orbifolds associa to arbitrary surfacegroups (orientable or non-orientable, compact or non-compact), in bothan exponential form and in an infinite product form. Geometrically,each factor of this infinite product corresponds to an isomorphismclass of a connec covering space of a manifold Sigma. The essentialingredient for the calculation is a structure theorem of thecentralizer of homomorphisms into wreath products described in termsof automorphism groups of Gamma-equivariant G-principal bundles overfinite Gamma-sets. As corollaries, we obtain many identities incombinatorial group theory. As a byproduct, we prove a simple formulawhich calculates the number of conjugacy classes of subgroups of givenindex in any group. Our investigation is motiva by orbifoldconformal field theory.Secondary: 57S17, 57D15, 20E22, 37F20, 05A15Keywords:Automorphism group, centralizer, combinatorial group theory, coveringspace, equivariant principal bundle, free group, Gamma-sets,generating function, Klein bottle genus, (non)orientable surfacegroup, orbifold Euler characteristic, symmetric products, twissector, wreath === lot of fixed-record length files fromall kinds of sources, and and I am looking for away to determine the record length. In some casesthe records are termina by several combinationsof ; those are pretty easy. The hard files,however, come from IBM mainframes, in EBCDIC, with no line(or record) terminator. The first thing I do is convertthose to ASCII (with the Unix dd conv=ascii) command).The files contain personnel information, such as SSN,employee's name, phone number, and in many cases thereare several fixed consecutive blanks which were reservedbut not used. I figure that if I divide the charactersin 3 main classes: alphabetic, numeric and blanks it wouldbe much mor eefficient than trying an exhaustive bruteforce approach.I one took a digital design class in which the bigrecent discovery was spectral techniques to minimizethe chip's real state use. I guess I am looking for analgorithm like those, but much simpler, since we aredealing with one-dimension only.Any suggestion or comments are welcome === locally compact groupLet $G$ be a locally compact group, and let $wG$ be its weakly almostperiodic compactification. The kernel $K(wG)$ of $wG$ is then a groupisomorphic to the almost periodic compactification of $G$.There are (non-compact) groups such that $wG = K(wG) cup G$ (the unionis necessarily disjoint). For such groups, the closed ideal $wGsetminus G$ of $wG$ then has an identity.Question: Let $G$ be a locally compact group (not compact) such that $wGsetminus G$ has an identity. Does that necessarily mean that $wGsetminus G = === L^2Originator: baez@math-cl-n01.math.ucr.edu (John Baez)What's an easy low-brow way to prove that polynomialfunctions times the Gaussian exp(-x^2) are dense in L^2(R)?I know a general-purpose L^2 Stone-Weierstrass theoremthat does the job, but it requires knowing the Fouriertransform is unitary, which takes a while to prove.I also know the eigenfunctions of the harmonic oscillatorHamiltonian are an orthonormal basis of L^2(R), and thisalso does the job, but again it relies on some stuff thattakes work to prove.For a class I plan to teach, I'd like a quick proofthat doesn't use much machinery, if one === in L^2> What's an easy low-brow way to prove that polynomial> functions times the Gaussian exp(-x^2) are dense in L^2(R)?Suppose that f is in L^2(R) and is orthogonal to all such products. Expanding exp(zx) in a power series and using Fubini's theorem, check thatint_R f(x) exp(zx) exp(-x^2) dx = 0,first for all complex z of sufficiently small modulus, then for all complex z by analytic continuation. In particular,int_R f(x) exp(-itx) exp(-x^2) dx = 0,for all real t. As f(x)exp(-x^2) is clearly an element of L^1(R),this last display means that the Fourier transform of f vanishes. Thus f === Gaussian are dense in L^2> What's an easy low-brow way to prove that polynomial> functions times the Gaussian exp(-x^2) are dense in L^2(R)?See chapter 1 of Igusa's (Springer Ergebnisse, 70's) bookTheta Functions for a low-brow proof. I suspect that thisbook will also have other things you may find useful === are dense in L^2 > What's an easy low-brow way to prove that polynomial functions times > the Gaussian exp(-x^2) are dense in L^2(R)? > I know a general-purpose L^2 Stone-Weierstrass theorem that does > the job, but it requires knowing the Fourier transform is unitary, > which takes a while to prove. > I also know the eigenfunctions of the harmonic oscillator Hamiltonian > are an orthonormal basis of L^2(R), and this also does the job, but > again it relies on some stuff that takes work to prove. > For a class I plan to teach, I'd like a quick proof that doesn't use > much machinery, if one exists. >For each compact interval I, polynomials times exp(-x^2) are dense inthe continuous functions on I with the sup norm; you can appeal toStone-Weierstrass, or its special case for polynomials. And continuousfunctions with compact support are dense in L^2(R).The above does not use Fourier transform being === times a Gaussian are dense in L^2> What's an easy low-brow way to prove that polynomial functions times> the Gaussian exp(-x^2) are dense in L^2(R)?I know a general-purpose L^2 Stone-Weierstrass theorem that does> the job, but it requires knowing the Fourier transform is unitary,> which takes a while to prove.I also know the eigenfunctions of the harmonic oscillator Hamiltonian> are an orthonormal basis of L^2(R), and this also does the job, but> again it relies on some stuff that takes work to prove.For a class I plan to teach, I'd like a quick proof that doesn't use> much machinery, if one exists.>For each compact interval I, polynomials times exp(-x^2) are dense in>the continuous functions on I with the sup norm; you can appeal to>Stone-Weierstrass, or its special case for polynomials. And continuous>functions with compact support are dense in L^2(R).>The above does not use Fourier transform being unitary.>Is this low-brow enough?Possibly a little too low-brow, since it doesn't _quite_ give aproof. Say f is in L^2. You can choose g continuous withcompact support so ||f-g||_2 < epsilon. And now you showthat one can approximate g _on_ the support of g by afunction of the required type, but you actually have toapproximate g in === times a Gaussian are dense in L^2>What's an easy low-brow way to prove that polynomial functions times>the Gaussian exp(-x^2) are dense in L^2(R)?>I know a general-purpose L^2 Stone-Weierstrass theorem that does>the job, but it requires knowing the Fourier transform is unitary,>which takes a while to prove.>I also know the eigenfunctions of the harmonic oscillator Hamiltonian> are an orthonormal basis of L^2(R), and this also does the job, but>again it relies on some stuff that takes work to prove.>For a class I plan to teach, I'd like a quick proof that doesn't use>much machinery, if one exists.>>For each compact interval I, polynomials times exp(-x^2) are dense in>>the continuous functions on I with the sup norm; you can appeal to>>Stone-Weierstrass, or its special case for polynomials. And continuous>>functions with compact support are dense in L^2(R).>>The above does not use Fourier transform being unitary.>>Is this low-brow enough?> Possibly a little too low-brow, since it doesn't _quite_ give a> proof. Say f is in L^2. You can choose g continuous with> compact support so ||f-g||_2 < epsilon. And now you show> that one can approximate g _on_ the support of g by a> function of the required type, but you actually have to> approximate g in === have pos this to amazon.com, and readers of this forumshould also find it to be of interest. Review of Edwin T. Jaynes' book, _Probability Theory: The Logic of Science_ (Five stars (out of five))To pure mathematicians, probability theory is measure theory inspaces of measure 1. To the extent to which you remain a puremathematician, this book will be incomprehensible to you.To frequentist statisticians, probability theory is the study ofrelative frequencies or of proportions of a population; those areprobabilities.To Bayesian statisticians, probability theory is the study ofdegrees of belief. Bayesians may assign probability 1/2 to theproposition that there was life on Mars a billion years ago;frequentists will not do that because they cannot say that therewas life on Mars a billion years ago in precisely half of allcases -- there are no such cases.To _subjective_ Bayesians, probability theory is about subjectivedegrees of belief. A subjective degree of belief is merely howsure you happen to be.Noninformative _objective_ Bayesians assign noninformativeprobability distributions when they deal with uncertainpropositions or uncertain quantities, and replace them withinformative distributions only when they update them because ofdata. Data, in this sense, consists of the outcomes of randomexperiments.Informative _objective_ Bayesians -- a rare species -- ask whatdegree of belief in an uncertain proposition is logicallynecessita by whatever information one has, and they don'tnecessarily require that information to consist of outcomes ofrandom experiments.Jaynes is an informative objective Bayesian. This book is hisdefense of that position and his account of how it is to be used.Pure mathematicians will not find that this book resembles thatbranch of pure mathematics that they call probability theory.Jaynes rails against those he disagrees with at great length.Often he is right. But often he simply misunderstands them. Forexample, writing in the 1990s, he said that pure mathematiciansreject the use of Dirac's delta function and its derivatives, andrela topics. That is nonsense; the delta function has longbeen considered highly respectable, and required material in thegraduate curriculum. Unfortunately Jaynes's misunderstandings maycause some others to misunderstand him when he is right.Statisticians are more informed than pure mathematicians andwill disagree with Jaynes for better reasons. _Some_statisticians will agree with him.This book many flaws, made all the more annoying by the factthat we need to overlook them in order to understand === Goldbach-Idea - probably dumb!Just thought I'd throw in my 2 cents on Goldbach. Not my area, but thethread made me think of the following.If you were to generate a set of false primes by insisting that inany interval they are distribu amongst the odd numbers with thesame frequency as the real ones (we can forget about 2),What is the probability that such a set has the Goldbach property(i.e. any even number can be expressed as the sum of two falseprimes)? Is it non-zero? What would the limit of this probability beif one only had to start looking after an arbitrarily large n?My really stupid idea was that if (subject to some reasonableverification method for sets of false primes being sta) theprobability was non-zero, then the Goldbach conjecture could be trueby accident.If however the probability does tend to zero, then surely theconjecture is equally fascinating!I'm no number theorist, but would love to === shift theoremsHi all.Let $L$ be a second-order strongly elliptic operator over a region $Omega$.Consider the problem of finding the solution $u$ of the problem $Lu=f$in $Omega$ (with $u=0$ on $partialOmega$), where $f$ is areal-valued function defined over $Omega$.Roughly speaking, the usual shift theorem for such problems says thatif $fin H^r(Omega)$, then $uin H^{r+2}(Omega)$, with$$|u|_{H^{r+2}(Omega)} le C |f|_{H^r(Omega)}.$$Can anybody point me to information about how the shift constant $C$depends on the coefficients of the differential operator $L$? Forexample, suppose that appropriate norms of these coefficients arebounded; is there === Jack,Luminet and friends analysis is incorrect...Cornish et al arguments are convincing. I think that Luminet and friends are seeing what they want (wishful thinking?).See more details below.Note also below American Institute of Physics mention of ofthe propulsive effects of the dark energythe vacuum energy is a constant independent of time ... the work done by the pressure [in the adiabatic expansion of the Universe] is just sufficient to maintain the [zero point] energy density constant ... the vacuum acts as a reservoir of unlimi energy, which can supply as much as is required to inflate a given region to any required size at constant energy density. This supply of energy is what is used in 'inflationary' theories of cosmology to create the whole universe out of almost nothing. p. 26 Cosmological Physics, John Peacock, Cambridge 1999Harness the cosmic energy General las MacArthur, Duty, Honor, Country West Point, 1962 ci in Destiny Matrix http://amazon.comNote the creative tension (Ray Chiao, UCB) between the themes of metric locality and global topology.-------------------------------PHYSICS NEWS UPDATEThe American Institute of Physics Bulletin of Physics NewsJames RiordonCOSMOLOGY THEORIES COME AND GO as new information becomesavailable. The geometry and nature of the universe must be one ofthe most fascinating questions for the human species. EarlyEgyptians thought the universe was a rectangular box. AlexandrianGreeks pictured the cosmos as a set of concentric crystallinespheres, a view adop by the medieval Catholic Church, whichexecuted Giordano Bruno for holding that the universe was infinitein extent. In the 20th century Hubble's surveys of recedinggalaxies suppor the idea of an expanding spacetime scaffolding.This model, now called the big bang, is generally the accepoverarching theory, but it has been amended several times to includean early inflationary phase and, more recently, the existence ofdark energy, an entity or mechanism which apparently allows theexpansion of the universe visible to our telescopes to be speedingup, and not slowing down. Also not slowing down is the list of newcosmological ideas. Last year's entrant was the ekpyrotic model(http://www.aip.org/enews/physnews/2002/split/588-2.html ),according to which our universe and all the energy and matterresiding therein arises from the collision of two immense membranesembedded in an even larger multi-dimensional volume. Last week'sinteresting new cosmology development was the suggestion that theuniverse is finite and has a dodecahedral (soccerball) geometryleading cosmology news, presen at a meeting in Cleveland,featured observations of very distant (8 to 10 billion light yearsaway) and unusually bright supernovas, recorded by the Hubble SpaceTelescope. This accords with the dark energy model which holds thatthe general expansion of the universe was relatively slow 10 billionyears ago and afterwards got much faster, owing to the propulsiveeffects of the dark energy winning out over the attractive andslowing effects of gravity (paper by Adam Reiss,http://www.phys.cwru.edu/events/cosmol03.php; also see === Science NewsOnline, 11 October ).Subject: Popular Press News Story: A Universe of Riddleshttp://msnbc.com/news/979271.asp?0sl=-41A Universe of RiddlesScientists have solved the deepest mysteries of the cosmos. So why are theystill so confused?By William UnderhillNEWSWEEK INTERNATIONALOct. 20 issue - Sean Paling clumps down the tunnel in hefty work boots, hispath lit only by his miner's lamp. The temperature is a sweaty 37 degreesCelsius; the underground air has a thick, recycled quality to it. This isthe bottom of Britain's deepest mine, 1,100 meters from daylight, asix-minute ride down a clanking elevator. This is not what I signed up for MAYBE NOT. BUT strange and wonderful things are happening beneaththe Yorkshire Moowhere scientists are chasing some of the great riddlesof the universe. A brisk 10-minute walk from the mineshaft lies thecavernous extension of the Boulby Underground Laboratory for Dark MatterResearch, which opened last summer at a cost of 3.1 million pounds. Inbrilliantly lit rooms hewn out of the rock, sheltered from the interferenceof cosmic rays that constantly rain down on the sunlit world aboveground,scientists tend supersensitive experiments. The task at hand: to search forworks.My prediction is that these experiments will be null like the Michelson-Morley experiment to detect the Earth's motion through the Galilean group aether. This is because dark matter, like dark energy, is a virtual exotic vacuum effect if my falsifiable theory is correct. There's been plenty of progress of late. But as things often go incosmology, answers only seem to lead to new, more disturbing questions. Inrecent years scientists have figured out how old the universe is (13.7billion years) and what it's made of (we'll get to that), they've confirmedthat it all star with the big bang (though much may have gone onbeforehand), and they've sent up the Wilkinson Microwave Anisotropy Probe(WMAP) to map the afterglow. If that sounds reassuring, consider a paper in last week's issue ofNature. Dr. Jeffrey Weeks, a mathematician in New York, uses these very mapsto argue that the universe may be shaped with 12 sides, and that passingthrough one side would only put you instantaneously on the opposite side,light years away from where you star.Here is the problem I am struggling with. What you see above is the Star Gate Cosmic Pong Video Game in The Sky, Hall of Teleporting Mirrors Fun House view, which does not appear to be identical physically in all measurable respects, to the multiply-connec closed finite no boundary curved 3D space picture like the bug crawling around a toroidal 2D surface. In a sense, any finite multiply-connec space is a Star Gate effect that needs exotic dark energy/matter vacuum region to sustain it. The issue here is whether or not there are edges or walls i.e. boundaries. Actual walls for the edge of the world would violate the large scale isotropy - homogeneity cosmological principle. Luminet et-al claim that their proposal without boundary obeys this cosmological principle. Of course, the cosmological principle embodied in the FRW metric does not work on smaller scales, e.g. your room has a wall.You can look through a very large traversable wormhole mouth wall or boundary and see the other side and this may be as if the space was without boundary. On the other hand, if there is really an exotic vacuum suppor traversable wormhole mouth you can tweak it for time travel to both past and future in global FRW cosmic time. I am supposing that Hawking's chronology protection conjecture is wrong since we do not need large quantum gravity metric fluctuations here.t'Hooft-Susskind picture that Lp*(t) is cosmic time dependent need not be true since the Universe is domina by exotic vacuum off-mass-shell stuff, not real on-mass-shell stuff, for both dark energy and dark matter, that implies that the FRW parametersOmega (dark energy) + Omega(dark matter) ~ 1 dominates the cosmic expansion, are both w = -1 independent of the FRW scale factor a(t) unlike ordinary on-mass shell w = 0 with a(t)^-3 scaling and radiation w = 1/3 with a(t)^-4 scalingHence total Omega(exotic vacuum) ~ 1 corresponding to /(exotic vacuum in large scale) ~ (Ho/c)^2 ~ (10^-28)^2 cm^-2 independent of cosmic expansion time t.Therefore Lp* = Lp^2/3(c/Ho)^1/3 = 1 fermi, i.e. 1 Gev energy scale is essentially t independent because the Universe stuff is almost totally exotic macro-quantum vacuum with only a very small amount of on-mass-shell stuff that we are made from.This notion of a finitehall-of-mirrors universe has elici skepticism from some astronomebuttheir alternative isn't much better: an infinite stretch of mini-universes,of which ours is only one. Before you get woozy, though, let's get back tobasics: How big is the universe? It's getting bigger all the time, but the shocking news of recentyears is that the pace of expansion is far greater than previously thought.In 1998, astrophysicists discovered that Type 1a supernovae-a class ofexploding stars-were fainter than they should be given their distance fromsince the big bang, it's accelera. What would make that happen? A new mystery force, called dark energy, may work against gravity,driving the galaxies faster and faster apart. (Einstein posi somethingsimilar, then changed his mind.) If dark energy is growing in strength, itcan't simply be leftover oomph from the big bang. Some answers might beforthcoming from NASA's Supernovae Acceleration Probe, to be sent up in2008. The giant telescope would allow astronomers to see farther away, andthus farther back in time, to earlier supernovae explosions. That willhopefully lead to a clearer sense of how dark energy behaved in the past. So what would the existence of dark energy mean? Trouble. Scientists used to think that the universe would eithercollapse back in upon itself as the momentum of the big bang finally peteredout (the big crunch), or it would continue to expand forever. This year ateam of scientists at Dartmouth College proposed another fate: the bigrip. If the power of dark energy continues to grow, eventually it willoverpower gravity and rip apart everything plans just yet. A big rip wouldn't comefor another 22 billion years.Harnessing anti-gravity dark energy on a small scale makes Star Trek Real i.e. warp drive and metric engineering smaller Star Gates. It also allows Doomsday WMD as described in Chapter 9 of Sir Martin Rees's Our Final Hour and also sets the peristent UFO allegations in a completely new context.. What is the universe made of? As scientists observe distant galaxies with ever more powerfultelescopes, they've calcula that the galaxies are too lightweight to beswirling around as violently as they do. They can tell by measuring thestarlight, a rough indication of how heavy the galaxies should be, and thespeed with which the galaxies spin; a denser galaxy should spin faster, likea ballerina who moves her arms to her side. What unseen material could bemaking these galaxies more massive than they appear? Mysterious, theoreticalstuff called dark matter. Recent data from WMAP suggests that dark matter infact accounts for 90 percent of creation.Newsweek International Oct. 20 Issue What exactly is dark matter made of?This is the big error the Pundits are now making IMHO. The Question is: What is The Question? (JA Wheeler) Let experiment decide. My bet, all dark matter detectors will never click with the right stuff.If dark matter exists,the big bang.Wrong IMHO.If so, it's unlike anything we've experienced: it doesn'treflect light and it barely interacts with anything else. (Hence the acronympassing through your body right now, and you wouldn't feel a thing. How do we know WIMPs really exist?This whole WIMP idea is wrong IMHO. It's asking the wrong question. Wrong idea. We don't, not with any certainty. That's what the Anglo-Americanteam at Boulby and others are seeking to remedy. But how do you spot aexperiments that won't show WIMPs themselves but will register theiroccasional collisions with the nuclei of atoms. It's sort of like playingbilliards with an invisible cue ball. Some detectors at Boulby containchemicals that give off a characteristic flash when a nucleus is hit by aWIMP. The detectors are placed underground so that the rock shields themfrom just about everything-except the WIMPs, which pass right through. Why should we care ? Because these are the most basic questions about the nature of ourworld. Says Prof. Carlos Frenk, an astrophysicist at Durham University inBritain: One of the greatest achievements of mankind would be to find outwhat is keeping the galaxies together. If nothing else, the quest will keepastronomers off the streets.