mm-1059 Not all such states are distinguishable by measurement, but they all produce different distributions of results under measurement, so the distinction between all such states is operationally significant for simulation. Since most points in a continuum cannot be written on a tape, regardless of the encoding chosen, not even a single atom can be simulated by a TM. Sure, you can approximate if you want, but that wasnÕt the claim, was it?snÕt this digitising the data? if at some point in time after the initial state the results a Ôfunctionof the initial state then sure, it canÕt be simulated. but we treathe term real-world-function as a function you can encapsulate.unction ~ same result given the same inputnalog === data ~ no inputs are the sameerc==Subject: Re: Poll: Are PCs Turing Machines? > encoding chosen, not even a single atom can be simulated by a TM.atter TMatt Timmermans was wondering when your destiny was going to kick in. The reduction of youruantum entanglement with Adam is now complete, you are mortal again.erc==ubject: Re: Poll: Are PCs Turing Machines?n sci.math, |-|erch@r.c>32983gF3jfrppU1@individual.net>:>> infinite people each ßip coins infinite times>=>> all sequenes of heads and tails have been tossed to infinite length> [rest snipped]> Produce the list. Read the assumptions. Herche assumptions are as follows. (If these are not correct pleaseomeone let me know. These arenÕt exactly according to Dedekindr Cauchy, but should be compatible with either.)1] N is constructed by PeanoÕs Axioms or similar. J is N union {0} union {-x: x in N}. Q = {j/k: j, k in J, k != 0}. All of these sets are provably denumerable and are totally ordered, though the denumeration wonÕt follow the ordering in the case of Q, or even in J.2] Each and every positive real number r in R can be approximated by a sequence q_i and another sequence qÕ_i, both in Q, such that 0 < q_i < qÕ_j for any i,j in N, and furthermore that: [a] 0 < abs(q_i - q_j) < epsilon for any i,j > a certain N(epsilon), i != j. [b] 0 < abs(qÕ_i - qÕ_j) < epsilon for any i,j > a certain NÕ(epsilon), i != j. [c] 0 < abs(q_i - qÕ_j) < epsilon for any i,j > a certain M(epsilon), i != j. These sequences are not necessarily unique; e.g. the number 1/3 can be approximated by the pair .3, .33, .333, ... and .4, .34, .334, ... or by 2/9, 3/12, 4/15, 5/18, ... and 4/9, 5/12, 6/15, 7/18, ... . Since 1/3 is in fact rational this isnÕt the best of examples, admittedly; however the idea is also applicable to other numbers such as pi, e, and sqrt(2). The details are left to the inquisitive reader. Note that it is NOT legal to define q = q_i = qÕ_i for any rational number q in Q; however all rational numbers are in fact real, since one can simply use the sequences q_i = q - 1/(i+k), qÕ_i = q + 1/(i+k) instead, for some integer k > 0. (The value k may be necessary if q is close to 0, to avoid q_i rossing over This is admittedly a technical restriction.)3] If r is defined by q_i and qÕ_i, then -r, the negative of r, is defined by -qÕ_i and -q_i. (The swapping is of course necessary.)4] 0 is less than all positive reals, greater than all negative reals.5] Addition of two positive reals can be defined in a fairly straightforward fashion; if r is bounded by the series q_i and qÕ_i, and s is bounded by the series p_i and pÕ_i, then r+s can be bounded by the series q_i+p_i and qÕ_i+pÕ_i. Subtraction is the addition of the negative, and has to be done with some care because of the zero crossover problem; the details are left to the reader (the author suggests chopping off the heads of the two defining series if necessary; if the two numbers are in fact equal things get interesting). Multiplication of two positive reals is similar; multiplication of a negative real with a positive, or two negatives, is done by asserting (-r)s = r(-s) = -rs and (-r)(-s) = rs. Multiplication by 0 is 0.6] r < s if one can find a rational t such that, if r is bounded by q_i and qÕ_i and s by p_i and pÕ_i, then qÕ_i < t for any i > a certain N, and t < p_i for any i > a certain M. r > s if one can find a rational u such that, if r is bounded by q_i and qÕ_i and s by p_i and pÕ_i, then q_i > u for any i > a certain L, and pÕ_i < u for any i < a certain K. If neither of the above hold true, then r = s.ote that TX_10 = {j/10^k: k >= 0, j, k in J} is a *proper*ubset of all real numbers -- it doesnÕt contain 1/3, forxample, or pi, e, or sqrt(2). However, it contains all finiterefixes.- 191, ewill3@earthlink.nettÕs still legal to go .sigless.==ubject: quotient ring confusion by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBG14tT31433; am trying to get a good grasp as to what this ring ooks like[x]/(x^2-5, 5x-10)essing with the relations x^2=5 and 5x=10 i get the dditional relation that 2x^2-5x=0. I do not suppose could be so lucky as to have that the ideal x^2-5,5x-10) is principal, and generated by 2x^2-5x lone? i feel that if that were the case, then the uotient ring would be something like the dyadic ationals Q_2 since we would be adjoining 5/2 to Z.n any case, i really need to be able to get my head round the structure in order to understand the rest f this example from my professor (i am trying to fill n the missing details).==ubject: Re: quotient ring confusion I am trying to get a good grasp as to what this ring looks like R = Z[x]/(x^2-5, 5x-10) Z[x]ote: 5 is in the ideal I = (x^2-5, 5x-10) Z[x]ia 5 = 5 ((x-2)(x+2) - (x^2-5))hus I = (x^2-5, 5x-10, 5) Z[x] = (x^2,5) Z[x]o R = Z[x]/(5,x^2) = Z/5[x]/(x^2) which ishe ring of ual numbers[1] over the 5-elt field.-Bill Dubuque==ubject: Permanent Gravity Fields in GR?n NewtonÕs gravity theory, gravity is a force like any other, e.g. the lectric force. Every crackpot scheme I have seen about how Einstein is rong (about gravity letÕs leave special relativity aside for now) is ased on the amateurÕs inability to get past this cognitive barrier. Z s no exception to this rule. Neither is the so-called PV theory pushed y Eric Davis in the recent USAF teleportation report.ttp://www.space.com/businesstechnology/technovel_ teleport_041103.htmlor example in NewtonÕs theory using GalileoÕs relativity of absolute imultaneity, which is OK when v/c << 1 we can approximate the EarthÕs f mass m described by the Lagrangian = (1/2)mv^2 - mgz is altitude from surface of the Lab inertial frame. typical solution is the parabola (for initial conditions only along z) n the x-t the problem because of GalileoÕs equivalence rinciple.f we put in an initial horizonal speed u(0) along x then(t) = x(0) + u(0)tolve for t and eliminate t in z(t) replacing it by x and one gets nother parabola for z(x).n NewtonÕs theory this parabola is a NON-INERTIAL motion in the aw Fg = ma is obeyed.he picture does a bait and switch Gestalt Shift in going from Newton to instein that most people simply do not get - including some physicists.n EinsteinÕs theory of gravity, called geometrodynamics, Fg is liminated. It is a orce without Force The parabola is an INERTIAL OTION called the eodesic i.e. the straightest possible path not in D space but in 4D CURVED spacetime. The EarthÕs surface is not an arth because of quantum pressures from PauliÕs exclusion principle and he electric force so that detectors at rest on EarthÕs surface move on on-geodesic world lines. What we think of as the ravity force of eightis really only the electrical force pushing us off the natural orce free weightless ree ßoatgeodesic. This is why Astronauts are ontinually weightless in their free ßoat elliptical orbits round Earth ith rockets off. Those closed orbits are the straightest possible paths n the curved space-time caused by the EarthÕs mass M.f the distance from center of Earth to surface is r, the curvature cale at the surface is ~ GM/c^2r^3 = g/c^2r with dimensions 1/(Area) here here (Area)^1/2 ~ 1AU ~ 10^13 cm ~ local radius of curvature of pace-time from the Earth at its surface.he -forceis only felt by local observers pushed off geodesics by on-gravity forces. Local observers on geodesics are weightless with ero g-force.he g-force is not a tensor because if a tensor is zero in one local rame it is zero in all other local frames connected to it. You cannot ransform a non-zero tensor to a zero tensor or vice versa within the ules of EinsteinÕs geometrodynamics from 1916. You cannot even say that he g-force has a tensor piece that is not identically zero. Such a ensor piece of g-force is called on-metricityand that is zero in insteinÕs 1916 theory.xamplechwarzschild approximation to the curved spacetime above the surface of arth. The local frame invariant differential separation between two eighboring events iss^2 = (1 - 2GM/c^2r)(cdt)^2 - (1 - 2GM/c^2r)^-1dr^2 - dl^2(ßat pherical polar)he REST LNIF time clicked off by a clock at rest at distance r isT(LNIF) = (1 - 2GM/c^2r)^1/2(cdt) AliceÕs REST LNIF clock at fixed rob free falling past Alice in contrast measuresT(LIF) = dT(LNIF)/(1 - 2GM/c^2r)^1/2ote thatGM/c^2r < 1his is the LOCAL GRAVITY RED SHIFThis same red shift is also seen by a distant REST LNIF observer at r -> nfinityhat isT(LIF at r) = dT(REST LNIF at r -> infinity)al PuthoffÕs PV theory only has half of EinsteinÕs equivalence rinciple in his ables I & IIwith an incorrect ontology and easurement theory. That is, he only has the RHS of the above equation. al Puthoff does not believe we need to use tensors in his approach to he etric engineeringfor propellantless propulsion, i.e. arp drivehat is the permanent gravity field?e need the idea of the Killing vector field.urved spacetimes can, but need not, have special symmetries. If the etric guv(LNIF) is invariant under {LC} parallel transport along a amily of worldlines, the field of tangent vectors to those world lines s called a Killing vector field.he Schwarzschild metric has the Killing vector field T^u = (1,0,0,0).u is timelike inside local light cones for r > 2GM/c^2u is lightlike i.e. T^uTu = 0 on the Event Horizon of the vacuum black ole at r = 2GM/c^2u is spacelike outside the local light cone for r < 2GM/c^2r where the oordinate patch for r > 2GM/c^2 breaks down.he permanent g-force, more precisely, the g-acceleration for REST LNIF bservers is theURFACE GRAVITY = a(1 - 2GM/c^2r)^1/2 = THE NORM OF THE PROPER 4-acceleration needed to keep the LNIF bserver at a fixed r. It cannot exist physically without some on-gravity force!he SURFACE GRAVITY at the EVENT HORIZON of the BLACK HOLE is ~ c4/GM.e. c^2/(Schwarzschild Radius)he Hawking black body radiation temperature is ~ hc^3/16piGMkBhe entropy S of the Event Horizon is/kB ~ (GM/c^2)^2/(hG/c^3)==ubject: Re: Permanent Gravity Fields in GR?-RFC2646: Format=Flowed; Response In NewtonÕs gravity theory, gravity is a force like any other, e.g. the electric force. Every crackpot scheme I have seen about how Einstein is wrong (about gravity letÕs leave special relativity aside for now) is based on the amateurÕs inability to get past this cognitive barrier.eah, I have the same trouble. Nobody can get past the cognitive barrierhat wormholes are nests for bright green ßying elephants to lay their ggs in. Z is no exception to this rule. Neither is the so-called PV theory pushed by Eric Davis in the recent USAF teleportation report. http://www.space.com/businesstechnology/technovel_teleport_ 041103.html For example in NewtonÕs theory using GalileoÕs relativity of absolute simultaneity, which is OK when v/c << 1ell, always OK then, because for Einstein he velocity of light in our heory plays the part, physically, of an infinitely great velocity. eference :ttp://www.fourmilab.ch/etexts/einstein/specrel/www/ut some people cant get past the cognitive barrier that he doesnÕt mean trißing 299,792,458 metres per second, taking hours to reach (say) aturne really means infinite, and v/(infinity) << 1, right? we can approximate the EarthÕs surface as a Newtonian Global Inertial Lagrangian L = (1/2)mv^2 - mgz z is altitude from surface of the Lab inertial frame. A typical solution is the parabola (for initial conditions only along z) in the x-t plane z(t) = z(0) + v(0)t - (1/2)gt^2 Note that principle. If we put in an initial horizonal speed u(0) along x then x(t) = x(0) + u(0)t Solve for t and eliminate t in z(t) replacing it by x and one gets another parabola for z(x). In NewtonÕs theory this parabola is a NON-INERTIAL motion in the law Fg = ma is obeyed.k. So what? The picture does a bait and switch Gestalt Shift in going from Newton to Einstein that most people simply do not get - including some physicists.eah, I have that same trouble with elephant eggs. In EinsteinÕs theory of gravity, called geometrodynamics, Fg is eliminated. It is a orce without Force The parabola is an INERTIAL MOTION called the eodesic i.e. the straightest possible path not in 3D space but in 4D CURVED spacetime. The EarthÕs surface is not an Earth because of quantum pressures from PauliÕs exclusion principle and the electric force so that detectors at rest on EarthÕs surface move on non-geodesic world lines. What we think of as the ravity force of weightis really only the electrical force pushing us off the natural force free weightless ree ßoatgeodesic. This is why Astronauts are continually weightless in their free ßoat elliptical orbits round Earth with rockets off. Those closed orbits are the straightest possible paths in the curved space-time caused by the EarthÕs mass M.es, but the trouble is this chair is pushing my arse up. call that a force, not a LNIF. If the distance from center of Earth to surface is r, the curvature scale at the surface is ~ GM/c^2r^3 = g/c^2r with dimensions 1/(Area) where here (Area)^1/2 ~ 1AU ~ 10^13 cm ~ local radius of curvature of space-time from the Earth at its surface. The -forceis only felt by local observers pushed off geodesics by non-gravity forces. Local observers on geodesics are weightless with zero g-force. The g-force is not a tensor because if a tensor is zero in one local frame it is zero in all other local frames connected to it. You cannot transform a non-zero tensor to a zero tensor or vice versa within the rules of EinsteinÕs geometrodynamics from 1916. You cannot even say that the g-force has a tensor piece that is not identically zero. Such a tensor piece of g-force is called on-metricityand that is zero in EinsteinÕs 1916 theory. Example Schwarzschild approximation to the curved spacetime above the surface of Earth. The local frame invariant differential separation between two neighboring events is ds^2 = (1 - 2GM/c^2r)(cdt)^2 - (1 - 2GM/c^2r)^-1dr^2 - dl^2(ßat spherical polar) The REST LNIF time clicked off by a clock at rest at distance r is dT(LNIF) = (1 - 2GM/c^2r)^1/2(cdt) AliceÕs REST LNIF clock at fixed r Bob free falling past Alice in contrast measures dT(LIF) = dT(LNIF)/(1 - 2GM/c^2r)^1/2 Note that 2GM/c^2r < 1 This is the LOCAL GRAVITY RED SHIFT This same red shift is also seen by a distant REST LNIF observer at r -> infinity That is dT(LIF at r) = dT(REST LNIF at r -> infinity) Hal PuthoffÕs PV theory only has half of EinsteinÕs equivalence principle in his ables I & IIwith an incorrect ontology and measurement theory. That is, he only has the RHS of the above equation. Hal Puthoff does not believe we need to use tensors in his approach to the etric engineeringfor propellantless propulsion, i.e. arp drive What is the permanent gravity field? We need the idea of the Killing vector field. Curved spacetimes can, but need not, have special symmetries. If the metric guv(LNIF) is invariant under {LC} parallel transport along a family of worldlines, the field of tangent vectors to those world lines is called a Killing vector field. The Schwarzschild metric has the Killing vector field T^u = (1,0,0,0). Tu is timelike inside local light cones for r > 2GM/c^2 Tu is lightlike i.e. T^uTu = 0 on the Event Horizon of the vacuum black hole at r = 2GM/c^2 Tu is spacelike outside the local light cone for r < 2GM/c^2r where the coordinate patch for r > 2GM/c^2 breaks down. The permanent g-force, more precisely, the g-acceleration for REST LNIF observers is the SURFACE GRAVITY = a(1 - 2GM/c^2r)^1/2 a = THE NORM OF THE PROPER 4-acceleration needed to keep the LNIF observer at a fixed r. It cannot exist physically without some non-gravity force! The SURFACE GRAVITY at the EVENT HORIZON of the BLACK HOLE is ~ c4/GM i.e. c^2/(Schwarzschild Radius) The Hawking black body radiation temperature is T ~ hc^3/16piGMkB The entropy S of the Event Horizon is A/kB ~ (GM/c^2)^2/(hG/c^3)eah yeah...reat yarn, but IÕve never seen a black hole and neither has anyone lse.ormholes are more fun, but you wonÕt find one unless it has a bright reen ßying elephants egg in it.ndrocles.==ubject: Re: How to visualize limits in category theory? posting-account=tdYrvA0AAACTm02P3kgbrEwRzJI8b08Sn addition, may I ask if anyone knows any nice way ofescribing/explaning colimits?==ubject: Re: How to visualize limits in category theory? posting-account=tdYrvA0AAACTm02P3kgbrEwRzJI8b08SÕd be interested in hearing how other people interpret limits allright I find myself that the yramid-type-thing with commuting faceswayf thinking about them is to my satisfaction.oseph Goguen gives his interpretations of concepts in category theoryn his nicely written manifesto (which is fun to read anyway).ww.cs.ucsd.edu/users/goguen/pps/manif.pse says about limits:A diagram D in a category C can be seen as a system of constraints,nd then a limit of D represents all possible solutions of the system. donÕt know if these are in any way close to the answer to youruestion (of course I hope they touch on it anyway).==ubject: Re: How to visualize limits in category theory?I have previously asked the newsgroup about a very simple way of defining a complete category and there was a person who gave me a very nice definition.. that it has limit for all small diagram. I understand that perfectly but I cannot visualize it like I would like to visualize limits say in analysis. Is there really no way of visualizing categorical limits?In the book I am reading (and I hate to say it, but I am not finding it to my liking!) I keep on reading about complete category this, complete category that (not to mention well-poweredness, cowell-powerdness and cocompletness).. makes me crazy. Simple algebraic stuffs being proven by category theory is just not my style. And worst is, my supervisor keeps telling me that it is really required for me to read the book and be able to comfortably work with categories (I donÕt very much get it, we work on algebraic geometry and I have never expected that I had to do so much of category theory.).ecall that a functor F:J->A has a limit iff there is an isomorphismsignaled by =) natural in a in AA) A^J(Delta(a), F) = A(a, Lim F)here Delta:A->A^J is the diagonal functor sending a in A to theonstant functor J->A.ou can view Lim F as a solution to an soquation- an equation upo isomorphism:B) A^J(Delta(a), F) = A(a, ---- )onedaÕs lemma guarantees that if there is a solution to (B) then its unique up to (uniquely determined!) isomorphism. In other words, asith all representable functors, Lim F is a solution to anisoquationor, as somebody else has written in this thread, a systemf constraints *for the category* itself.f this view helps you or not I donÕt know, but IÕll proceed any way-) The fundamental idea is the main principle of category theorywhich I saw first explicitly stated in a paper by Kapranov andoevodsky)In a category it is unnatural and undesirable to speak about equalityf two objects n the other hand it is not only desirable, but healthy and natural,o speak about isomorphic objects. Isomorphic objects are, for allategorial matters, indistinguishable. The ideal situation would be toave a first-order language in which the relation of equality betweenbjects (remember: ndesirable and unnatural is replaced by theelation of isomorphic in such way that if a is an object, f(a) is arue formula and b = a then f(b) is a true formula (IÕm being sloppyn my description, I hope I can get my point across)f you are interested in this, work has been done by M. Makkai. Startith voiding the axiom of choice in general category theory(go tottp://www.math.mcgill.ca/makkai/). I quote the abstract hereThe notion of anafunctor is introduced. An anafunctor is, roughly, afunctor defined up to isomorphism Anafunctors have a general theoryaralleling that of ordinary functors; they have naturalransformations, they form categories, they can be composed, etc.nafunctors can be saturated, to ensure that any object isomorphic to possible value of the anafunctor is also a possible value at theame argument object. The existence of anafunctors in situations whenrdinarily one would use choice is ensured without choice; e.g., for aategory which has binary products, but not specified binary products,he anaversion of the product functor is canonically definable, unlikehe ordinary product functor that needs the axiom of choice. When theomposition functors in a bicategory are changed into anafunctors, onebtains anabicategorics. In the standard definitions of bicategoriesuch as the monoidal category of modules over a ring, or theicategory of spans in a category with pullbacks, and many others, oneses choice; the anaversions of these bicategories have canonicalefinitions. The overall effect is an elimination of the axiom ofhoice, and of non-canonical choices, in large parts of generalategory theory. To ensure the Cartesian closed character of theicategory of small categories, with anafunctors as 1-cells, one uses weak version of the axiom of choice, which is related to A. BlassÕxiom of Small Violations of Choice (1979). . Rodrigues== posting-account=uJhfTw0AAACZ85X1hg4ZQuYw9kXQVPPGol - hereÕs the quote that shows how hopeless the problem truly is:The American teacher] was especially taken aback by the textbook. Byrades seven andight, kids in the Singapore program are doing high-school-levellgebra. thought, wow, thatÕs complicated -- even for me,says Mr.eating. He was eventually won over when he saw how enthusiastic hiswn students became about math.ur own math teachers canÕt even do 7th grade math. Christ.==lol - hereÕs the quote that shows how hopeless the problem truly is:[The American teacher] was especially taken aback by the textbook. Bygrades seven andeight, kids in the Singapore program are doing high-school-levelalgebra. thought, wow, thatÕs complicated -- even for me,says Mr.Keating. He was eventually won over when he saw how enthusiastic hisown students became about math.Our own math teachers canÕt even do 7th grade math. Christ.ou may be surprised, but I am not. ut it is worse than that; the ew math which succeededn teaching mathematical concepts, not just how to compute,o children, could not be learned by the teachers, evenith major attempts to teach them. The case is noifferent now. And the schoolsemphasis on memorizationnd routine is greater.- his address is for information only. I do not claim that these viewsre those of the Statistics Department or of Purdue University.erman Rubin, Department of Statistics, Purdue Universityrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558==You may be surprised, but I am not. But it is worse than that; the ew math which succeededin teaching mathematical concepts, not just how to compute,to children, could not be learned by the teachers, evenwith major attempts to teach them. olution: Fire those that canÕt, hire those that can. Simple enough, no?ich==>You may be surprised, but I am not. >But it is worse than that; the ew math which succeeded>in teaching mathematical concepts, not just how to compute,>to children, could not be learned by the teachers, even>with major attempts to teach them. Solution: Fire those that canÕt, hire those that can. Simple enough, no?o.equires: Pay enough, and provide enough job satisfaction, to attract andetain those who can.hose who cannot are cheaper and easier to find and keep than those who can.