-----------------------------them which need to be critically examined as well as some clearly sta ideas and how they connect to the WMAP data. Now for a fresh second look. E for Ellis's writing. S for mine and L for Luminet et-al. First with John Ellis's writing:E1: In thinking about the large-scale shape of the Universe, three interlinked questions must be confron. First, what is its spatial curvature? ... flat ... negatively curved ... positively curved ... ?S1: flat 3D space curvature means discrete parameter k = 0 & continuous parameter Omega0 = 1.0 on the nose in a canonical form of the isotropic - homogeneous large-scale FRW metric solution for guv(x) of Einstein's local scale-dependent geometrodynamic field equationGuv(x,L) + /zpf(x,L)guv(x,L) = -(8piG*(L)/c^4)Tuv(x,L)at scale L when L - c/Ho ~ 10^28 cmG*(L) - G(Newton)/zpf(x,L) - / = Einstein's dark energy cosmological constant.negative 3D space curvature is k = -1, Omega0 < 1positive 3D space curvature is k = +1, Omega0 > 1FRW metric isds^2 = c^2dt^2 - R^2(t)[(1 - kr^2)^-1dr^2 + r^2d(spherical polar angle line element)^2]r is dimensionless co-moving radial coordinate.Einstein's field equation determines a differential equation for R(t).t is a universal global cosmic time ~ 1/T where T is the absolute Kelvin temperature of the cosmic blackbody radiation remnant from the Big Bang.Unlike special relativity, general relativity under these special large-scale conditions of isotropy and homogeneity permits the measurement of a global cosmic time so that the comoving frame is, practically speaking a pragmatically useful global preferred frame of reference for navigating to the stars and beyond. The global Diff(4) covariant solution spontaneously breaks the local Lorentz symmetry in a way familar with the More is different (PW Anderson) principle of emergence in condensed matter physics e.g. superconductors with generalized phase rigidity - Andrei Sakharov's metric elasticity for the emergence of Einstein's gravity with unified dark energy/matter field from the zero point micro-quantum vacuum fluctuations of all fields in the unstable globally flat Minkowski space-time. Einstein's gravity emerges from this inflationary micro-quantum vacuum instability domina by the BCS attraction between virtual electrons and positron holes near the negative energy edge of the -mc^2 Dirac-Fermi surface.H(t) = R(t)^-1R(t),t,t is ordinary time derivativeUniversal cosmic redshift z isemit frequency/observed frequency = 1 + z = R(t observed)/R(t emit)Critical density of on mass shell ordinary stuff on large-scale isrhoc(t) = 3H(t)^2/8piG(Newton)Omega = rho/rhoc = 8piGrho/3H^2Einstein's local field equations implyR(t),t,t = -4piG(Newton)R(t)c^2rho(1 + 3w)w = pressure/energy densityFor ordinary on mass shell matter v/c << 1 w ~ 0rho(ordinary matter w ~ 0) ~ R(t)^-3For electromagnetic far field radiation w = -1/3rho(electromagnetic radiation w = 1/3) ~ R(t)^-4For the random micro-quantum zero point vacuum fluctuations of all quantum fields w = -1rho (all virtual random zero point micro-quantum vacuum field fluctuations) is independent of the cosmic scale expansion factor R(t), where on large-scale FRWthe vacuum energy is a constant independent of time ... the work done by the pressure [in the adiabatic expansion of the Universe] is just sufficient to maintain the [zero point] energy density constant ... the vacuum acts as a reservoir of unlimi energy, which can supply as much as is required to inflate a given region to any required size at constant energy density. This supply of energy is what is used in 'inflationary' theories of cosmology to create the whole universe out of almost nothing. p. 26 Cosmological Physics, John Peacock, Cambridge 1999rho(zero point fluctuations) = c^2/zpf(x,L)/G*(x,L)This follows from Einstein's equivalence principle together with Heisenberg's uncertainty principle.Ho is the present value where c/Ho ~ 10^28 cmacceleration parameter = q = -R,t,tR/R,t^2For a Universe domina by exotic vacuum residual zero point dark energy/matter Grho is domina by c^2/zpfUse the normalized scale factor a(t) = R(t)/R(now) = R(t)/RoEinstein's local field equation implies in FRW case8piG(Newton)rho(total)/3 = Ho^2(Omega(exotic vacuum) + Omega(w ~ 0 ordinary matter now)a(t)^-3 + Omega(real photons)a(t)^-4]kc^2/H(t)^2R(t)^2 = (Omega(exotic vacuum) + Omega(w ~ 0 ordinary matter now)a(t)^-3 + Omega(real photons now)a(t)^-4 - 1Note that as the Universe expands with a(t) getting larger and larger, the cosmic scale independent exotic vacuum zero point energy dominates the cosmic dynamics in the FRW large-scale regime.The exotic vacuum stress-energy density local tensor field istuv(zpf) = (c^4/8piG)/zpfguv/zpf = (Renormalized Planck Length)^-2[1 - (Renormalized Planck Length)^3|Macro-Quantum Vacuum Coherence Order Parameter|^2]This explains why Einstein's cosmological constant / in the FRW limit is so small and not ~ (Newtonian Planck Length)^-2 - it is the Macro-Quantum Vacuum Coherence missing from previous theories.Einstein's local gravity of curved space-time emerges from the ripples in the coherent Goldstone phase of the vacuum coherence order parameter like FM radio.The unified dark energy/matter residual micro-quantum random exotic vacuum zero point fluctuation field emerges from the ripples in the Local Vacuum Coherence Order Parameter = |Higgs Intensity|^1/2e^i(Goldstone Phase)E2: The second question is whether the Universe is 'open' or 'closed' -- that is, is it spatially infinite, containing an infinite amount of matter, or is it spatially finite, containing a finite amount of matter? Positively curved space sections are necessarily closed, but the converse does not necessarily follow: both flat and negatively curved space sections can be finite if their connectivity is more complica than in Euclidean space, meaning that their topology is quite unusual.E3: So the third issue is, what is the large-scale topology of the Universe?S2: Closed means without boundary i.e. a cycle dual to a closed form in Cartan-DeRham co-homology of integrating exterior differential forms on multiply connec manifolds or co-forms or chains with Betti numbers that are equivalent to sums of skeletal simplices (special kinds of graphs - ways of connecting the dots). Wheeler used this in the 50's with his geons of Mass without mass, Charge without charge and Spin without spin for Bohm's hidden variables fit to a Kerr-Newman exotic vacuum solution. This needs a very large gravity coupling 10^40 large than Newton's on the scale of 1 fermi to work.Ellis in E2 notes that a simply-connec space of constant positive curvature must be closed with no boundary and finite. Flat and negatively curved spaces can be closed with no boundary and finite if they are multiply-connec.Why no boundary? Can we physically imagine space with edges, i.e. walls or boundaries? Yes.Is formal topological equivalence physically complete? No.E4: in a flat toriodal space, as you exit right you enter left, and space is finiteS3: This is like the old Pong Video Game. Is this space with boundaries always physically equivalent to a closed space without boundaries? Jack,Luminet and friends analysis is incorrect...Cornish et al arguments are convincing. I think that Luminet and friends are seeing what they want (wishful thinking?).Any way the topology issue served a good purpose. It showed that the majority of physicists confound curvature of a metric compatible connection defined on a manifold (of course, equipped with a metric), with the bending of that manifold as a sub-manifold embedded in a (pseudo)Euclidean space with a large number of dimensions to account for some non trivial topologies. How many dimensions need the large (pseudo)Euclidean space have?Well, it is a well known result (Campbell, J.E., A Course on Differential Geometry, Clarendon Press, Oxford, 1926) that locally any four manifold (here, the spacetime manifold) can be embedded in E^5, the fact is that a four-dimensional manifold equipped with a Lorentzian metric, supposed to satisfy Einstein's field equations (without cosmological term) cannot be embedded even locally in E^5. It is a classical result (see, Eddington, A. S., The Mathematical of Relativity, third edition Chelsea Publ. Co., New York, 1975) that the embedding can be done locally only in a (pseudo)euclidean space E^10. Also, isometric embeddings of general Lorentzian spacetimes would require a lot of extra dimensions (Clarke, C. J. S., On the Global Isometric Embedding of Pseudo-Riemannian Manifolds, Proc. Roy. Soc. A 314, 417-428 (1970)). More precisely, as far as we know, one only gets *all* Lorentzian 4-manifolds as sub-manifolds of a E^91 spacetime with 2 time dimensions. Clarke also showed that Clarke showed that we can get all globally hyperbolic 4-dimensional Lorentzian manifolds as submanifolds of E^89 Minkowski spacetime. Of course, when why use the word (pseudo)Euclidean I mean that any spacetime of this kind is equipped with the usual affine structure that make all them flat. Another issue. Einstein's equations do not fix the topology of the world manifold. In that way, accord to my view all statements that exotic matter, black energy, etc. can change the topology of spacetime creating wormholes is simply wishful thinking since that statement cannot be proved by any logical inference. Indeed, energy momentum produce curvature according to Einstein's equation and not change of topology. Of course, I'm not saying here that nature forbids wormholes. Maybe the universe has indeed an exotic topology and such an object exist naturally or even that such an object can be engineered. What I said, I state again: there is no theory that describes how to build a wormhole. I apologize, if you did not like my comments, and to say the above I found inspiration in Aristotle who in his Nicomachean Ethics, book 1, chapter 6 said in a similar situation where he could not agree with the presentations of some of his friends on a given subject that: `...piety requires us to honor truth above our friends'], and also in my (late) friend Pertti Lounesto that enlightened us for many years with his posters on errors and counterexamples to `theorems' found in the literature on Clifford algebras. I'm sending also a copy of a new paper (it can be found at the IMECC website as RP40/03) that originally I intended to present at Vigier Conference (I spent a whole month on that paper, working 12 hours per day). But, as I told you before I decided that it would be fair from my to Vigier generous offer to pay my ticket to attend that conference.I will submit RP 40/03 for publication in a leading Physics journal. Comments are welcome. Of course, if I someone will prove that I'm wrong (an infinitesimal probability event) in my comments above and concerning RP 40/03 I will be the first to recognize the fact. Jack says, Part of the problem here is what is meant by 'illusion'physically? The relevant analogy here is the gravitational lens effect predic by Einstein.There is no problem seeing that multiple images of a distantgalaxy or quasar may be observed due to an intervening mass (such as alarge galaxy) bending the light rays. We say that the multiplicity of theimages is an illusion --really there is only one source of the image, i.e.,the distant galaxy or quasar. If we see multiple images of some patch of the cosmic backgroundradiation (CMB), we can reason that this is an illusion that impliesthat the universe is a finite manifold -- but of complica topology. Way back when Eddington was writing about cosmology [Cf. *TheExpanding Universe* Cambridge, 1933] he said:hemisphere of the spherical world we have finished the count, and thethis view we are said to use elliptical space (though the name 'elliptical'does not seem very appropriate).' What Eddington had in mind was that the standard 3-sphere S^3 is adouble cover of elliptical space which is also called 3D projectivespace P^3. That is P^3 = S^3/{+1,-1}. And since SU(2) is geometrically S^3,we can write: SO(3) = SU(2){+1,-1} = SU(2)/Z2, where Z2 is the cyclic 2 group.Eddington explains that the elliptical space volume would be half thevolume of the standard 3-sphere--i.e., (pi)^2 R^3, rather than 2(pi)^2 R^3. In terms of the A-D-E classification of spherical spaces, SU(2)/Z2corresponds to the A1 Coxeter graph which is a single node *, the barebeginning of the series. The idea that the universal 3D manifold mightbe SU(2)/ID, with volume 2(pi)^2 R^3/120 (corresponding to the E8 Coxetergraph) is merely an extreme--and very interesting--version ofEddington's idea way back in 1933.OK I think you are saying that there is physically only one closed multiply connec 3-dimensional space with Betti number 6 (I think) with no boundary not 120 of them. Luminet's Fig 3 is confusing in that regard.Your SU(2)/ID collapses all the above Fig 3 pictured 120 dodecahedrons into one with no real faces but a kind of 3D hyper-torus with 6 Star Gate equivalent handles or traversable wormholes that obey the large scale isotropy -> homogeneous cosmological Copernican principle of no preferred location in space. The Star Gate picture applies both for real edges or faces and without them, i.e. multiple connections in a closed finite manifold with no boundary. The essential physical idea I think is this:Any physical 3D space that is finite multiply-connec with no boundary obeying the large-scale FRW Copernican Cosmological Principle cannot physically exist unless it is domina by the local exotic macro-quantum coherent vacuum residual micro-quantum random zero point energy density field /zpf =/= 0. BTW, I just got back from the Science Library here with a huge pileof very relevant papers most of which are in *Classical & Quantum Gravity*Volume 15 (1998). For example, Ringing the eigenmodes from compactmanifolds, by Neil J Cornish and Neil G Turok (pp 2699-2710). It's asif the compact universe rings like a bell, and the topology of the bellhas a big effect on the harmonics--the eigenmodes, which should show === the UniverseIf the universe is sort of like a big soccer ball except when you come out oneside you come in another, could not some of the acceleration being seen(attribu to dark energy) be actually due to the universe pulling on itselfthrough gravity? That is, could gravity c the soccer === mmeron@cars3.uchicago.eduX-NewsOnePostHost: CCHKHKELIKGCDCLBGPOKELKJKKNDPCBCMNBMJLFKAn essay watering an oasis in a wasteland! All the usual suspects>>wandered in to take a cool drink and chat.In the spirit of Mati's advisor's dictum about all the easy stuff>>always having already been done, I think there must be another about>>all worthwhile advances being easily discouragable. If their value>>were obvious from where we stand, then they would already have been>>discovered.>Well, the exception being stuff which appears very difficult right >from the beginning. But yes, I get your meaning here.>A kind of statistical falacy bodes here: while 99% of worthwhile>>programs will have been discouraged at early stages, 99% of easily>>discouragable programs in fact well deserve to be discouraged -- so>>claiming that discouragment is evidence of probable merit (the crank's>>sylogism), is spurious.Yes, right.But it does show that the person on the right path will have to be>>able to persevere in the face of discouragement.Question is, how do you know that you're persevering on the right path >instead of stubbornly clinging to a wrong one. As you said, >statistically the odds are in favor of the secone (by a huge margin). >So, there should be something convincing you to continue.Mati Meron | When you argue with a fool,>meron@cars.uchicago.edu | chances are he is doing just the sameYeah, but what? Confidence in yourself? Do you allow just three strikes or awhole spreader load of dead ends? A person can waste a whole lifetimechasing rainbows(;^)Many a good idea has driven the originator to a pauper's grave. ----- Pos via NewsOne.Net: Free (anonymous) Usenet News via the Web ----- http://newsone.net/ -- Free reading and anonymous posting to 60,000+ groups NewsOne.Net prohibits users from === Re: consecutive composite integersThen, these should be counter-examples:148992 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 97> 148993 = 13 * 73 * 157> 148994 = 2 * 23 * 41 * 79> 148995 = 3 * 3 * 5 * 7 * 11 * 43> 148996 = 2 * 2 * 193 * 193740600 = 2 * 2 * 2 * 5 * 5 * 7 * 23 * 23> 740601 = 3 * 3 * 19 * 61 * 71> 740602 = 2 * 29 * 113 * 1131474244 = 2 * 2 * 29 * 71 * 179> 1474245 = 3 * 3 * 5 * 181 * 181> 1474246 = 2 * 83 * 83 * 1071674434 = 2 * 31 * 113 * 239> 1674435 = 3 * 5 * 7 * 37 * 431> 1674436 = 2 * 2 * 647 * 647106891244 = 2 * 2 * 59 * 673 * 673> 106891245 = 3 * 3 * 3 * 3 * 5 * 19 * 29 * 479> 106891246 = 2 * 7 * 7 * 11 * 229 * 433All these sequences are maximal, i.e. tightly bounded by two> prime numbers.Helmut RichterWhat's your point? In every example, there exists a prime factorlarger than any other factor in the sequence; for example, in the lastsequence, 673 is the largest prime === Then, these should be counter-examples: 148992 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 97> 148993 = 13 * 73 * 157> 148994 = 2 * 23 * 41 * 79> 148995 = 3 * 3 * 5 * 7 * 11 * 43> 148996 = 2 * 2 * 193 * 193 740600 = 2 * 2 * 2 * 5 * 5 * 7 * 23 * 23> 740601 = 3 * 3 * 19 * 61 * 71> 740602 = 2 * 29 * 113 * 113 1474244 = 2 * 2 * 29 * 71 * 179> 1474245 = 3 * 3 * 5 * 181 * 181> 1474246 = 2 * 83 * 83 * 107 1674434 = 2 * 31 * 113 * 239> 1674435 = 3 * 5 * 7 * 37 * 431> 1674436 = 2 * 2 * 647 * 647 106891244 = 2 * 2 * 59 * 673 * 673> 106891245 = 3 * 3 * 3 * 3 * 5 * 19 * 29 * 479> 106891246 = 2 * 7 * 7 * 11 * 229 * 433 All these sequences are maximal, i.e. tightly bounded by two> prime numbers. Helmut Richter What's your point? In every example, there exists a prime factor> larger than any other factor in the sequence; for example, in the last> sequence, 673 is the largest prime factor.> Don CoolAnd it's raised to a power. Once you relax that you a) lose yourproof of the Erdos-Selfridge theorem and b) have a meaninglessresult. Among all the (finitely many) prime factosurely therehas to be a === composite integersThen, these should be counter-examples:148992 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 97> 148993 = 13 * 73 * 157> 148994 = 2 * 23 * 41 * 79> 148995 = 3 * 3 * 5 * 7 * 11 * 43> 148996 = 2 * 2 * 193 * 193740600 = 2 * 2 * 2 * 5 * 5 * 7 * 23 * 23> 740601 = 3 * 3 * 19 * 61 * 71> 740602 = 2 * 29 * 113 * 1131474244 = 2 * 2 * 29 * 71 * 179> 1474245 = 3 * 3 * 5 * 181 * 181> 1474246 = 2 * 83 * 83 * 1071674434 = 2 * 31 * 113 * 239> 1674435 = 3 * 5 * 7 * 37 * 431> 1674436 = 2 * 2 * 647 * 647106891244 = 2 * 2 * 59 * 673 * 673> 106891245 = 3 * 3 * 3 * 3 * 5 * 19 * 29 * 479> 106891246 = 2 * 7 * 7 * 11 * 229 * 433All these sequences are maximal, i.e. tightly bounded by two> prime numbers.Helmut RichterWhat's your point? In every example, there exists a prime factor> larger than any other factor in the sequence; for example, in the last> sequence, 673 is the largest prime factor.> Don CoolAnd in each case, that largest prime factor appears squared, contrary to the sta hypothesis that that prime should not appear to a power higher than 1.If it were only that there is a largest prime, the whole thing would be trivial, as every non-empty set of primes contains a largest === decompostion.(EVD)Is it possible to do EVD on a hermitian === Q: help with Eigen vector decompostion.(EVD)> Is it possible to do EVD on a hermitian matrix that is singular or close> to singular.octave:5> x = [1 2]x = 1 2octave:6> A = x'*xA = 1 2 2 4octave:7> [U,V] = eig(A)U = -0.89443 0.44721 0.44721 0.89443V === support1.mathforum.org (8.11.6/8.11.6/The Math Forum, surprise, you all seem correct about the metric part>>of the in the definition itself.>>I also appreciate the reader pointing out the regular shapes.>>I had seen these shapes before- and would caution that the>>shapes are perhaps as not as regular as they may seem. The octogon>>that comes out at n=3 does not seem to be a true octogon (mabie I'm>>missing something again...) and for n = 4 the situation appears even>>worse, with spaces way out on the perepherie being reached in (a>>minimum of) 4 moves as well as spaces close to the starting square.>>This was already poin out by another reader- take x=(1,1) and >y=(1,2), then d(x,y)=4 and y surely cannot lie on the outer edge of>>the octogon. Of course, it seems correct to point out that the>>octogons keep turning up as part of the picture again and again for>>higher n.About the even / odd properties I mentioned before... at least that>>does appear to be regular for all n, doesn't it?>> Can you nevertheless learn to post properly? Your entire post appears to>be a quote, even though it comes from you. It's hard to follow your>writing.>-- >Ioannis>http:// users.forthnet.gr/ath/jgal/Eventually, _everything_ is understandable.I star posting here for the first time 3 weeks ago and everything I pos kept coming out as one long line. Thoughtthe >`s somehow preven that, but as I see now the reason seems (?) to come from not hitting the enter key at the end of a line (even though the messaging program will automatically scrolldown to the next line). By the way, Can you please post properly?would === convection-diffusion by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9CJJeW11864;Bonjour,Je cherche une liste: les differences entre sequencielles nombres premiep.ex: Nombres premiers: 2,3,5,7,11,13,17,....., Les differences entre eux: === Fin.Differences and convection-diffusion>Bonjour,>Je cherche une liste: les differences entre sequencielles nombres>premiep.ex:> Nombres premiers: 2,3,5,7,11,13,17,.....,> Les differences entre eux: 1,2,2,4,2,4,.... aurevoir, Marian MatuszykVoyez l'Encyclop.8edie en ligne des suites de nombres entierssuite A001223Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === Integrate[x^2*e^x^2,x] by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9CNGnf28134;.snip>Did you look it up in a table?>Another Contour integration>No we were supposed to be able to do this using integration by parts with>>some manipulation of e^x^2. I tried changing it to (e^x)^x, the only>>manipulation I could think of, and can't get an answer that way either. So>>integration tables are out.>Not necessarily. Knowing the answer might give you the hint you need to >get the solution method.There are only 6 choices that I'd call in any way natural. (I like >int(u dv) = uv - int(v du), it's suggestive to me. My wife likes >something else, int(fg) = f int(g) - int (int(g) f'), it's suggestive to >her. I'll use my version.)u = 1 dv = x^2e^(x^2)dx. I can't integrate dv, so I can't do this.>u = x dv = x e^(x^2)dx. I can integrate this dv, so this is >possible.>u = x^2 dv = e^(x^2)dx. I can't integrate this dv.u = x^2e^(x^2) dv = dx. du is a mess, I'd rather put this off for later.>u = xe^(x^2) dv = x dx. Same.>u = e^(x^2) dv = x^2 dx. Well, now du = 2xe^(x^2)dx and v = x^3/3. > v du is 2/3x^4e^(x^2)dx, and we're going the wrong way (the exponent of >the x part is increasing).So the only thing to try is u = x. And you get a formula with the >integral of e^(x^2)dx, which is at least somewhat of an improvement. > Especially === about optimal short sorts by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9D0aJe01615;Very nice code,but could you please explain teh strategy used for this algorithm?gouri>
 sequence
of n items where optimal means (a) the number of comparisons>
required for any permutation is either m or m+1 where m is the
largest> integer less than log2(n!) and (b) the average number
of comparisons> averaged over all permutations of the ordering
is a minimum...This is nearly equivalent to the standard
optimal sorting problem >>of finding the sort that reduces the
worst-case number of comparisons. It's nearly
equivalent....You're right. Not quite the same. If you could
guarantee that every input>permutation required either
m=log_2(n!) comparisons or m+1 comparisons,>then the number of
permutations which require m comparisons, and the>number that
require>m+1 comparisons, would be uniquely determined, hence
would have to be a>solution to your variation of the optimal
sorting problem. This fact is>true for the standard optimal
sort of 5 elements, but is not true of the>standard optimal
sort for 6, 7, or 8 elements.>The minimum sort for 5 elements
is something I often assign as homework. Must be a fun class.
I have the impression that the minimum sort for 5>> elements
is unique with respect to the decision tree.Yes it is. They
demonstrate that along the way. In your first posting,>you
also asked for good code that does this sort. The following is
what>I distribute as part of the homework answer, but it's not
particularly>elegant. It does concentrate on simple nes
conditionals, as opposed>to the natural implementation of the
decision tree; but it does this>at the expense of a larger
number of assignments than would be ideal: if (b < a)
swap(a,b);> if (d < c) swap(c,d);> if (d < b) { swap(a,c);
swap(b,d); }> if (e < b) {> swap(e,b); swap(e,d);> if (b < a)
swap(a,b);> if (c < b) { swap(b,c); if (b < a) swap(a,b); }>
else if (d < c) { swap(c,d); }> } else if (c < b) {>
swap(b,c);> if (b < a) swap(a,b);> if (e < d) swap(e,d);> }
else if (c < e) {> if (e < d) swap(e,d);> } else { // The one
case with only 6 comparisons.> swap(e,c); swap(e,d);> }> //
Assert: a <= b <= c <= d <= e--Darrah Chavey Department of
Math & Computer Science> chavey@beloit.edu Beloit College; 700
College St; Beloit, Wisc.> (608)-363-2220 http://www.beloit.edu/~
chavey 53511-- The main reason Santa is so jolly is
===
existence of an aperiodic monotile - still an open
Math Forum, $Revision: 1.9 primary) with ESMTP id
h9D1F7o04224Has a single polygon that tiles the plane only
aperiodically been found,or proved not to exist, or is this
===
Kepler's Equation (was: How do you do this integral?) by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9D3wMY14665;> Now you can see why
an attempt to solve for
theta,>> even when reduced to an attempt to solve for u,>>
is just as hard as solving Kepler's
equation E - e * sin(E) = M (given M, e, and find E) which
has been studied in great detail.Has anybody ever done detailed
numerical studies of it directly, and not >via the brute force
methods of pushing f=ma and dh/dt=0 ( h = angular >momentum )?
Have the solutions ever been tabula and trea as known >functions?> If
you insist on a formula for E in terms of>> M and e, (0 < e <
1), The Encyclopedic>> Dictionary of Mathematics offers a
Fourier series E = M + sum[n=1 to inf] (2/n)*J_n(n*e) *
sin(n*M) where J_n is the Bessel function
of order n.>> In practice, one uses numerical methods to>>
approximate E.Has anybody ever done a study of how quickly
this expansion converges, as a >function of
===
Kepler's Equation by support1.mathforum.org (8.11.6/8.11.6/The
Math Forum, $Revision: 1.9 primary) id h9DBmRr12050;>John
Schutkeker>> Does anybody else agree with me that a
mathematical solution to Kepler's>> Equation, in closed fom,
is a sufficiently important unsolved mathematical>> problem to
merit inclusion in the Clay Foundation's list of million
dollar>> prizes for mathematics?>I guess it would depend on
who's paying :) Would _you_ stump up US$1 million>for this
problem?>The equation in question is>x = y - a sin y>which we
would like to solve for y in terms of x. I can't swear to it,
but I>think it has been shown that the solution in closed form
(which needs>clarification) is impossible.>Larry>in case y is
known and a is a constant it's simple to slove x.but in case y
is known how could you find the answer to y=x+a sin
===
Misule !! by support1.mathforum.org (8.11.6/8.11.6/The Math
Forum, $Revision: 1.9 primary) id h9DBmSM12054;Salut, Misule
!!Pai da ce nu spui direct, ma, ce te framanta !!ca te ajuta
mosu !!>I spent many-many hours on the Net looking for some
material that I am highly interes in. Can anyone give me some
links or recommend any kind of sources? I am specifically
trying to get a better understanding of the Equations of
Mathematical Physics, of the mathematical modelling of the
physical world. Why some phenomena are described by certain
types of equations (with specific examples)? How various
mathematical tools fit specific fields in Physics? I am also
looking for a description of maths and its equations in
general, as intuitive and advanced in the same time as
===
metric? by support1.mathforum.org (8.11.6/8.11.6/The Math
Forum, $Revision: 1.9 primary) id h9DBmSE12061;I mentioned
earlier... (x_1-y_1, x_2-y_2)= (even number, even number),
then d(x,y) = even number. If (x_1-y_1, x_2-y_2)= (even
number, odd number), then d(x,y) = odd number. Finally, if
(x_1-y_1, x_2-y_2)= (odd number, odd number), then d(x,y) =
even number. In other words, the same rules for adding natural
numbers.Also notice that if d(x,y)= even number, then there
seems to be no path leading from x to y in an odd number of
moves. In the same manner, if d(x,y)= odd number, then there
exists no path leading from x to y in an even number of moves;
this no matter how far out on the board you go and come back.
Let I_n(xy) be the number of knight paths from x to y
consisting of exactly n moves. Take d_2(x,y) = min (over n)
I_n(xy), I_n(xy) not emtpy. Is d_2 a metric? (fairly certain
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9DBmTk12081;dear sirPlease help me
solving this problemTwo trains 200 miles apart are moving
toward each other; each one is going at a speed of 50 miles
per hour. A fly starting on the front of one of them flies
back and forth between them at a rate of 75 miles per hour. It
does this until the trains collide and crush the fly to death.
What is the total distance the fly has flown?And the number of
iteration made by the fly?The solution is simple, using s=t*v,
but is ther another wayof solving it by trough limit?fromprem
===
Please help me solving this problem> Two trains 200 miles
apart are moving toward each other; > each one is going at a
speed of 50 miles per hour. > A fly starting on the front of
one of them flies back and forth > between them at a rate of
75 miles per hour. It does this until the > trains collide and
crush the fly to death. What is the total distance > the fly
has flown?And the number of iteration made by the fly?> The
solution is simple, using s=t*v, but is ther another way> of
solving it by trough limit?from> premYes, it can be solved by
===
in integral calculus - PLEASE HELP by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h9DIvGD10305;I shall be glad if someone will assist me in
finding an answer to the problemI = integral (2b*y^2*dy) /
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problem in integral calculus - PLEASE HELP> I shall be glad if
someone will assist me in finding an answer to the problemI =
integral (2b*y^2*dy) / (a+by+cy^3)a,b,c are constants.>
partial fractions:begin by factoring the denominator...Maple
says: / ----- | / 2 | | ) |_R ln(y - _R)|| 2 b | /
|--------------|| | ----- | 2 || | b + 3 c _R /| _ /where the
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Maple and Pollard-rho by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h9DIvHl10315;I don't know what Maple does, but if p=k*m+1 for
known k,we could iterate the function f(a)=a^k+1rather than
f(a)=a^2+1.Then there would be only 1+(p-1)/k possible values
for f(x) mod prather than (p=1)/2. That should save a factor
sqrt(k).>In Maple, the 'ifactor' command has the option of
using the>Pollard rho method. The help window contains the
comment:The pollard base method accepts an additional optional
integer: >ifactor(n,pollard,k), which increases the efficiency
of the method >when one of the factors is of the form
k*m+1.I'm wondering what the change in algorithm behind the
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subtended angle given chord and arc lengths by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9E1F9v05448;>
 Is it possible
to calculate by a direct, non iterative method, the angle>> at
the centre of a circle subtended by a chord and minor arc of
known>> lengths? A result to any accuracy may be found by
iteration using the following>> equality where A is the length
of the arc, B is the length of the chord>> and the angle is
measured in radians: Ratio A/B=(1/2*angle subtended by B)/(Sin
(1/2*angle subtended by B)) Richard Elderton.>>
woodman@cix.co.ukFind the perpendicular to the chord at its
midpoint by setting>the dot product to the chord and the
perpendicular equal to >zero. Its exact distance to the center
is still unknown.>Find the derivative of the equation of the
chord, which>would be that of a typical circle. This is the
slope to the>line segments tangent to the chord. Then the dot
product>between different angles of the radius, and the
segments>always equal zero due to their perpendicularity.The
Ferret-Serret formulas esp. the radius of curvature>may be
helpful.Once you find the length of the radius to the arc,
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proof of Goldbach Conjecture by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h9E1F8I05416;I have tried the Goldbach once, only ending up
with the realizationsuch that I need to learn.Is there any
problem if you would describe the methods you
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9E1FAn05467;Can anyone help me,
the question is what amount of fertilizer is needed to provide
1lb of phosphorus(P) per 1000 plants to a yard plan with ground
cover on a 6 * 6in spacing? The yard measures 300*500ft. The
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problem in integral calculus - PLEASE HELP by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9E2ho511448;>I shall be glad if
someone will assist me in finding an answer to the problemI =
integral (2b*y^2*dy) / (a+by+cy^3)a,b,c are constants.Try it
===
Intersection area of 2 circles by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
cx1, cy1 and cx2, cy2 and>radiuses r1, r2.>Now if dist(c1, c2)
< r1+r2, the circles intersect. What I need is
the>intersection's center of mass (if 2d can have mass), and
>________________________________________>Looking for a good
game? Do it yourself!>GLBasic - you can
do>www.GLBasic.com>what is the shape of the intersection of 2
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9ECjLM18979;I am not an expert on
what is being talked about here,but nevertheless wan to add
something to this discussion. If X is a metric space and A is
any closed set, then take [x]=[y] if x, y in A. This gives a
quotient spaceX/A. Define the following function on X/A times
X/A:d([x],[y])= min( d(x,y) , d(x,A)+ d(y,A)) ) whered(x,A) =
inf { d(x,a): a in A }. Then d is a metric on X/A (I showed
this a few years back and had the idea for it when I was still
in high school, ~10 yrs ago).I refer to it as the phone metric,
because ifA is chosen as the location of conglomerate of points
(telephones) on a map, for example, and x and y are the
locationsof two people, and d(x,y) is the time/distance it
takes for a person at x to communicate with a person at y
(say, by walking), then d([x],[y]) gives the correc time for a
person at x to communicate with a person at y taking into
account thatthe person at x can walk to the nearest telephone
(same for y)...Of course, A can also be a 12-sided object in
euclidean space. Inmy drawings from high school, I wasnt`t
concerned with the metric side of the problem, but took the
following route:step one, take a line segment and pretendthat
its two end points are joined; then one gets an
abstractcircle. step two, take a disc and pretend that its
edges are actuallyone point; then one gets an abstract
sphere.step three, take a ball and pretend...One question, if
A is supposed to be a 12-sided object, what happenswhen a
large object runs into one of the corners? Does it come
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proper mapping Thm by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h9EHS9e19515;Let F:M->N be a holomorphic function between
complex manifolds,X analytic subset of M. If f|X :X->N is a
closed map does it followthat f(X) analytic subset of N? OR,
if only f(X) is assummed a closed subset, doesit follow that
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overlapping circles. by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h9ELBNg03717;>
>Say I have two circles, A and B, or radii
RA, adn RB. Their centre points>>are loca at LA and LB. If the
distacen between LA and LB is less than>>RA+RB, then they
overlap. What i want to do is calculate the area of
their>>overlap. Is this possible?>This is a standard calculus
problem. Just change the coordinates so >that their centers
have the same x-coordinates, calculate the points of
>intersection, and integrate top minus bottom from point a to
point b.It's also true that anything you can do with calculus
you can do with >just algebra and geometry. Good luck.I don't
know if this is that similiar to the question that was just
asked. I am also looking for the overlap area. What if you
have 2 circles with a radius of 30 (feet for example) and they
are spaced 50' apart from each other, overlap of 10', how do
you calculate the overlap area? In calculating the total area:
A=pi*r(squared) - overlap area. 5,654.87 SQ Ft - overlap area =
total surface area. Please show how the overlap area is
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have two circles, A and B, or radii RA, adn RB. Their centre
points>are loca at LA and LB. If the distacen between LA and
LB is less than>RA+RB, then they overlap. What i want to do is
calculate the area of their>overlap. Is this possible?>This is
a standard calculus problem. Just change the coordinates so
>>that their centers have the same x-coordinates, calculate
the points of >>intersection, and integrate top minus bottom
from point a to point b.It's also true that anything you can
do with calculus you can do with >>just algebra and geometry.
Good luck.>>What if you have 2 circles with a radius of 30
(feet for example) and they are spaced 50' apart from each
other, overlap of 10', >how do you calculate the overlap area?
In calculating the total area: A=pi*r(squared) - overlap area.
>5,654.87 SQ Ft - overlap area = total surface area. Please
show how the overlap area is calcula. Chris.Put one circle's
center at coordinate (0,0) and the other on the xaxis at the
distance you specify, so in your example it would be at(50,0)
assuming this distance is between the centers.Now solve the
system(x-0)^2+(y-0)^2=30^2(x-50)^2+(y-0)^2=30^2This results in
the equation -100x=-2500 so x=25, this is the xcoordinate of
the intersection points of the 2 circles. Draw a
segmentbetween these 2 points, this is a chord. For each
circle, the areabetween the circle and the chord is called a
circular cap (orsegment). The area of overlap is the sum of
these 2 caps.The formula for the area of a circular cap
is:A=(r^2)*[theta-sin(theta)]/2where theta=2*arccos(1-h/r) and
h is the heigth of the cap and thetamust be in radians.So in
you example h=30-25 =5 so A=about 112.5 for the first circle
andsince the 2 circles have the same radius, the area of
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circles.> I don't know if this is that similiar to the
question that was just> asked. I am also looking for the
overlap area. What if you have 2> circles with a radius of 30
(feet for example) and they are spaced 50'> apart from each
other, overlap of 10', how do you calculate the overlap>
area?Use formula (12)
atHow could I find the
general solution to this homogeneous equation with no
===
zero represent all integers.Why? by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h9ENqhX15336;>
|> Right, except that aleph_null is more
properly called aleph_zero.>|> (In English, we count zero,
one, two, not null, one, two; I think>|> the null comes from
German.)So actually in English it's aleph-nought. >But I'm
sure you're right about American. ;-)>|> Have you read the
classic _One Two Three... Infinity_ by the great>|>
*physicist* George Gamow? It was from that book that I first
learned>|> about infinite cardinals, 45 years ago. Your
question reminds me of>|> the following rather misinformed
sentence from Dr. Gamow's otherwise>|> excellent book: We know
that [aleph_zero] represents the number of>|> all
integealeph_one represents the number of all geometrical>|>
points, and aleph_two the number of all curves, but nobody as
yet has>|> been able to conceive any definite infinite
collection of objects that>|> should be described by
aleph_3.Yes, Gamov was a good author for high-school kids, but
got a few too>many things a bit wrong. A bit of a hit-and-run
author. The comment>about being aleph-two curves was way off.
Bothered me for years.>Apart from being wrong about alephs in
general, (vis-a-vis CH), it was>*really* pushing it to call
all functions *curves*, as he did. Curves>definitely carries
with it the idea of continuity at least; and then >there's
only aleph-1 (i.e. c) again.Gamov was also a bit off with his
relativity homilies with Mr Tompkins.>Sure, just for kids, but
one may as well get the basics right...>|> No problem, I'm in a
good mood. I'll even forgive George Gamow *his*>|> amateurish
and incorrect use of terminology.I'll try to be as generous,
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9FChkZ01655;I don't know of a
special name for q, but a chain of suchprimes q_{i+1}=2q_i -1
is called a Cunningham chain of thesecond kind; see
http://mathworld.wolfram.com/CunninghamChain.html>I know that
if both p and 2p+1 are primes, then p is a Sophie
Germain>prime. What is the proper name for a prime q when 2q-1
is also prime?>I couldn't find any references of this. I
apologize if this is too>naive a question for this group. Any
ref's would be apprecia. >Perhaps we could call q a (SG)'
aid,>Conrad F. Connie Eaton>P.S., Is there a name for
===
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h9FFmdn14459; I really need some help on this question. I've
not been studying matrices for more than a couple of weeks and
I am getting in a muddle. Let lamda = L and let epsilon = e. Im
is the identity matrixThe elementary matrices E(r,s;L) and
P(r,s) are defined as follows:E(r,s;L) = Im + Le(r,s)P(r,s) =
Im - e(r,r) - e(s,s) + e(r,s) +e(s,r)Where r does not =
se(r,s)e(u,t) = { e(r,t) s=u and 0 where s doesn't = uIf the
indices i, r, s are all different find the elementary matrix X
such thatP(i,r)E(r,s;L) =
XP(i,r)--------------------------------I star on the answer
and substitu the formal definitions for P(i,r) and E(r,s;L)
and multiplied out the bracketsAfter I collec all my terms
back up I was left withP(i,r) + Le(i,s) I don't know if that
it right or not and I am stuck on how to turn that into the
XP(i,r)I'd really appreciate any help that anyone can give me.