==aul Murray wites:>>You may be surprised, but I am not. >>But it is worse than that; the ew math which succeeded>>in teaching mathematical concepts, not just how to compute,>>to children, could not be learned by the teachers, even>>with major attempts to teach them. > Solution: Fire those that canÕt, hire those that can. Simple enough, no?No.Requires: Pay enough, and provide enough job satisfaction, to attract andretain those who can.rue enough. But how many of the teachers Dr. Rubin refered to above wereired? My guess is: none! Given the lack of real consequences for failing toearn the ew mathconcepts well enough to teach them, is the outcomeuprising? Those who cannot are cheaper and easier to find and keep than those who can. would only add that they are also, clearly, harder to get rid of.ich==here are further (ridiculus!) obstacles.1) One often needs a teaching certificateo teach high school. Note that youonÕt need to prove you know the subject you teach; you just need theertificate.sarcasm intended)2) people who really know the subjecto not want to be required to follow aorced curriculum. 3) Unmotivated students. This is perhapshe main problem. There are sub-cultures,thnic groups etc. who simple DO NOTALUE EDUCATION. And this is whats creating the low test scores.You can lead a horseÕs ass to knowledge, but you canÕt make him think. == (3) Unmotivated students. This is perhaps the main problem. There are sub-cultures, ethnic groups etc. who simple DO NOT VALUE EDUCATION. And this is what is creating the low test scores. would not try to teach differential equations to my pet dog. He simply oes not have the necessaries to comprehend. Similarly trying to teach ny more math to slum dwellers past what they need to score dope deals s a waste of time.ob Kolker== posting-account=WZMvOwwAAAC_B1TaayrVN99BJgDWQkUc Similarly trying to teach any more math to slum dwellers past what they need to score dopeeals is a waste of time.s is trying to teach sanity to Republicans. predict the slum dwellers get out of the slumefore RJK figures out how to help.== posting-account=XkryvQwAAABxyOU1-w32yMV3IiEuQovZ There are further (ridiculus!) obstacles. (3) Unmotivated students. This is perhaps the main problem. There are sub-cultures, ethnic groups etc. who simple DO NOT VALUE EDUCATION. And this is what is creating the low test scores. ou can lead a horseÕs ass to knowledge, but you canÕt make himhink. s a laid off (and now apparently careerless engineer) who is workingn becoming a high school math teacher perhaps I can provide someractical insight (1) One often needs a teaching certificate to teach high school. Note that you donÕt need to prove you know the subject you teach; you just need the certificate. (sarcasm intended)ell, in Massachusette you need to provide transcripts ( I have mathegree) and take the MTEL ( I forget what the acronym means ) test. Theest can handled easily by anyone who has finished high school math uphrough trig and probably has a year of AP calculus. The Massepartment of Education is an amazing example of an out of controlovernment bureaucracy and navigating through the hoops is easilyquivalent to a masters level thesis (2) people who really know the subject do not want to be required to follow a forced curriculum.n this state you have standardised state wide tests you have to teacho (3) Unmotivated students. This is perhaps the main problem. There are sub-cultures, ethnic groups etc. who simple DO NOT VALUE EDUCATION. And this is what is creating the low test scores.e are talking public education, not expensive private schools andnmotivated students will almost by definition be part of the problem.here is also an mageproblem for mathematics. How often have youeen interviews of prestigiou people who state, proudly, that they onlyot up to {1,2} year algebra or trig before giving up on math. Math,ccording to popular culture, is for social incompetents, for those whoommunicate only to blackboards or computers and not successfully tony member of the opposite sex.TW Bob..I believe our paths have crossed at MITRE and Security Dynamics== lol - hereÕs the quote that shows how hopeless the problem truly is: [The American teacher] was especially taken aback by the textbook. By grades seven and eight, kids in the Singapore program are doing high-school-level algebra. thought, wow, thatÕs complicated -- even for me,says Mr. Keating. oor Mr. Keating and his class. Well you know teaching doesnÕt pay very uch. And itÕs run by the government. So it doesnÕt work very well. At a hopping center near where I live, thereÕs one of those commercial laces where they teach the kids math. Every Saturday morning the place s full of little kids. Their parents pay taxes to run the schools, and hen the ones who can afford to buy their kids a decent education on the eekends.==ubject: Wantzel trisection impossibility proof posting-account=-VGfHgwAAACWfwXnBk4_GXRSqowmxH_fÕve been searching the Ônet for a proof of the impossibility oßassical angle trisecion problem. So far IÕve come up short. Theirst proof was done by Pierre Wantzel. Using search terms likewantzel, trisection, impossibility , proofresults mostly initations. Surely someone must have an english translation of theaper or another subsequent explaination out there. Can anyone pointe in the right direction?he original paper in french is...ANTZEL : Recherches sur les moyens deeconnaitre si un probleme de geometrie peut se resoudre a la regle etu compas, in : Journal de Mathematiques Pures et Appliquees, 1837 pp66-372hich I got from a previous thread on sci.mathttp://makeashorterlink.com/?V28D1290Areg Buchholz==ubject: Re: Wantzel trisection impossibility proof posting-account= y3wZYhMAAABYsCtaDBjCWE5oFd14ElQZbfvQjxC1czdFUKdrfKUl4ghe short version is that ruler and compass constructions involve thequivalent of arithmetic plus square roots, if the Cartesianoordinates of constructible points are considered. This, from thetandpoint of Galois theory, means that with such constructions one canharacterize the field extension over the rational numbers in which theoordinates lie as being of degree a power of two.risection of an arbitrary angle (or for that matter, duplication of aube) involves finding a real root of an irreducible monic cubicolynomial. Hence the corresponding degree of the field extensionould have a factor of three, which powers of two are known not toave. QEDauss pretty well laid the groundwork for this by showing how toonstruct by ruler and compass those regular polygons which can be.==ubject: Re: Can anybody prove(?) the multinomial distribution? posting-account=3UqHHwwAAADBNY2J8a7gGWZjrJQudVdo Can anybody help me on this? ItÕs urgent.. ItÕs intuitively understandable, but how do you prove itathematically? Suppose that an experiment can result in one of r possible outcomes,he i_th outcome having probabilty p_i, i=1,...,r, (sum: i=1~r)p_i=1. If n of these experiments are performed, and if the outcome of anyne of the n does not affect the outcome of the other n-1 experiments,hen show that the probability that the first outcome appears x1 times,he second x2 times, and the r_th xr times is [n!/(x1!x2!...xr!)] * (p1^x1*p2^x2*...pr^xr) when x1+x2+...+xr=n (known as the multinomial distribution, ref. Introduction torobability Models, 7th ed., Sheldon Ross, p84 #17 )ereÕs a 2 year late follow-up.he more interesting problem is exercise #19, to find the probabilityistribution of the sum of the outcomes.==ubject: Fibonacci problem, not easy 2 3 5 8 13 ... 6 10 16 26 42 ... 12 19 31 50 81 ... 14 23 37 60 97 .....ach row satisfies the recurrence relation of the Fibonacci sequence. Theop row, call it row one, is the Fibonacci sequence itself. The first numbern each row is the smallest number not appearing in any previous row. If(N)denotes the first number in row N, then the second number in that row,all it Y(N), is- if N is even, phi*X rounded down- if N is odd, phi*X rounded uphere phi = (1+sqrt(5))/2.how Y(N) is even for every N.his is problem #6 on this page:ttp://faculty.evansville.edu/ck6/integer/ unsolved.htmlpparently the problem has been open for ten years now.==ubject: Maximal Idealet S be the set of all continuous functions from R to R with the operationsefined on S by (f + g) (x) = f(x) + g(x) and (fg) (x)= f(x)g(x) for any f, in S. had no trouble showing that S is a commutative ring with identity.et I ={f in S s/t f(1) = 0}. I had no trouble showing that I is an idealf S. am having trouble showing that I is a maximal ideal. This is what I haveo far. supposed that J is an ideal os S s/t I in J in S.f I = J we are done-- I is a maximal ideal in S.o suppose I /= J , J in S.o there exists f in J , F not in I. So f(1) = k /=0.f I could get 1 in J they since J absords S. But I canÕt seem to show thisr come up with another technique. Why is this not obvious?? (yeah, I knowtÕs just probably me)==ubject: Re: Maximal Ideal Let S be the set of all continuous functions from R to R with the operations defined on S by (f + g) (x) = f(x) + g(x) and (fg) (x)= f(x)g(x) for any f, g in S. I had no trouble showing that S is a commutative ring with identity. Let I ={f in S s/t f(1) = 0}. I had no trouble showing that I is an ideal of S. I am having trouble showing that I is a maximal ideal. = ker e, e = eval at 1 m e = field R => ker e maximalr, concretely: f = f1 + (f - f1)-Bill Dubuque==ubject: Re: Maximal Ideal posting-account= y3wZYhMAAABYsCtaDBjCWE5oFd14ElQZbfvQjxC1czdFUKdrfKUl4gick any f not in I, ie. a continuous function such that f(1) isonzero.erhaps you can think of a way to find a continous function g in I suchhat: + gs a nonzero constant function? What would this imply about an idealontaining both f and g?==ubject: Re: A periodic matrix representation of 17^(1/3)?-RFC2646: Format=Flowed; Original Suppose A=[324,126,49;833,324,126;2142,833,324] and B=[0,1,2;0,3,5;1,7,13]. Let C_n=A^n*B. How can I show that lim C_n[2,3]/C_n[1,3]=17^(1/3)? Richn general, if you are trying to figure out a power of some matrix, such as our A in C_n, you want to diagonalize A using a similarity transformation..e., there exists D such that A = S*D*S^(-1)... ofcourse its not always rue, but its worth a try.o, I will not explain how to get D and S as you should look it up, but its bvious that A^n = S*D^n*S^(-1)nd D^n, since it is diagonal id D^n = diag(l1^n, l2^n, ..., lm^n)hat is, you just raise the i,jth entry top the nth power(ofcourse, Dij = 0 or i!=j).o, luckly for you, ify ou diagonalize it, you just have to worry about m, th powers instead of something like m*m nth powers. Also, lim A^n = S*lim ^n *S^(-1) so it involes just finding D. Once you get D, your life is much asier. ==ubject: Re: A periodic matrix representation of 17^(1/3)?Suppose A=[324,126,49;833,324,126;2142,833,324] and B=[0,1,2;0,3,5;1,7,13]. Let C_n=A^n*B. How can I show that lim C_n[2,3]/C_n[1,3]=17^(1/3)?int: Express C_n in terms of eigenvectors and eigenvalues of A.obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: fERMAT 420ermatÕs Last Theorem en Tsutomu Ito 2-16-04 will show that FermatÕs n=4 proof is invalid. In FermatÕs n=4 proof,ermat initially states that the variable d2 does not form integer solutions of X, Y and Z. Inddition, Fermat formed a n=4 and area proof. Wiles formed an allegedroof of FermatÕs equation using elliptical curves that originatedrom FematÕs area proof. will show that FermatÕs n=4 and WilesÕ proofs are invalid then proveermatÕs last theorem using mathematical quantitative analysis. . FermatÕs n=4 Proof ermatÕs n=4 proof is described. I will show that FermatÕs n=4 proofhat is based on a contradiction that does not exist.(Case k>1) Here is the key step: suppose that x and y are wholeumbers, with x^4 + y^4 a perfect square; that is ^4 + y^4 = c^2, (equ 5)here c is a whole number. Then we can find new whole numbers u,v,ith^4 + v^4 = d^2, (equ 6)here d is a whole number smaller than c.(Osserman, p. 19)ermat forms the contradiction by stating that: any solution of ^4 + y^4 =c^2 (equ 7)eads to another solution^4 + v^4 = d^2 (equ 8)ith d1 and similarily for n=1 does not exist.ermat implies that by proving that, ^4 + Y^4 = Z^2 (equ 9). oes not form integer solutions also proves that ^4 + Y^4 = Z^4 (equ 10) oes not form integer solutions (Shanks, p. 144). Squaring both sidesf equation 9,X^4 + Y^4]^2 = [Z^2]^2 = Z^4 = X^8 + 2X^4Y^4 + Y^8. (equ 11)nd equating equation 11 and equation 10,^4 + Y^4 =/ X^8 + 2X^4Y^4 + Y^8, (equ 12)quations 9 and 10 are not equivalent. FermatÕs n=4 prove that usesquation 9 to prove n=4 is invalid since equations 9 and 10 are notquivalent. n addition, the integer solution equations of n=2 are, = 2uv, b = u^2 - v^2, and c = u^2 + v^2. (equ 13a,b,c), ermat uses, ^2 = 2uv, Y^2 = u^2 - v^2, and Z = u^2 + v^2 (equ 14a,b,c), Shanks, p.141) to prove n=4. Using equations 13a,b,c and 15a,b,c = X^2, b = X^2 and c = Z. (equ 16a,b,c) nserting equations 16a,b,c in equation 9, ^2 + b^2 = c^2. (equ 17) ermat is stating that since he has found the a solution to equation 9hat he can used those equation to prove n=4. Using = 9^(1/4), b = 16^(1/4) and c = 25^(1/4) (equ 18a,b,c) n ^4 + b^4 = c^4----------------> 9 + 16 = 25 (19a,b) ermat is assuming that the solutions of equation 19a,b imply aelationship between solutions of n=2 and integer solutions of n=4hich there is not. n addition, the n=2 equation, ^2 + Y^2 = Z^2 (equ 20) n=2) is completely different from, ^4 + Y^4 = Z^2. (equ 21). sing Z = 6, equation 20 forms a circle of radius 6,^2 + Y^2 = 36.(equ 22)sing Z = 6 in equation 21, ^4 + Y^4 = 36,(equ 23)quation 23 is not a circle. Equations 20 and 21 are completelyifferent equations. The integer solutions of n=2 cannot be used torove n=4 since equations 20 and 21 are completely differentquations. Therefore, FermatÕs n=4 proof is invalid.ermatÕs n=4 proof does not testing all integer combinations of X and. FermatÕs proof is only proving equations 14a,b,c do not formnteger solutions of n=4 (equ 21). There are an infinite number ofnteger combinations that are not tested in FermatÕs n=4 proof;xample, the integers X = 22 and Y = 231 are not included inermatÕs n=4 contradiction proof. FermatÕs n=4 prove does not includell integers.xample, using n=4 X^4 + Y^4)^(1/4)=C (equ 24) et Y = X + 1, (equ 25) nserting equation 25 in equation 24, (X + 1)^4 + X^4]^(1/4) = C (equ 26) here are an infinite number of equations that describe integerombinations that are not included in FermatÕs n=4 proof. Example, = X + 2, Y = X^2 + 1, Y = X^2 + X, Y = X^2 + X + 3tc.............(equ 27) he proof for n=4 must include all possible integer combinations of Xnd Y. There are an infinite number of integer combinations that areot included in FermatÕs n=4 proof. Fermat is justifying theon-existence of integer solutions of n=4 using a contradictionethod where only a single group of equations (equ 5a,b,c) are used.ermatÕs n=4 proof in incomplete. . Wiles Proof iles uses elliptic curve equation to prove FermatÕs last theorem for>2. Wiles proof of FermatÕs last theorem is based on the ellipticurve equation (Poorten, p. 196-7), ^2 = X(X - a^n)(X + b^n) (equ 28). owever, FermatÕs equation is ^n + b^n = c^n (equ 29). t is not mathematical possible to derive equation 29 from equation 28ince equation 28 becomes, ^2 = X^3 + X^2(b^n - a^n) - X(ab)^n. (equ 30) ilesproof of FematÕs theorm is invalid since equation 30 does notnclude the c^n and (ab)^n and (b^n - a^n) are sums in FermatÕsquation. Wiles proof ( Wiles, p. 448) that uses equation 30 isnvalid. . Proof of FLT. will prove FermatÕs last equation using the pattern formed by theolutions formed by n=1 and n=2, and the near solutions of n=3 and=4. Using n=1 in equation 1, + Y = Z (equ 31) ll integers of equation 31 form solutions. The first integer setccurs at (1,1,2).he integer solutions of n=2 are described with the followingquations, = 2uv, Y = u^2 - v^2 and Z = u^2 + v^2 (equ 32) ot all integers form integer solutions for n=2. The first integeret occurs at (3,4,5).he n=1 and n=2 solutions show a pattern where integer solutions areormed when a solution set is forms close to the origin.n addition, when n=1, the strength of the solution set formation isaximum since all integers form solutions; when n=2 not all integersorm solutions. The strength of the solution set formation iseakening as n increases. When n=3, ^3 + 8^3 = 728 where 9^3 = 729 off by 1. (equ 33)^3 + 10^3 = 1729 where 12^3 = 1728 off by 1.(equ 34)n integer solution set does not form close to the origin for n=3 yetear solutions occur close to the origin equations 33 and 34. The n=3quation does not from solutions near the origin. The integer sequencends at n = 3. The solution set is weakening as n increases; at n=4^4 + 8^4 = 6497 where 9^4 = 6561(equ 35)herefore, only n=1 and n=2 form integer solutions.. Conclusion have shown that FermatÕs n=4 proof is invalid. In FermatÕs n=4roof, Fermat initially states that the variable d FermatÕs Last Theorem> Ben Tsutomu Ito> 12-16-04> I will show that FermatÕs n=4 proof is invalid. ...> Fermat is assuming that the solutions of equation 19a,b imply a> relationship between solutions of n=2 and integer solutions of n=4> which there is not. Where is James Harris when we need him?hen has *anyone* ever needed James Harris for *anything*? Not once inis entire life, I suspect.- ayne Brown (HPCC #1104) | hen your tailÕs in a crack, you improvisewbrown@bellsouth.net | if youÕre good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, ilverlock ==ubject: Re: fERMAT 420>en ito >>FermatÕs Last Theorem>>Ben Tsutomu Ito>>12-16-04>>I will show that FermatÕs n=4 proof is invalid.>...>>Fermat is assuming that the solutions of equation 19a,b imply a>>relationship between solutions of n=2 and integer solutions of n=4>>which there is not.>Where is James Harris when we need him? When has *anyone* ever needed James Harris for *anything*? Not once in his entire life, I suspect.hat sounds a little meaner than deserved! Sure the guy is a athematical annoyance. (Good use of solved that for me) But I am fairly sure that heÕs not 100% useless for everything in very aspect of his life. I mean, maybe the guyÕs good at his day job. aybe he even has kids that like him. Maybe he takes out the trash for is wife without being asked. I donÕt know...r maybe the holidays are giving me a bleeding heart. *shrug* wallace==ubject: Re: quotient ring confusion by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBG5cVD21100;I am trying to get a good grasp as to what this ring looks likeZ[x]/(x^2-5, 5x-10)messing with the relations x^2=5 and 5x=10 i get the additional relation that 2x^2-5x=0. I do not suppose i could be so lucky as to have that the ideal (x^2-5,5x-10) is principal, and generated by 2x^2-5x alone? i feel that if that were the case, then the quotient ring would be something like the dyadic rationals Q_2 since we would be adjoining 5/2 to Z.o, itÕs not principal. But we have equalities of ideals (x^2 - 5, 5(x - 2)) = (x^2 - 5, 5) = (x^2, 5)here the second equality is trivial and the first arises ssentially by the calculation 5 = 5(x - 2)(x + 2) - 5(x^2 - 5). ence the quotient ring is Z[x]/(x^2, 5) ~ Z_{5}[x]/(x^2), hich is isomorphic to a ring whose underlying additive group s Z_5 x Z_5, and where the multiplication is given by (a, b)*(c, d) = (ac, bc + ad) [a, b, c, d in Z_5]. odd Trimble==ubject: Re: Why did you choose math? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBG5cVK21117;I would like to know the reason you chose mathematics. It was quitepossible that you could not write any research paper.I am pondering about whether I choose physics or mathematics. Thisdoes not mean at all that I have talent enough to even make thispondering worth a while.f you arenÕt talented in math, the best advise you can receive iso steer away from it - especially if you are considering it forareer purposes other than teaching in public schools. ditto forhyzics.It seems to me that this kind of problem is one that will be solvedonly after trying. A specific reason I am asking this is thatI got about 5/2 years to be away from school. Within this period,I think I will pursue either physics or math.ow will you be able to accomplish such feat? the feat being that oftudying math or phyzics while you are away from school...To me, doing math can really make my personality more unique. Buta problem with concentrating on math is that I might not learn anything by reading a book independently. I could learncollege algebra. But as for advanced math, I am not sure. I triedPrinciples of Mathematical Analysis, which, according to a reviewin Amazon com, seems very famous. I somewhat understood it, butI can not remember much. Another problem is that if I become acomplete math geek, my social life will be gone forever.ow old are you? seriously.As for physics, if I were to study physics independently, my planwould be to read a calculus-based textbook and get a sense ofphysics. A problem with this is that this is not tooprogressive. Since advanced math is necessary for more advancedphysics courses (I guess), this is all I can do.Why did you choose mathematics? chose math because i wanted to be chronically unemployed...ell, that was a joke - the reason i chose math is because i liket, correct that, love it...delta01211==ubject: Re: Why did you choose math?if you arenÕt talented in math, the best advise you can receive isto steer away from it - especially if you are considering it forcareer purposes other than teaching in public schools. ditto forphyzics.ou could be surprised at what is possible. Learning Mathematics academicallys one thing; actually applying it in realistic situations is another thing. lgebryonic==ubject: Which do you prefer? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBG5cVq21130;or abb. of physics courses, which of PHSX and PHYX do you prefer? personally prefer the former. The latter makes me feel that itÕsoo soft; the Y makes me feel that way. S, on==ubject: Multiple SVD solutions posting-account=aI6g3Q0AAABNpUZFq495l-Zgoa1GZKNT read that for a non-square matrix, A, eg: mxn, m posting-account=aI6g3Q0AAABNpUZFq495l-Zgoa1GZKNTf complex nos, there could be infinite combinations of e^(-jx); butor real nos., there exist two possibilities (+1 & -1). Generallypeaking if we choose [v] such that [v] = [V]*e^(jx), the correspondingu] = [U]*e(-jx).. A more specific question would then be: f AAhas distinctigenvalues; and if A is an under-determined system (ie # rows< #olumns), knowing only U & D, can one determine a/set of {v} such thathen put together we can obtain:H]*e^(jx) = [U]*[D]*[v]Õ. Can one arbitrarily choose [v] as a unitary matrix of appropriateimension; and subsequently determine a [u], assuming knowledge of theystem [H]? ==ubject: Re: Multiple SVD solutions I read that for a non-square matrix, A, eg: mxn, m THE STATE-OF-THE-ART IN PHYSICS> There is similarity and distinction between mathematics and physics.> They are similar in the sense that they both study the structure,> properties and behavior of their respective subject matter. The> difference lies in their subject matter: that of mathematics is the> representation of thought, making it a language, and that of physics> is nature, making it a science. Both suffer from similar defects:> mathematical spaces and their concepts are ill-defined and nature and> physical concepts are also ill-defined. ThatÕs one of the reasons he Smart Model Theory was developed. Physicss full of ill-defined things like, electron orbitals, and electron change ofrbital to show various lines on the line spectra, and uncertainty etc... . The Smart Model has accurately predicted the structures now observed fromeal atomic images, like for example oxygen. The Smart Model predicted oxygenas a ring. QM said it was a sphere. Now they observe oxygen rings on, PdO. Itooks like this from real atomic images showing oxygen attached to Pd like thisn the outer layers of Pd,ooooooooooooooooooooooooooo oooooooooooooo ooooooooooooooooooo ooooooooo oooooooooooooo oooooooooooooooooooooooooooooooooomartÕs Alt. Physics News Groupttp://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1. Enterprize (Science Journal)ttp://smart1234.s-enterprize.com/==ubject: Re: quotient ring confusion by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBG6uM027109;>I am trying to get a good grasp as to what this ring >looks like>Z[x]/(x^2-5, 5x-10)>messing with the relations x^2=5 and 5x=10 i get the >additional relation that 2x^2-5x=0. I do not suppose >i could be so lucky as to have that the ideal >(x^2-5,5x-10) is principal, and generated by 2x^2-5x >alone? i feel that if that were the case, then the >quotient ring would be something like the dyadic >rationals Q_2 since we would be adjoining 5/2 to Z.No, itÕs not principal. But we have equalities of ideals (x^2 - 5, 5(x - 2)) = (x^2 - 5, 5) = (x^2, 5)where the second equality is trivial and the first arises essentially by the calculation 5 = 5(x - 2)(x + 2) - 5(x^2 - 5). Hence the quotient ring is Z[x]/(x^2, 5) ~ Z_{5}[x]/(x^2), which is isomorphic to a ring whose underlying additive group is Z_5 x Z_5, and where the multiplication is given by (a, b)*(c, d) = (ac, bc + ad) [a, b, c, d in Z_5]. Todd Trimble==ubject: Re: quotient ring confusion (a,b) *(c,d) = (ac, ad+bc)-> (a+bx)*(c+dx) = ac + (ad+bc)x via x^2 = 0 in R[x]/(x^2) can this multiplication law be derived directly from the construction of the quotient ring?es, see mt remark above.ince n>1 => x^n = 0 in D = R[x]/(x^2), every elt of Ds equivalent to a linear poly a + bx. Now it shoulde clear from above how R^2, with the above product *,s isomorphic to the ring of dual numbers D.