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9FFmil14490;>
 A THEORY ABOUT
AEONs > (Auto confined ElectrOmagnetic eNtites). Wheeler
published in 1955 in Physical Review, Vol. 97.2, an>which
describes the confinement of electromagnetic radiation
by>published in Physics Essays, titled: A continuous model
of>matter based on AEONs, which describes the auto confinement
of>light by its own gravitational field or a
coherent>electromagnetic field. An interesting aspect is that
it turns out>that AEONs obey the quantum mechanical
Schrodinger wave equation>as well as the relativistic Dirac
equation like elementary>matter is built of compressed light
(electromagnetic radiation).>This compression is caused by the
gravitational or electromagnetic forces, >genera by the light
wave itself. Due to this auto confinement, the >frequency of
the electromagnetic radiation increases to the values of For
further information contact by E-mail: J.W.Vegt@tm.tue.nl
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h9FJ1Li28758;>hot-girl
=6 let x : argument of z find max * min of 6cos(x) + 8sin(x)
-------------------------- i wait your hot-advice...please...
thank in advance.|z-10i | =6It is a circle with centre in 10i
and radius 6. Let P the point on the>circle, and Q = 10i.
Then, when OP is tangent to the circle, OP _|_PQ and>|OP| =
sqrt(10^2 - 6^2) = 8.Let a = arcsin(3/5). Obviously, a <
<= x <= pi/2 + aDividing by 10,10((3/5)cos(x) + (4/5)sin(x)) =
10(sen(a)cos(x) + cos(a)sin(x)) = 10sin(x+a)Thenpi/2 <= x + a
<= pi/2 + 2aThe maximum of 10sin(x+a) is then 10, when x + a =
pi/2. The minimum happen>when x + a = pi/2 + 2a. Thensin(pi/2 +
2a) = cos(2a) = cos^2(a) - sin^2(a) = 16/25 - 9/25 = 7/25and
the minimum is 14/5. The product max*min is then 28, a perfect
resultI hope you cite it when you present your hot-work ...>--
>Saludos,Ignacio Larrosa Ca.96estro>A Coru.96a
(Espa.96a)>ilarrosa@matematicas.net>Is it kinder to hot-girl
to give the person a hint and let herthink about it, or to
===
Antidiagonal, InfinityEuclid's postulates are extended by
Hilbert, but they're not castaside, except for one of them. It
becomes not a necessary conditionbut a descriptive condition.
The point is that mathematicians didn'tlet the parallel
postulate stop them from developing non-Euclideangeometry.Say
there is a collection of n many integers. Knowing that half
ofthem are even is the same thing as knowing that given one at
uniformlyrandom, there are even (1:1) odds that it's even (a
multiple of two).I don't have a method to select a random
integer uniformly, but forany such method that there ever
could be the probability of its resultbeing an even integer is
one half.It's not much easier to define a method to get a
random (and in thiscontext implicitly of a uniform
distribution) real number from theunit interval, but the
probability of it being less than one half isone half.Knowing
that probability is the same thing as knowing the fact
aboutthe population that its elements can be divided into two
groups ofequal size, and that these are different groups than
those of thirdsof the unit interval of different sizes. The
probability of a realnumber from the unit interval being from
that interval is preciselytwice that of from two equally sized
half intervals and is exactlyequal to one.Half the integers are
even: represent the non-negative integers as anaturally or
canonically ordered sequence by an infinite binarysequence
(10)..., that sequence has a density of one half. The
otherhalf of the non-negative integers is odd, the odd
integers are theother half of the integers to the even
integers. The multiples ofthree are not the other half of the
integers to non-multiples ofthree, they're the other third to
multiples of three plus one andmultiples of three plus two,
there are three equal categories, thirds. To demand otherwise
in ignorance of that plain fact is ridiculous. To be sure
there is a sophistica enough understanding of these asinfinite
sets that a function is easy to define among the multiples
oftwo and the multiples of three, but a closer examination
that aspecification of the function as being in the form x+C
gives a muchmore intuitive match to these well known sequences
of odd and evennumbers definitely showing that being in line
with expectation. Thetrue sophisticate demands more to
reconcile the intuition withsuccinct provability.Comparing the
cardinal numbers of two sets is one way to compare theirsize,
qualitatively for infinite sets. There are other ways
toconsider the comparative size of two infinite sets. If a set
is aproper superset of another, it is in that sense larger. If
a set hasinfinitely many proper subsets that are proper
superset of anotherset, it is in that sense larger. If the
sets are partially disjointor disjoint, compare each to their
union. Determine if there is aleast polynomial relating them,
with absolutely minimum order andcoefficients. Sets can
generally reduce to sets of numbers orordinals, or composites
of the empty set. These methods bythemselves don't work on all
sets, but on the ones they do they havesome meaning.How is it
proven that half the integers are even? One method toconsider
is the asymptotic density of the positive even integers
amongthe positive integeit is 1/2, and has a equivalent
meaning to thathalf of the positive integers are even.How
about proving that half of the non-zero integers are positive?
The integers are, after all, symmetric about the origin. This
bringsinto consideration more than the elements of the
positive and negativeintegers as sets, but also that for each
element of one set itsnegative exists in the other set and
that that's negative is again theelement. This is different
than noting that -2x exists in the otherset, as -2x of that is
4x of the original, there is a function betweenthe two, and its
inverse is the same function, f(x)= -x. Anotherconsideration is
that if there exists a random distribution over thepositive
integethen flipping a coin for the result and calling itthus
positive or negative gives a random distribution over
thenon-zero integers.I read the other day that the probability
in selecting two integers atrandom that they are coprime is 6 /
pi^2.I want to say that the set of real numbers comprising the
interval [0,1) can be split into two lesser subsets equal in a
size to themselves,[0, 1/2) and [1/2, 1). Then, I expect to say
that the size of each ofthose subsets is less than that of
[0,1), and the size of the set ofthe elements comprising
[-1/2, 1/2) is equal to that of [0,1), theyare rela via
f(x)=x-1/2.The linear translation f(x)=x+C being defined is
===
Factorial/Exponential Identity, InfinityI'll clarify that: the
current state of opinion appears to be thatalmost all real
numbers are absolutely normal and uncountably many(irrational)
real numbers are absolutely abnormal. Those are notnecessarily
my conclusions nor premises, although many reals
areabnormal.No rational number is normal. Many rational
numbers have zero-densityof one half, as a number normal to
base two or absolutely normal does.Even Homer nods. Wake each
sleeping Homer. Assume the Homer's arein a line and have each
awake Homer awaken the Homer to his left. Ifonly the even
Homers doze then half of them are asleep. Pleaseexplain what
you meant by Even Homer nods. Is it a nod of assent?A number
normal to base two is normal to base 2^x for positiiveinteger
x.It might seem more intuitive from our finitist understanding
that veryfew numbers would be normal. While that might be the
case is is yetapparent that many irrational numbers appear to
have zero-density ofone half, a requisite condition of them
being normal.Why ask? Why not? In this thread we've discussed
an expression of n!/ (n/2)!^2 2^n as ndiverges to infinity.
That would seem to imply that that many of thesequences have
equal densities of ones and zeros. Yet, in light ofthat fact
and possibly contradictorily to it while that might be
thecase, any absolutely normal number has equal densities of
zeros andones, and Borel says the Lebesgue measure of abnormal
numbers is zero. This is where I just thought that half the
binary numbers had equaldensities of ones and zeros.This leads
back to varying definitions of measure, and as well
theconsideration of divergent sums.Assuming that we have some
mutual comprehension of what density iswithin an infinite
binary sequence, is it so that a number normal tobase two has
===
Naive Q: Set theory, logic - which comes first?When
mathematicians say that a theorem has been proven, they
mean>that its truth has been established beyond all doubt. No.
That is not either a Mathematical concept or an achievable
goal.You have a point, two points actually, but I did not say
that proofis a mathematical concept, and it may be that it is
impossible for usto really prove anything (I do not think so,
though).The integers have more properties than can be captured
by any axiom>system.And what would those be? Any that follow
from the definition of natural number I give below.> How does
one distinguish an axiom system in> which those proerties are
true from one in which they are not?In some cases it is easy
to tell; in others it is not (cases may evenexist where it is
impossible for us humans).In his paper G.9adel showed how, for
any sound formal system for naturalnumbers (that is, one in
which all provable statements are true), onecan find a true
statement about natural numbers not provable in theformal
system. If one adds that statement as an axiom to the
originalsystem one gets a new sound formal system, but if
instead one adds itsnegation one gets an unsound formal
system.You said that there are statements about natural
numbers that can>neither be proven nor disproven,The term
Natural Numbers usually refers to the axiom system known as>
the Peano Postulates.This is false. G.9adel showed how to find
true statements about naturalnumbers not provable from the
(first-order) Peano Postulates.which may or may not be true,
but>does not follow from what G.9adel showed.It certainly
does, unless you are using the term natural numbers as> some
vague metaphysical concept having nothing to do with any
axiom> system. What do *you* mean by natural numbers?I do not
know what you mean by a metaphysical concept. The
naturalnumbers are the numbers0, 1, 2, 3, 4, ....When spelled
out this becomes0, s(0), s(s(0)), s(s(s(0))), s(s(s(s(0)))),
... .The natural numbers can be obtained from two
ingredients:1) The natural number 0.2) The successor operation
s, which applied to a natural number yieldsa new natural
number.Natural numbers are perfectly well-defined, despite the
fact that theycannot be axiomatised in a first-order system.
There is nothingstrange about natural numbers. We encounter
them every day.G.9adel did show that in a given>formal system
for natural numbers there are statements about natural>numbers
that can neither be proven nor disproven.Which means that if
you stick to Mathematics my statement [there are statements
about natural numbers that can neither be proven nor
disproven] > is true. It> doesn't matter which axiom system
you pick, as long as it includes> Peanos' Postulates among its
consequences, there will be statements> that can neither be
proven nor disproven.Your use of the word mathematics is
===
generating function>By the way, I might as well add
this:>exp(sum{p=primes} 1/p^x) = sum{k=1 to oo}
A(k)/k^x,>where A(k) = product{p=primes} 1/(a(p,k))! ,and
where each a(p,k) is a nonnegative integer such thatp^a(p,k)
is the highest power of the prime p which divides k.(I
think...)I think so, too. In other words the coefficient of
n^{-x} in exp(sum_p p^{-x}) is 1/prod_i k_i! if n=prod_i
p_i^{k_i}.However, returning to the OP, today I made a few
other calculationsand found out that *formally*sum_p p^{-x} =
sum_{k=1}^{infty} mu(n)/n log(zeta(nx)).There do not *seem* to
be any issues suggesting that this equalityshould not hold also
in the analytical sense, but what interests memost is that as I
had guessed initially this function once again seemsto be
independent of zeta in the sense that it cannot be expressedin
terms of zeta by means of (a finite number of)
algebraicoperations, composition with elementary functions
and/or derivation.Am I right?Michele-- > Comments should say
_why_ something is being done.Oh? My comments always say what
_really_ should have happened. :)- Tore Aursand on
===
(was: Python syntax in Lisp and Scheme)>>I am not claiming that
it is a counterexample, but I've always met>>with some
difficulties imagining how the usual proof of Euler's>>theorem
about the number of cornesides and faces of a
polihedron>>(correct terminology, BTW?) could be formalized.
Also, however that>>could be done, I feel an unsatisfactory
feeling about how complex it>>would be if compared to the
conceptual simplicity of the proof itself.Well it certainly
_can_ be formalized. (Have you any experience>with _axiomatic_
Euclidean geometry? Not as in Euclid - no pictures,No, I have
no experience with it. Just heard it exists: I presumeyou're
talking about the work of Hilbert... but then I'm not sure
thatit provides a full formalization! It's clear though that
it brings onestep forward towards formalization.>nothing that
depends on knowing what lines and points really
are,>everything follows strictly logically from explictly sta
axioms.>Well, I have no experience with such a thing either,
but I know>it exists.) [should have read ahead...]>Whether the
formal version would be totally incomprehensible>depends to a
large extent on how sophistica the formal>of Euler's theorem
in the language of set theory, with no>predicates except is an
element of it would be totally>incomprehensible. Otoh in a
better formal system, for>example allowing definitions, it
could be just as comprehensible>as an English version. (Not
that I see that this question hasIn any case I'm sure too that
*that particular proof* can beformalized! But, even if I am
under the impression that most proofs,for some meaning of
most, would be affec by the concern expressedabove, somehow I
feel like the effect with this one would be oneorder of
magnitude stronger, to say the least!>any relevance to the
existence of alleged osophical>inconsistencies that haven't
been specified yet...)In fact, it has no relevance to that
matter: I thought that myintroduction should have made that
clear. My comment is something Ihappend to think about
sometimes; your words just reminded me of itand this seemed to
be the right chance to talk about it. No more thanthat, no more
than a naive comment...Michele-- > Comments should say _why_
something is being done.Oh? My comments always say what
_really_ should have happened. :)- Tore Aursand on
===
little silly, but is there a (sub)map somewhere of integersy
and n for which y^n - 1 is prime? (where y will probably
always be aprime number) I'd like to experiment with these
couples. I guess it's easyenough to create a small table
yourself but most divergrions seem to happenin a range where
my computer would become kind of slow..and perhaps youhave
interesting theories.I also wonder about y^n + 1 and about y!
- 1 or y! + 1 being prime.And eh, another thing I wonder is if
there have been experiments wihtrepresenting primes in a
different system than the decimal system (forinstance a kind
of system where each digit is 2*previous_digit^2 in value.And
lastly I also wonder if there have been any theories on
correlationsbetween prime number dispersion (I can't think of
the right word) andfractals, or any complex iteration model.
sorry for all the questions, they star oozing
===
be a little silly, but is there a (sub)map somewhere of
integers> y and n for which y^n - 1 is prime? (where y will
probably always be a> prime number) I'd like to experiment
with these couples. I guess it's easy> enough to create a
small table yourself but most divergrions seem to happen> in a
range where my computer would become kind of slow..and perhaps
you> have interesting theories.> I also wonder about y^n + 1
and about y! - 1 or y! + 1 being prime.> And eh, another thing
I wonder is if there have been experiments wiht> representing
primes in a different system than the decimal system (for>
instance a kind of system where each digit is
2*previous_digit^2 in value.> And lastly I also wonder if
there have been any theories on correlations> between prime
number dispersion (I can't think of the right word) and>
fractals, or any complex iteration model. sorry for all the
questions, they star oozing out.....> QuaternionThere are no
integers y > 2 and n > 1 for which y^n - 1 is prime. And 2^n -
1 can be prime only if n is also prime, but not always then.
For example, n = 11, with 2^11 - 1 = 2047 = 23*89, is the
first of many prime n's where 2^n-1 is composite.There are no
odd integers n, with y > 1, for which y^n + 1 is prime.For
even integers n, y^n + 1 can be either prime or
===
about y^n + 1 and about y! - 1 or y! + 1 being prime.[...]>
....>
QuaternionSeehttp://www.research.att.com/projects/OEIS?Anum=
A002981http://www.research.att.com/projects/OEIS?Anum=
A002982http://research.microsoft.com/~pleyland/factorization/
===
silly, but is there a (sub)map somewhere ofintegers> y and n
for which y^n - 1 is prime? (where y will probably always be
a> prime number) I'd like to experiment with these couples. I
guess it's easy> enough to create a small table yourself but
most divergrions seem tohappen> in a range where my computer
would become kind of slow..and perhaps you> have interesting
theories.y^n - 1 = (y - 1)(y^(n-1) + y^(n-2) + ... + y +
1)Then, in order to y^n - 1 to be prime, it must be y - 1 = 1
==> y = 2.Also, if n = p*q, with p, q > 12^p*q - 1 = (2^p)^q -
1 = (2^p - 1)((2^p)^(q-1) + ... + 2^p + 1)Then in order to 2^n
- 1 to be prime, n must be prime, but it isn'tsufficient. 2^11
- 1 = 23*89. If 2^n - 1 is prime, it is calle a
Mersenneprime.(see http://www.mersenne.org/prime.htm)> I also
wonder about y^n + 1 and about y! - 1 or y! + 1 being prime.>
And eh, another thing I wonder is if there have been
experiments wiht> representing primes in a different system
than the decimal system (for> instance a kind of system where
each digit is 2*previous_digit^2 in value.> And lastly I also
wonder if there have been any theories on correlations>
between prime number dispersion (I can't think of the right
word) and> fractals, or any complex iteration model.>By
similar arguments, if y^n + 1 is prime, then y = 2 and n =
2^k. Thenumbers 2^2^k + 1 are Fermat numbeand only five of
them are provedprimes, for k = 0, 1, 2, 3 and 4. For k = 5 to
25 or more (tha last a bignumber), they are composite.You
probably enjoy with
http://www.utm.edu/research/primes/index.html.-- Ignacio
Larrosa Ca.96estroA Coru.96a
===
This might be a little silly, but is there a (sub)map
somewhere of> integers>> y and n for which y^n - 1 is prime?
(where y will probably always be a>> prime number) I'd like to
experiment with these couples. I guess it's>> easy enough to
create a small table yourself but most divergrions seem to>
happen>> in a range where my computer would become kind of
slow..and perhaps you>> have interesting theories.y^n - 1 = (y
- 1)(y^(n-1) + y^(n-2) + ... + y + 1)Then, in order to y^n - 1
to be prime, it must be y - 1 = 1 ==> y = 2.Okay.. I'm so
stupid..> Also, if n = p*q, with p, q > 12^p*q - 1 = (2^p)^q -
1 = (2^p - 1)((2^p)^(q-1) + ... + 2^p + 1)Then in order to 2^n
- 1 to be prime, n must be prime, but it isn't> sufficient.
2^11 - 1 = 23*89. If 2^n - 1 is prime, it is calle a Mersenne>
prime.(see http://www.mersenne.org/prime.htm)>> I also wonder
about y^n + 1 and about y! - 1 or y! + 1 being prime.>> And
eh, another thing I wonder is if there have been experiments
wiht>> representing primes in a different system than the
decimal system (for>> instance a kind of system where each
digit is 2*previous_digit^2 in>> value. And lastly I also
wonder if there have been any theories on>> correlations
between prime number dispersion (I can't think of the right>>
word) and fractals, or any complex iteration model.> By
similar arguments, if y^n + 1 is prime, then y = 2 and n =
2^k. The> numbers 2^2^k + 1 are Fermat numbeand only five of
them are proved> primes, for k = 0, 1, 2, 3 and 4. For k = 5
to 25 or more (tha last a big> number), they are
composite.Yeah, how on earth did they found out 2^2^25 is
composite..you might as wellbreak RSA encryption while you're
at it..> You probably enjoy with
===
Visiting Assistant Professor at the University of Montana.
[.snip.]>> By similar arguments, if y^n + 1 is prime, then y =
2 and n = 2^k. The>> numbers 2^2^k + 1 are Fermat numbeand only
five of them are proved>> primes, for k = 0, 1, 2, 3 and 4. For
k = 5 to 25 or more (tha last a big>> number), they are
composite.Yeah, how on earth did they found out 2^2^25 is
composite..you might as well>break RSA encryption while you're
at it..There are many ways to prove a number is composite
without having toexhibit a factorization of it.A strong break
for RSA would require you to actually find the
primefactorization of the modulus (although there are other
ways ofbreaking RSA that do not involve factoring the
modulus).For example, one can prove that a number n is not
prime by finding aninteger a such that n does not divide
a^n-a; Fermat's Little Theoremstates that if p is a prime,
then a^p-a is a multiple of p for eveyrinteger
a.http://www.prothsearch.net/fermat.htmlIn addition, the kinds
of factors that a Fermat number (numbers of theform 2^{2^{n}} +
1) can have is restric, so one does not have totry all sorts of
possible factors; it is believed that is how Eulerfound the
factorization of 2^{2^5}, for example. Acoording to the page
quo above, the largest Fermat number known tobe composite is
2^{2^{2478782}}+1, which has a factor of 3*2^{2478785} +
===
this greatly simplifies formalizing Peano's Axioms>(Peano
Arithmetic) from 5 complex wffs to the single wff TRUE(x).