-Bill Dubuque==ubject: Re: Random Rational?...]>How about this:>for each positive integer N let>P_N = N^{-2}|{(m,n): m,n integers, 1<= m,n <= N, m/n has even denominator}|>and let P = lim_{N->infinity} P_N (if that limit exists).>Is P a good interpretation of he probability that a random rational>number has an even denominatorNo. Or more precisely, nobody has shown it.t seems reasonably common for mathematicians to describe results aseing about robabilitieswhen they are not trying to be precise, andhe result is plainly not literally a probability. The probability ofn integerÕs being squarefree is 6/pi^2; the probability of two numberÕseing relatively prime is likewise 6/pi^2, and so on. Really itÕs someind of density result.n number theory, what we call eightfunctions are common. Thelogarithmic) height of a rational m/n in lowest terms could be defineds (log) (max{|m|,|n|}). These functions have nice approximate behaviorsnder rational maps.o it would be considered *relatively* natural to consider the densityf a set of rationals under the height ordering. Hence it would makeome sense to suppose that the result was meant to say that the fractionf the rationals expressible with numerator and denominator <=N inbsolute value that had an even denominator in lowest terms tends toward/3 as N goes to infinity.The distribution implied by your definition is not a uniform distributionbecause it is weighted heavily toward low denominators. For example, m/n=1happens N times in your m*n space, m/n=1/2 happens ßoor(N/2) times,m/n=1/10000 happens ßoor(N/10000) times. es, these should be counted just once each, although it doesnÕt actuallyatter in this case. Including all the forms of each rational that areot reduced as separate cases amounts roughly to giving each rational aeight inversely proportional to its height (without the log). This isretty much liable to give the same result... I guess I donÕt rememberffhand what conditions are needed to ensure that it does. If the densitypproaches a limit speedily enough, either definition will do.We all know that niform distributionover a countably infinite set isnonsense. Therefore, for the phrase andom rational numberto meananything, some other distribution must be specified.>Is P = 1/3?For what itÕs worth, I think the answer is yes.lmost certainly yes. If we take Robin ChapmanÕs definition, the numberf pairs where n is odd already is [(N+1)/2] where [x] is the functioniving the greatest integer <=x. The number of pairs where the denominators divisible by exactly k powers of 2 and the numerator is divisible byt least k powers of 2 is [([N/2^k]+1)/2][N/2^k] <= N^2/2^{2k+1}. Alsohe limit of [([N/2^k]+1)/2][N/2^k]/N^2 as N goes to infinity is/2^{2k+1}. So the fraction of pairs for which m/n reduces to araction with an odd denominator is <=1/2+1/8+1/32+...=1/3, and is aum of terms which approach 1/2, 1/8, 1/32,... respectively as N goeso infinity. For any epsilon>0, truncate the series 1/2+1/8+... at aoint where the remaining terms add to > We all know that niform distributionover a countably infinite set is> nonsense. Therefore, for the phrase andom rational numberto mean> anything, some other distribution must be specified.Let niform distributionover a countably infinite set be nonsense.But how about a robability densityover a countably infinite set?ure, thatÕs reasonable. Even probability distribution is reasonable,rovided you specify a function which sums to 1 over its domain.Then the probability density of the perfect squares within the naturalintegers - as I understand it - depends on where you are. And in theneighbourhood of x it is equal to 1/sqrt(x) . he density of the squares on the interval [1,x] is approximately 1/sqrt(x). he density in the neighborhood around x is 1/(2*sqrt(x)), I think. onsider 100 +-10. It contains approximately 1 square for every 20 numbers.o get that answer, take the derivative of the function sqrt(x), whichpproximates the quare counting functionßoor(sqrt(x)).The probability densityof the primes - as I understand it - is 1/ln(x) approximately (: PrimeNumber Theorem. Here x is a sufficiently large natural number.) Boththese probability densities become zero (infinitesimal) for x -> oo .gain, your density 1/ln(x) is the correct density for the interval [1,n],ut is it right for the small neighborhood around x? The prime countingunction PI(x) can be approximated by Li(x), and the derivative of that is/ln(x), so... I think thatÕs right.Nevertheless, the primes are more numerous than the squares, becauseln(x) < sqrt(x) ==> 1/ln(x) > 1/sqrt(x) for x > 2 .O yeah, I find that the density of the cubes goes like x^(-2/3) .aking the derivative of x^(1/3) I get x^(-2/3)/3.Question: do these findings bear any resemblance with results obtainedin fficialmathematics? Since IÕm rather ignorant in number theory. donÕt represent fficialmath, except for the one time I revoked a manÕsight to use the term nfinitybecause he was abusing the concept.-Keith Lewis klewis {at} mitre.orghe above may not (yet) represent the opinions of my employer.==ubject: Re: Random Rational?>Let niform distributionover a countably infinite set be nonsense.>But how about a robability densityover a countably infinite set? Sure, thatÕs reasonable. Even probability distribution is reasonable, provided you specify a function which sums to 1 over its domain.hatÕs difficult, though, with distributions which are quite dependentn here you arein, say, the natural numbers. It seems that youÕlleed then things like a unning meanand unning spread Are theseeasties well-known concepts in, say Number Theory, or Probability? The density of the squares on the interval [1,x] is approximately 1/sqrt(x). The density in the neighborhood around x is 1/(2*sqrt(x)), I think. ur answers differ by a constant factor; left it out everywhere. To get that answer, take the derivative of the function sqrt(x), which approximates the quare counting functionßoor(sqrt(x)).eah, thatÕs how you do it! Again, your density 1/ln(x) is the correct density for the interval [1,n], but is it right for the small neighborhood around x? The prime counting function PI(x) can be approximated by Li(x), and the derivative of that is 1/ln(x), so... I think thatÕs right.Õve found, experimentally, that the unning spreadvaries accordingo an estimate by Koch, as: sqrt(x).ln(x) . But theoretically? IÕve goto clue whatsoever ... Taking the derivative of x^(1/3) I get x^(-2/3)/3.onfirmative again. That is, apart from the constant factor I left out.an de Bruijn==ubject: Re: The consise CantorÕs proof ?he post was left unreplied to. Yes its one for DCProof I think, who canake the shortest formal proof. Not that I accept Er e R outside of the Eb e N.ereÕs my new slant/very rational you can represent lines up with some natural on a single list.very real you can represent lines up with some natural on a single list.o contradicitons, no need for hyperinfinity.----------------ach row is a real from 0 to 0.999..g. r3(4) = UTM(3, 4) mod 10, the 4th digit of the 3rd real.o represent the anti-diagonal of UTM(x, y) mod 10 (ignoring halt issues)ou can only write such as r(x) = UTM(x, x)+1 mod 10 (mod 9 removes the 0.999.. issue)TM is a program itself TM-u for some natural number u.hereforeTM(x,x) + 1 mod 10 = TM-u(x,x) + 1 mod 10 = UTM(u, (x,x)) +1 mod 10etM-v(d) = TM-u(d,d) - 1henTM(x,x) + 1 mod 10 = UTM(v, x)hen x = vTM(x,x) + 1 = UTM(x,x)ONTRADICTIONherefore there is no valid representation of a real UTM(x,x)+1 mod 10erc-ab jab jab quote IÕm a critical thinker > Of course the free b in (r = !L(b)) puts doubt > on such an algebraic derivation of CantorÕs > proof. Such an algebraic derivation AS WHAT? Proofs ARE derivations. There is no such thing as a derivation, ALGEBRAIC OR otherwise, of a proof. There is rather a derivation OF the THEOREM proved, FROM the axioms and hypotheses, and that derivation IS the proof. As somebody else has already pointed out, =!L(b)is not even a valid step in your proof, let alone a reason why somebody might find an algebraic definition difficult. Your proof does require one to make infinitely many choices in going from > 4a. not(exists a, forall b, L(a,b) != L(b,b) > (negation of 3) to > 4b. not(exists a, forall b, L(a,b) =!L(b,b) > (! is some suitable digit change function) but even THAT infinity does not involve anything suspect (like the axiom of choice), because it is easy to stipulate a finitary rule governing the choice, because there is only a constant finite number of different numerals for DIGITS in these representations. For maximum clarity it should just be 2, in which case there is no choice because there is only 1 digit-change function. You are making infinitely many choices from infinitely many pairs of shoes, not pairs of socks. The end result of this is that you finally believe CantorÕs theorem after all. So why canÕt you just admit you were stupid for not seeing it in the first place instead of claiming (falsely) that you have invented something insightful and concise that might be of future value? The concise version of this proof is the set-theoretical one showing that any function has a subset of its domain set has a subset of that set (an element of its domainÕs powerset) that is NOT in its range. Af[y(f)cDom(f) & Ab[~f(b)=y(f)]]. That there really does exist such a y(f) for every f really is a theorem. You donÕt even have to arrange the domain in a list. This whole problem is much simpler and deeper than that and the infinite-list version WORKS for reasons having NOTHING to do with infinity OR list-hood! Dave Seaman posted a concise proof of this in ZF to the newsgroup years ago.==ubject: Re: The consise CantorÕs proof ? <32d0qnF3jjaesU1@individual.net> posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAghought the post was left unreplied to. Yes its one for DCProof I think, whoan make the shortest formal proof. Not that I accept Er e R outside ofhe Eb e N. HereÕs my new slant/ every rational you can represent lines up with some natural on aingle list. every real you can represent lines up with some natural on a singleist. no contradicitons, no need for hyperinfinity. ----------------- Each row is a real from 0 to 0.999.. eg. r3(4) = UTM(3, 4) mod 10, the 4th digit of the 3rd real. To represent the anti-diagonal of UTM(x, y) mod 10 (ignoring haltssues) you can only write such as r(x) = UTM(x, x)+1 mod 10 (mod 9 removeshe 0.999.. issue) UTM is a program itself TM-u for some natural number u.his is incorrect. Therefore UTM(x,x) + 1 mod 10 = TM-u(x,x) + 1 mod 10 = UTM(u, (x,x)) +1 mod 10 Let TM-v(d) = TM-u(d,d) - 1 Then UTM(x,x) + 1 mod 10 = UTM(v, x) When x = v UTM(x,x) + 1 = UTM(x,x) CONTRADICTION Therefore there is no valid representation of a real UTM(x,x)+1 mod0 Herc -- jab jab jab quote IÕm a critical thinker > Of course the free b in (r = !L(b)) puts doubt > on such an algebraic derivation of CantorÕs > proof. > Such an algebraic derivation AS WHAT? > Proofs ARE derivations. There is no such > thing as a derivation, ALGEBRAIC OR otherwise, > of a proof. There is rather a > derivation OF the THEOREM proved, > FROM the axioms and hypotheses, and > that derivation IS the proof. > As somebody else has already pointed > out, =!L(b)is not even a valid step > in your proof, let alone a reason why > somebody might find an algebraic > definition difficult. Your proof does > require one to make infinitely many > choices in going from > 4a. not(exists a, forall b, L(a,b) != L(b,b) > (negation of 3) > to > 4b. not(exists a, forall b, L(a,b) =!L(b,b) > (! is some suitable digit change function) > but even THAT infinity does not involve > anything suspect (like the axiom of choice), > because it is easy to stipulate a finitary > rule governing the choice, because there is > only a constant > finite number of different numerals for > DIGITS in these representations. > For maximum clarity it should just be 2, > in which case there is no choice because > there is only 1 digit-change function. > You are making infinitely many choices from > infinitely many pairs of shoes, not pairs of socks. > The end result of this is that you finally > believe CantorÕs theorem after all. > So why canÕt you just admit you were > stupid for not seeing it in the first place > instead of claiming (falsely) that you have > invented something insightful and concise > that might be of future value? > The concise version of this proof is the > set-theoretical one showing that > any function has a subset of its domain > set has a subset of that set (an element > of its domainÕs powerset) that is NOT in its range. > Af[y(f)cDom(f) & Ab[~f(b)=y(f)]]. > That there really does exist such a y(f) for > every f really is a theorem. > You donÕt even have to arrange > the domain in a list. This whole problem > is much simpler and deeper than that and > the infinite-list version WORKS for reasons > having NOTHING to do with infinity OR > list-hood! > Dave Seaman posted a concise proof > of this in ZF to the newsgroup years ago.==ubject: Re: How big is the electron?<41bdfafa$1@news.accesscomm.ca>The electron is a spiral-shaped cloud extendingoutward from the nucleus *in exactly the sameway* that a galaxyÕs arm is made.They are the same thing.uppose you were shooting a 3D stereoptic movie for a classroometting, would it be sufficient to put a few drops of milk on ainyl record, spin it a little bit on a phonograph player, andhen ray-trace it up and down so that it has the appearance ofepth and height? Would that be suitable for your basic hydrogentom? Using the same setup, would two drops work for a helium atom?==ubject: Re: What is a proof, exactly?IÕve heard that before, but is it really required.ouÕve heard *what* before? You didnÕt quote anything.Õm guessing from the context that you are referring to the versionf the axiom of extensionality that one uses when equality is treateds a defined term. Certainly this version of the axiom is a consequencef the other one, and thereÕs no way of proving it from just axioms ofF besides extensionality.ou seem to be trying to show here that itÕs irrelevant somehow,ut your reasoning is very handwavy. Perhaps the crux of it ishis:...]I think the idea of whether there is one empty set or many is silly andunecissary, since a FO language with no logical equality and just abinary relation canÕt distinguish between any sets x that make thesentance z (z in x <=> S[z])true,ncorrect. ThereÕs no guarantee that such sets are indistinguishabley first-order sentences. They may have the same members, but theyay still be distinguishable by not being members of the same otherets themselves.f the form of the axiom of extensionality that I gave holds true,hen while in principle itÕs still possible for there to be nonequalets having the same members, itÕs not very relevant, since one canimply identify them. By identify them, I mean consider insteadquivalence classes under the relationship of extensional equalityhaving the same members). These equivalence classes would thenatisfy the usual axiom of extensionality.hat construction relies upon the axiom holding, though. If thequivalence relation is ~, and a~b, c~d, then we know by theefinition of ~ that a is a member of c iff a is a member of d,nd b is a member of c iff b is a member of d. But we also needo know that a is a member of c iff b is a member of c, whichoesnÕt automatically follow unless the axiom holds. With thexiom, a is a member of c iff b is a member of d and the simplequivalence-class construction goes through.f one leaves this axiom out, however, things become moreomplicated. There is a way of converting a set theory whichoesnÕt satisfy extensionality into one that does, but it involvesore than simply identifying sets having the same members. Not onlyo you identify sets having the same members, but also sets whichriginally had different members, but were caused to have the sameembers once their members whose members were the same were identifiedith each other. And so on-- it can be thought of as a transfinitenduction.lease try a little harder to think this kind of thing throughefore concluding that what most people say about it is illy...]And IÕve been personally ßamed a bunch for asking questions that canÕtbe stated in the FO language of set theory, find this implausible.eith Ramsay==ubject: Re: What is a proof, exactly? posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KUJ.E. |IÕve heard that before, but is it really required. YouÕve heard *what* before? You didnÕt quote anything.hat you need eqality/extension at all. But I never understoof theationale. We have an iff, so the first thing one can with the axiomf equality/extension is replace all FOL sentences and formulas withquality with equivalent (given axiom of equality/extension) sentencesnd formulas without equality. And once youÕve done that you couldake that axiom and throw it in the garbage, never needing it again.or instance the efinitionof S being a bijection becomes (As (seS> Ea Eb (a,b)eS))&AaAbAcAd( ((a,b)eS & (c,d)eS) => ( (Af (fea <=>ec)) <=> (Af (feb <=> fed)) ) ) instead of the original (As (seS => Eab (a,b)eS))&AaAbAcAd( ((a,b)eS & (c,d)eS) => (a=c <=> b=d) ) IÕm guessing from the context that you are referring to the version of the axiom of extensionality that one uses when equality is treated as a defined term. Certainly this version of the axiom is aonsequence of the other one, and thereÕs no way of proving it from just axiomsf ZF besides extensionality.Õm asking hy do we have to prove it at all if you had a FOLithout equality, how can the question even e askedwithin theanguage, maybeEy (yez Ax ~xey) & Ay (yez => Ax ~xey)could be true of some z buthe formula y (yez <=> Ax ~xey)could be false? Which leads to myootstraping example where I consider ONLY the sentences of the formEz Ay yez <=> S(y,z)to be candidates for sethood. If I bootstrapike that, can the axiom of equality be redundant based on theyntactical form of the candidate sets? Which leads to the question ofhether there is a syntactical form for the candidate sets which thenequires no axioms, and just the rules of logic.ithout considering the models without the axiom of equality, it woulde rather difficult for me to find a way to make it superßuous. You seem to be trying to show here that itÕs irrelevant somehow, but your reasoning is very handwavy. Perhaps the crux of it is this: [...] |I think the idea of whether there is one empty set or many is sillynd |unecissary, since a FO language with no logical equality and just a |binary relation canÕt distinguish between any sets x that make the |sentance z (z in x <=> S[z])true, Incorrect. ThereÕs no guarantee that such sets are indistinguishable by first-order sentences. They may have the same members, but they may still be distinguishable by not being members of the same other sets themselves. If the form of the axiom of extensionality that I gave holds true, then while in principle itÕs still possible for there to be nonequal sets having the same members, itÕs not very relevant, since one can simply identify them. By identify them, I mean consider instead equivalence classes under the relationship of extensional equality (having the same members). These equivalence classes would then satisfy the usual axiom of extensionality. That construction relies upon the axiom holding, though. If the equivalence relation is ~, and a~b, c~d, then we know by the definition of ~ that a is a member of c iff a is a member of d, and b is a member of c iff b is a member of d. But we also need to know that a is a member of c iff b is a member of c, which doesnÕt automatically follow unless the axiom holds. With the axiom, a is a member of c iff b is a member of d and the simple equivalence-class construction goes through. If one leaves this axiom out, however, things become more complicated. There is a way of converting a set theory which doesnÕt satisfy extensionality into one that does, but it involves more than simply identifying sets having the same members. Not only do you identify sets having the same members, but also sets which originally had different members, but were caused to have the same members once their members whose members were the same weredentified with each other. And so on-- it can be thought of as a transfinite induction. Please try a little harder to think this kind of thing through before concluding that what most people say about it is illy think youÕve missed the point entirely. I know that itÕs not sasyas making equivalence classes of sets, because I consider thatery idea a week ago long before posting the idea to usenet. I didnÕtay that I wanted an equivalent theory. Consider the qeustion to bebout intended models if thatÕs easier. You can say that your intendedodel was such that a nonlogical = relation can be defined such thatxAy (x=y <=> (Az (xez <=> yez))&(Az (zex <=> zey)) ) is true a truetatement of your model (and then you might as well have had an axiomo make that be true). But I can just ask hy that model or hyhat axiom which is what I was trying to do. What I meant aboutilly was n unrelective assertion a on-sequitor or a cause itoeslevel of response. Being silly by being cavalier, not by beingtupid. If someone is clear about only wanting models where thattatement is true as intended models, then thatÕs not silly at all, butt will lead to me asking about these intended models, what they areike, why they like them, and what do they not like about other models,tc. So for instance: if one had a model where here was no suchelation = that could make AxAy (x=y <=> (Az (xez <=> yez))&(Az (zex=> zey)) ) true then why is that not a good model? That would be auestion for someone that wasnÕt merely being silly about anting thexiom So with the usual axiom or the equivalent new one that assertshe consistency of asserting the existence of a relation like =, anyet such that all of itÕs memebrs are empty is equal to the others, but priori I donÕt see why we should choose to either disallow or fail toistinguish a set that contains AN empty set from a set that containsLL the empty sets, from one that contains ome but not allemptyets, and I donÕt see any reason to care which one we have forlassical math, since classical math assume that all those setsescribed where the same anyway, so having more in a consistent mannerust gives a more plentiful universe without breaking anything. One ofhe those set satisifies Az (zey <=> Ax ~xez), but there is no reasono expect that only that set was useful, but assering your non-logicalxiom would restrict us to models where the other two sets donÕt exist.hey all are naive correspondances from the syntactic form {{}}, and Ian reason about all three sets with a FO language with a binaryelation and no equality to ask questions like Ax Ey S[x,y].niqueness, e.g. S[x,y]=>(x=y) can be rephrased from the logical levelo the phrase S[x,y]=>(Az (zex <=> zey)), at the level of the binaryelation and classical mathematics works just as well, it wonÕt havehe same intended meaning, but the theorems of mathematics hold overust fine for the intended sets. The point is that there is a level athich people can distinguish sets (contains all empty sets) versuscontains some empty sets but not all) that can be different than theevel at which the FO sentences dinstinguish between them. People caneßect on many levels. Consider the following case that may or mayurn out to be useful: if one considered the class of all sets k suchhat ~kex, then you could have a subuniverse that treats x like thempty set, and if the subclass of that subuniverse is appropriatelyrounded like the whole universe is sometimes considered to be groundedo the empty set, then maybe applying a theorem about the whole setniverse to that isomorphic class would produce an interesting result.ow am I supposed to know a priori that that is never useful? I am Iust dumb and itÕs obvious to everyone else?