>Actually, TRUE(x) is one of only 3 axioms needed to formalize
the>Theory of Computation, as I describe below.1. TRUE(x)>2.
YIT(I,J,K)>3. -~YES(x,x)Say what? How about giving an actual
first-order axiomatization of your> Theory. What you appear to
have here are 3 constants (I, J, K), 2> functions (~, YES), and
3 predicates (TRUE, YIT, -), all unreala. So> there must be a
lot of background machinery in addition to your only 3>
axioms.Variables I, J, K represent input to the relation and X
representsoutput. P,Q means that P and Q are logically
equivalent and so wedefine set TRUE(a) as P(a),P(a)^TRUE(a).
YIT(a,b,c) means Program ahalts yes on input b after c
iterations. so that YIT(I,J,K) meansthat is recursive.
YES(a,b) means Program a halts yes on input b.so that
YES(a,b),(eA)YIT(a,b,A) and -~YES(x,x) means that the
set~YES(a,a) is not recursively enumerable.I don't use normal
Logic because, for various reasons, it is not wellsui for
representing metamathematical assertions. There are
eightuniversal rules of inference that apply to any
programming system. The same rules are used for program
synthesis as for the Theory ofComputation. They also prove
various Set Theory axioms (e.g. ZFC)based on very simple
primitive relationships between various bases ofcomputing.
Developing axioms for Set Theory is then analogous todefining
a programming language. A program halting yes on an input
isanalogous to a set containing an element or a wff with a
free variablebeing provable when a number is substitu for that
variable.I give a number of formal derivations in the papers
that I ciearlier.The ratio of theorems to axioms is much
larger than typical texts(which define one concept to prove
===
& Godel> ... But didn't Hilbert actually claim that a lot more
than that is> possible? Didn't he ask for (and claim that it
must exist) a decision> procedure to determine if an arbitrary
predicate calculus wff is valid More than that, an arbitrary
wff of mathematics.Where did he ever claim that there was a
decision procedure?Wir Mussen wisen, wir werden wissen. Speech
in Konigsberg in 1930,> now on his tomb in Gottingen.Wir werden
wissen is We will know that might express determination> or
prediction. (I have trouble with will in English, never mind>
German!)But does this constitute a claim for a decision
procedure? --that would surely entail an algorithmic or
uniform way of reachingconclusions, rather than a claim that
there is *some* way ofresolving a given problem. I don't know
his writings well enoughto know which interpretation is suppor
===
Hilbert & Godel> ...But didn't Hilbert actually claim that a
lot more than that is> possible? Didn't he ask for (and claim
that it must exist) a decision> procedure to determine if an
arbitrary predicate calculus wff is valid More than that, an
arbitrary wff of mathematics. Where did he ever claim that
there was a decision procedure? Wir Mussen wisen, wir werden
wissen. Speech in Konigsberg in 1930,> now on his tomb in
Gottingen. Wir werden wissen is We will know that might
express determination> or prediction. (I have trouble with
will in English, never mind> German!)But does this constitute
a claim for a decision procedure? --> that would surely entail
an algorithmic or uniform way of reaching> conclusions, rather
than a claim that there is *some* way of> resolving a given
problem. I don't know his writings well enough> to know which
===
Resolvants & Galois groupsDoes anyone know any books covering
===
Abelian groups, this leaves 3 with 3 generatoand>6 with 2
generators. Z_2 x D_4 and Z_2 x Q have 3, leaving only
one>other with 3 generators.That's right. The other one with
three generators is the central> product of Z_4 and D_4, which
is defined as the direct product of> Z_4 and D_4 but with their
central elements of order 2 amalgama.> This turns out to be
isomorphic to the central product of Z_4 and Q> and is group
number 13 in the GAP list.D_8 has 2 generatoleaving 5 more
with 2 generators.Yes. Of these, 3 have elements of order 8,
and two do not.> I can answer any specific queries about these
groups.> I lost part of this post--can't recall what was here
though.I was talking about grp #36 in Wavrick's numbering.In
36, all cyclic subgroups of order 4 are normal, whileno cyclic
subgroup of order 4 is normal in #37.#36 and 37 both have 8
elements of order 44 and7 of order 2.In #36, there are 4
cyclic normal subgroups of order 4, genera byx_i, i = 1,2,3,4.
Z =  is the center, and the other 3are also normal. Also,
all 4 squares x_i^2 are equal and inthe center. (See previous
posts). There are 6 other elements oforder 2, call them y_j.
The relation of the x_i and y_j is as follows;y_j x_i y_j =
x_i^r, where r = +/- 1 (+ 1 means they commute).The following
values of (i,j) commute. Others have r = -1.i = 2, j = 1,2 ; i
= 3, j = 3,4 ; i = 4 , j = 5,6.Recall the x_1 generates the
center.The group is genera by (x_1,y_2,y_4).I have to say that
this is not a very good description,but I will post it
anyway--not one of my best posts though.>>D_8 has 2
generatoleaving 5 more with 2 generators.Yes. Of these, 3 have
elements of order 8, and two do not.>I can answer any specific
it and it worked fine.Would someone post a definition of the
semi-direct product?I have a vague memory of it--I liked
Hungerford's def.I recall the product involving an
automorphism, and asI said, I have done the case of
automorphisms of cyclicnormal subgroups many times.If C_n =
 is cyclic of order n, then the automorphismsA(C_n) = {f_r;
C_n --> C_n : x --> f_r(x) = x^r}is isomorphic to Z_n* = units
in Z_n.These are all the automorphisms, right?Yes, they must
===
interesting program for doing things> with groups of order <
32 at http//math.ucsd.edu/~jwavrick .> program (or using
telnet);> [...]> ORDERS for Groups Number 35 and 38> Group
number 35 of Order 16> 1 elements of order 1: A> 3 elements of
order 2: C E G> 12 elements of order 4: B D F H I J K L M N O
P> 0 elements of order 8:> 0 elements of order 16:> Both 35
and 38 have the same center; Z = {A E C G}.> This is the same
distribution of orders as in Z_2 x Q, so> 35 or 38 is Z_2 x
Q.> For 35 and 38, there are 6 = 12/2 elements of order 4,You
mean 6 *cyclic subgroups* of order 4, not 6 elements. Yes. x
and x^(-1) = x^3 are not the same element, of course.> and if
we call these x_i; i=1,6, then let> x_1^2 = x_2^2, x_3^2 =
x_4^2, x_5^2 = x_6^2,> which gives 3 elements of order 2, all
of which are in the center.No. All elements of order 4 in Z_2
x Q square to the same> element, so 2 of the 3 involutions in
the center are not the> square of an element of order 4.Right
again. I noticed this after posting. If Z_2 = (0,1) andx,y,and
z = xy are in Q, write X = (m,q) in Z_2 x Q,where m is
additive, q multiplicative. Then (1,1)^2 = (0,1) = E =
identity= (1,-1)^2 = (0,-1)^2 and x^2 = y^2 = z^2 = -1,X0 =
(0,x); X1 = (1,x), etc., x^3 = -x, etc. give the 12 elementsof
order 4. They all square to (0,-1).> Likewise, only 2 of the 3
involutions in the center of the> other non-abelian group with
this distribution of element> orders are the squares of
elements of order 4. This group is the> semidirect product Z_4
x| Z_4 with presentation  b^4 = 1, a^b = a^-1>.I
will have to look at the semidirect product again.I guess it
can cover more than automorphisms of normal subgroups,which is
all I am familiar with.What do you mean by a^b? I have seen
some call this the conjugateof a by b, though this is new to
me.I'm sure you are right.This must be group 38, which I will
===
TransversalityContent-Length: 952Originator: rusin@vesuvius>
but my question relates the following situation:g:A->B ,
f:[0,1]->B such that f is transverse to the set of critical>
values of g.my question is if g^{-1}(im(f)) is a smooth
sub-manifold of A. Not necessarily. Let z=x+iy and
h(z,t)=(z^3,t), then h:R^3->R^3 has acritical set z=0. Let
further p be the projection p(x,y,t)=(x,t) andconsider the
composition g=ph. Then g:R^3->R^2 has a critical valuesset x=0
(a line). Now, let L be another line in R^2 transverse, butnon
orthogonal to x=0. (It may be realized as the image of a
mapf:R->R^2 transverse to line x=0.) Then it is easy to see
that g^{-1}(L) consists of 3 planes intersecting in a single
point, so itis not a manifold. Maybe this is true for stable
singularities, for example in case offold type singularity
===
Re: Regular TransversalityContent-Length: 2593Originator:
rusin@vesuvius> but my question relates the following
situation:g:A->B , f:[0,1]->B such that f is transverse to the
set of critical>>values of g.my question is if g^{-1}(im(f)) is
a smooth sub-manifold of A.> Not necessarily. Let z=x+iy and
h(z,t)=(z^3,t), then h:R^3->R^3 has a> critical set z=0. Let
further p be the projection p(x,y,t)=(x,t) and> consider the
composition g=ph. Then g:R^3->R^2 has a critical values> set
x=0 (a line). Now, let L be another line in R^2 transverse,
but> non orthogonal to x=0. (It may be realized as the image
of a map> f:R->R^2 transverse to line x=0.) Then it is easy to
see that > g^{-1}(L) consists of 3 planes intersecting in a
single point, so it> is not a manifold.I don't quite believe
this. Take the diagram h p R^3 ---> R^3 ---> R^2and consider
p^(-1) L. It's simply a plane that sits obliquelyto the t-axis
(in your terminology). Off that axis, h is a 3-foldcovering
map, so h: h^(-1) (p^(-1) L {t-axis}) --> L {t-axis}is a
3-fold covering map. However, it can't be a set of three
puncturedplanes, since there is no arrangement of three planes
that meet at asingle point in R^3. Instead, if you take a right
circular Z cylinderabout the t-axis, the intersection of L with
Z is an ellipse; thepreimage h^(-1) Z is again a right circular
cylinder surrounding the t-axis, and the lift of that ellipse
is a curve that oscillates upand down (i.e., parallel to the
axis) in 3 periods as it encirclesthe cylinder once. The
amplitude of these oscillations (for the mapyou've described,
(z,t) |--> (z^3,t), appears to be proportional tor^(1/3),
where r is the radius of the cylinder h^(-1)Z. If I nowdraw
radii in the plane p^(-1)L, they pull back (via h^(-1)) to
curvesthat look like r^(1/3) in the appropriate plane
containing the t-axis.In particular, it appears that (with the
exception for the two pointsin the ellipse that sit on the
plane orthogonal to the t-axis, orrather their preimages via
h^(-1)), all those radii pull back tocurves that are tangent
to the t-axis.should be a single surface, with a point on the
t-axis where thereis no tangent plane, due to the oscillation
of nearby points I'vejust described.> Maybe this is true for
stable singularities, for example in case of> fold type
===
Epigone-thread: glabeusnayContent-Length: 153Originator:
rusin@vesuviuswhat are those stable singularities?, my map is
a quadratic g:R^n->r^m ; (n>m)Is there a simple way of knowing
===
between Orthogonal VectorsContent-Length: 2913Originator:
rusin@vesuvius>Consider two Sets A and B of n-dimensional
vectors. A contains ALL the>vectors with exactly 'i' 1s (and
'n-i' 0s), and B contains all the>vectors with 'j' 1s (and
'n-j' 0s). WLOG, assume i < j <= n.More to the point, assume i
+ j <= n so it is possible for vectorsin A and B to be
orthogonal.>Now consider the folowing game: at each round, you
remove one random>vector from A (if it is not exhaus yet), and
all of its orthogonal>vectors (vecotrs whose position wise
boolean AND yeild the vecotr>000...000) from B (if there is
any left). For example. for i=2, j=3,>and n=6, if you remove
110000 from A, you also remove 001110, 000111,>001011, 001101
from B. Note that some of these vectors from B might>have
already removed by some earlier removal from A.>Is it possible
to find a closed form of the function F(r), # of>vectors
remaining in the set B as a function of the number of
removals>r from A?Of course not, since this depends on which
vectors are removed from A,not just the number of them.> In
case it is not possible, a reasonably tight upper bound>would
suffice.Consider the case where r distinct members of A are
selec at random(with equal probabilities). Any given b in B
survives if and only if allthe selec members of A are
non-orthogonal to it. The number of vectorsin A that are
non-orthogonal to b is (n choose i) - (n-j choose i),so the
probability that b survives isp = binom(binom(n,i) -
binom(n-j,i),r)/binom(binom(n,i),r)(where binom(x,y) = (x
choose y)) and the expec number of b remaining is p
binom(n,j). Thus any upper bound U(r) on the number remaining
must satisfy U(r) >= ceil(p binom(n,j)), and any lower bound
L(r) must satisfy L(r) <= floor(p binom(n,j)). For example,
with n=6,i=2,j=3,r=5, we find L(r) <= 5 and U(r) >= 6.This
case is small enough to enumerate all possibilities, and I
find(with Maple's help) that the least possible number of
survivors is 3 (e.g. choosing (1,2), (1,6), (2,6), (3,4) and
(3,5) from A)and the largest possible is 10 (e.g. choosing
(1,2),(1,3),(1,4),(1,5),(1,6)).It seems likely that (as in
this example) a larger numberof survivors will occur when the
chosen members of A share as many members as possible. Take
the largest m<=i such that binom(n-m,i-m) >= r. Then it is
possible to choose r members of Awhich all contain {1,...,m}.
Any member of B that intersects {1,...,m} will survive. There
are binom(n,j) - binom(n-m,j) membersof B that intersect
{1,...,m}. Thus U(r) >= binom(n,j) - binom(n-m,j)whenever
binom(n-m,i-m) >= r. This bound is tight in the example
above(n=6,i=2,j=3,r=5,m=1).Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
===
Grad schoolContent-Length: 696Originator: rusin@vesuvius>>
budget makes severe cuts to UC funding. ...Yes, there are some
cuts scheduled, but state spending on the Univ. of>California
nearly doubled between 1995 and 2002. I wouldn't worry>about
it too much. You can get some more info about the budget
here.Yes, things were horrible in 1995, and they're horrible
again.See the graph on page 14 of op. cit.--Paul Vojta,
===
schoolContent-Length: 3465Originator: rusin@vesuvius>I'd like
some info about math PhD programs, if anyone can please help.
A>concern I have are rankings, in particular: how much do they
matter when you>make *final* decisions?My admitly biased
opinion is that the NRC rankings are themselvesgly overra.
Don't take them too seriously. They are okay if youjust squint
at them and take them only as a rating of research reputation.I
don't think that the rankings are reliable for other attributes
ofthe graduate program such as funding or attention from the
faculty.In the physics list, for example, one school that
doesn't even have aPhD program slipped in by mistake. Even as
a strict research ranking,the NRC numbers show some grievous
misunderstandings. They are also10 years out of date. The
single most egregious and damaging mistakemay be that Oklahoma
State University, which is really a fine researchdepartment,
was left out entirely.I've also been irrita with Davis'
position in the NRC rankings sinceI came here. It was tied for
83rd. Maybe it was because UC Davishad been improving quickly,
or because the raters got tired of tippingtheir hat to the
numerous UC schools, or because someone at Davis
didn'tdescribe the department very well to the NRC in 1993. In
any case itdidn't square with my impression when I interviewed
here. Moreover,half of the department has been hired since
1993. Actually I was alot more irrita when I first arrived; I
haven't cared much lately.But the rankings still do complicate
graduate recruiting here. I thinkthat our most successful
recruiting strategy is to impress a few studentsto the point
that they disbelieve the rankings entirely.Of course there are
also the US News graduate program rankings.Those are little
more than simple opinion polls and I think that theyreflect
people's memories of the NRC rankings. It has also been
saidthat mathematicians generally admire their own graduate
alma maters.That would give historically large programs a big
advantagein the rankings.You expressed interest in operator
algebras. I can be a bit moreobjective about that area since
we don't have much of it here at Davis.Texas A&M does, and I
think that that department is also notablyunderra in the old
NRC list. One way that you can see what someschools like Texas
A&M have been doing is with a fielded search at myfront end for
the math arXiv:
http://front.math.ucdavis.edu/search/from:tamu.eduAlthough
this is good additional information, I won't say that it's
anymore fair than the rankings. It might be less fair, by
itself.We do have a lot of applied harmonic analysis and PDEs
here at Davis,if that also interests you. And I don't agree
with either PeterMontgomery's or Roger Schlafly's portrayals
of UC's budget situation.UC isn't either starving or living
fat. It has gotten a lot of new moneyfrom enrollment
expansion. Now there is a famous state budget crisis, butthe
university is a skillful lobbyist and I think that it is
relativelywell-prepared to ride out the storm. Out-of-state
tuition is high, andthat does affect graduate recruiting. But
it's like medical coverage:the university is always finding
ways to foot its own bill.-- / Greg Kuperberg (UC Davis) / /
Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ /
===
inlineContent-Length: 22239Originator: rusin@vesuviusHere are
this week's titles in the mathematics arXiv, available at:
http://front.math.ucdavis.edu/
http://front.math.ucdavis.edu/submissionsThis week in the
mathematics arXiv may be freely redistribuwith attribution and
16 Jan)-------------------------------------------------AG:
Algebraic Geometry----------------------math.AG/0401185
 Gabriela
Schmithuesen: An algorithm for finding the Veech group of an
origamimath.AG/0401173
 Alexander
Schmitt: Moduli for decora tuples of sheaves and
representation spaces for quiversmath.AG/0401167
 Paolo Aluffi:
Chern classes of birational varietiesmath.AG/0401166
 Pramathanath
Sastry: Duality for Cousin Complexesmath.AG/0401159
 Sean Keel,
Eugene Tevelev: Chow Quotients of Grassmannians
IImath.AG/0401147
 Jean Valles:
Invariant multidimensional matricesmath.AG/0401140
 Stephane
Zahnd: Descente de torseugerbes et points rationnels - Descent
of torsogerbes and rational pointsmath.AG/0401107
 Sonia Brivio,
Gian Pietro Pirola: Alternating groups and rational functions
on surfacesmath.AG/0401100
 Euisung Park:
On higher syzygies of ruled surfacesmath.AG/0401092
 Ania
Otwinowska: Sur les varietes de HodgeAP: Analysis of
PDEs--------------------math.AP/0401129
 Valeria
Banica: Remarks on the blow-up for the Schrodinger equation
with critical mass on a plane domainAT: Algebraic
Topology----------------------math.AT/0401178
 Gregory
Lupton, Samuel Bruce Smith: Rationalized Evaluation Subgroups
of a Map II: Quillen Models and Adjoint Mapsmath.AT/0401176
 Michael
Joswig: Computing Invariants of Simplicial
Manifoldsmath.AT/0401168
 Soren
Galatius: Mod p homology of the stable mapping class
groupmath.AT/0401160
 Johannes
Huebschmann: Homological perturbations, equivariant
cohomology, and Koszul dualitymath.AT/0401132
 Daniel C.
Isaksen: Flasque model structures for simplicial
presheavesmath.AT/0401130
 Jouko
Mickelsson: Twis K theory invariantsmath.AT/0401081
 P. Hu: Higher
string topology on general spacesCA: Classical Analysis and
ODEs-------------------------------math.CA/0401086
 Alexei
Zhedanov: Biorthogonality of the Lagrange interpolantsCO:
Combinatorics-----------------math.CO/0401179
 William P.