Õm worried that we are unecissarily cutting off validity preservingroof techniques like that isomorphism of classes. In that particularase, I expect that a standard proof would exist, but that leads to thehole idea of nonstandard proofs. In nonstandard analysis, anytatement without the term tandardthat has a nonstandard proof,lso has a standard proof. But what if someone made a technique suchhat instead of saying xistence of weird proof implies existence oformal proofbut instead said xistence of weird proof implies truthf statement even if statement has no normal proof then this would be bona fide useful new thing. I donÕt know if new axioms, less axioms,r a stronger logic (like IF logic) could do these things. But I alsoonÕt know that they canÕt, and IÕd like to either know that they canÕtr have the supposition that they might be taken seriously. ThatÕs all.==ubject: Please advice on this math - motion of roulette balli all, hope iÕm posting to the right forum, sorry if not. have dug out some equations that relate to the prediction of aoulette balls drop off point on the wheel.Õve need help verifing if these equations are correct or nonsense. have copied a section below for anyone to have a look at: he following mathematics deals with the X,Y and Z axis within theonfines of a Roulette Wheel enviroment. axisN1*COS(a)-(mg)*COS(a)=0 axisN2+N1[Capita lIDoubleDot]*SIN(a)+mg*sin(@)*COS(Y)= m*@entre)=m*V^2/R=m*[YÕ)^2]*R=Linear Velocity=Ball Track Radius=Centripedal acceleration axisFfr+Air Drag=m*@tan=m*YÕÕ *Rriction Force a This is negative as it is opposing the Z axisir Drag is the force that is equal too:Air Drag= - 0.5*CD*P*TT*r^2*V^2 (TT is pie) this is also a minusalue!D is Drag Coefficiaent is AIr density is the balls Radius axis is always tangentially directed.-----------------------------------------fter some very simple Algerbraic Transformation and incorporating thebove formulas we get the next differential equation:ÕÕ=(a+air*R)*(YÕ)^2=b* SIN(Y)+c*COS(Y)+d (*)here is the determining friction factor(Ie 0.004)ir =-[0.5*CD*P*TT*r^2*V^2]/m=a*g*SIN(@)/R=b/a=a.g.COS(@)SIN(a)+1 )/(R.COS(@)he ball movement sters to this equation only till the moment when itoses the contact with the vertical side of the ball track or:N2]=0o the Drop off condition is:(YÕ)^2]*R+g*COS(@)*tg(a)-g*SIN(@)*COS(Y)=0 (**)ow lets introduce some real values into the equations and see theredicted results:T=3.14=9.807=0.4=16.7.TT/180 inner slope of statorD=0.47=0.5.21.10^-3 Radius of ball=1.22=9.10^-3 Mass of Balla)= 0.004 Friction factor for rolling between the ball and the track=0.8 grad Tilt Angle0= 0 sec1=30 secondshese values determin the time interval of 30 sec since the start ofpinning! also calcualted the time the ball loses contact with the verticalall of the ball track, this is when(**) becomes true!ime till drop off is 17.04 secondsy this time the ball passes 4935 Grad or 13.7 revolutions from startoint!t this moment the ball has a velocity of 2.7 Rads/Sec or 0.43 Revser/Sec! -----------------------* www.GroupSrv.com-----------------------*==ubject: Re: Please advice on this math - motion of roulette ball posting-account=CfSJ5AwAAAD1yt3VP50q913IBHikxMCd Hi all, I hope iÕm posting to the right forum, sorry if not. I have dug out some equations that relate to the prediction of a roulette balls drop off point on the wheel. IÕve need help verifing if these equations are correct or nonsense. I have copied a section below for anyone to have a look at: The following mathematics deals with the X,Y and Z axis within the confines of a Roulette Wheel enviroment. Y axisN1*COS(a)-(mg)*COS(a)=0 X axisN2+N1[Capita lIDoubleDot]*SIN(a)+mg*sin(@)*COS(Y)= m*@ centre)=m*V^2/R=m*[YÕ)^2]*R V=Linear Velocity R=Ball Track Radius @=Centripedal acceleration Z axisFfr+Air Drag=m*@tan=m*YÕÕ *R Friction Force a This is negative as it is opposing the Z axis Air Drag is the force that is equal too: Air Drag= - 0.5*CD*P*TT*r^2*V^2 (TT is pie) this is also a minus value! CD is Drag Coefficiaent P is AIr density r is the balls Radius Z axis is always tangentially directed. ------------------------------------------ After some very simple Algerbraic Transformation and incorporatinghe above formulas we get the next differential equation: YÕÕ=(a+air*R)*(YÕ)^2=b*SIN(Y)+c *COS(Y)+d (*) Where a is the determining friction factor(Ie 0.004) Air =-[0.5*CD*P*TT*r^2*V^2]/m b=a*g*SIN(@)/R c=b/a d=a.g.COS(@)SIN(a)+1)/(R.COS(@) The ball movement sters to this equation only till the moment when itoses the contact with the vertical side of the ball track or: [N2]=0 So the Drop off condition is: [(YÕ)^2]*R+g*COS(@)*tg(a)-g*SIN(@)*COS(Y)=0 (**) Now lets introduce some real values into the equations and see the predicted results: TT=3.14 g=9.807 R=0.4 a=16.7.TT/180 inner slope of stator CD=0.47 r=0.5.21.10^-3 Radius of ball P=1.22 m=9.10^-3 Mass of Ball (a)= 0.004 Friction factor for rolling between the ball and the track @=0.8 grad Tilt Angle t0= 0 sec t1=30 seconds These values determin the time interval of 30 sec since the start of spinning! I also calcualted the time the ball loses contact with the vertical wall of the ball track, this is when(**) becomes true! Time till drop off is 17.04 seconds By this time the ball passes 4935 Grad or 13.7 revolutions from start point!At this moment the ball has a velocity of 2.7 Rads/Sec >r 0.43 Revs per/Sec!t is a good attempt at modeling Roulette Ball motion simulation.asically it modifies banking curve formula tan(th)= V^2/(R g) fortationary ball on conical track. You can run ODEs onaple/Mathematica/Matlab or other simulation software, plug in theonstants and produce a lot of graphs describing the motion. But results as good as the constants plugged in, cannot be used to bet as theres no means to know velocity, or a guaranteed friction value, may besed to calibrate a given Roulette :)... scientific attitude is toelieve that chances of bet under controlled or experimental conditionsre better than pure blind shots. All that if you can carry laptopsnto a Casino..==ubject: Re: Please advice on this math - motion of roulette ball posting-account=CfSJ5AwAAAD1yt3VP50q913IBHikxMCd Hi all, I hope iÕm posting to the right forum, sorry if not. I have dug out some equations that relate to the prediction of a roulette balls drop off point on the wheel. IÕve need help verifing if these equations are correct or nonsense. I have copied a section below for anyone to have a look at: The following mathematics deals with the X,Y and Z axis within the confines of a Roulette Wheel enviroment. Y axisN1*COS(a)-(mg)*COS(a)=0 X axisN2+N1[Capita lIDoubleDot]*SIN(a)+mg*sin(@)*COS(Y)= m*@ centre)=m*V^2/R=m*[YÕ)^2]*R V=Linear Velocity R=Ball Track Radius @=Centripedal acceleration Z axisFfr+Air Drag=m*@tan=m*YÕÕ *R Friction Force a This is negative as it is opposing the Z axis Air Drag is the force that is equal too: Air Drag= - 0.5*CD*P*TT*r^2*V^2 (TT is pie) this is also a minus value! CD is Drag Coefficiaent P is AIr density r is the balls Radius Z axis is always tangentially directed. ------------------------------------------ After some very simple Algerbraic Transformation and incorporatinghe above formulas we get the next differential equation: YÕÕ=(a+air*R)*(YÕ)^2=b*SIN(Y)+c *COS(Y)+d (*) Where a is the determining friction factor(Ie 0.004) Air =-[0.5*CD*P*TT*r^2*V^2]/m b=a*g*SIN(@)/R c=b/a d=a.g.COS(@)SIN(a)+1)/(R.COS(@) The ball movement sters to this equation only till the moment when itoses the contact with the vertical side of the ball track or: [N2]=0 So the Drop off condition is: [(YÕ)^2]*R+g*COS(@)*tg(a)-g*SIN(@)*COS(Y)=0 (**) Now lets introduce some real values into the equations and see the predicted results: TT=3.14 g=9.807 R=0.4 a=16.7.TT/180 inner slope of stator CD=0.47 r=0.5.21.10^-3 Radius of ball P=1.22 m=9.10^-3 Mass of Ball (a)= 0.004 Friction factor for rolling between the ball and the track @=0.8 grad Tilt Angle t0= 0 sec t1=30 seconds These values determin the time interval of 30 sec since the start of spinning! I also calcualted the time the ball loses contact with the vertical wall of the ball track, this is when(**) becomes true! Time till drop off is 17.04 seconds By this time the ball passes 4935 Grad or 13.7 revolutions from start point!At this moment the ball has a velocity of 2.7 Rads/Sec >r 0.43 Revs per/Sec!t is a good attempt at modeling Roulette Ball motion simulation.asically it modifies banking curve formula tan(th)= V^2/(R g) fortationary ball on conical track. You can run ODEs onaple/Mathematica/Matlab or other simulation software, plug in theonstants and produce a lot of graphs describing the motion. But results as good as the constants plugged in, cannot be used to bet as theres no means to know velocity, or a guaranteed friction value, may besed to calibrate a given Roulette :)... scientific attitude is toelieve that chances of bet under controlled or experimental conditionsre better than pure blind shots. All that if you can carry laptopsnto a Casino..==ubject: Re: Please advice on this math - motion of roulette ball Hi all, I hope iÕm posting to the right forum, sorry if not. I have dug out some equations that relate to the prediction of a roulette balls drop off point on the wheel. IÕve need help verifing if these equations are correct or nonsense. I have copied a section below for anyone to have a look at: The following mathematics deals with the X,Y and Z axis within the confines of a Roulette Wheel enviroment. Y axisN1*COS(a)-(mg)*COS(a)=0 X axisN2+N1[Capita lIDoubleDot]*SIN(a)+mg*sin(@)*COS(Y)= m*@ centre)=m*V^2/R=m*[YÕ)^2]*R V=Linear Velocity R=Ball Track Radius @=Centripedal acceleration Z axisFfr+Air Drag=m*@tan=m*YÕÕ *R Friction Force a This is negative as it is opposing the Z axis Air Drag is the force that is equal too: Air Drag= - 0.5*CD*P*TT*r^2*V^2 (TT is pie) this is also a minus value! CD is Drag Coefficiaent P is AIr density r is the balls Radius Z axis is always tangentially directed. ------------------------------------------ After some very simple Algerbraic Transformation and incorporating the above formulas we get the next differential equation: YÕÕ=(a+air*R)*(YÕ)^2=b*SIN(Y)+c *COS(Y)+d (*) Where a is the determining friction factor(Ie 0.004) Air =-[0.5*CD*P*TT*r^2*V^2]/m b=a*g*SIN(@)/R c=b/a d=a.g.COS(@)SIN(a)+1)/(R.COS(@) The ball movement sters to this equation only till the moment when it loses the contact with the vertical side of the ball track or: [N2]=0 So the Drop off condition is: [(YÕ)^2]*R+g*COS(@)*tg(a)-g*SIN(@)*COS(Y)=0 (**)t would help if you mentioned what these variables represent. This lso looks like a physics problem more than a math problem. The ormulas are just that, formulas. Whether they are valid is a question f physics. Now lets introduce some real values into the equations and see the predicted results: TT=3.14 g=9.807 R=0.4 a=16.7.TT/180 inner slope of stator CD=0.47 r=0.5.21.10^-3 Radius of ball P=1.22 m=9.10^-3 Mass of Ball (a)= 0.004 Friction factor for rolling between the ball and the track @=0.8 grad Tilt Angle t0= 0 sec t1=30 seconds These values determin the time interval of 30 sec since the start of spinning! I also calcualted the time the ball loses contact with the vertical wall of the ball track, this is when(**) becomes true! Time till drop off is 17.04 seconds By this time the ball passes 4935 Grad or 13.7 revolutions from start point! At this moment the ball has a velocity of 2.7 Rads/Sec or 0.43 Revs per/Sec! *-----------------------* www.GroupSrv.com *-----------------------*- ill Twentymanmail: wtwentyman at copper dot net==ubject: Re: PROOF that 0.99999... = 1In sci.math, S. Enterprize CompanyIn sci.math, S. Enterprize Company>>I made a typing error here, >>10x = 9 + x>> ItÕs still useless.>> let x = 2>> 10( 2) = 9 + 2>> 20 =/= 11>Congratulations! YouÕve proven that x=2 is not a root of>>the above equation.>>Care to try another number, though? :-)>>[.sigsnip]>>-- >>#191, ewill3@earthlink.net>>ItÕs still legal to go .sigless.> Ok, letÕs look at it in this point of view.> 10x = 9 + x> x can equal 1 or .999... .> There are two roots to the equation.> x1 = 1> x2 = .999...> but,> x1 =/= x2If x1 != x2, is 10*x1 - 9 - x1 == 10*x2 - 9 - x2?1 =/= x2wo different roots.[rest snipped]-- #191, ewill3@earthlink.netItÕs still legal to go .sigless.martÕs Alt. Physics News Groupttp://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1. Enterprize (Science Journal)ttp://smart1234.s-enterprize.com/==ubject: Re: PROOF that 0.99999... = 1megasnip]/3 = .33333... = 3x10^-1+3x10^-2+3x10^-3+3x10^-4+3x10^-5+...hus:*1/3 =*(3x10^-1+3x10^-2+3x10^-3+3x10^-4+3x10^-5+...)hich is:x10^-1+9x10^-2+9x10^-3+9x10^-4+9x10^-5+... = .99999...herefore .99999... = 3*1/3 = 3/3 = 1]==ubject: Re: PROOF that 0.99999... = 1n sci.math, S. Enterprize Companysmart1234@aol.com>>In sci.math, S. Enterprize Company>>In sci.math, S. Enterprize Company>>I made a typing error here, >>10x = 9 + x> ItÕs still useless.>> let x = 2>> 10( 2) = 9 + 2>> 20 =/= 11>>Congratulations! YouÕve proven that x=2 is not a root of>the above equation.>>Care to try another number, though? :-)>>[.sigsnip]>>-- >#191, ewill3@earthlink.net>ItÕs still legal to go .sigless.>> Ok, letÕs look at it in this point of view.>> 10x = 9 + x>> x can equal 1 or .999... .>> There are two roots to the equation.>> x1 = 1>> x2 = .999...>> but,>> x1 =/= x2>If x1 != x2, is 10*x1 - 9 - x1 == 10*x2 - 9 - x2? x1 =/= x2 Two different roots.h, so an equation of polynomial degree n (n=1 in this case)an have more than n roots?rest snipped]- 191, ewill3@earthlink.nettÕs still legal to go .sigless.==ubject: Re: JSH: But what if it works? <876535qrfk.fsf@phiwumbda.org> posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY >You people spend a lot of time criticizing me for the creative part, >when I brainstorm out ideas and have the GALL to actually talk about >them! You may find brainstorming works better if you donÕt do the following to the people youÕre brainstorming with: - call them liars - threaten to call their employers - suggest that youÕre going to set the US army on them. After all, youÕre the one asking for help, from people who have no obligation to you. -- Richard am POSTING on Usenet, and some of those posts arenÕt very serious.ut you people act like every one is being handed down like one of theen Commandments, like a missive from God, and then get on my casebout them for YEARS after the fact.nd the jab about calling someoneÕs employer is just dishonest.he specific case is David Ullrich, a professor at Oklahoma Stateniversity, where you can see tatein the name of the school, whoade a post talking about racial slurs. Specifically HE introduced theubject of racial slurs, and HE talked about thinking a racial slur washe appropriate response to one of my posts saying heÕd acted as myapdog in an instance.ow if you people were sane, a professor introducing the subject oface in a backhanded attempt at insulting someone by claiming to havethought* about insulting them but changing his mind wouldnÕt beefended.hy defend racism?ut instead you people come after me, year, after year, after year,epeating the same old lies and refusing to tell the entire story.ow the reality is that taken aback by UllrichÕs comments I contactedis school, and was told by someone in the Human Resources departmenthat I should call the head of his department, and THEY even gave mehe number. called, talked to the guy, who was quite nice, and heard some weirdtuff that Ullrich had been telling him--twisting the story. And thatould have been it!ut ULLRICH came back on sci.math and started talking about meegistering a complaint!!!ext thing I know posters are coming in and not just talking abouthinking about racial slurs, theyÕre delivering them, and Ullrich keepslaying the victim the whole time.ut IÕm the one who was getting called nigger on the newsgroup, andllrich is the one who started it all by introducing the subject ofacial slurs.hose of you who defend him and his behavior are just racist.ou twist the story because thatÕs what racists do, as why else supportt and keep bringing it up?nd just so you know I DO NOT KEEP BRINGING IT UP! Other posters do.ow, eventually, you losers might succeed in getting someone interestedn the full story who doesnÕt like the idea of a professor at a stateniversity playing such games, and doesnÕt care if itÕs just on Usenet,o Ullrich may finally get bit by his machinations.tÕs not like thereÕs a statute of limitations on impact on him for hisomments.ventually he can still face the music on it, and talking about it overnd over again, and brining it up repeatedly could cause that, as, inact, here in America, his statements could be considered hate speech.nd his tricky way of delivering of talking about thinking aboutelivering a racial slur and changing his mind, is not really verylever.ike IÕm thinking about telling a joke about your mother and someailors.h well, changed my mind!ames Harris==ubject: Re: JSH: But what if it works?riginator: richard@cogsci.ed.ac.uk (Richard Tobin)> You may find brainstorming works better if you donÕt do the following> to the people youÕre brainstorming with:> - call them liars> - threaten to call their employers> - suggest that youÕre going to set the US army on them.The specific case is David Ullricho, that is not the case I was referring to. I was thinking of the And I want you to realize that I *will* send the promised letters to universities for those of you who behave so egregiously that it must be addressed.ut you are missing the point. I gave you some reasons why yourbrainstormingdoesnÕt work. Defending your actions wonÕt donything to make it work.And the jab about calling someoneÕs employer is just dishonest.ee, you did it again.- Richard==ubject: Re: JSH: But what if it works?>>You people spend a lot of time criticizing me for the creative part,>>when I brainstorm out ideas and have the GALL to actually talk about>>them!>You may find brainstorming works better if you donÕt do the following>to the people youÕre brainstorming with:> - call them liars> - threaten to call their employers> - suggest that youÕre going to set the US army on them.>After all, youÕre the one asking for help, from people who have no>obligation to you.>-- Richard I am POSTING on Usenet, and some of those posts arenÕt very serious. But you people act like every one is being handed down like one of the Ten Commandments, like a missive from God, and then get on my case about them for YEARS after the fact.erhaps if you labelled your posts as either peculation erious nd ff-the-cuffit would help. And the jab about calling someoneÕs employer is just dishonest. The specific case is David Ullrich, a professor at Oklahoma State University, where you can see tatein the name of the school, who made a post talking about racial slurs. Specifically HE introduced the subject of racial slurs, and HE talked about thinking a racial slur was the appropriate response to one of my posts saying heÕd acted as my lapdog in an instance.ereÕs the thing. The thought police are not in the USA yet. He can hink whatever he wants without consequence as long as he *doesnÕt say t*. He expressed aggrevation to the point of *considering* the use of racial slur, but thatÕs like saying I wanted to say something else hen I said arn Now if you people were sane, a professor introducing the subject of race in a backhanded attempt at insulting someone by claiming to have *thought* about insulting them but changing his mind wouldnÕt be defended. Why defend racism? But instead you people come after me, year, after year, after year, repeating the same old lies and refusing to tell the entire story. Now the reality is that taken aback by UllrichÕs comments I contacted his school, and was told by someone in the Human Resources department that I should call the head of his department, and THEY even gave me the number. I called, talked to the guy, who was quite nice, and heard some weird stuff that Ullrich had been telling him--twisting the story. And that could have been it!onsidering how you normally present the actsof a situation about senet posts, saying that he twisted the story is not particularly credible. But ULLRICH came back on sci.math and started talking about me registering a complaint!!!nd this surprises you? Next thing I know posters are coming in and not just talking about thinking about racial slurs, theyÕre delivering them, and Ullrich keeps playing the victim the whole time.hen complain the employers of those who actually make the slurs. But IÕm the one who was getting called nigger on the newsgroup, and Ullrich is the one who started it all by introducing the subject of racial slurs.ncle Al does a nice job of introducing all manner of vulgarities ithout prompting. Is it possible he started this trend? Those of you who defend him and his behavior are just racist.Õll be honest with you, DavidÕs comment was not appropriate in my ppinion. However, I think your reaction was less appropriate. It is ossible for two people to be wrong, and one to be more wrong than the ther. If you had not responded, you would have a much stronger case gainst David. You twist the story because thatÕs what racists do, as why else support it and keep bringing it up?erhaps not everyone here feels that one thought indicates racism. If a an watches a single move with female nudity, is he a male chauvinist? And just so you know I DO NOT KEEP BRINGING IT UP! Other posters do.nd you jump on it rather than letting it drop. Now, eventually, you losers might succeed in getting someone interested in the full story who doesnÕt like the idea of a professor at a state university playing such games, and doesnÕt care if itÕs just on Usenet, so Ullrich may finally get bit by his machinations. ItÕs not like thereÕs a statute of limitations on impact on him for his comments. Eventually he can still face the music on it, and talking about it over and over again, and brining it up repeatedly could cause that, as, in fact, here in America, his statements could be considered hate speech. And his tricky way of delivering of talking about thinking about delivering a racial slur and changing his mind, is not really very clever. Like IÕm thinking about telling a joke about your mother and some sailors.s it a funny joke? Oh well, changed my mind!arn! I was getting curious about it.- ill Twentymanmail: wtwentyman at copper dot net==ubject: Re: JSH: But what if it works?-Spam-This: SpamCopies@YahooGroups.Com IÕm investigating this REALLY SIMPLE idea for factoring integershere already is a really simple way to factor integers, namely to tryll odd integers from 3 to the square root of the number. This way isery slow for large numbers. The key is not to have a really simpleay, but a really fast way. If you ever achieve that, please write aomputer program to implement your method, and put it up as a CGIerver application for us to test to see if it really works. Untilhen, I suggest you stop posting garbage whereby you boast aboutomething you havenÕt yet achieved.==ubject: Re: JSH: But what if it works?-RFC2646: Format=Flowed; Originaluck you....... ==ubject: estimation of this functioni All,I have 3 equations: L = aX + bY + cZ M = cX + dY + eZ N = fX + gY + hZ where a,b,c,d,e,f,g,h are constants and are between and 1. L,M,N and X,Y,Z are variables and they can be between 0 and 100.Now, at some point I know the csts a-i and M and N. X,Y,Z and L arenknown. I have to estimate the value of L with minimum error. I knowhere is a way but could not figure it out. Please help.Sorry for theotation.==ubject: Re: estimation of this functionHi All, I have 3 equations: L = aX + bY + cZ M = cX + dY + eZ N = fX + gY + hZ where a,b,c,d,e,f,g,h are constants and are between0 and 1. L,M,N and X,Y,Z are variables and they can be between 0 and 100. Now, at some point I know the csts a-i and M and N. X,Y,Z and L areunknown. I have to estimate the value of L with minimum error. I knowthere is a way but could not figure it out. Please help.Sorry for thenotation.ou have 3 equations with 4 unknowns. ou could start by trying min and max values for {X,Y,Z} -- just 1 value at time, then solve for the other 3 and see if they are in range. r maybe just try possible values of L, then solve for X, Y, Z. See whichalues of L are legal, then use the center of that range as your estimate. hoosing the center of the range minimizes the orst caseerror.-Keith Lewis klewis {at} mitre.orghe above may not (yet) represent the opinions of my employer.==ubject: JSH: Those Ullrich defenders posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUYepeatedly over some years IÕve kept seeing posters claiming that Iried to get David Ullrich fired, or that I call peopleÕs employersbout newsgroup postings, as these Ullrich defenders try to make a casegainst me, by never telling the full story.he real story is that YEARS ago, in the usual back-and-forth, andctually rather dumb arguments that have ßared up around my posts, Iaid that David Ullrich acted as my lapdog in an instance.ver a year later in yet another one of the ßare-ups, Ullrichesponded that more than a year before when IÕd said that heÕd acted asy lapdog in an instance, heÕd gotten angry, thought a racial slur washe appropriate reply, but been talked out of delivering it.mportantly, back when I had said heÕd acted as my lapdog in annstance, I picked up from his postings--as they got even weirder thanormal--that he was very upset, so I APOLOGIZED to Ullrich about theapdog comment.t was over a year later from the time of the comment and my apologyhen Ullrich made his comments, where HE introduced the subject ofacial slurs.ow I know that many of you donÕt want to be troubled with the facts,ut here they are anyway, though I find it amazing how little many ofou seem to care about facts.o I thought about it more and more, and eventually complained tollrichÕs school as heÕs a professor at Oklahoma State University, and happen to think that his behavior reßected badly on his school andn academics. think those who defend Ullrich wish Usenet to be a place where youan say anything you want without any consequences.y making a complaint scares them because it challenges their anarchistosition.hy else would anarchist types defend a racist statement for so longnd with so much effort?s make no mistake, UllrichÕs delivery doesnÕt leave much doubt.inting at calling someone by a racial slur by talking about wanting toeliver it, calling it an appropriate reply, and then saying you werealked out of it, isnÕt even really subtle.hy would Usenet posters react so consistently in UllrichÕs defense?hy do posters keep bringing it up even now? say becaue they fight to be able to say anything on Usenet, withoutonsequences.ow hereÕs where it gets really interesting, as a complaint is onehing, but Ullrich is the one who informed the newsgroup.e has tenure people. donÕt know how stupid many of you are, but professors with tenureonÕt typically need to worry about a single complaint.F Ullrich were actually worried, why would HE bring it up on Usenet?hy would Ullrich be the one to publicize a complaint, if he actuallyad concerns about his job?nstead he *claimed* to be worried, which fueled a firestorm on theewsgroup, with lots of people calling me names, including some actualacial slurs.Õm sure Ullrich was quite satisfied and probably thought he had hisevenge on me for daring to say heÕd acted as my lapdog in an instance.ow think about that, as Ullrich has acted as if saying someone acteds your lapdog in an instance is a grievous insult. And I actuallypologized soon after making the comment!ou people are used by this person in stuff that isnÕt even verylever, and definitely isnÕt really defensible.he reality is that Ullrich used you as his lapdogs in a bizarre needo come after me, where he picked race, and many of you defended him,nd STILL defend him.upposedly, itÕs nothing if he didnÕt actually deliver a racial slur.ust talking about thinking about delivering a racial slur but notoing it, shouldnÕt be ground for a complaint.ut hey, UllrichÕs thoughts were his own, until HE gave them in aublic post.onÕt think your thoughts matter if you publicize them?hy donÕt you go tell some Secret Service agent that youÕd beenhinking about killing President Bush, but been talked out of it?sk yourselves one question, why, over a year after the event, wouldllrich feel a need to inform me and the newsgroup what he says hethought* a year earlier?ames Harris==ubject: Re: JSH: Those Ullrich defenders posting-account=45mQHQ0AAABhEGyZbLqUdwPik-gRPwwC Repeatedly over some years IÕve kept seeing posters claiming that I tried to get David Ullrich fired, or that I call peopleÕs employers about newsgroup postings, as these Ullrich defenders try to make aase against me, by never telling the full story. The real story is that YEARS ago, in the usual back-and-forth, and actually rather dumb arguments that have ßared up around my posts, I said that David Ullrich acted as my lapdog in an instance. Over a year later in yet another one of the ßare-ups, Ullrich responded that more than a year before when IÕd said that heÕd acteds my lapdog in an instance, heÕd gotten angry, thought a racial sluras the appropriate reply, but been talked out of delivering it.or someone so concerned about racism, you certainly display annusual amount of your own sexism concerning the gender of some ofour correspondents. He has tenure people.eally? How many enure peopledoes he have?==ubject: Re: JSH: Those Ullrich defenders !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i/yuIsqbLLCh/!U#X[S~( 5eZ41to5f%E@ÕELIi $t^VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^] PfkwWhining and crap] For someone so concerned about racism, you certainly display an unusual amount of your own sexism concerning the gender of some of your correspondents. refer you toor a few choice quotes from Harris (even though this posting was not direct reply to Harris himself).- avid Kastrup, Kriemhildstr. 15, 44793 Bochum==ubject: Re: JSH: Those Ullrich defenders posting-account=jj3O3QwAAABZLagqq9jnjQLaOd53yftT For someone so concerned about racism, you certainly display an unusual amount of your own sexism concerning the gender of some of your correspondents.r, for that matter, racism.las, the original is gone, but here you can find it quoted, apparentlyn full:he original message had IDrturo Magidin, sans .sig==ubject: Re: JSH: Those Ullrich defenders Repeatedly over some years IÕve kept seeing posters claiming that I tried to get David Ullrich fired, or that I call peopleÕs employers about newsgroup postings, as these Ullrich defenders try to make a case against me, by never telling the full story.he full story is readily found through a Google search. The first time heard about this I looked it up. For all that David Ullrich is not lways the most courteous, he is clearly a skilled mathematician. He ells you what your errors are and when you prove yourself uninterested, e also tells you what your personal faults are. Such bluntness is arely appreciated these days, especially by the hearer. donÕt know if this is behavior in person, but he says he works on eing polite with his students, and IÕve no reason to doubt him. Many eachers make extra efforts to treat students with respect.snippage of rant] My making a complaint scares them because it challenges their anarchist position. Why else would anarchist types defend a racist statement for so long and with so much effort?erhaps you missed the point. He didnÕt make a racist statement. He considered* making one and decided against it. As make no mistake, UllrichÕs delivery doesnÕt leave much doubt. Hinting at calling someone by a racial slur by talking about wanting to deliver it, calling it an appropriate reply, and then saying you were talked out of it, isnÕt even really subtle. Why would Usenet posters react so consistently in UllrichÕs defense?erhaps because your behavior is significantly worse, yet you fail to ee it. His behavior may not be perfect, but you are the last person to e throwing stones. When you stop insulting people, you will be more ikely to get sympathy about insults received. IF Ullrich were actually worried, why would HE bring it up on Usenet?o reveal your character. Why would Ullrich be the one to publicize a complaint, if he actually had concerns about his job?o reveal your character. Supposedly, itÕs nothing if he didnÕt actually deliver a racial slur. Just talking about thinking about delivering a racial slur but not doing it, shouldnÕt be ground for a complaint. But hey, UllrichÕs thoughts were his own, until HE gave them in a public post.hat, exactly, were his thoughts? I certainly donÕt know. DonÕt think your thoughts matter if you publicize them?hen they arenÕt thoughts, but speech. Why donÕt you go tell some Secret Service agent that youÕd been thinking about killing President Bush, but been talked out of it?nd wouldnÕt that be different from telling a Secret Service agent that ouÕre thinking about killing him? Yes. One is a threat, the other is n admission of temporary poor judgement. Ask yourselves one question, why, over a year after the event, would Ullrich feel a need to inform me and the newsgroup what he says he *thought* a year earlier?o make a point. Apparently, you missed the point.- ill Twentymanmail: wtwentyman at copper dot net==ubject: Re: JSH: Those Ullrich defenders Perhaps you missed the point. He didnÕt make a racist statement. He *considered* making one and decided against it.ould it have mattered if he _had_ made a racist statement? People arear too sensitive about such things.==ubject: Re: JSH: Those Ullrich defenders>Perhaps you missed the point. He didnÕt make a racist statement. He>*considered* making one and decided against it. Would it have mattered if he _had_ made a racist statement? People are far too sensitive about such things. would have felt that James had a valid complaint. Then again, IÕve een reminded that JamesÕs stated purpose in talking with DavidÕs uperiors was to try to silence David, not necessarily get him fired. unny, James is the one who always complains about censorship.ow, I also think that a single comment would be better ignored than sed as the basis of starting problems. I find some of Uncle AlÕs stuff xtremely offensive, but choose to ignore it rather than seek out his mployer. A kill-file/filter is great for that sort of thing.unny thing, if I didnÕt know James uses Google IÕd think I was in his ilter.- ill Twentymanmail: wtwentyman at copper dot net==ubject: Re: JSH: Those Ullrich defendersDiscussion, linux) Ask yourselves one question, why, over a year after the event, would Ullrich feel a need to inform me and the newsgroup what he says he *thought* a year earlier?ecause you brought the incident up in a way that annoyed him?---- >But remember, I did so after his feelings got hurt while playing >he game he gameindeed. I recall the incident you refer to. I donÕt recall the details. But I do recall that the ameyou were playing was such that it seemed to me a perfectly appropriate reply would be some sort of racial slur - someone talked me out of that. Like you referred to me as your lap dog or something. Boy was I mad about that. he gameindeed.----- Yup, as far as IÕm concerned, if you live out your lives smiling thentire time full of pride in your *believed* accomplishments, when youever had any, well thatÕs ok with me. --James Harris, a man of remarkable accomplishments.==ubject: Re: JSH: Those Ullrich defenders posting-account=BoVIJQwAAABWQmiBreBpIBK6U9cCf57f Repeatedly over some years IÕve kept seeing posters claiming that I tried to get David Ullrich fired, or that I call peopleÕs employers about newsgroup postings, as these Ullrich defenders try to make aase against me, by never telling the full story.his old stuff James? BORING. More operators and objects please! If youeed inspiration, just get drunk and sing to the walls. It alwaysorked before, didnÕt it?- arry Lardeplies to group please==ubject: Re: JSH: Those Ullrich defenders posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs Repeatedly over some years IÕve kept seeing posters claiming that I tried to get David Ullrich fired, or that I call peopleÕs employers about newsgroup postings, as these Ullrich defenders try to make aase against me, by never telling the full story. The real story is that YEARS ago, in the usual back-and-forth, and actually rather dumb arguments that have ßared up around my posts, I said that David Ullrich acted as my lapdog in an instance.his is ancient history.erhaps of more interest is your more recent threat tory to get a school to withhold a Ph.D. from a grad studentho did nothing more than correspond with you and try toelp you straighten out your arguments. No apdogs nolurs, not even any thinking about slurs.ou seem to have deleted your original message, butortunately the text is still available in the thread.> IÕve mentioned a Cornell math student who emailed me, of his own> volition, offering help, and the possibility that if I explained my> work to him that Cornell itself might back me, which would make a> tremendous difference.> I carefully and patiently talked him through the basic argument, and> he re-worked it, until reaching the conclusion where he tried tount.> That was August of last year. He is a graduate student at Cornell> University.> What do you think are the chances that he will ever get a math Ph.D?> IÕm telling you they are nil to none. Obviously I keep his name> hidden now for my own purposes but as a mathematician I would not be> doing my duty to the field to sit back and allow him to ever get a> Ph.D if I could stop it.> That might sound unfair but think about the responsibility that goes> with the field. still get furious reading this and thinking that thereight even be the remotest possibility you could destroyomeone this way. It is one of the slimiest things youÕve everaid, and thereÕs a lot of competition in that department. Randy==ubject: Re: JSH: Those Ullrich defenders[...]IF Ullrich were actually worried, why would HE bring it up on Usenet?Why would Ullrich be the one to publicize a complaint, if he actuallyhad concerns about his job?Õve explained this many times. No, I had no concern about myob. I thought people might be interested in the sort of tacticsou use to try to win arguments. (And I had some concern for theob of the next person you tried this sort of crap on.)...]Ask yourselves one question, why, over a year after the event, wouldUllrich feel a need to inform me and the newsgroup what he says he*thought* a year earlier?ade perfectly clear in the post where I made the comment.or those of you with short memories: YouÕd just said somethingotally inappropriate. I was trying to point out that there wasuch a thing as ou just donÕt talk to people that way ands an illustration I mentioned something that I never saidears ago, even though you had me mad enough to say thingshat IÕd normally never say.James Harris***********************avid C. Ullrich==ubject: Re: JSH: Those Ullrich defenders posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAgawn.asically, itÕs like this: You contacted someoneÕs employer with a viewowards intimidating them into stopping replying to your posts.ut, not only that, apparently you want to do this without anyonehinking youÕre a bastard.he simple fact is, if you take a step like that some people will thinkouÕre a bastard.eal with it. Repeatedly over some years IÕve kept seeing posters claiming that I tried to get David Ullrich fired, or that I call peopleÕs employers about newsgroup postings, as these Ullrich defenders try to make aase against me, by never telling the full story. The real story is that YEARS ago, in the usual back-and-forth, and actually rather dumb arguments that have ßared up around my posts, I said that David Ullrich acted as my lapdog in an instance. Over a year later in yet another one of the ßare-ups, Ullrich responded that more than a year before when IÕd said that heÕd acteds my lapdog in an instance, heÕd gotten angry, thought a racial sluras the appropriate reply, but been talked out of delivering it. Importantly, back when I had said heÕd acted as my lapdog in an instance, I picked up from his postings--as they got even weirderhan normal--that he was very upset, so I APOLOGIZED to Ullrich about the lapdog comment. It was over a year later from the time of the comment and my apology when Ullrich made his comments, where HE introduced the subject of racial slurs. Now I know that many of you donÕt want to be troubled with the facts, but here they are anyway, though I find it amazing how little many of you seem to care about facts. So I thought about it more and more, and eventually complained to UllrichÕs school as heÕs a professor at Oklahoma State University,nd I happen to think that his behavior reßected badly on his school and on academics. I think those who defend Ullrich wish Usenet to be a place where you can say anything you want without any consequences. My making a complaint scares them because it challenges theirnarchist position. Why else would anarchist types defend a racist statement for so long and with so much effort? As make no mistake, UllrichÕs delivery doesnÕt leave much doubt. Hinting at calling someone by a racial slur by talking about wantingo deliver it, calling it an appropriate reply, and then saying you were talked out of it, isnÕt even really subtle. Why would Usenet posters react so consistently in UllrichÕs defense? Why do posters keep bringing it up even now? I say becaue they fight to be able to say anything on Usenet, without consequences. Now hereÕs where it gets really interesting, as a complaint is one thing, but Ullrich is the one who informed the newsgroup. He has tenure people. I donÕt know how stupid many of you are, but professors with tenure donÕt typically need to worry about a single complaint. IF Ullrich were actually worried, why would HE bring it up on Usenet? Why would Ullrich be the one to publicize a complaint, if he actually had concerns about his job? Instead he *claimed* to be worried, which fueled a firestorm on the newsgroup, with lots of people calling me names, including somectual racial slurs. IÕm sure Ullrich was quite satisfied and probably thought he had his revenge on me for daring to say heÕd acted as my lapdog in annstance. Now think about that, as Ullrich has acted as if saying someone acted as your lapdog in an instance is a grievous insult. And I actually apologized soon after making the comment! You people are used by this person in stuff that isnÕt even very clever, and definitely isnÕt really defensible. The reality is that Ullrich used you as his lapdogs in a bizarre need to come after me, where he picked race, and many of you defended him, and STILL defend him. Supposedly, itÕs nothing if he didnÕt actually deliver a racial slur. Just talking about thinking about delivering a racial slur but not doing it, shouldnÕt be ground for a complaint. But hey, UllrichÕs thoughts were his own, until HE gave them in a public post. DonÕt think your thoughts matter if you publicize them? Why donÕt you go tell some Secret Service agent that youÕd been thinking about killing President Bush, but been talked out of it? Ask yourselves one question, why, over a year after the event, would Ullrich feel a need to inform me and the newsgroup what he says he *thought* a year earlier? James Harris==ubject: Re: GCH vs. Axiom of Choice. posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAgG is a version of ZF with classes. Obviously you need to have classesn your language to even assert the existence of L, let alone prove it.on(ZF) is a sentence in the first-order language of number theory, andhe first-order language of number theory can be embedded in theirst-order language of set theory.==ubject: Re: GCH vs. Axiom of Choice. posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU > Does Shoenfield address whether for two sets of axioms A and B iftÕs > the case that Con(ZF)=>Con(ZF+A) and Con(ZF)=>(ConZF+B) togethermply > that (ZF+A|-S => ~(ZF+B|-~S) for statements S about L)? I haveooks > on mathematical logic to reference for things about Con and |-,tÕs > the stuff about L that I need help with because I donÕt know what Ls > and itÕs not in my books, thatÕs why I turn to sci.math. Shoenfield addresses everything you need to know to answer all your questions for yourself. In particular, he explains what Con(ZF)eans. already have books (on mathematical logic) that define weakonsistency (no S such that there are two derivation from axioms andnding on S and the other on ~S) and strong consistency (exists an Such that there is no derivation from axioms and ending on S), but IonÕt know if thatÕs what people are talking about regarding L, thetatements I hear about L seem highly ualifiedin the sense of beingeserved with many qualifications and it makes me nervous that I donÕtnderstand things, when itÕs quite possible that I do. Before spending40 on yet another book, I want to know if it will mention L, I mean Iould just track down GoedelÕs original consistency proof of GCH ifhere really are no other sources.==ubject: Re: GCH vs. Axiom of Choice. posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU > I do know how to write or all n either n is not a natural > number or A|- S when S is the sentence n corresponds towhen A is > fixed finite set of axioms and |- is a fixed meachnical proof > technique, but even if I could let A be an infinite conjuntion of > axioms, then this is still proof theoretical, itÕs A|=S that Ihought > consistency was supposed to talk about, am I wrong about that? Hard to say, since I need to guess at your meaning. Why not justut all this vague mumbling aside and study the subject in some standard exposition, say ShoenfieldÕs classic book, which is cheaply available in a recent printing? Would you wafße on in this style about algebraic geometry or number theory or differential topology?oes Shoenfield address whether for two sets of axioms A and B if itÕshe case that Con(ZF)=>Con(ZF+A) and Con(ZF)=>(ConZF+B) together implyhat (ZF+A|-S => ~(ZF+B|-~S) for statements S about L)? I have booksn mathematical logic to reference for things about Con and |-, itÕshe stuff about L that I need help with because I donÕt know what L isnd itÕs not in my books, thatÕs why I turn to sci.math.==ubject: Re: GCH vs. Axiom of Choice. Does Shoenfield address whether for two sets of axioms A and B if itÕs the case that Con(ZF)=>Con(ZF+A) and Con(ZF)=>(ConZF+B) together imply that (ZF+A|-S => ~(ZF+B|-~S) for statements S about L)? I have books on mathematical logic to reference for things about Con and |-, itÕs the stuff about L that I need help with because I donÕt know what L is and itÕs not in my books, thatÕs why I turn to sci.math. Shoenfield addresses everything you need to know to answer all youruestions for yourself. In particular, he explains what Con(ZF) means.