Orrick: The maximal {-1,1}-determinant of order
15math.CO/0401175
 Nicholas
Eriksson: Toric ideals of homogeneous phylogenetic
modelsmath.CO/0401157
 Guangyue Han:
Generalized PSK in Space Time Codingmath.CO/0401154
 Boaz Tsaban:
Random strategies with memory for the Robin Hood
gamemath.CO/0401117
 Michael Huber:
Classification of flag-transitive Steiner quadruple
systemsmath.CO/0401089
 F. Hivert,
J.-C. Novelli, J.-Y. Thibon: The Algebra of Binary Search
TreesCV: Complex Variables---------------------math.CV/0401188
 Dmitry
Khavinson, Genevra Neumann: On the number of zeros of certain
rational harmonic functionsmath.CV/0401171
 John Erik
Fornaess, Nils Ovrelid, Sophia Vassiliadou: Semiglobal results
for $barpartial$ on a complex space with arbitrary
singularitiesmath.CV/0401170
 John Erik
Fornaess, Nils Ovrelid, Sophia Vassiliadou: Local $L^2$
results for $barpartial$: The isola singularities
casemath.CV/0401165
 Byung-Geun Oh:
Zeros of the Derivatives of Faber Polynomials Associa with a
Universal Covering Mapmath.CV/0401150
 James Silipo:
L'amibe d'un syst`eme de sommes d'exponentielles `a
fr'equences r'eellesmath.CV/0401142
 Joel Merker,
Egmont Porten: Characteristic foliations on maximally real
submanifolds of C^n and envelopes of holomorphymath.CV/0401111
 Charles Favre,
Mattias Jonsson: Valuative analysis of plurisubharmonic
functionsDG: Differential
Geometry-------------------------math.DG/0401182
 C. A. i: The
division map of principal bundles with groupoid structure and
generalized gauge transformationsmath.DG/0401180
 C. A. i:
Holonomy and parallel transport in the differential geometry
of the space of loops and the groupoid of generalized gauge
transformationsmath.DG/0401162
 Mohamad A.
Hindawi: On the Filling Invariants at Infinity of Hadamard
Manifoldsmath.DG/0401161
 Johannes
Huebschmann: Relative homological algebra, homological
perturbations, equivariant de Rham theory, and Koszul
dualitymath.DG/0401152
 Jean-Baptiste
Butruille: Classification des varietes approximativement
kahleriennes homogenesmath.DG/0401136
 Jih-Hsin
Cheng, Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang: Minimal
surfaces in pseudohermitian geometry and the Bernstein problem
in the Heisenberg groupmath.DG/0401125
 Jason Lotay:
2-Ruled Calibra 4-folds in R^7 and R^8math.DG/0401123
 Jason Lotay:
Constructing Associative 3-folds by Evolution
Equationsmath.DG/0401112
 Paul-Emile
Paradan: Cohomologie equivariante et quantification
geometriquemath.DG/0401110
 Masatoshi
Kokubu, Wayne man, Kentaro Saji, Masaaki Umehara, Kotaro
Yamada: Singularities of flat fronts in hyperbolic
3-spacemath.DG/0401097
 Alexander I.
Nesterov: Jacobi fields and odular structure of affine
manifoldsmath.DG/0401096
 Joel B.
Mohler, Ron Umble: Minimal Paths on Unicone and Bicylinder
Boundariesmath.DG/0401094
 J. F. Barraud,
O. Cornea: Lagrangian Intersections and the Serre Spectral
Sequencemath.DG/0401080
 Matthias
Weber, David Hoffman, Michael Wolf: An embedded genus-one
helicoidDS: Dynamical
Systems---------------------math.DS/0401184
 Sebastien
Gouezel: Decay of correlations for nonuniformly expanding
systemsmath.DS/0401146
 D. Wilczak, P.
Zgliczynski: Heteroclinic Connections between Periodic Orbits
in Planar Restric Circular Three Body Problem - Part
IImath.DS/0401145
 D. Wilczak, P.
Zgliczynski: Topological method for symmetric periodic orbits
for maps with a reversing symmetrymath.DS/0401104
 E. Jerome
Benveniste, David Fisher: Nonexistence of invariant rigid
structures and invariant almost rigid
structuresmath.DS/0401093
 J.-R.
Chazottes, E. Ugalde: Entropy estimation and fluctuations of
Hitting and Recurrence Times for Gibbsian sourcesFA:
Functional Analysis-----------------------math.FA/0401187
 R. H. Levene,
S. C. Power: Reflexivity of the translation-dilation algebras
on L^2(R)math.FA/0401151
 A.G. Smirnov:
Fourier transformation of Sato's hyperfunctionsmath.FA/0401127
 E. Ournycheva,
B. Rubin: An analogue of the Fuglede formula in integral
geometry on matrix spacesmath.FA/0401122
 Narutaka
Ozawa: A note on non-amenability of B(ell_p) for
p=1,2math.FA/0401091
 Victor
Shulman, Lyudmila Turowska: Operator synthesis II. Individual
synthesis and linear operator equationsGM: General
Mathematics-----------------------math.GM/0401099
 Omran Kouba:
Elementary evaluation of $int_0^infty{sin^p t/t^q}
dt$math.GM/0401082
 A. K.
Kwasniewski, B. K. Kwasniewski: On trigonometric-like
decompositions of functions with respect to the cyclic group
of order nGN: General
Topology--------------------math.GN/0401155
 Boaz Tsaban:
SPM Bulletin 7GR: Group Theory----------------math.GR/0401133
 Graham Niblo,
Michah Sageev, Peter Scott, Gadde A. Swarup: Minimal
cubingsGT: Geometric
Topology----------------------math.GT/0401186
 David T. Gay,
Robion Kirby: Constructing symplectic forms on 4-manifolds
which vanish on circlesmath.GT/0401183
 J. Scott
Carter, Masahico Saito: Generalizations of Quandle Cocycle
Invariants and Alexander Modules from Quandle
Modulesmath.GT/0401174
 Craig Jensen,
John Meier: The cohomology of right angled Artin groups with
group ring coefficientsmath.GT/0401163
 Greg Friedman:
Cobordism of disk knotsmath.GT/0401135
 Khaled
Qazaqzeh, M. Gilmer: The parity of the Maslov index and the
even cobordism categorymath.GT/0401120
 Nathan
Broaddus: Noncyclic covers of knot complementsmath.GT/0401084
 Hitoshi
Murakami, Yoshiyuki Yokota: The colored Jones polynomials of
the figure-eight knot and its Dehn surgery spacesKT: K-Theory
and Homology-------------------------math.KT/0401158
 Hans-Joachim
Baues, Teimuraz Pirashvili: Shukla cohomology and additive
track theoriesLO: Logic---------math.LO/0401134
 Ilijas Farah,
Jindrich Zapletal: Between Maharam's and von Neumann's
problemsmath.LO/0401095
 Hugo Luiz
Mariano: Profinite Structures are Retracts of Ultraproducts of
Finite StructuresMP: Mathematical
Physics------------------------quant-ph/0311159
 Vasily E.
Tarasov: Quantization of non-Hamiltonian and Dissipative
Systemsmath-ph/0401032
 S. Baskal, Y.
S. Kim: Rotations associa with Lorentz boostsmath-ph/0401031
 Alan Carey,
Hendrik Grundling: Amenability of the Gauge
Groupmath-ph/0401030
 M. Lorente:
Raising and lowering operatofactorization and
differential/difference operators of hypergeometric
typecond-mat/0401245
 S. Mitra, B.
Nienhuis, J. de Gier, M. T. Batchelor: Exact expressions for
correlations in the ground state of the dense O(1) loop
modelmath-ph/0401029
 Edwin
Langmann: An algorithm to solve the elliptic
Calogero-Sutherland modelmath-ph/0401028
 Gerald Kaiser:
Energy-momentum conservation in pre-metric electrodynamics with
magnetic chargeshep-th/0401080

G.Sardanashvily, G.Giachetta: What is geometry in quantum
theoryhep-lat/0401017
 Fumihiko
Sugino: Super Yang-Mills Theories on the Two-Dimensional
Lattice with Exact Supersymmetrymath-ph/0401027
 Michael K.-H.
Kiessling, Carlo Lancellotti: On the master equation approach:
linear and nonlinear Fokker--Planck equationsmath-ph/0401026
 Karmadeva
Maharana: Group analysis of Schroedinger equation with
generalised Kratzer type potentialmath-ph/0401025
 Giuseppe
Gaeta: Symmetry of stochastic equationsmath-ph/0401024
 E. Ragoucy:
Integrable systems with impuritymath-ph/0401023
 Fabian Brau:
Necessary and sufficient conditions for existence of bound
states in a central potentialmath-ph/0401022
 Fabian Brau:
Upper limit on the number of bound states of the spinless
Salpeter equationmath-ph/0401021
 Fabian Brau,
Francesco Calogero: A class of ($ell$-dependent) potentials
with the same number of ($ell$-wave) bound
statesmath-ph/0401020
 Fabian Brau,
Francesco Calogero: Upper and lower limits on the number of
bound states in a central potentialmath-ph/0401019
 Michel Bauer,
Denis Bernard: SLE, CFT and zig-zag
probabilitiesmath-ph/0401018
 N. Bazunova,
A. Borowiec, R. Kerner: Universal differential calculus on
ternary algebrashep-th/0202069
 Cecilia
Albertsson, Ulf Lindstrom, Maxim Zabzine: N=1 supersymmetric
sigma model with boundaries, IIcond-mat/0401102
 Franco
Ferrari: Topologically Linked Polymers are Anyon SystemsNA:
Numerical Analysis----------------------cs.MS/0401008
 Amparo Gil,
Javier Segura, Nico M. Temme: Algorithm xxx: Modified Bessel
functions of imaginary order and positive
argumentmath.NA/0401131
 Amparo Gil,
Javier Segura, Nico M. Temme: Integral Representations for
Computing Real Parabolic Cylinder Functionsmath.NA/0401128
 Amparo Gil,
Javier Segura, Nico M. Temme: Computing solutions of the
modified Bessel differential equation for imaginary orders and
positive argumentsmath.NA/0401116
 Amparo Gil,
Wolfram Koepf, Javier Segura: Numerical algorithms for the
real zeros of hypergeometric functionsNT: Number
Theory-----------------math.NT/0401181
 Alireza
Sarveniazi: Ramunajan $(n_1,n_2,...,n_{d-1})$-regular
hypergraphs based on Bruhat-Tits Buildings of type
$tilde{A}_{d-1}$math.NT/0401156
 Michel L.
Lapidus, Machiel van Frankenhuijsen: Fractality,
Self-Similarity and Complex Dimensionsmath.NT/0401149
 Andrew
Pollington, Sanju Velani: Metric Diophantine approximation and
'absolutely friendly' measuresmath.NT/0401148
 Victor
Beresnevich, Detta Dickinson, Sanju Velani: Diophantine
approximation on planar curves and the distribution of
rational pointsmath.NT/0401141
 Zongduo Dai,
Kunpeng Wang, Dingfeng Ye: Multidimensional continued fraction
and rational approximationmath.NT/0401126
 S. M. Gonek:
Three Lectures on the Riemann Zeta-Functionmath.NT/0401124
 Adebisi
Agboola, Benjamin Howard: Anticyclotomic Iwasawa theory of CM
elliptic curves IIhep-th/0401052
 Simon Davis: A
Proof of the Odd Perfect Number Conjecturemath.NT/0401118
 Victor
Beresnevich, Detta Dickinson, Sanju Velani: Measure theoretic
laws for lim sup setsmath.NT/0401085
 Yoichi
Motohashi: A note on the mean value of the zeta and
L-functions. XIVOA: Operator
Algebras---------------------math.OA/0401139
 Remus Nicoara,
Sorin Popa, Roman Sasyk: Some remarks on irrational rotation
$HT$ factorsmath.OA/0401138
 Sorin Popa: A
Unique Decomposition Result for HT Factors with Torsion Free
Coremath.OA/0401121
 Narutaka
Ozawa: A Kurosh type theorem for type II_1 factorsOC:
Optimization and
Control----------------------------math.OC/0401109
 Amnon
Yekutieli: On the Structure of BehaviorsPR: Probability
Theory----------------------math.PR/0401144
 D.N. Zhabin:
Stochastic Processes with Short Memorymath.PR/0401143
 Christina
Goldschmidt, James Norris: Essential edges in Poisson random
hypergraphsmath.PR/0401115
 David J.
Aldous, Gregory Miermont, Jim Pitman: Weak convergence of
random p-mappings and the exploration process of inhomogeneous
continuum random treesmath.PR/0401114
Skorokhod Problem and its offsringQA: Quantum
Algebra-------------------math.QA/0401164
math.QA/0401137
 Evgeny Mukhin,
Alexander Varchenko: Discrete Miura Opers and Solutions of the
Bethe Ansatz Equationsmath.QA/0401119
 Alexander
Kirillov Jr: On $G$--equivariant modular
categoriesmath.QA/0401108
 Erik Backelin,
Kobi Kremnitzer: Quantum flag varieties, equivariant quantum
D-modules and localization of quantum groupsmath.QA/0401088
 N.A. Gromov,
V.V.Kuratov: Cayley--Kein Contractions of Quantum Orthogonal
Groups in Cartesian Basismath.QA/0401087
 Ayumu Hoshino,
Toshiki Nakashima: Polyhedral Realizations of Crystal Bases for
Modified Quantum Algebras of Type Amath.QA/0401083

A.K.Kwasniewski: On extended umbral calculus, oscillator-like
algebras and Generalized Clifford AlgebraRA: Rings and
Algebras----------------------math.RA/0401177
 Lars Eld'en: A
Note on the Eigenvalues of the Google Matrixmath.RA/0401113
 Ibrahim Assem,
Manuel Saorin: Abelian exact subcategories closed under
predecessorsmath.RA/0401103
 Michael
Pinsker: Maximal clones on uncountable sets that include all
permutationsmath.RA/0401102
 Michael
Pinsker: Clones containing all almost unary
functionsmath.RA/0401101
 Michael
Pinsker: The clone genera by the median
functionsmath.RA/0401098
 Z. Reichstein,
D. Rogalski, J. J. Zhang: Projectively simple ringsRT:
Representation Theory-------------------------math.RT/0401106
 Pavle
Pandv{z}i'c: Equivariant analogues of Zuckerman
functorsmath.RT/0401105
 Pavle
Pandv{z}i'c: A simple proof of Bernstein-Lunts
equivalencemath.RT/0401090
 Paola Cellini,
Pierluigi Moseneder Frajria, Paolo Papi: ad-nilpotent ideals
containing a fixed number of simple root spacesSG: Symplectic
Geometry-----------------------math.SG/0401172
 Denis Auroux,
Viktor S. Kulikov, Vsevolod V. Shevchishin: Regular homotopy
of Hurwitz curvesmath.SG/0401169
 Ko Honda:
3-dimensional methods in contact geometrySP: Spectral
Theory-------------------math.SP/0401153
 Lachieze-Rey
Marc: Laplacian eigenmodes for the three-Sphere-- / Greg
Kuperberg (UC Davis) / / Visit the Math ArXiv Front at
http://front.math.ucdavis.edu/ / * All the math that's fit to
===
23390Originator: rusin@vesuviusHere are this week's titles in
the mathematics arXiv, available at:
http://front.math.ucdavis.edu/
http://front.math.ucdavis.edu/submissionsThis week in the
mathematics arXiv may be freely redistribuwith attribution and
23 Jan)-------------------------------------------------AC:
Commutative Algebra-----------------------math.AC/0401301
 B. Zilber:
Covers of the multiplicative group of an algebraically closed
field of characteristic zeromath.AC/0401292
 Morten Brun,
Tim Roemer: Subdivisions of toric complexesmath.AC/0401220
 Christopher J.
Hillar: Cyclic Resultantsmath.AC/0401192
 Joseph
Gubeladze, Zaza Mushkudiani: Commutator automorphisms of
formal power series ringsAG: Algebraic
Geometry----------------------math.AG/0401290
 Martin
Moeller: Variations of Hodge structures of a Teichmueller
curvemath.AG/0401260
 Yi Hu: Stable
Configurations of Linear Subspaces and Quotient Coherent
Sheavesmath.AG/0401254
 Alessandra
Sarti: A geometrical construction for the polynomial
invariants of some reflection groupsmath.AG/0401244
 Cindy De
Volder, Antonio Laface: Base locus of linear systems on the
blowing-up of P^3 along at most 8 general
pointsmath.AG/0401225
 Adrien
Dubouloz: Generalized Danielewski Surfacesmath.AG/0401204
 Andrea
D'Agnolo, Pierre Schapira: On twis microdifferential modules
I. Non-existence of twis wave equationsmath.AG/0401201
 F. Pakovich:
On trees that cover chains or starsmath.AG/0401190
 Alexis G.
Zamora: On the number of singular fibers of a semistable
fibration: Further Consequences of Tan's InequalityAP:
Analysis of PDEs--------------------math.AP/0401291
 Alessio
Pomponio: Schrodinger equation with critical Sobolev
exponentmath.AP/0401278
 Andreas
Wannebo: A higher order Weierstrass approximation theorem - a
new proofmath.AP/0401255
 Andreas
Wannebo: A remark on the history of Hardy inequalities fot
domainsmath.AP/0401253
 Andreas
Wannebo: Hardy and Hardy PDO type inequalities in domains.
Part Imath.AP/0401234
 Herbert Koch,
Daniel Tataru: Dispersive estimates for principally normal
pseudodifferential operatorsmath.AP/0401198
 Gianni Dal
Maso, Gilles A. Francfort, Rodica Toader: Quasi-static
evolution in brittle fracture: the case of bounded
solutionsmath.AP/0401196
 Gianni Dal
Maso, Gilles A. Francfort, Rodica Toader: Quasistatic crack
growth in finite elasticityAT: Algebraic
Topology----------------------math.AT/0401283
 Zhi-Ming Luo:
Homotopy theory of presheaves of simplicial
groupoidsmath.AT/0401274
 Timothy
Porter: $mathcal{S}$-categories, $mathcal{S}$-groupoids, Segal
categories and quasicategoriesCA: Classical Analysis and
ODEs-------------------------------math.CA/0401299
 Ilia Krasikov:
Turan inequalities and zeros of orthogonal
polynomialsmath.CA/0401281
 Liviu I.
Nicolaescu: Derangements and asymptotics of Laplace transforms
of polynomialsmath.CA/0401271
 Y. Chen, A.
Its: A Riemann-Hilbert Approach to the Akhiezer
Polynomialsmath.CA/0401249
 Laura
Wisewell: Families of surfaces lying in a null
setmath.CA/0401243
 Bernhard
Kroetz, Sundaram Thangavelu, Yuan Xu: The heat kernel
transform for the Heisenberg groupCO:
Combinatorics-----------------math.CO/0401300
 Ira M. Gessel,
Guoce Xin: A Combinatorial Interpretation of The Numbers
$6(2n)! /n! (n+2)!$math.CO/0401293
 Yonatan Bilu,
Nati Linial: Monotone Maps, Sphericity and Bounded Second
Eigenvaluequant-ph/0401126
 Gerard
Duchamp, Karol A. Penson, Allan I. Solomon, Andrej Horzela,
Pawel Blasiak: One-parameter groups and combinatorial
physicsmath.CO/0401252
 M. Ishikawa,
F. Jouhet, J. Zeng: A generalization of Kawanaka's identity
for Hall-Littlewood polynomials and
applicationsmath.CO/0401247
 Jeong Han Kim,
Oleg Pikhurko, Joel Spencer, Oleg Verbitsky: How Complex are
Random Graphs in First Order Logic?math.CO/0401239
 Tim Boykett,
Peter Mayr: Difference Methods and Ferrero
Pairsmath.CO/0401237
 Frederic
Chapoton: Enumerative properties of generalized
associahedramath.CO/0401235
 Ilse Fischer:
Another refinement of the Bender-Knuth
(ex-)Conjecturemath.CO/0401224
 Mike Develin:
The space of $n$ points on a tropical line in
$d$-spacemath.CO/0401218
 Toufik
Mansour: Counting occurrences of 3412 in an
involutionmath.CO/0401217
 Eric Egge,
Toufik Mansour: Involutions Restric by 3412, Continued
Fractions, and Chebyshev Polynomialsmath.CO/0401216
 David Callan:
A combinatorial proof of Sun's curious identityCV: Complex
Variables---------------------math.CV/0401302
 Vincent Guedj,
Ahmed Zeriahi: Intrinsic capacities on compact Kahler
manifoldsmath.CV/0401264
 Steven R.