==ubject: Re: GCH vs. Axiom of Choice. posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU |Ah, I should have been more clear. L is a model (right?), ItÕs a proper class of sets, the sets satisfying a certain condition. (Being onstructible) Another way to describe it is that itÕs the minimal model of ZF containing all the ordinals.s L a set? Obvious L is not in L, but if seems like definitions of say# seem to say that it exists if L exists. Is that because thexistence of L is undecideable because otherwise ZF could prove itÕswn consistency by proving the existence of L?snip Q&A I understood, yeah!] | For instance if we had to assume ZFC+GCH |to prove it, then it wouldnÕt mean much IMO. Heh, itÕs sort of amusing when to prove that some fact holds true when relativized to a model one needs to use the same fact not relativized to the model. I can see how that might seemnderwhelming, but itÕs not necessarily trivial or uninteresting. For instance, one had to prove that the axioms of ZF hold in L, too, and obviously we need to assume some of them in order to prove they hold in L. meant that if we prove GCH in L without assuming GCH then itÕs a bitike getting GCH for free, and if we have to assume it, then since L is standard model (x models x, and xey is true inside the model iff xeys true outside the model) then I wasnÕt sure the enterprise was reallyenerating anything. | I assume that ZF can |prove only limited things about L, since ZF is incomplete, but if I |add GCH to ZF then can ZF+GCH prove something new about L? IÕm guessing it probably can, but I donÕt know of a good example. My guess is that some assumption intermediate between ZF and ZF+V=L is sufficient to prove (say) that aleph_1 is the same as aleph_1 relative to L. Certainly V=L implies that!aybe my original question wasnÕt clear. If you take every sentance Xuch that Con(ZF)=>Con(ZF+X) & Con(ZF+~X), then can there be a sentence about L (which to more clear f L is a set IÕll define as aentence such that (Ex T) is equivalent to (Ex xeL & T) for everyormulas (Ex T) and (T) that are subformulas of S and such that (Ax T)s equivalent to (Ax (~xeL)vT) for every formulas (Ax T) and (T) thatre subformulas of S) such that ZF+X|- S and ZF+~X|- ~S. And moreenerally, how about any chain of X,Y,Z, ... such that for every subset or B of of X,Y,Z,... such that Con(ZF)->Con(ZF+A) andon(ZF)->Con(ZF+B) is it possible that ZF+A |- S and ZF+B |- ~S. IÕmrying to figure out if the superconsistent axioms of ZF together proofhe alleged fixed truths of L. FOL is complete, so it might work. On the other hand, there arenÕt any first order properties of L that can be proven in ZF+GCH but not in ZF. In fact, for an arbitrary sentence X in the language of ZF, the following are equivalent: (a) ZF |- (V=L->X) (b) ZF + V=L |- X (c) ZF |- L |= X. can read (a) and (b), but (c) and (d) must have some operator orderreference I donÕt know, or is ZF |- L |= X only to be read as ZF |-L|=X), which I donÕt even know what it means unless L is a sentence,nd even so how one proves a validity based on axioms I donÕt know soven then it doesnÕt make sense to me. (d) ZF + V=L |- L |= X (a)->(b) is a case of the deduction theorem. (b)->(c) holds because itÕs a theorem of ZF that V=L relativized to L is true Given a model M of ZF, we can consider constructibility as relativized to M. The elements of M satisfying this relativized constructibility form a submodel L_M, which is another model of ZF. If we repeat thisrocess, to get L_{L_M}, itÕs possible to prove that we donÕt shrink the model any; L_{L_M} = L_M. So L_M is a model of V=L, the axiom stating that every set is in L. So anything thatÕs a consequence of V=L holds relative to L. If X is a theorem of ZF+V=L, then that fact can be proven in ZF, and then applied to L. (c)->(d) is obvious. To prove (d)->(a), assume first that (a) is false. By GoedelÕs completeness theorem, there would exist a model of ZF+V=L+~X. Suppose M is a model of ZF+V=L+~X. Since the L relativized to M is the same as M, we get that M |= ~(L|=X). Hence M is also a model of ZF+V=L in which L|=X fails. Since GCH is a consequence of V=L, (cÕ) ZF + GCH |- L|= X is also equivalent. | How about |ZF+(~GCH), can that also prove things about L? Since ~GCH is so weak, I donÕt think there are many theorems about L in ZF+(~GCH) that arenÕt also theorems in ZF. We can deduce from ~GCH the existence of sets not in L, of course.Õm trying to find out if new relativity consistent axioms added to ZFrove things about L consistently and whether taken as a whole they doo completely. | Do the truths of L |vary depending on what other axioms L is defined with? IÕm tryingo |be more clear, but I apologize in advance if I failed again. How would whether something is true of L depend on axioms? donÕt even get the definition of L, so to me itÕs all provisional. Of course, what can be proven to be true of L from a set of axioms does depend on the axioms! IÕm not familiar with the details, but I gather that some large cardinal axioms (I think the existence of a measurable cardinal is strong enough) imply a relatively detailed account of the structure of L.ut are there other axioms that imply details that contradict thoseetails? Note that the existence of a measurable cardinal is inconsistent with V=L.es, of course. IÕm just using them to increase the deductive power of system to prove the alleged fixed truths of L. Axioms need not berue, as long as IÕm only using them in a way to prove truths of L. This real 0# that is believed to be outside of L encodes some of the structure of L (if it exists).s it L v 0# whose existence is debated?==ubject: Re: GCH vs. Axiom of Choice. posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KUodel numbering is a correpondance between sentences and naiventegers. I can easily make a naive injecttion from sentences toonnegative integers, and I can even extend it to be mapping onto theonnegative integers. But I donÕt know how to write or all n either is not a natural number or n is true when n is interpreted as aentence I do know how to write or all n either n is not a naturalumber or A|- S when S is the sentence n corresponds towhen A is aixed finite set of axioms and |- is a fixed meachnical proofechnique, but even if I could let A be an infinite conjuntion ofxioms, then this is still proof theoretical, itÕs A|=S that I thoughtonsistency was supposed to talk about, am I wrong about that?==ubject: Re: GCH vs. Axiom of Choice. I do know how to write or all n either n is not a natural number or A|- S when S is the sentence n corresponds towhen A is a fixed finite set of axioms and |- is a fixed meachnical proof technique, but even if I could let A be an infinite conjuntion of axioms, then this is still proof theoretical, itÕs A|=S that I thought consistency was supposed to talk about, am I wrong about that? Hard to say, since I need to guess at your meaning. Why not just putll this vague mumbling aside and study the subject in some standardxposition, say ShoenfieldÕs classic book, which is cheaply availablen a recent printing? Would you wafße on in this style aboutlgebraic geometry or number theory or differential topology?==ubject: Re: GCH vs. Axiom of Choice. Would you wafße on in this style about algebraic geometry or number theory or differential topology?lease donÕt give him ideas! :-)- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_-mailer: xrn 9.02==ubject: Re: GCH vs. Axiom of Choice.ail-To-News-Contact: abuse@dizum.com> At the risk of sounding like Pilate, hat is truth?I always thought> that athematical Truthwas only definable with respect to a given> set of axioms. On the contrary, there is no such definition. Mathematical truth, asdefined in mathematics, is not relative to any set of axioms. still want to know hat is truth?If itÕs not in relation to a setf axioms, how is it determined? What does it mean to be true? What ishe decision procedure for mathematical truth?hat is the truth value of: For any set, k, there exists a set, j, such that |k| < |j| < |P(k)|hat is the truth value of: For any line, L, and any point outside that line, P, there exist infinitely many lines, Ln, through P and parallel to L.ow do you determine the truth or falsity of these?- ichael F. Stemperinclude isualize whirled peas!==ubject: Re: GCH vs. Axiom of Choice. I still want to know hat is truth?If itÕs not in relation to a set of axioms, how is it determined? What does it mean to be true? What is the decision procedure for mathematical truth? Demonstrably, there is no such decision procedure. Why should there be?==ubject: Re: GCH vs. Axiom of Choice.ail-To-News-Contact: abuse@dizum.com> I still want to know hat is truth?If itÕs not in relation to a set> of axioms, how is it determined? What does it mean to be true? What is> the decision procedure for mathematical truth? Demonstrably, there is no such decision procedure. Why should there be? never thought that there was. I always thought that mathematicsorked on the basis that truth was defined relative to a set ofxioms. Different axioms, different statements become true. Thatas how I thought that we ended up with Euclidean and non-Euclideaneometries. If you assume that there are zero, one, or n lines through point outside a line parallel to that line, you end up with aifferent set of statements that are considered true.vcbwtvkll4r.fsf@beta19.sm.ltu.se>, you made the statement that:athematical truth, as defined in mathematics, is not relative toany set of axioms. he only interpretation that I can put on this statement is thattatements arenÕt true or false based upon what axioms you choose.s far as I can tell, that only leaves us with two options:. Statements are never true or false.. Statements are true or false, without the need to choose any axioms.f course, itÕs possible that I misunderstood what you said. But, ifhatÕs the case, what did your statement mean?- ichael F. Stemperinclude lways use apostropheÕs and uotation marksproperly.==ubject: Re: GCH vs. Axiom of Choice. The only interpretation that I can put on this statement is that statements arenÕt true or false based upon what axioms you choose. As far as I can tell, that only leaves us with two options: 1. Statements are never true or false. 2. Statements are true or false, without the need to choose any axioms.tatements are true or false *in a particular model*.or example, in the axioms for Group Theory, the proposition: very element has order 2. s neither true nor false. But in certain groups, sayhe Klein 4-group, that proposition is true. Andn other groups, say the cyclic group of order 4,hat proposition is false.- . A. Edgar http://www.math.ohio-state.edu/~edgar/==ubject: Re: GCH vs. Axiom of Choice. The only interpretation that I can put on this statement is that statements arenÕt true or false based upon what axioms you choose. As far as I can tell, that only leaves us with two options: 1. Statements are never true or false. 2. Statements are true or false, without the need to choose any axioms. Of course, itÕs possible that I misunderstood what you said. But, if thatÕs the case, what did your statement mean? The interpretation 2 is perfectly correct! Let us consider, foroncreteness, statements in arithmetic. Their truth or falsity isefined without reference to any set of axioms whatever.==ubject: Re: GCH vs. Axiom of Choice.>> At the risk of sounding like Pilate, hat is truth?t the risk of sounding like Tarski, t is rainingis true if andnly if it is raining.>>I always thought>> that athematical Truthwas only definable with respect to a given>> set of axioms.> On the contrary, there is no such definition. Mathematical truth, as>defined in mathematics, is not relative to any set of axioms.I still want to know hat is truth?If itÕs not in relation to a setof axioms, how is it determined? What does it mean to be true? What isthe decision procedure for mathematical truth?ast things first: why should there *be* a(n infallible) ecisionrocedure for mathematical truth What, to your mind, is he ecision procedure forsome other exemplary kind of ruthf an example exists, what is the obstacle to your adapting ito the needs of mathematics? If none exists, why should athematicalruthbear a unique burden?ee Rudolph==ubject: Re: GCH vs. Axiom of Choice.>ItÕs certainly not *my* position. I think the honest>way of describing the position would be to say you think>the powerset axiom is really false--that there isnÕt any completed>whole that contains, say, all the reals. I guess thatÕs not>directly inconsistent with thinking that you *can* collect>(say) all the countable ordinals into a completed whole. I think thatÕs it. As a physicist I donÕt like saying that a set exists that contains all the reals, because itÕs not strongly falsifiable in the way that IÕd want it to be,nly you can speak to whether itÕs tronglyfalsifiable inhe way you want. But it *is* falsifiable. Specifically, itould prove an inconsistency, and thereÕs your falsification. not even well defined enough.tÕs completely well defined. Given your putative set of alleals, is there a real thatÕs not in it, or not? If there is,hen thatÕs not the set of all reals. If there are any questions about what is or isnÕt a set, then itÕs unclear to have all the subsets, because we havenÕt described what is or isnÕt a set.here are no such questions about subsets of an existing set. TheyÕrell sets.==ubject: Re: GCH vs. Axiom of Choice. <3228t8F3eejn4U1@individual.net> <32a1uaF3jdl14U1@individual.net> posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU > not even well defined > enough. ItÕs completely well defined. Given your putative set of all reals, is there a real thatÕs not in it, or not? If there is, then thatÕs not the set of all reals.ow do I distinguish the set of all reals from a set of arbitraryeals? I can verify that the countably many definable reals are theirr not, but as long as it has all those and at least one more, whatlse can I do? Nothing else can be exhibited to be in there or not inhere. > If there are any questions about what is or isnÕt a set, then > itÕs unclear to have all the subsets, because we havenÕt describedhat > is or isnÕt a set. There are no such questions about subsets of an existing set.heyÕre all sets.ays who? Why have an axiom schema of separation then if all thehings asserted to exist by the axiom schema were already known toxist (i.e. be sets)?==ubject: Re: GCH vs. Axiom of Choice. How do I distinguish the set of all reals from a set of arbitrary reals? What do you mean istinguish What action or proof or other kindf procedure are you referring to, and how is the set given to you?==ubject: Re: GCH vs. Axiom of Choice. > It > quantifies over sentences, something I donÕt know how to do within a > formal sentence. Google odel numberingnfinite sets are equivalent.oss F.==ubject: Counting set coversello all. I was wondering if there is some relatively simple way to ount the number of set covers of a set of size n that satisfy the ollowing properties:. No set in the cover is a subset of another set in the cover. For all sets A(i) in the cover: A(i) (union of all A(k) (k != i)) = {}he second property could also be formulated as: There is no set in the over that contains an element not contained in another set in the cover.here are still a huge amount of such covers, but I would be interested n comparing the amount of such covers to the amount of all possible set overs.- aniel Sjblom==ubject: Re: How to visualize limits in category theory? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGDiPt30069;In addition, may I ask if anyone knows any nice way ofdescribing/explaning colimits?quote]urthermore, notice that this is essentially the subject matter oßgebraic geometry! that is, the concept of imit of a diagramisore or less equivalent to the concept of lgebraic variety carvedut by a system of equations the nodes of the diagram correspond toariables, while the arrows correspond to equations (or onstraintss goguen puts it) of the form =f(x) where the variable xorresponds to the node at the tail and the variable y to the ode at the head of the arrow.unquote]or colimits, the nodes correspond to generators, while the arrows orrespond to relations of the form = f(x) In more technical anguage, a colimit is reducible to coproducts [the generator part] nd coequalizers [the relation part]. odd Trimble==ubject: Re: How to visualize limits in category theory? posting-account=tdYrvA0AAACTm02P3kgbrEwRzJI8b08Shem out and was going to ask for verification if my thoughts wereorrect...but they seem to be (being exactly what you said).I am finding understanding these structures immensely satisfying)==ubject: Re: Prime numbers problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGDiQR30080;This is easily proved as a consequence of BonseÕs Inequality:If p_r denotes the rÕth prime, then(p_{r+1})^2 < p_1 * p_2 * ... * p_rfor r >= 4....]y the way, I noticed that BonseÕs inequality was absent from ric WeissteinÕs Mathworld, so would you mind sending him email directly to mathworld@wolfram.com ] a contribution about it ? .Astanoff==ubject: Difference Sets in the Affine Group of GF(2^m) ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGDiQA30103;e denote by GF(2^m) the Galois field of 2^m elements (i.e. the extention of GF(2) = {0,1} of degree m).et G be the non Abelian affine group of GF(2^m)(i.e. the group of the nonsingular affine transformations from GF(2^m) to GF(2^m)). It is a non Abelian group. want to know if there exist difference sets in this group and in particular the so-called Hadamard difference sets.aurent==ubject: Re: State-of-the-Art in Physics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGDiQn30108;>> THE STATE-OF-THE-ART IN PHYSICS> There is similarity and distinction between mathematics and physics.>> They are similar in the sense that they both study the structure,>> properties and behavior of their respective subject matter. The>> difference lies in their subject matter: that of mathematics is the>> representation of thought, making it a language, and that of physics>> is nature, making it a science. Both suffer from similar defects:>> mathematical spaces and their concepts are ill-defined and nature and>> physical concepts are also ill-defined. ThatÕs one of the reasons he Smart Model Theory was developed. Physicsis full of ill-defined things like, electron orbitals, and electron change oforbital to show various lines on the line spectra, and uncertainty etc... . The Smart Model has accurately predicted the structures now observed fromreal atomic images, like for example oxygen. The Smart Model predicted oxygenwas a ring. QM said it was a sphere. Now they observe oxygen rings on, PdO. Itlooks like this from real atomic images showing oxygen attached to Pd like thison the outer layers of Pd, ooooooooooooooooooooo oooooo oooooooooooooo oooooooooooo ooooooo ooooooooo oooooooooooooo oooooooooooooooooooooooooooooooooooSmartÕs Alt. Physics News GroupS. Enterprize (Science Journal)asic constituent of matter?. E. Escultura==ubject: Re: State-of-the-Art in Physics> THE STATE-OF-THE-ART IN PHYSICS> There is similarity and distinction between mathematics and physics.> They are similar in the sense that they both study the structure,> properties and behavior of their respective subject matter. The> difference lies in their subject matter: that of mathematics is the> representation of thought, making it a language, and that of physics> is nature, making it a science. Both suffer from similar defects:> mathematical spaces and their concepts are ill-defined and nature and> physical concepts are also ill-defined.> ThatÕs one of the reasons he Smart Model Theory was developed.Physics>is full of ill-defined things like, electron orbitals, and electron change>orbital to show various lines on the line spectra, and uncertainty etc... .> The Smart Model has accurately predicted the structures now observed from>real atomic images, like for example oxygen. The Smart Model predictedoxygen>was a ring. QM said it was a sphere. Now they observe oxygen rings on, PdO.>looks like this from real atomic images showing oxygen attached to Pd likethis>on the outer layers of Pd,> ooooooooooooooooooooo> oooooo oooooooooooooo> oooooooooooo> ooooooo ooooooooo> oooooooooooooo > ooooooooooooooooo>oooooooooooooooooo>SmartÕs Alt. Physics News Group>http://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1>S. Enterprize (Science Journal)>http://smart1234.s-enterprize.com/basic constituent of matter?E. E. Esculturaields, in an intelligent logic way to keep the proton perpetually moving forn upper mean life of about 10^30 years.ith Helix Spiral Spinning Fields, CW spin ( ) sub-proton ( ) ((((( ))))) (((((((((( )))))))))) Neutral Zone ((((( ))))) made of 3 quarks ( ) ( ) sub-anti-proton CCW spin This is the internal view. The external view is another Helix Spiralrequency. When additional energy is absorbed, a photon is released within theeutral zone. The Neutral Zone prevents the matter and anti-matter fromolliding. The quarks oscillate producing a constant line spectra. Thisonstant resonance also keeps the mass constant, which is just a highub-atomic pressure zone in the Neutral Zone. You may ask what keeps the proton spinning. The EM fields produced withinigh pressure neutral zone sets up a sub-atomic resistance. The quarks sets up sub-atomic capacitance, and the main internal Helix Spiral Field sets up aub-atomic inductance. So what you have is basically a RCL sub-atomic electricscillator circuit. The basic sub-atomic voltage is the potential differenceetween the matter and anti-matter side which is for a proton at rest about 938eV. I made calculations comparing a mitochondria proto-motive potential withhis proton set up and I end up with the same basic current found in aitochondria, which is the power source for a cell and the RNA and DNA. The oscillator field produces the same line spectra for hydrogen. The linepectra is really the convergence of three oscillator interactions at theeutral Zone core. They produce certain lines when a cfft ( complex fastourier tranform) is taken for these oscillators. Using these oscillators fft,he total line spectra can be produced for hydrogen. And the same goes for helium and all the other elements. Except there is aeriodic Table of Elements. The Smart Model is really more accurate than the QM Model or the Standardodel and is in agreement at all levels of analysis, including chemistry, math,hysics, classicle physics, stellar, galactic, and universal levels. The Smart Model also has a unified universal field theory that unifies allields in the universe, not just the atomic level that Einstein was working on.martÕs Alt. Physics News Groupttp://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1. Enterprize (Science Journal)ttp://smart1234.s-enterprize.com/==ubject: [no subject] by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGDiRf30122;UANTUM ALGEBRAODAY, QUANTUM physicists dont know the structure of the electron. Computation fails. It took qualitative mathematics to know it. Recall that the electron is a left primum that induces a vortex ßux of superstrings to form a magnetic field of left polarity. A right primum has right polarity. The poles of a primum are extremities of its cylindrical eye, a region of low pressure. Therefore, they suck, de-agitate and accumulate superstrings inside it. Their agitation and conversion to kinetic energy prima or photons provide the awesome power of nuclear explosion. I apply qualitative mathematics to primal interaction. By ßux compatibility, two prima of opposite toroidal ßux spins attract and two prima of the same ßux spin repel at their equators. However, prima of the same torodal ßux spin, although repulsive at their equators, are joined by a primum of opposite toroidal ßux spin called connector. Quantum algebra determines the charge of coupled or composite prima (algebraic sum of component charges). Being neutral, the neutrino cannot be simple primum; it is equatorially coupled pair of prima of opposite but numerically equal charges, say, +q and -q. Therefore, the sum is 0 and there is no coherent ßux around it. The proton is a pair of right quarks joined equatorially by a left primum as connector. Their centers are collinear, by energy conservation, since the two right prima are repulsive. Therefore, the charge of the proton is +2/3 -1/3 +2/3 = +1. It has counterclockwise net ßux around it. The neutron is a pair of right quarks joined by an electron and a left quark between them. Being of the same charge, these connectors are far as apart as possible. The coherent ßuxes at the center between the electron and left quark and the two right quarks sucks a neutrino (being neutral), by ßux-low-pressure complementarity. Therefore, the neutrons charge is: 2/3 -1/3 +2/3 -1 +0 = 0. The photon is simple primum that has broken away from its loop when scooped up by a basic cosmic wave (propagated by vibration of atomic nuclei). Dark viscosity ßattens the photons cycles; it becomes sinusoidal. As long as its forward toroidal ßux speed equals the speed of its carrier basic cosmic wave, the photon is stable. Otherwise, it breaks and its toroidal ßux remains non-agitated superstring (dark). Thus, the photon has no rest mass. These prima have known masses: 1.674754 x 1/1024 grams = 1.0087 atomic mass units (amu); Proton: 1.672648 x 1/1024 grams = 1.0073 amu; electron: 0.0009109535 x 1/1024 grams = 0.0005486 amu. Note that the proton is 1836 times as massive as the electron. When the electron gets suitably close to its anti-matter they collide and convert and project each other into two photons in opposite directions (I shall discuss anti-matter interaction later). Thus, photon from electron has the mass of the electron because it carries its toroidal ßux (true of photons from other prima). Physicists previously assumed the neutrino had 0 mass. Detecting it is quite difficult. They place a pool of water about two miles underground. When neutrino strikes the pool, bubble forms and that requires energy. Therefore, the neutrino has mass but that would be difficult to find without a theory. However, the few natural laws I have discovered suffice to calculate it. The same laws allowed me to determine the neutrons component prima and how they are joined together. It consists of a proton, an electron and a neutrino. Since the masses of the neutron, proton and electron are known I calculate the mass of the neutrino: 0.0013515 x 1/1024 grams = 0.0008514 amu obtained by subtracting from the neutrons mass the sum of the masses of the electron and proton. Comparing it with the mass of the electron, it turns out to be heavier, that is, 1.55 times the electrons mass. One can see here the power of a theory. Earlier, we saw how the search for the basic constituent shifted away from the atom since it is not there. The shift had the benefit of insights from the laws of nature. Huge resources were spent in that futile search. I eventually discovered it in dark matter at the cost of a few minutes of reßection under the guidance of the laws of nature. The prima were discovered at no material cost; only the right scientific orientation, a traine! mind and analysis of data provided by the Hubble . E. Esculturaniversity of the Philippines==ubject: INTEGRAL by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGDiRk30161;ello!ould you be so kinde to help me with my problem :roblem is INTEGRAL: Finde the area under vault of curve that is y^2= x^4(4-x) . The resault is fraction.ristina!y e-mail is : bezinakristina@yahoo.com==ubject: Re: INTEGRAL Hello! Would you be so kinde to help me with my problem : Problem is INTEGRAL: Finde the area under vault of curve that is y^2= x^4(4-x) . The resault is fraction. Kristina! My e-mail is : bezinakristina@yahoo.coman you sketch the curve?an you write the problem as an integral?an you evaluate the integral?s your answer 16/105 ?ive us more information!