Bell: Quadrature domains and kernel function
zippingmath.CV/0401200
 Han Peters:
Non-autonomous dynamics of holomorphic mappings in projective
spaceDG: Differential
Geometry-------------------------math.DG/0401294
 Adrian
Andrada, Isabel Dotti: Double products and hypersymplectic
structures on $R^{4n}$math.DG/0401275
 Joel Fine:
Constant scalar curvature Kahler metrics on fibred complex
surfacesmath.DG/0401270
 Anton Deitmar:
A conjectural Lefschetz formula for locally symmetric
spacesmath.DG/0401267
 Pascal Collin,
Robert Kusner, William H. Meeks, III & Harold Rosenberg: The
topology, geometry and conformal structure of properly
embedded minimal surfacesmath.DG/0401257
 Oliver Baues,
William M. Goldman: Is the deformation space of complete
affine structures on the 2-torus smooth?hep-th/0111043
 Christian van
Enckevort: Note on mirror symmetry and coisotropic D-branes on
torigr-qc/0401081 
M. Anderson, P.T. Chrusciel, E. Delay: Non-trivial, static,
geodesically complete space-times with a negative cosmological
constant II. $nge 5$math.DG/0401230
 Francois
Labourie: Anosov Flows, Surface Groups and Curves in
Projective Spacemath.DG/0401221
 Marco
Gualtieri: Generalized complex geometrymath.DG/0401210
DS: Dynamical
Systems---------------------math.DS/0401250
 Francois
Berteloot, Christophe Dupont: Linearisation d'endomorphismes
holomorphes de CP(k) et caracterisation des exemples de Lattes
par leur mesure de Greenmath.DS/0401206
 Antonios
Zagaris, Hans G. Kaper, Tasso J. Kaper: Fast and Slow Dynamics
for the Computational Singular Perturbation
Methodmath.DS/0401202
 Andreas Knauf,
Iskander A. Taimanov: On the integrability of the n-centre
problemmath.DS/0401194
 P'eter
B'alint, Serge Troubetzkoy: Rotor interaction in the annulus
billiardFA: Functional
Analysis-----------------------math.FA/0401258
 I. V.
Krasovsky: Gap probability in the spectrum of random matrices
and asymptotics of polynomials orthogonal on an arc of the
unit circlemath.FA/0401256
 I. V.
Krasovsky: Asymptotics for Toeplitz determinants on a circular
arcmath.FA/0401205
 Torsten
Ehrhardt: The asymptotics of the Fredholm determinant of the
sine kernel on an intervalGM: General
Mathematics-----------------------math.GM/0401279
 M. Andrle, L.
Rebollo-Neira, E. Sagianos: Backward Optimized Orthogonal
Matching PursuitGR: Group
Theory----------------math.GR/0401305
 George M.
Bergman, Saharon Shelah: Closed subgroups of infinite
symmetric groupsmath.GR/0401304
 George M.
Bergman: Generating infinite symmetric groupsmath.GR/0401280
 Bahls:
Relative hyperbolicity and right-angled Coxeter
groupsmath.GR/0401269
 Donghi Lee: A
tighter bound for the number of words of minimum length in an
automorphic orbitmath.GR/0401266
 Richard P.
Kent IV: Achievable ranks of intersections of finitely genera
free groupsmath.GR/0401236
 Wolfgang
Bertram, Karl-Hermann Neeb: Projective completions of Jordan
pairs Part II. Manifold structures and symmetric
spacesmath.GR/0401213
 Joerg
Winkelmann: A Lie Group without universal
coveringmath.GR/0401193
 Michael
Aschbacher, Michael K. Kinyon, J.D. lips: Finite Bruck
LoopsGT: Geometric
Topology----------------------math.GT/0401284
 Stefano
Vidussi: Symplectic Tori in Homotopy E(1)'smath.GT/0401282
 Ismar Volic:
Configuration space integrals and Taylor towers for spaces of
knotsmath.GT/0401259
 Oliver Baues:
Infra-Solvmanifolds and Rigidity of Subgroups in Solvable
Linear Algebraic Groupsmath.GT/0401251
 Emmanuel
Auclair, Christine Lescop: Algebraic version of the clover
calculus for homology 3-spheresmath.GT/0401211
 Lewis Bowen:
An Isometry Between Measure Homology and Singular HomologyKT:
K-Theory and Homology-------------------------math.KT/0401295
 Joachim Cuntz:
Bivariant $K$-theory and the Weyl algebraLO:
Logic---------math.LO/0401303
 B.Zilber:
Analytic and pseudo-analytic structures (a survey)MG: Metric
Geometry-------------------math.MG/0401219
 Semyon
Alesker: Valuations, non-commutative determinants, and
quaternionic pluripotential theorymath.MG/0401191
 Mathieu
Dutour, Frank Vallentin: Some six-dimensional rigid formsMP:
Mathematical Physics------------------------hep-th/0401142
 Aristophanes
Dimakis, Folkert Muller-Hoissen: Extension of Noncommutative
Soliton Hierarchiesgr-qc/0401067
 A.M. Gavrilik:
Applying the q-algebras U'_q(so_n) to quantum gravity: towards
q-deformed analog of SO(n) spin networksnlin.SI/0401029
 Olaf
Lechtenfeld, Alexander S. Sorin: A note on fermionic flows of
the N=(1|1) supersymmetric Toda lattice
hierarchymath-ph/0401039
 E. Caliceti,
S.Graffi: Canonical Expansion of PT-Symmetric Operators and
Perturbation Theorymath-ph/0401038
 Steven H.
Simon, Aris L. Moustakas: Eigenvalue Density of Correla
Complex Random Wishart Matricescond-mat/0401211
 P. Contucci,
C. Giardina', J. Pule': Thermodynamic Limit for Finite
Dimensional Classical and Quantum Disordered
Systemsquant-ph/0401098
 S. Baskal, E.
Georgieva, Y. S. Kim, M. E. Noz: Lorentz Group in Ray
Opticsmath-ph/0401037
 Alexander G.
Abanov, Maxim Braverman: Topological calculation of the phase
of the determinant of a non self-adjoint elliptic
operatormath-ph/0401036
 Peter B.
Weichman: Surface modes and multi-power law structure in the
early-time response of magnetic targetsmath-ph/0401031
 Alan Carey,
Hendrik Grundling: Amenability of the Gauge
Grouphep-th/0401098
 O.P.Santillan,
A.G.Zorin: Toric hyperkahler manifolds with quaternionic
Kahler bases and supergravity solutionshep-th/0401072
 Giovanni Landi,
Fedele Lizzi, Richard J. Szabo: Matrix Quantum Mechanics and
Soliton Regularization of Noncommutative Field
Theorymath-ph/0401035
 Natig M.
Atakishiyev, Anatoliy U. Klimyk, Kurt Bernardo Wolf: Finite
q-oscillatormath-ph/0401034
 David B.
Fairlie: Implicit Solutions of PDE'smath-ph/0401033
 Bozhidar Z.
Iliev: Generalized Doppler effect in spaces with a transport
along pathscond-mat/0401287
 Yan V.
Fyodorov: Complexity of Random Energy Landscapes, Glass
Transition and Absolute Value of Spectral Determinant of
Random Matricesquant-ph/0312203
 Marco Frasca:
1/N-expansion for the Dicke model and the decoherence
programmath-ph/0401024
 E. Ragoucy:
Integrable systems with impurityNT: Number
Theory-----------------math.NT/0401289
 N. Ishii:
Trace of Frobenius endomorphism of an elliptic curve with
complex multiplicationmath.NT/0401285
 Apoloniusz
Tyszka: A discrete form of the theorem that each field
endomorphism of R or Q_p is the identitymath.NT/0401276
 Hilmar Hauer,
Ignazio Longhi: Teitelbaum's exceptional zero conjecture in
the function field casemath.NT/0401265
 David Helm: On
maps between modular Jacobians and Jacobians of Shimura
curvesmath.NT/0401262
 Ernie Croot:
k-term Arithmetic Progressions in Sumsetsmath.NT/0401238
 Habiba Kadiri:
Une region explicite sans zero pour la fonction Zeta de
Riemannmath.NT/0401231
 Jan-Hendrik
Evertse, Umberto Zannier: Linear equations with unknowns from
a multiplicative group in a function fieldmath.NT/0401228
 Zhi-Wei Sun:
Binomial Coefficients and Quadratic Fieldsmath.NT/0401223
 Kevin Ford:
The distribution of integers with a divisor in a given
intervalmath.NT/0401215
 Kevin Ford: On
Bombieri's asymptotic sievemath.NT/0401209
 Michael
Larsen: Rigid Lattices are Mordell-Weilmath.NT/0401197
 Rolf Soeren
Krausshar: Generalized Analytic Automorphic Forms for some
Arithmetic Congruence subgroups of the Vahlen group on the
n-Dimensional Hyperbolic Spacemath.NT/0401195
 Manfred
Kuhleitner, Werner Georg Nowak: The lattice point discrepancy
of a body of revolution: Improving the lower bound by
Soundararajan's methodOA: Operator
Algebras---------------------math.OA/0401242
 Huaxin Lin:
Full extensions and approximate unitary
equivalencesmath.OA/0401241
 Huaxin Lin:
Extensions by simple $C^*$-algebras -- Quasidiagonal
extensionsmath.OA/0401240
 Huaxin Lin:
Simple nuclear $C^*$-algebras of tracial topological rank
onemath.OA/0401227
 Heath Emerson,
Ralf Meyer: Dualizing the coarse assembly mapOC: Optimization
and Control----------------------------math.OC/0401297
 Jorge Cortes,
Sonia Martinez, Francesco Bullo: Spatially-distribu coverage
optimization and control with limi-range
interactionsmath.OC/0401199
 Somdeb Lahiri:
Stable Outcomes For Contract Choice ProblemsPR: Probability
Theory----------------------math.PR/0401248
 Paolo Dai Pra,
Gustavo Posta: Logarithmic Sobolev Inequality for Zero-Range
Dynamicsmath.PR/0401233
 Endre
Cs'{a}ki, Ant'{o}nia F{o}ldes, P'al R'ev'esz: Maximal local
time of a d-dimensional simple random walk on
subsetsmath.PR/0401229
 Alice
Guionnet, Mylene Maida: Character expansion method for the
first order asymptotics of a matrix integralmath.PR/0401208
 Christina
Goldschmidt: Critical random hypergraphs: the emergence of a
giant set of identifiable verticesmath.PR/0401189
 Martin
Hildebrand: A Survey of Results on Random Random Walks on
Finite GroupsQA: Quantum
Algebra-------------------math.QA/0401272
 E. Celeghini,
P. P. Kulish: Deformation of orthosymplectic Lie superalgebra
osp(1|2)math.QA/0401268
 Mikhail
Khovanov, Lev Rozansky: Matrix factorizations and link
homologymath.QA/0401246
 Damien
Calaque, Pavel Etingof: Lectures on tensor
categoriesmath.QA/0401245
 V.Tarasov:
Duality for Knizhnik-Zamolodchikov and Dynamical Equations,
and Hypergeometric Integralsmath.QA/0401232
 Gaston Andres
Garcia: On Hopf algebras of dimension $p^3$math.QA/0401226
 L. Feher, I.
Marshall: The non-Abelian momentum map for Poisson-Lie
symmetries on the chiral WZNW phase spacemath.QA/0401207
 A.Chakrabarti:
A nes sequence of projectors and corresponding braid matrices
$hat R(theta)$: (1) Odd dimensionsRA: Rings and
Algebras----------------------math.RA/0401263
 V. Bavula:
Gelfand-Kirillov dimension of commutative subalgebras of
simple infinite dimensional algebras and their quotient
division algebrasRT: Representation
Theory-------------------------math.RT/0401298
 Kendra Nelsen,
Arun Ram: Kostka-Foulkes polynomials and Macdonald spherical
functionsmath.RT/0401296
 J.M.
Landsberg, L. Manivel: A universal dimension formula for
complex simple Lie algebrasmath.RT/0401287
 David
Goldberg: Reducibility for $SU_n$ and generic elliptic
representationsmath.RT/0401286
 David
Goldberg: On Isotropic Bessel Models for Principal Series
Representations of $SO_{n,n+1}(Bbb R)$math.RT/0401261
 Dubravka Ban:
Symmetry of Arthur parameters under Aubert
involutionmath.RT/0401222
 I. Mirkovic,
K. Vilonen: Geometric Langlands duality and representations of
algebraic groups over commutative ringsmath.RT/0401214
 Xiao-Wu Chen,
Hua-LIn Huang, Pu Zhang: Dual Gabriel Theorem with
applicationsmath.RT/0401203
 Stephan
Mohrdieck: A Steinberg C-Section for Non-Connec Affine
Kac-Moody GroupsSG: Symplectic
Geometry-----------------------math.SG/0401277
 Bernhard
Kroetz, Michael Otto: A refinement of the complex convexity
theorem via symplectic techniquesmath.SG/0401273
 Rui Loja
Fernandes, ippe Monnier: Linearization of Poisson
bracketsmath.SG/0401212
 Paul Seidel:
Exact Lagrangian submanifolds of $T^*S^n$ and the graded
Kronecker quiverSP: Spectral
Theory-------------------math.SP/0401288
 Michael
Hitrik: Boundary spectral behaviour for semiclassical
operators in one dimension-- / Greg Kuperberg (UC Davis) / /
Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ /
===
<3FF8937F.839F8940@hw-itc.de>Epigone-thread:
traldzemqueiContent-Length: 1358Originator: rusin@vesuviusIn
reply to Holger Walliser's question --If I've understood your
sta question correctly (andalso guessed some unsta parts
correctly), then whatyou're looking for cannot exist. It is
not possibleto have a measure on an infinite dimensional
normedvector space that possesses some of the most
basic,familiar properties of n-dimensional Lebesgue
measure.More specifically: On any infinite-dimensionalnormed
vector space, it is not possible to have a measurethat is
nonnegative, translation-invariant,finitely or countably
additive, and takes positivefinite values on balls of positive
radius.That follows easily from Kottman's theorem:If X is any
infinite-dimensional normed vector space(with reals or complex
numbers as scalar field), thenthere exists an infinite sequence
of points, allhaving norm 1, and all having distance greater
than1 from each other. Draw balls of radius 1/2 aroundthose
points, and you'll have infinitely many disjointballs of
radius 1/2 all contained inside a ball ofradius 3/2.A short
proof of Kottman's Theorem can be found Diestel's book
Sequences and Series in Banach Spaces. That proofis also
given, with slightly more detail, in my bookHandbook of
Analysis and its Foundations.I'm not sure what happens if you
change to some otherfield; I've never thought about that.Eric
===
pamthawhuContent-Length: 272Originator: rusin@vesuviusCan
anyone give me any information on the possible galois
extensions of the Laurent Series field k((x)), where k is
algebraically closed? I know that all such extensions are
cyclic if characteristic of k is 0, but what about positive
===
numbersContent-Length: 736Originator: rusin@vesuviusConsider
the following number, where m and n are positive integers:B(m,
n) = (5m)!(5n)!/(m!n!(3m + n)!(3n + m)!)I am trying to show it
is an integer (something that must be truesince it is true for
n and m less than 1000, pari says it).My problem is: how to
show this (and for similar combinatorial numbersinvolving
quotients)?.I tried to show that [5m/k] + [5n/k] - [m/k] -
[n/k] - [(3n+m)/k] - [(3m+n)/k] >= 0where [] is the integer
part, and m, n y k are positive integers,but I didn't succed
(there are too many cases and subcases); from thisit is
deduced the statement by using the well-known formula for
thep-adic valuation of n!.Any reference to a book or paper
===
partitions into intervalsContent-Length: 729Originator:
rusin@vesuviusHi. Consider the following.Let X be a finite
set. If A and B are subsets of X,and Asubseteq B, the interval
[A,B] is the collectionof subsets C such that Asubseteq C
subseteq B.A partition into intervals is a partition
{I_1,ldots I_n} of 2^X such that each I_i is an interval. Of
course,there are at least two such partitions: {[A,A] :
Asubseteq X} and {[emptyset, X]}.The object I'm interes in is
the collection of partitions into intervals of 2^X. I have two
questions 1) do you know of referencesdealing with this stuff
(or more generally, partitionsinto intervals of PO sets) 2)
I'm trying to find thecardinality of the set of partitions
into intervals.All the best,FedericoFederico
===
continuesContent-Length: 1917Originator:
rusin@vesuviusPublication of the Kyoto Proceedings continues
with:Geometry and Topology Monographs, Volume 4
(2002)Invariants of knots and 3-manifolds (Kyoto 2001)Paper
no. 22, pages
337--362URL:http://www.maths.warwick.ac.uk/gt/GTMon4/paper22.
abs.htmlTitle:The algebra of knot trivalent graphs and
Turaev's shadow worldAuthor(s):Dylan P. ThurstonAbstract:Knot
trivalent graphs (KTGs) form a rich algebra with a few
simpleoperations: connec sum, unzip, and bubbling. With these
operations,KTGs are genera by the unknot tetrahedron and
Moebiusstrips. Many previously known representations of knots,
including knotdiagrams and non-associative tangles, can be
turned into KTGpresentations in a natural way. Often two
sequences of KTG operations produce the same output on
allinputs. These `elementary' relations can be subtle: for
instance,there is a planar algebra of KTGs with a
distinguished cycle. Studyingthese relations naturally leads
us to Turaev's shadow surfaces, acombinatorial representation
of 3-manifolds based on simple 2-spinesof 4-manifolds. We
consider the knot trivalent graphs as theboundary of a such a
simple spine of the 4-ball, and to consider aMorse-theoretic
sweepout of the spine as a `movie' of the knotgraph as it
evolves according to the KTG operations. For every
KTGpresentation of a knot we can construct such a movie. Two
sequences ofKTG operations that yield the same surface are
topologicallyequivalent, although the converse is not quite
true.Secondary: 57M20, 57Q40Keywords:Knot trivalent graphs,
shadow surfaces, spines, simple 2-polyhedra, graph
operationsAuthor(s) address(es):Department of Mathematics,
Harvard University Cambridge, MA 02138, USAEmail:
===
sha1:tnl01CJPQkrIqEOXuInKSOHIpXY=> Hello Everyone, Question:
Let f: [0,1] -> R (where R is the set of real>> numbers) and
is infinitely differentiable. How can I show>> that f has an
anti-derivative? I'd figured the Fundamental Theorem of
Calculus would help>> me in answering this question, but
unfortunately it assumes>> the existence of anti-derivatives.
How can I logically>> show that the function f above has an
anti-derivative? Look at the _other_ Fundamental Theorem - the
one about the derivative> of the integral.There is more than
one? (That is not a rewrite of the firstfundamental TM) My
book, Adams R. Fundamentals of Calculus, only stateone. What
===
Everyone, Question: Let f: [0,1] -> R (where R is the set of
real>> numbers) and is infinitely differentiable. How can I
show>> that f has an anti-derivative? I'd figured the
Fundamental Theorem of Calculus would help>> me in answering
this question, but unfortunately it assumes>> the existence of
anti-derivatives. How can I logically>> show that the function
f above has an anti-derivative? Look at the _other_
Fundamental Theorem - the one about the derivative> of the
integral.There is more than one? (That is not a rewrite of the
first> fundamental TM) My book, Adams R. Fundamentals of
Calculus, only state> one. What is the other?With suitable
hypotheses:(1) int(f(x), x = a..b) = F(b) - F(a) where the
derivative of F(x) isf(x).(2) The derivative of int(f(t), t =
a..x) is f(x).Some authors call (1) the Fundamental Theorem;
others call (2) theFundamental Theorem. Some speak of a First
and Second FundamentalTheorem. They are easy consequences of
each other.OP was referring to (1) and I was suggesting (2)
would be more helpful.-- Paul SperryColumbia, SC
===
Permutation Question>A question for any with time to burn
about permutations.1. using the numerals 0,1,2,3,4,5,6,7 how
many different 4 digit>numbers can be made greater than
3400?please explain the process not just the answer if
to 7777 in base 8.Subtracting the two numbers gives 4377 (base
8).Converting 4377 base 8 to base 10:4 * 8^3 = 20483 * 8^2 =
===
Permutation QuestionA question for any with time to burn about
permutations.1. using the numerals 0,1,2,3,4,5,6,7 how many
different 4 digit>numbers can be made greater than 3400?please
The problem looks like your counting from 3400 to 7777 in base
8.> Subtracting the two numbers gives 4377 (base 8).>
Converting 4377 base 8 to base 10:> 4 * 8^3 = 2048> 3 * 8^2 =
192> 7 * 8^1 = 56> 7 * 8^0 = 7> Total is 2303.BillThis result
assumes that a digit can be used more than once in a number.