==ubject: Re: INTEGRAL>Hello!>Would you be so kinde to help me with my problem :>Problem is INTEGRAL: Finde the area under vault of curve that is y^2=>x^4(4-x) . The resault is fraction.>Kristina!>My e-mail is : bezinakristina@yahoo.com Can you sketch the curve? Can you write the problem as an integral? Can you evaluate the integral? Is your answer 16/105 ?if so, your answer is too small by a factor of 256 or12, depending on how nder vault of curveis interpreted]ick==ubject: Re: INTEGRAL >>Hello! >Would you be so kinde to help me with my problem : >Problem is INTEGRAL: Finde the area under vault of curve that is y^2= >>x^4(4-x) . The resault is fraction. >Kristina! >>My e-mail is : bezinakristina@yahoo.com >> Can you sketch the curve? > Can you write the problem as an integral? > Can you evaluate the integral? > Is your answer 16/105 ? [if so, your answer is too small by a factor of 256 or 512, depending on how nder vault of curveis interpreted]orry, I mistakenly used y^2=x^4(1-x) for that.==ubject: Re: CantorÕs diagonal proof wrong? <41ad4f31$14$fuzhry+tra$mr2ice@news.patriot.net> <41b2432e$17$fuzhry+tra$mr2ice@news.patriot.net> posting-account=htRwYA0AAACUC1yg4djqvdjZ_SB9JXGq >If you accept that the whole set of natural numbers is created byhe >Peano axioms, Axioms donÕt create, they describe.hat is a matter of personal opinion. For me, PeanoÕs axioms create theet of natural numbers. >but if you simultaneously deny that induction is capable of reaching >all natural numbers (all lines of the list), issue is not any imaginary claims that there is an unreachableatural number; the issue is CantorÕs antidiagonal argument, which neither makes nor requires any such claim.y induction I have shown that the distance of any natural number fromhe origin 0 is n - 0 = n. Hence finite. Any natural number has ainite distance from the origin means: The set IN of all naturalumbers ist not infinite. (This does not imply that IN was finite. INisnot at all!)==ubject: Re: CantorÕs diagonal proof wrong? That is a matter of personal opinion. For me, PeanoÕs axioms create the set of natural numbers. What, if anything, does this statement mean?==ubject: Re: CantorÕs diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> posting-account=htRwYA0AAACUC1yg4djqvdjZ_SB9JXGq >If you think, that then his list would be impossible, we do agree. No, we donÕt agree; he proved that no such list is possible. Or, to put it another way, he proved that for any list of real numbers there is a real number not on the list.e proved that a complete list of all real numbers IR is impossible.is result is correct, because no complete set IR does exist. But hisroof is wrong.ou can see this best by considering the following Cantor-list:.0.1.11.111..f you start to construct the diagonal number 0.111..., you will seehat it is always contained in the next line, how far you ever proceed.ou can never keep up, except in a nice dream.==ubject: Re: CantorÕs diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs >If you think, that then his list would be impossible, we do agree. > No, we donÕt agree; he proved that no such list is possible. Or, to > put it another way, he proved that for any list of real numbershere > is a real number not on the list. He proved that a complete list of all real numbers IR is impossible. His result is correct, because no complete set IR does exist. But his proof is wrong. You can see this best by considering the following Cantor-list: 0.0 0.1 0.11 0.111 ... If you start to construct the diagonal number 0.111..., you will see that it is always contained in the next line,eally?s 0.111... a number? Let us call it x.o you agree with the following statement: Either x is inhe list, or x is not in the list. Both can not be true.hat about the following statement: If x is in the list,t occurs at position n where n is a natural number. Randy==ubject: Re: CantorÕs diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> posting-account=htRwYA0AAACUC1yg4djqvdjZ_SB9JXGqM: >Every natural number is an ordinal number and is simultaneously the >cardinal number of the sequence of all its successors including >itself: 1,2,3,...,n. The sequence up to a natural number n can never >have the cardinal number aleph_0. CantorÕs proofs do not involve finite sequences. Omega is not a natural number.ence the set of natural numbers cannot be actually infinite. It isotentially infinite, i.e., there is no threshold but there is nonfinite number. And the same holds for the cardinal number of eachequence 1,2,3,...,n.ou see it best, if you imagine Card{1,2,3,...,n} as a function f(n) of. The sloop is 1 and that will never change. FOR ALL FINITE NUMBERS nand there are no others)==ubject: Re: CantorÕs diagonal proof wrong? You see it best, if you imagine Card{1,2,3,...,n} as a function f(n) of n. The sloop is 1 and that will never change. FOR ALL FINITE NUMBERS n (and there are no others) The sloop is in fact Sloop John B, and we hope it will neverhange. Hence the well-known exhortation: So hoist up the John BÕs sails, see how the main sail sets, Call for the captain ashore, and let me go home. Let me go home, I want to go home, Well I feel so break up, I want to go home.==ubject: Re: CantorÕs diagonal proof wrong?> You see it best, if you imagine Card{1,2,3,...,n} as a function f(n) of> n. The sloop is 1 and that will never change. FOR ALL FINITE NUMBERS n> (and there are no others) The sloop is in fact Sloop John B, and we hope it will neverchange. Hence the well-known exhortation: So hoist up the John BÕs sails, see how the main sail sets, Call for the captain ashore, and let me go home. Let me go home, I want to go home, Well I feel so break up, I want to go home.uh, it was a typo. He meant lop==ubject: Force to be calculatedt is simple question apparently. But it seems to me not so easy toor out, high maths are probably needed. Here is the song: One ball of Kg, held 1 meter above the surface of perfect balance. We let theall fall without any initial speed. Question: what is the value ofhe force that will be indicated by needle of the balance in the exactoment of the choc, which is suposed to be 100% elastic (no loss ofnergy)?==ubject: Re: Force to be calculatedThis question really should be in sci.physics) Question: what is the value of the force that will be indicated by needle of the balance in the exact moment of the choct the exact time of first contact, the needle will either readnfinite force or zero, depending upon the model.f the impact is a mathematically idealised model that takes zeroime, then the function of force with time is a elta functionhich has an nfinitevalue at zero and zero value everywhere else.f the impact is modelled in a more physically reasonable way, thenhe force starts at zero, increases to a maximum, and then drops backo zero. In this case, the important feature for your problem is thatt starts at zero. Tim==ubject: Re: Force to be calculatedIt is simple question apparently. But it seems to me not so easy towor out, high maths are probably needed. Here is the song: One ball of1 Kg, held 1 meter above the surface of perfect balance. We let theball fall without any initial speed. Question: what is the value ofthe force that will be indicated by needle of the balance in the exactmoment of the choc, which is suposed to be 100% elastic (no loss ofenergy)?his is a physics question; it really belongs in sci.physics. But thereÕsath involved, so...f the ball and scale are completely rigid, the force at the moment ofontact will be of infinite force for an infinitessimal amount of time, ast it were a Dirac delta function. egardless of how rigid it is, the area under the force curve will be enoughmpulse to accellerate the ball from its downward velocity to an equalpward velocity -- 8.8 kg-m/s.-Keith Lewis klewis {at} mitre.orghe above may not (yet) represent the opinions of my employer.==ubject: Re: Force to be calculated posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs >It is simple question apparently. But it seems to me not so easy to >wor out, high maths are probably needed. Here is the song: One ballf >1 Kg, held 1 meter above the surface of perfect balance. We let the >ball fall without any initial speed. Question: what is the value of >the force that will be indicated by needle of the balance in thexact >moment of the choc, which is suposed to be 100% elastic (no loss of >energy)? This is a physics question; it really belongs in sci.physics. ButhereÕs math involved, so... If the ball and scale are completely rigid, the force at the momentf contact will be of infinite force for an infinitessimal amount ofime, as it it were a Dirac delta function. Regardless of how rigid it is, the area under the force curve will benough impulse to accellerate the ball from its downward velocity to anqual upward velocity -- 8.8 kg-m/s.es. As you say, conservation of energy dictates the areander the force curve, but not its shape or its maximum value.o say what the needle reads, you need a more detailed modelhat can tell you how the transient event translates into reading. - Randy==ubject: Re: Force to be calculated posting-account=CfSJ5AwAAAD1yt3VP50q913IBHikxMCd But it seems to me not so easy to work outut it is easy .. high maths are probably needed.o, you need to realize that spring deformation x is to be found from auadriatic equation given below, from combining potential energies ofpring in the balance and mgh.. 1 Kg, held 1 meter(h) above the surface of perfect balance.We let the ball fall without any initial speed. Question: what is the value of the force that will be indicated by needle of the balance in the exact moment of the choc, which is suposed to be 100% elastic (no loss of energy)?INT: Equate maximum potential energy to springÕs potential energy byeformation through x. mg(h+x)=kx^2/2, F= k*x. Note that even if h=0,he balance registers double weight at impact time due to sudden loadntroduction.==ubject: Re: Force to be calculatedverybody who has done a bungee jump will correctly guess that one will ndergo a 2mg apparent force (i.e. twice oneÕs weight) at the deepest oint of the jump.he balance will still register zero weight at the moment of impact; == ot until the falling body has reached == its deepest point the balance ill register double weight.TW, I never did a bungee jump. In my boyhood and studentÕs times I erformed this kind of experiments as well as one can do with common ousehold and university physics stuff.appy experimenting: Johan E. Mebius>But it seems to me not so easy to work out> but it is easy ..>high maths are probably needed.> No, you need to realize that spring deformation x is to be found from aquadriatic equation given below, from combining potential energies ofspring in the balance and mgh.>1 Kg, held 1 meter(h) above the surface of perfect balance.We let>the ball fall without any initial speed. Question: what is the>value of the force that will be indicated by needle of the balance>in the exact moment of the choc, which is suposed to be 100%>elastic (no loss of energy)?> HINT: Equate maximum potential energy to springÕs potential energy bydeformation through x. mg(h+x)=kx^2/2, F= k*x. Note that even if h=0,the balance registers double weight at impact time due to sudden loadintroduction.==ubject: Online equation system solver need it to solve systems of 3 or more quadratic equations.==ubject: Re: Deep Thoughts # 17: Liar Paradox is a Formal Metamathematical Theorem <4ujtr05euj9l4q0fcpb8q2sbp141r399op@4ax.com> posting-account=9wVPIwwAAAAonf5Dj39AQaTL2sJYvErF >> You _really_ need to work on the irony detector. Truth >> and provability are _different_ things - when you define >> one to be the other you exhibit amazing ignorance of the >> most basic issues. >> ************************ > David C. Ullrich >The definitions of Turing Machines and Lambda Calculus are very >different, too, arenÕt they? For heavenÕs sake, at some point you should just accept that youÕve made a fool of yourself and drop it. I mean you have plenty of experience with this phenomenon...Only at parties.) You are the one who claimed that what Smullyan and are doing is ignorant. I only agreed with Smullyan. (His 4 textsere the source of many of the theorems that my formalism represents.)our statement above concerning the gnoranceof defining truthnd provability to coincide makes sense only in the context of somearticular system - the system in which they are being defined. Thats exactly what Smullyan is doing. (It sounds to me like you arerying to retract the statement by substituting an alternatexplanation of what it means - but havenÕt come up with a meaningfullternative.) Ok. To answer your question: Yes, those two definitions are very different. I canÕt imagine what your point is.wo points:. Having different definitions doesnÕt mean that two things areifferent.. Truth and provability need not be defined with different systems.hey are defined with different systems only because those who describehem as such just havenÕt developed a single system for both. Theotion of recursive functions is the one system developed to define theunctionality of both Turing Machines and Lambda Calculus. Likewise,ruth and provability (two sets of sentences) can be formalized within single system.oth your premise (2) and reasoning (1) are wrong. (I could fault youess for mistake 2 because that, of course, is current conventionalisdom.) (i) If someone asked for the definition of a Turing Machine and someone else replied with the definition of the Lambda Calculus that person would indeed be exhibiting blithering confusion.es! And if they say that any person who is wrong about question Aust have property B, in all fairness, does that mean that if they arerong about A then they themself possess property B? (ii) On the other hand, itÕs true that the set of functions computable by TMÕs is the same as the set of functions computable by the LC. _Guessing_ what your point might be here, I have to point out that truth and provability are _not_ equivalent.hat does it mean to say that truth and provability are or are notquivalent? It depends on the system in which they are being defined.t is meaningless to talk about truth and provability outside of somearticular system. They coincide in some systems and they donÕtoincide in others. (This sounds like more of the ill-formed alternatexplanation of your original statement.) (Validity and provability are equivalent. And truth _in that one particular model_ is equivalent to provability _in that one particular formal system_. A very curious formal system, btw, since thereÕs no procedure to recognize whether a proof is valid.)ow could something be a proof but not be a valid proof? You seem toe trying to say either:. We cannot recognize if a given sequence of formulas is a proof.. We cannot recognize if a given proof proves a given sentence.. We cannot recognize if a given sequence of formulas is a proof of aiven sentence.ut none of these is definitely true anyway. Given that the set ofheorems is not r.e. (system N), any of these can be false if the otherwo are true.n any case, it is not curious to Smullyan (or me.) Everyon-axiomatizable theory has a non-recursive proof predicate, assuminghat the sets of (the Godel numbers of) formulas and of proofs are r.e.s PA ery curiousbecause thereÕs no procedure to decidehether a sentence is a theorem?y your reasoning, the user of system N, which is non-axiomatizable andomplete, could equally well claim that PA is very curious because itss axiomatizable but not complete.he true lesson, of course, is that there is a trade-off. (The samerade-off that I described in my 2 MetaMathematical Theorems,ctually.) ItÕs not that the true and provable sentences canÕt behe same. It is the fact that there are consequences - both good andad - if they are the same. After all, determining trade-offs inormal systems is central to MetaMathematics. f a system hasroperty A then it must have property B. Nothing curious abouthat.-BWe call a formula F(v1,. . .,vn) correct if for every n-tuplea1,...,an), the sentence F(a1,. . .,an) is true. We let N be theirst-order system whose axioms are all the correct formulas (thisncludes all logically valid formulas.) Thus, the provable formulas of are nothing more than the axioms of N. ecursion Theory for Metamathematics - Raymond M. Smullyan, 1993In applications to semantical systems, T will be the truth set, andn syntactical systems, T will be the set of theorems.- Theory oformal Systems, Chapter III. Incompleteness and Undecidability ************************ David C. Ullrich==ubject: Category Theory posting-account=Hf4h4w0AAAADyBPYPG-OsVhCAUnaxJy6 set exists and its cause is the categorified set. Meaning the nextbstracted set is always in relation to the set of interest.hat is debated is the exact relation of set to its abstracted form.nd to call the symbol in predicate the next abstraction withoutebating, is a common means of learning incorrectly, category theory. foundational relation is postulated to exist, such that all relationsay be stated.nd in the realm of science, objective relations are used to solve thisong standing mystery. Only relations testable in science are to beiven to mathematics.N predicate, set A -> a.nd here the relation denoted by, ->, is the correct one. good teacher will also allow the relations actual cognition, exceptt is as a trained knowledge, without hardly any symbolic expression toupport. Meaning, a good teacher is critical to the good user of thecience of category theory.n order to give words to the symbol, ->, here is its definition.->, a set exists independently of its contents, and the cause toontents is likewise independent of existence. aking the complex relationship of existence to value.nd the meaning of dependency and independency identical. relation of true objective causality exists in nature such that theelation of existence to value is found always independent!!!!! foundational nonlinear relation is called the, ->.ouglas Eagleson- Gaitherbsurg, MD USA==ubject: Help Neededi everyone,Õm currently in the process of forming a web-based security company,hat will offer a wide range of services to clients, and IÕmearching for people that would be interested into going intoartnership with me.m looking for people who specialize, or are interested in any of theollowing areas:etworkingystem vulnerability/ penetration testing / C++ / C# programmingava Programmingerl Programmingython Programminglectronics/ Hardware designryptographyelematicsf youre interest, please get in touch with me at ebmaster@afis.tkarry-----------------------* www.GroupSrv.com-----------------------*==ubject: Solving g( x+3y ,3x + y) = x*y , g :R * R->R by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGHHIG19584;ow do you take this?.an you generalize to : g(m(x,y),n(x,y))=h(x,y) g,m,n,h R*R->R functions, only g(x,y) unknown ? Explain the necessary conditions...lain.==ubject: Re: Solving g( x+3y ,3x + y) = x*y , g :R * R->R How do you take this?. Can you generalize to : g(m(x,y),n(x,y))=h(x,y) g,m,n,h R*R->R functions, only g(x,y) unknown ? Explain the necessary conditions... Alain.or the problem given in the subject line, let u = x+3y and v = 3x +. Then you can write x and y as functions of u and v (IÕll let you dohis) and g(u, v) = x * y._______________________________ric J. Wingler (wingler@math.ysu.edu)ept. of Mathematics and Statisticsoungstown State Universityne University Plazaoungstown, OH 44555-000130-941-1817==ubject: Re: Binary field; trace; New(?) Conjecture for fast determination of the trace of an element of a binary field. I would have put your question as follows: Suppose f(t) is an irreducible polynomial of degree d over k = F_p. Let K = k[t]/(f(t)) = F_q where q = p^d. Then the characteristic equation of c = t mod f(t) is f(t) = 0. Now you are asking for the trace of c^j as an element of the extension K/k. If the eigenvalues of c are lambda_1,...,lambda_d then the coefficients of f(t) = t^d - h_1 t^{d-1} + h_2 t^{d-2} + ... are the basic symmetric functions h_1 = sum lambda_i, h_2 = sum_{i==ubject: weighted arithmetic and geometric meansellouppose x_1,....x_n and w_1,...w_n are postive numbers and define = (Sum(i=1,n)(w_i*x_i))/(Sum(i=1,n)(w_i)) and = (Product(i=1,n)(x_i)^(w_i))^(1/Sum(i=1,n)(w_i))) want to prove that there holds an inequalty similar to that relatedo the arithmetic and geometric means, that is, a>= g, with equalityf, and only if, x_1 = ....x_n.ince I already now that the arithmetic/geometric means inequalty isrue, I tried to do as follows.irst, if all the w_iÕs are positive integers, then we see readily seehat a is the arithmetic mean of numbers x_1,...x_n if if we we takeach x_i w_i times. Since a similar conclusion is true of g, we applyhe a/g means inequality to conclude that, if all the w_iÕs arenteger then the propostion is true for a and g.f all the w_iÕs are rational, then, representing each w_i as theatio between 2 positive integers and doing some elementary algebraicransformations, we see a =aand g =gÕ, where aand gare weightedeans similar to a and g corresponding, now, to integer weights.herefore, we are sent back to the integer case, which shows theroposition still holds if all the w_iÕs are positive rationals.f the w_iÕs are real positive integers, then, for a fixed butrbitray (x_1,...x_n) , the functions (w_1,...w_n) -> a(w_1,...w_n)nd (w_1,...w_n) -> g(w_1,...w_n,)defined on the subset of R^nomposed of their points with positive coordinates, are continuous.ince a>=g in the subset of R^n composed of their points withositive and rational coordinates and since this latter subset isense in the former, it follows that a(w_1,...w_n) >= g(w_1,...w_n)or every positive w_1,...w_n. Since this holds for arbitray positive_1,...x_n, we have proved that, in fact, a>=g. But we are not done,ecause these arguments do not imply that equality occurs if and onlyf x_1...