The use of the word permutation in the statement of the
===
Permutation QuestionA question for any with time to burn about
permutations.1. using the numerals 0,1,2,3,4,5,6,7 how many
different 4 digit> numbers can be made greater than
3400?please explain the process not just the answer if
than once in any one > number. Since you refer to
permutations, I will presume not.How many starting with 4,or
5, or 6, or 7?How many starting with 34, or 35, or 36 or
trying to help outa student) but I came up with 1080 being 7P3
x 4 for the numbersstarting with either 4,5,6 or 7 and 6P2 x 4
for the numbers startingwith 34,35,35 or 37. Have I got it
===
Permutation QuestionA question for any with time to burn about
permutations.1. using the numerals 0,1,2,3,4,5,6,7 how many
different 4 digit> numbers can be made greater than
3400?please explain the process not just the answer if
than once in any one > number. Since you refer to
permutations, I will presume not.How many starting with 4,or
5, or 6, or 7?How many starting with 34, or 35, or 36 or
trying to help out> a student) but I came up with 1080 being
7P3 x 4 for the numbers> starting with either 4,5,6 or 7 and
6P2 x 4 for the numbers starting> with 34,35,35 or 37. Have I
your arithmetic is a bit off. Do you understand how to
calculate mPn? It is the product of n integestarting with m
and successively decreasing by one, so, e.g., 7P3 = 7*6*5. I
get (7*6*5)*4 =210*4 = 840 and (6*5)*4 =30*4 = 120adding up to
===
values of x for which2/3-x=1/3-1/x?After factoring by the LCD
I getx^2+6x-9=0 It seems that this trinomial has no solutions,
===
real values of x?>What are all real values of x for
which2/3-x=1/3-1/x?Therefore -x + 1/3 + 1/x = 0and multiplying
by -3x: 3x^2 - x - 3 = 0>After factoring by the LCD I
getx^2+6x-9=0 >It seems that this trinomial has no solutions,
or did I get it wrong?It would seem that one of us got it
wrong.The equation you gave has solutions of 1/6 +-
sqrt(37)/6.-- Stan Brown, Oak Road Systems, Cortland County,
New York, USA http://OakRoadSystems.comAn expense does not
have to be required to be considered necessary. -- IRS Form
===
values of x?>What are all real values of x for
which2/3-x=1/3-1/x?Therefore> -x + 1/3 + 1/x = 0> and
multiplying by -3x:> 3x^2 - x - 3 = 0>After factoring by the
LCD I getx^2+6x-9=0 >It seems that this trinomial has no
solutions, or did I get it wrong?It would seem that one of us
got it wrong.The equation you gave has solutions of 1/6 +-
sqrt(37)/6.I agree with Stan.And I think your error is in
===
going from '3x^2 - x - 3 = 0' to 'x^2+6x-9=0 '.Subject: Re:
x for which2/3-x=1/3-1/x?After factoring by the LCD I
getx^2+6x-9=0 > It seems that this trinomial has no solutions,
is not a solution)2x - 3x^2 = x - 3 after multiplying both
sides by 3x (valid if x <> 0)0 = 3x^2 - x - 3 Solve by
===
density functions> If I have two PDFs, X and Y> Can I find the
probability of (X AND Y) by just multiplying the tow>
===
density functions by support1.mathforum.org (8.11.6/8.11.6/The
===
Proof HelpI am not quite sure how to prove this problem. Can
someone give me alittle help, It would greatly be apprecia.
The question is asfollows: prove that if a and m are positive
integeand if thereexits an integer s with sa=1 mod m then a
===
HelpLets change this to a and b are positive integers and
there exists aninteger n with na=1 mod b then a and b are
relatively prime. (I am changingthe name of the variables
around to make the proof clearer - nobody callstwo variables
that are otherwise identical a and m)Lets assume a and b are
not relatively prime, and so they have a commonfactor k>1. Let
a=ka' and b=kb'Assume that there is an integer n with the
property you want.Thenna = 1 mod bna = mb+1 (for some m)nka' =
mkb' + 1k(na'-mb') = 1So k=1. But this contradicts k>1.
Therefore they have no common factor.> I am not quite sure how
to prove this problem. Can someone give me a> little help, It
would greatly be apprecia. The question is as> follows:> prove
that if a and m are positive integeand if there> exits an
integer s with sa=1 mod m then a and m are relatively prime.
===
weighing the sum?This is very off-thread but I am getting
desperate.Packing boxes into a TIR container and weighing the
sum?A customer of mine needs a program which packsas many
boxes into a standard TIR container, andwhich also calculates
the wieght of the packedload.Does such a *program* exist? Free
or to buy....The boxes are of course different sizes and
weightsetc.--http://www.ransen.com/Repligator - Easy graphics
===
Number Project updateThe bi-annual update of the Erdos Number
Project web site has beencomple. The URL is
http://www.oakland.edu/~gman/erdoshp.htmlBriefly, this project
studies the subject of collaboration inmathematical research in
general and the collaborations of Paul Erdos(1913-1996) in
particular. On the web site are the list of the 509people who
have written joint papers with Erdos (they have Erdos number1)
and the lists of their other coauthors (a total of 6984 of them
--the people with Erdos number 2). The data are organized in
severalways, and various statistics are summarized. We also
include an updateto the complete bibliography of Erdos
(numbering more than 1500 papers),preprints of papers about
this subject, and dozens of links to relamaterial, including
the flurry of recent serious research activity oncollaboration
graphs.Further information is contained in the README file,
available at thesite.As always, we want to know about
corrections, additions, and relainformation.-- Jerrold W.
Gman, Professor VOICE: (248) 370-3443 Department of
Mathematics and Statistics FAX: (248) 370-4184 Oakland
University FLESH: 346 SEB Rochester, MI 48309-4485 E-MAIL:
gman@oakland.edu WEB: http://www.oakland.edu/~gman/--
===
trying to do what looks like a simple integral but can't find
anytrigonometric relationships to do it. Does anyone know how
to do it?I= (sin3x)/(1+cosx)Any help would be much
===
repition my not be evil concepts in your classroom, butas a
high school math teacher of students who have no concept of
themultiplication facts, they must be evil concepts in the
elementaryclassrooms in my district. I do think there is a way
to make learningthe facts rewarding and not just a boring rote
process. I my room,many rewards are used to encourage learning
new concepts, because mostof my students did not find the joy
of knowing these facts.> there aren't any strategies per say
just have them stand in class and> say them other than
rewarding them with a trinket it is just something> they need
to do. You do them a great disservice by not fully> impressing
on them the joy in knowing these facts. Rote is not a bad>
word. Practice and repition are not evil concepts.--
===
Xavi,I wonder where you got these numbers from. I saw the
inequalitieslike [5m/k] + [5n/k] - [m/k] - [n/k] - [(3n+m)/k]
- [(3m+n)/k] >= 0in a very different context. They are rela to
Q-factorial terminal andcanonicaltoric singularities (or,
combinatorially, to simplicial lattice cones withrelativelyfew
points inside, in some sense). If I am not mistaken, your
particularcase should be rela to a family of stable
5-dimensional canonicalsingularities. Theoretically, one can
classify all such singularities, inany given dimension, but in
reality there are too many of them. Still,a lot is known.Here
are some references:1) Mori, Shigefumi; Morrison, David R.;
Morrison, Ian.On four-dimensional terminal quotient
singularities.Math. Comp. 51 (1988), no. 184, 769--786.2)
Sankaran, G. K. Stable quintuples and terminal quotient
singularities.Math.Proc. Cambridge os. Soc. 107 (1990), no. 1,
91--101.3) You may also want to check my papers and a survey on
this topic. They areavailable at
http://www.math.psu.edu/borisov/researchpage.html(see Toric
Varieties and Lattice Polytopes)I hope some of this will be
useful to you. Please let me know where thesenumbers are
coming from. Any references or pointers would be
greatlyapprecia.Alex> Consider the following number, where m
and n are positive integers: B(m, n) = (5m)!(5n)!/(m!n!(3m +
n)!(3n + m)!) I am trying to show it is an integer (something
that must be true> since it is true for n and m less than
1000, pari says it). My problem is: how to show this (and for
similar combinatorial numbers> involving quotients)?. I tried
to show that> [5m/k] + [5n/k] - [m/k] - [n/k] - [(3n+m)/k] -
[(3m+n)/k] >= 0> where [] is the integer part, and m, n y k
are positive integers,> but I didn't succed (there are too
many cases and subcases); from this> it is deduced the
statement by using the well-known formula for the> p-adic
valuation of n!. Any reference to a book or paper will be very
===
Algebraic and Geometric TopologyContent-Length:
2025Originator: rusin@vesuviusThe following paper has been
published:Algebraic and Geometric
TopologyURL:http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-4
.abs.htmlTitle:The boundary-Wecken classification of
surfacesAuthor(s):Robert F. Brown, Michael R.
KellyAbstract:Let X be a compact 2-manifold with nonempty
boundary dX and let f: (X,dX) --> (X, dX) be a
boundary-preserving map. Denote by MF_d[f] theminimum number
of fixed point among all boundary-preserving maps thatare
homotopic through boundary-preserving maps to f. The
relativeNielsen number N_d(f) is the sum of the number of
essential fixedpoint classes of the restriction f-bar : dX -->
dX and the number ofessential fixed point classes of f that do
not contain essential fixedpoint classes of f-bar. We prove
that if X is the Moebius band withone (open) disc removed,
then MF_d[f] - N_d(f) < 2 for all maps f :(X, dX) --> (X, dX).
This result is the final step in theboundary-Wecken
classification of surfaces, which is as follows. If Xis the
disc, annulus or Moebius band, then X is boundary-Wecken,
thatis, MF_d[f] = N_d(f) for all boundary-preserving maps. If
X is thedisc with two discs removed or the Moebius band with
one disc removed,then X is not boundary-Wecken, but MF_d[f] -
N_d(f) < 2. All othersurfaces are totally non-boundary-Wecken,
that is, given an integer k> 0, there is a map $f_k : (X, dX)
--> (X, dX) such that MF_d[f_k] -N_d(f_k) >= k.Secondary:
54H25, 57N05Keywords:Boundary-Wecken, relative Nielsen number,
punctured Moebius band, boundary-preserving mapReceived: 21
November 2002Author(s) address(es):Department of Mathematics,
University of California Los Angeles, CA 90095-1555, USA and
Department of Mathematics and Computer Science, Loyola
University 6363 St. Charles Avenue, New Orleans, LA 70118,
===
Epigone-thread:
traldzemqueiContent-Length: 1091Originator: rusin@vesuvius>If
I've understood your sta question correctly (and>also guessed
some unsta parts correctly), then what>you're looking for
cannot exist.I'm afraid you're right with this part>It is not
possible>to have a measure on an infinite dimensional
normed>vector space that possesses some of the most
basic,>familiar properties of n-dimensional Lebesgue
measure.I'm not interes in infinite dimensional spaces>I'm not
sure what happens if you change to some other>field; I've never
thought about that.But this is exactly what I'm interes in! I'm
working in an-dimensional Banachspace over such a field. But I
think I can learn from your explanations that I will have
exactly the same problem. Thisis because in the fields I'm
interes in the absolute value is sothat you can find
infinitely many elements with an absolute value of 1having
pairwise distance 1 and so I think I can use the
constructionyou explained even in the 1-dimensional
Banachspace which is the fielditself. Am I right with
===
spacesContent-Length: 538Originator: rusin@vesuvius> Let J: X
to X'' be the natural (isometric) injection.> Then J^*: X'''
to X' is a (norm 1) projection. > QED (modulo checking the
sufficiently clear, sorry.I do agree that the fact is simple,
but I amin which it is presen with a proofor at least as an
exercise.Probably, you mean that the fact is
conventionallyregarded as a folklore and does not requireany
===
complemen Banach spacesContent-Length: 919Originator:
rusin@vesuvius>> Let J: X to X'' be the natural (isometric)
injection.>> Then J^*: X''' to X' is a (norm 1) projection. >>
message was not sufficiently clear, sorry.>I do agree that the
fact is simple, but I am>in which it is presen with a proof>or
at least as an exercise.Probably, you mean that the fact is
conventionally>regarded as a folklore and does not require>any
reference when used. Is it so?It's simple enough that you could
reasonably justuse it without comment, or you could just givean
even shorter version of the proof above.Or you could cite [n]
Zak Thargle-Whoosh (personal communication). I'd love to see
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subset A in a real linear space B .Further w is a fixed
(rational) number in (0,1),e.g. w=1/2 . The set A is called
,,w-internal if and only ifx,y in A implies w*x +(1-w)*y is
also in A .Which are properties of such w-internal sets ?Such
sets were investiga ?
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how do we do a simulator using MATLAB for an example BJT, what
kind of parameters should be consider and what inputs should
beasking the user to enter? like this things what do you guys
think! hope someone helps me to find a
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comparisons of the> speed of different algorithms in MATLAB
6.5 for Windows.Maybe you already know, but there's a command
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'cputime'.Subject: Need help finding stationary segments of
week ago, but didn't get oneresponse. I had hoped that some of
you would know all about this, so I willtry again. If I am
seeking help from the wrong group, could you please point me
to theright group? I have road profile data (elevations),
measured at constant interval. Isuspect that the whole data
set contains a number of segments which arestastically
different (ie different spectral content, variance etc). My
aim is to be able to extract measures of roughness, and hence
a roughnessdistribution. To be meaningful, roughness measures
should only be taken fromstationary profiles. If one was to
assume some error tolerance, are there methods of
identifyingthe start and end of segments that could be
considered weakly stationary? An algorithm would be nice, but
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stationary segments of signal> Hullo All,> I asked for help on
this subject about a week ago, but didn't get one> response. I
had hoped that some of you would know all about this, so I
will> try again. I replied to your question -- look for the
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message with the headerbelow, using Google.Subject: Re:
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there a set of numbers between the rationals and reals?If the
real numbers consist of the following:1) rationals2) numbers
that can be represen by finite algorithms (example: eand pi)3)
numbers that are neither rational nor representable by a
finitealgorithmThe last item would include numbers like
this:0.123245223643792134....(infinite sequence of digits that
areimpossible to represent with any algorithm.)Why is there not
a well known set of numbers that does not include #3? It would
be more dense than the rationals but less than than thereals.
What is the purpose of describing numbers like #3?Sorry if I
am wrong about any of the above assumptions. I was readingan
online calculus tutorial and I was just curious.
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integralHello everyone.I have difficult with solving following
integral.Could somebody help me find correct way ?Question>
Int(-00,00), x / (x^3-8) dx=??To find Int(-00,00), x / (x^3-8)
dx ,Since x / (x^3-8) is not an even funtion,Int(-00,00), x /
(x^3-8) dx = { lim_R1->00 Int(-R1, 0) x / (x^3-8) dx } +{
lim_R2->00 Int(0, R2) x / (x^3-8) dx }.So far I'm pretty
confident.But,According to my book solution,if f(z)=z over
z^3-8,f(z) = z / z(z+1+sqrt(3)*i)(z+1-sqrt(3)*i)
andInt(-00,00), x / (x^3-8) dx = 2pi*i Res[f, -1+sqrt(3)*i] +
pi*i Res[f,2]=sqrt(3)pi over 6.I have 2questions about this
solution,First,I think this solution use that Int(-00,00), x /
(x^3-8) dx = lim_R->00Int(-R, R) x / (x^3-8) dx.But I can't
accept this, because x / (x^3-8) is not an even funtion.Is my
thinking incorrect ?Second, why pi*i Res[f,2] instead of 2pi*i
Res[f,2] ?I seek my complex analysis book to know this
reason,But there is no sufficient explanation about this.If
somebody can explain above my suspicion or have any
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Integral using complex integralHello everyone.I have difficult
with solving following integral.> Could somebody help me find
correct way ?Question> Int(-00,00), x / (x^3-8) dx=??To find
Int(-00,00), x / (x^3-8) dx ,Since x / (x^3-8) is not an even
funtion,> Int(-00,00), x / (x^3-8) dx = { lim_R1->00 Int(-R1,
0) x / (x^3-8) dx } +> { lim_R2->00 Int(0, R2) x / (x^3-8) dx
}.> So far I'm pretty confident.But,> According to my book
solution,> if f(z)=z over z^3-8,> f(z) = z /
z(z+1+sqrt(3)*i)(z+1-sqrt(3)*i) and> Int(-00,00), x / (x^3-8)
dx = 2pi*i Res[f, -1+sqrt(3)*i] + pi*i Res[f,2]> =sqrt(3)pi
over 6.I have 2questions about this solution,First,> I think
this solution use that Int(-00,00), x / (x^3-8) dx =
lim_R->00> Int(-R, R) x / (x^3-8) dx.> But I can't accept
this, because x / (x^3-8) is not an even funtion.> Is my
thinking incorrect ?Second, why pi*i Res[f,2] instead of 2pi*i
Res[f,2] ?> I seek my complex analysis book to know this
reason,> But there is no sufficient explanation about this.If
somebody can explain above my suspicion or have any
you have written it, the integral only makessense as a Cauchy
principal value integral. It has a simplepole on the real
z-axis at z = 2. So what you must do is draw acontour that
goes around that pole on a small semi-circle. Thenyou close it
with a large semi-circle of radius R, say in the upperhalf
z-plane. The part that is strictly along the real axis
willgive you the Cauch PV integral, as R -> infty and as the
radiusof small semicircle goes to 0.The rest is simple.--
^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ God
is not willing to do everything and thereby take away our free
will and that share of glory that rightfully belongs to us. --
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interpolationOriginator: rhn@mauve.rahul.net (Ronald H.
Nicholson Jr.) You can find the Lagrange interpolation formula
in Abramowitz & Stegun,>> Handbook of Mathematical Functions.
It is just a simple way to write a>> polynomial of degree N
that goes through N+1 points of a function. For example,
suppose you know f(x) at the points x0, x1 and x2. Call>>
these values f0, f1 and f2. Then you write > f0 *
(x-x1)*(x-x2) f1 * (x-x0)*(x-x2) f2 * (x-x0)*(x-x1)>f(x) ~
------------------ + ------------------ + ------------------>
(x0-x1)*(x0-x2) (x1-x0)*(x1-x2) (x2-x0)*(x2-x1) You see this
is quadratic in x. Higher-order polynomials are essentially>>
the same. >> This formula directly gives the value of f at any
x. What I'm actually>after is the kernel. Ie. | g(x) 0 <= |x|
<= 1>k(x) = | h(x) 1 <= |x| <= 2> | 0 |x| >= 2>What is g and
h?Can you measure the impulse response of an interpolation
formula?Is there a relationship between the impulse response
and the convolutionkernal?IMHO. YMMV.-- Ron Nicholson rhn AT
nicholson DOT com http://www.nicholson.com/rhn/ #include
 // only my own opinions,
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bE5eBYZSre5EAwvtcIp4CJd2yelMks46gtjBO8KuujuzjgDedoIXsxit could
also be riccatti! maybe that's the reason for the
difficultsolving?!> I wonder if there is anyone who could help
me solve this equation step bystep. y(x) = y(x)/x + 3(y(x))^2 I
think that this is Bernoullis equation, But not sure. Jaakko