= x_n.ollowing my reasoning, can anyone suggest how I can complete theroof? Or, maybe, itÕs better to start at the very beginning, withoutupposing the a/g means inequalty is known.manda==ubject: Re: weighted arithmetic and geometric means Hello Suppose x_1,....x_n and w_1,...w_n are postive numbers and define a = (Sum(i=1,n)(w_i*x_i))/(Sum(i=1,n)(w_i)) and g = (Product(i=1,n)(x_i)^(w_i))^(1/Sum(i=1,n)(w_i))) I want to prove that there holds an inequalty similar to that related to the arithmetic and geometric means, that is, a>= g, with equality if, and only if, x_1 = ....x_n. Since I already now that the arithmetic/geometric means inequalty is true, I tried to do as follows. First, if all the w_iÕs are positive integers, then we see readily see that a is the arithmetic mean of numbers x_1,...x_n if if we we take each x_i w_i times. Since a similar conclusion is true of g, we apply the a/g means inequality to conclude that, if all the w_iÕs are integer then the propostion is true for a and g. If all the w_iÕs are rational, then, representing each w_i as the ratio between 2 positive integers and doing some elementary algebraic transformations, we see a =aand g =gÕ, where aÕ and gare weighted means similar to a and g corresponding, now, to integer weights. Therefore, we are sent back to the integer case, which shows the proposition still holds if all the w_iÕs are positive rationals. If the w_iÕs are real positive integers, then, for a fixed but arbitray (x_1,...x_n) , the functions (w_1,...w_n) -> a(w_1,...w_n) and (w_1,...w_n) -> g(w_1,...w_n,)defined on the subset of R^n composed of their points with positive coordinates, are continuous. Since a>=g in the subset of R^n composed of their points with positive and rational coordinates and since this latter subset is dense in the former, it follows that a(w_1,...w_n) >= g(w_1,...w_n) for every positive w_1,...w_n. Since this holds for arbitray positive x_1,...x_n, we have proved that, in fact, a>=g. But we are not done, because these arguments do not imply that equality occurs if and only if x_1...= x_n. Following my reasoning, can anyone suggest how I can complete the proof? Or, maybe, itÕs better to start at the very beginning, without supposing the a/g means inequalty is known. Amandan induction proof (I discovered it for myself, but I could hardly be theirst) - OK, it uses Calculus:enote W(n) = w(1) + ... + w(n) A(n) = (w(1)*x(1) + ... + w(n)*x(n)) / W(n) G(n) = (x(1)^w(1) * ... * x(n)^(w(n))^(1/W(n))nd similarly for W(n+1), A(n+1), G(n+1).hen F(t), defined as(G(n+1)/A(n+1))^W(n+1) where x(n+1) is replaced with t (t>0),s a differentiable function of t. Use Calculus to establish thatF(t) attains its strict maximum when t=A(n), the maximum being (G(n)/A(n))^W(n).o, since (G(1)/(A(1))^W(1) = 1, you have an induction proof,ith necessary and sufficient conditions for equality.==ubject: review topic suggestions (ODEÕs)?Õm signed up for a class (College of Engineering) that deals with nalytical and numerical methods to solve mechanical engineering roblems. my ODE class was a long time ago now. Any advice on what articulars to review? I have an ODE textbook, but donÕt want to tackle t randomly in the next two weeks. I recall fairly well how to set up VP problems, use separation of variables, and use the method where you ake an integral(e ^tx) and apply it to both sides (donÕt recall what hat methodÕs called), and I recall how to (generally ) deal with on-homogeneous cases, though IÕd have to review some particular olution cases. Any advice is welcome.h, and IÕm not even exactly sure what nalytical and numerical ethodsmeans specifically. The catalog specifies diff eqÕs as a rereq, thatÕs about it. wallace==ubject: Re: review topic suggestions (ODEÕs)? IÕm signed up for a class (College of Engineering) that deals with analytical and numerical methods to solve mechanical engineering problems. my ODE class was a long time ago now. Any advice on what particulars to review? You need to review your reasons for undertaking this boring andost likely fruitless training.==ubject: Re: review topic suggestions (ODEÕs)?>IÕm signed up for a class (College of Engineering) that deals with >analytical and numerical methods to solve mechanical engineering >problems. my ODE class was a long time ago now. Any advice on what >particulars to review? You need to review your reasons for undertaking this boring and most likely fruitless training.tÕs a required class for a mechanical engineering degree. I have worked in the fieldfor some time, my company went under last year, I found hat the piece of paper is what will get me in the door, rather than my ngineering experience. At least, with a company that will pay me better than a TAÕs salary.lthough...I have found some of my classes to be fascinating and have earned a lot in areas outside my expertise, (ßuid mechanics and ynamics for example), there are also a lot of things that are totally seless to me. As a designer/product& process analyst, I did not need to pend the last 4 months learning in excruciating detail the solubility imits of metals on the atomic level. If I need an alloy with particular haracteristics, I have 4 websites of suppliers I can order it from.y professor, however, wasnÕt interested in my explanation, so I took he class, memorized a s**tload of stuff, took the final, and have romptly forgotten 90% of it.he structural analysis, and FEA stuff, though- thatÕs very useful to e, and I like it a lot. So thereÕs good and bad about being back in chool, 15 years after I thought I was done with it.k wallace==ubject: Question about Surreal Numbers understand that the surreal numbers are a proper class which some peopleind a bit problematic and I would like to ask the following question. Ife generate sets of surreal numbers using the usual recursive methodfrom http://en.wikipedia.org/wiki/Surreal_number) S0 = { 0 } S1 = { -1 < 0 < 1 } S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 < -1 < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } S4 = ... nd then transfinitely Sw, S(w+1), etc.ouldnÕt we define (for any ordinal x) F(x) as {s|s is a field (under the usual surreal operations ) and s is a subset of Sx}.his F(x) is the set of fields generated by ordinal xhen M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)}so this is a set of maximum sized fields generated with x).k one question is: is this set a singleton set?hen we can define N(x) as *some* element of M(x). This will beuperßuous if M(x) is a singleton set.o why canÕt we work with sets of the form N(x) for some ordinal?his way we donÕt worry about the surreals being a proper class, and wonÕtost interesting fields be captured if we set x right? Or what is wrongith my thinking?==ubject: Re: Question about Surreal Numbers posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg I understand that the surreal numbers are a proper class which someeople find a bit problematic and I would like to ask the followinguestion. If we generate sets of surreal numbers using the usual recursive method (from http://en.wikipedia.org/wiki/Surreal_number) S0 = { 0 } S1 = { -1 < 0 < 1 } S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 -1 < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } S4 = ... and then transfinitely Sw, S(w+1), etc. CouldnÕt we define (for any ordinal x) F(x) as {s|s is a field (under the usual surreal operations ) and s is subset of Sx}. This F(x) is the set of fields generated by ordinal x Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)} (so this is a set of maximum sized fields generated with x). Ok one question is: is this set a singleton set?tÕs an empty set for x I understand that the surreal numbers are a proper class which some people> find a bit problematic and I would like to ask the following question. If> we generate sets of surreal numbers using the usual recursive method> (from http://en.wikipedia.org/wiki/Surreal_number)> S0 = { 0 }> S1 = { -1 < 0 < 1 }> S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 < -1> < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 }> S4 = ...> and then transfinitely Sw, S(w+1), etc.> CouldnÕt we define (for any ordinal x) F(x) as> {s|s is a field (under the usual surreal operations ) and s is> subset of Sx}.> This F(x) is the set of fields generated by ordinal x> Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)}> (so this is a set of maximum sized fields generated with x).> Ok one question is: is this set a singleton set? ItÕs an empty set for x Then we can define N(x) as *some* element of M(x).> This will be> superßuous if M(x) is a singleton set. Another approach: In GonshorÕs n Introduction to the Theory of Surreal Numbers he shows how to define surreal numbers as well-ordered sequences of pluses and minuses. Then the surreal numbers of length less than an uncountable cardinal form a field (I think).o you know of any resources availale on the web (including journalave to actually get my hands on the book? Because his approach certainlyounds less hairy than mine.> So why canÕt we work with sets of the form N(x) for some ordinal? This> way we donÕt worry about the surreals being a proper class, and wonÕt> most interesting fields be captured if we set x right? Or what is wrong> with my thinking? Nothing wrong with this, but of course you wonÕt get all the surreal numbers this way.ell we could say that even the proper class of all surreals isnÕt all theurreals because....we havenÕt considered hyperordinals....and thisusiness of saying we havenÕt got all the surreals could go on for ever.nyway thank you very much for helping me out.==ubject: Implicit Function Theorem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGIRwJ25933;et (x*,y*) in S be a vector such h(x*,y*) = 0 and the matrix nabla_y h(x*,y*) is nonsingular, the implicit function states that here exist (x,y) in the neigborhood of (x*,y*) such that y = f(x) and h(x,f(x)) = 0. Also,abla f(x) = - nabla_x h(x,f(x)) * [ nabla_y h(x,f(x)) ]^-1. know that nabla_y h(x*,y*) is nonsingular, but why this applies to nabla_y h(x,f(x)) for x in the neigborhood of x* as well?==ubject: Re: Implicit Function Theorem Let (x*,y*) in S be a vector such h(x*,y*) = 0 and the matrix nabla_y h(x*,y*) is nonsingular, the implicit function states that there exist (x,y) in the neigborhood of (x*,y*) such that y = f(x) and h(x,f(x)) = 0.ouÕve badly mangled the IFT, omitting hypotheses, misstating the onclusion, ...Also, nabla f(x) = - nabla_x h(x,f(x)) * [ nabla_y h(x,f(x)) ]^-1. I know that nabla_y h(x*,y*) is nonsingular, but why this applies to nabla_y h(x,f(x)) for x in the neigborhood of x* as well?ell itÕs because of hypotheses youÕve left out in your sloppy statement of he IFT. h is C^1, so because det[dh/dy](x*,y*) is nonzero, det[dh/dy] is onzero in a neighborhood of (x*,y*) by continuity; your conclusion follows.==ubject: Re: More on the State-of-the-Art in Physics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGIRsp25873;1)/.../ calculated that 95% of our universe is dark matter.ou probably meen dark energy? As far as I know 20% is dark matter.2) /.../The discovery of the basic constituent of dark matter, the superstrings.here can you find this? To my knowledge this question is still highly open to debate and has been set to one of the 25 agendas to be set during the coming 25 years by KITP.occis==ubject: Re: More on the State-of-the-Art in Physics>1)/.../ calculated that 95% of our universe is dark matter.You probably meen dark energy? As far as I know 20% is dark matter.o 115% of our universe is dark? ee Rudolph==ubject: Re: More on the State-of-the-Art in Physics>>1)/.../ calculated that 95% of our universe is dark matter.>You probably meen dark energy? As far as I know 20% is dark matter. So 115% of our universe is dark? Lee RudolphtÕs only 115% dark if youÕre a mathematician.f youÕre a physicist itÕs 100% dark.f youÕre an engineer itÕs 99.99% dark +/- 5% dark.- eplace Roman numerals with digits to reply by email==ubject: Re: JSH: Those Ullrich defenders by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGIRse25855;> against me, by never telling the full story.This old stuff James? BORING. More operators and objects please! If youneed inspiration, just get drunk and sing to the walls. It alwaysworked before, didnÕt it?ll hail Operator Ambiguity. Ad Hominem==ubject: plane curve animations posting-account=OIaTGwwAAABntbwd95_ohTn1wTSa5sqoome 80 famous plane curves animations for didactic and estheticurposes are available here:ttp://xahlee.org/SpecialPlaneCurves_dir/ specialPlaneCurves.htmlXahxah@xahlee.orghttp://xahlee.org/ PageTwo_dir/more.html==ubject: Positive functionalsello.et M denote the set of nxn-matrices with complex entries, and let f:M->Ce a linear functional. It is not hard to prove that there is a matrix huch that f(m)=trace(hm) for all m in M.f f(m^*m) is real and positive for all m in M, why must h beelf-adjoint? That is, h^*=h. I donÕt ask for a complete solution, only aint :-)- ichael Knudsen==ubject: nth term of the seriesan anybody find the general term of the following set series:1) 1, 4, 10, 20, 35, 56, 84, ...2) 4, 15, 36, 70, 120, 189, 280, ...3) 6, 20, 45, 84, 140, 216, 315, ...atya==ubject: Re: nth term of the series Can anybody find the general term of the following set series: (1) 1, 4, 10, 20, 35, 56, 84, ... (2) 4, 15, 36, 70, 120, 189, 280, ... (3) 6, 20, 45, 84, 140, 216, 315, ...orm sequences from the difference. For first sequence: 6 10 15 21ow repeat:, 4, 5, 6nd again:,1,1,1equence is cubic.ou can form 4 simultaineous equations, as cubic sequence are all of theorm:n^3 + Bn^2 + Cn + Dnd solve but that is pain.nother approach is to think in algebra.ubic sequences are of the form An^3 + Bn^2 + Cn + Dse this to generate the nth terms for the sequences of differences you havebove:irst difference sequence is:A(n+1)^3+B(n+1)^2+C(n+1)+D] - [An^3+Bn^2+Cn+D] = A(3n^2+3n+1) + B (2n+1) +otice, that weÕve ÔlostD.epeat this process, using A(3n^2+3n+1) + B (2n+1) + C, to get the ntherm for the second difference: (6n+6) + 2Botice weÕve lost Cnd again:hird difference is the sequence: 6AThe above are good for all cubic sequences.)ow, work back:or our sequence, 6A = 1 , so A =1/6ub in A(6n+6) + 2B = 3 (first term in sequence of 2nd differences) = 1/2ub for A and B in A(3n^2+3n+1) + B (2n+1) + C = 3 (first term in sequencef 1st differences) = 1/3ow look at original sequence to get D, which is 0.o: T(n) = (n^3)/6 + (n^2)/2 + n/3heck for n=1, 2, 3 against original sequence.- rian Reayww.g8osn.org.ukww.amateurradiotraining.org.ukP#898== ubject: Re: nth term of the series Can anybody find the general term of the following set series: (1) 1, 4, 10, 20, 35, 56, 84, ... (2) 4, 15, 36, 70, 120, 189, 280, ... (3) 6, 20, 45, 84, 140, 216, 315, ...int: take first differences, second difference etc until you get a onstant.ob Kolker==ubject: Re: Santa Claus is the god of math !!!! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGKBUw03032;>There is NOTHING that Santa cannot prove !!!!>Wish for a proof for Christmas and youÕll get it!Can he solve the Continuum hypothesis?[In the sense that Paul Erdos said that ChristÕs answer to the question an we solve the CH?was aul Cohen gave all youcan know.h. This didnÕt make sense, I think. I read that Erdos said thataybe, maybe the CH can be solved, our brains being the reason weanÕt solve it. (i.e., Some creature might have a brain who can sayf itÕs true or not.) So the question is, does He have such anntelligence?elta01211==ubject: Re: Which do you prefer? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGKBUB03024;For abb. of physics courses, which of PHSX and PHYX do you prefer?I personally prefer the former. The latter makes me feel that itÕstoo soft; the Y makes me feel that way. S, on, on the other hand, is sharp. was talking with a friend of mine, and so clicked the Send Messageutton without thought.elta01211==ubject: Good intro statistics good? posting-account=PdOU8Q0AAAA6zo4zvKJ1KNRCbY5ITBWM did a quick search of good statistics books and came across theollowing: Problem Solving: A StatisticianÕs Guide by Chatfield, Christopher Statistics a New Approach [Hardcover] by Wallis, Wilson Allen Introduction to Probability Theory & Statistical Inference by Harold. Larsonowever, IÕm not sure if these are at the right level and/or cover theopics IÕm interested in. The level needs to be at the AP statisticsxam level (which although HS is probably more advanced first yearndergrad), and should give a good discussion of sample design andxperiments (and also regression and statistical inference). Also, IÕmot really looking for an AP exam review book (but of course,uggestions are welcome). What IÕm mainly looking for is a book thatives a good, yet accessible discussion of hatÕs going on under theoodin the AP topics (sample design, experiments, regression,tatistical inference), If any one knows of any books like this orlose to this...==ubject: Re: Good intro statistics good? I did a quick search of good statistics books and came across the following: * Problem Solving: A StatisticianÕs Guide by Chatfield, Christopherased on Amazon, this may fit your bill. * Statistics a New Approach [Hardcover] by Wallis, Wilson Allenmazon offered no info on this. * Introduction to Probability Theory & Statistical Inference by Harold J. Larsonalculus. ItÕs probably not what youÕre looking for. However, IÕm not sure if these are at the right level and/or cover the topics IÕm interested in. The level needs to be at the AP statistics exam level (which although HS is probably more advanced first year undergrad), and should give a good discussion of sample design and experiments (and also regression and statistical inference). Also, IÕm not really looking for an AP exam review book (but of course, suggestions are welcome). What IÕm mainly looking for is a book that gives a good, yet accessible discussion of hatÕs going on under the hoodin the AP topics (sample design, experiments, regression, statistical inference), If any one knows of any books like this or close to this...- ill Twentymanmail: wtwentyman at copper dot net==ubject: Is there at least woprimes in the closed interval [N,2N] for all N>1? posting-account=4za6cQwAAACWvvpq0N1XsH01GJ8y5PEDy BertrandÕs Postulate there is at least one prime number in thelosed interval [N,2N] for all N>1. Now the following question:s there at least woprime number in the closed interval [N,2N] forll N>1?lireza Abdollahi==ubject: Re: Is there at least woprimes in the closed interval [N,2N]for all N>1? By BertrandÕs Postulate there is at least one prime number in the closed interval [N,2N] for all N>1. Now the following question: Is there at least woprime number in the closed interval [N,2N] for all N>1?ased on the commentary in ttp://mathworld.wolfram.com/BertrandsPostulate.html the answer would be yes Assuming IÕm reading it right.- ill Twentymanmail: wtwentyman at copper dot net==ubject: Submitting a paper posting-account=uLDeKgwAAADCrJpczE-sPlRuCEKmJQCcy last post was probably lost in some server...y simple question is: Is it possible to submit a paper to one journalven if my e-mail address doesnÕt end with .edu?s there some of you who have post some any paper and is not arofessor or a teacher, but is only a simple math graduated ?iero Giacomelli==ubject: Re: Submitting a paper My last post was probably lost in some server... My simple question is: Is it possible to submit a paper to one journal even if my e-mail address doesnÕt end with .edu?es, it is: for instance, my e-mail address ends with .fr :-)Besides, having an e-mail address isnÕt (yet?) mandatory to submit papers,FAIK.)D==ubject: Re: Submitting a paper My last post was probably lost in some server... My simple question is: Is it possible to submit a paper to one journal even if my e-mail address doesnÕt end with .edu?es. IÕve published lots of papers, and my email doesnÕt end in edu.mail suffixes are not used to judge mathematical papers, their mathematicalontent is.- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: In what space a pentagram... posting-account=yKf8UwwAAABFL8I0LVVxaqT3lOBPEWSq..would have all lines orthogonal (or nearly so)? The usual pentagram,ith circle and all. Our drwng would be a plane representation (shadow)f some object in a higer dimension space. Is there any space where itould be a cube, a pyramid, other regular polyhedra? Can somebodymagine a `translation` where angles are chemical links, like saying its the ßat representation of a protein? What (known) molecules woulde equivalent to a pentagram under some transformation (or projectionoo)? What if we add two circles around? And if we consider differentolors as representative of a dimension (not geometrical, maybenterpreted as electromagnetism)? Say, the star one color, the circlesther colors... The goal is to find equivalent regular polyhedra inigh dim spaces, though it would be interesting to find figures withnown mechanical properties in higher spaces equivalent to a pentagrammbedded on a 2-plane. Can you think of other religious symbolismsnterpreted this way? Like the maltese cross or the swastika. It wouldlso be interesting to try to interpret them as electrical circuitseployed in higher dimensions, again with known properties as a key too a search...nother idea, ritual movements might spell the vertices of otherigures having direct interpretations in higher dimensional spaces.anilo, the Forgotten, the Forsaken, the Ignored, the Wanderer, theeeker, the Unnamed, the Misconceived, the Karmik, the Betrayed...==ubject: Who won this competition?n his antorian set theory and limitation of sizeMichael Hallettuotes this:he usefulness of mathematics, the esteem in which it is held, and theonourable name of Ôexact science par excellencerightly given it arell due to the clarity of its principles, the rigour of its proofs andhe precision of its theorems. In order to ensure the perpetuation of these precious merits in soeautiful a part of our knowledge, we seek a clear precise theory ofhat is called Infinite in mathematics. Competition announcement of the Berlin Academy of Science for 1786ho won, and with what?==ubject: Existence of countable Hamel basis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGLYee10428;ll normed spaces as vector spaces have basis,so calledamel basis.amel bases in infinite dimensional Banach spacesre uncountable.This follows from Baire category theorem.hat if the infinite dimensional space isnÕt complete?oes it always contain countable Hamel basis?he space d={(x_n):only finitely many x_n =/=0} has_n=(0,..,0,1,0,...0) as Hamel basis. would like to see other examples as this is the only one can think of in the moment.==ubject: Re: Existence of countable Hamel basisAll normed spaces as vector spaces have basis,so calledHamel basis.Hamel bases in infinite dimensional Banach spacesare uncountable.This follows from Baire category theorem.What if the infinite dimensional space isnÕt complete?Does it always contain countable Hamel basis?f course not; it might, for instance, contain a complete nfinite dimensional subspace even though it itself isnÕt omplete.ee Rudolph==ubject: Re: nth term of the series by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGLkGk11443;Can anybody find the general term of the following set series:se difference tables. (1) 1, 4, 10, 20, 35, 56, 84, ...n 3) [ choose 3, n = 3, 4, 5, ...(2) 4, 15, 36, 70, 120, 189, 280, ... 11, 21, 34, 50, 69, 91, ... 10, 13, 16, 19, 22, ... 3, 3, 3, 3, ... 4 + 11(n 1) + 10(n 2) + 3(n 3) n = 0, 1, 2, ...(3) 6, 20, 45, 84, 140, 216, 315, ... 14, 25, 39, 56, 76, 99, ... 11, 14, 17, 20, 23, ... 3, 3, 3, 3, ... 6 + 14(n 1) + 11(n 2) + 3(n 3) n = 0, 1, 2, ...ey fact is that for fixed k, the successive differences of n k) are given by (n k-1). Cf. finite difference calculus e.g. in Graham, Knuth, PatashnikÕs _Concrete Mathematics_).odd Trimble==ubject: Re: String Theory: Good, Bad and Bogus posting-account=KnAwYw0AAABy6E2ßLZOMBwwi4CQRTEg find myself having to review my solid state physics to try to followour post. Could you fill in more steps/background. IÕm especiallynterested in the derivation of HawkingÕs entropy formula for blackoles.hatÕs your take on SUSY, which is built into the alebraic structure oforentz/Poincare algebra? It does no problem should arise.B==ubject: Re: Existence of countable Hamel basis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGMAHx14180;All normed spaces as vector spaces have basis,so calledHamel basis.Hamel bases in infinite dimensional Banach spacesare uncountable.This follows from Baire category theorem.What if the infinite dimensional space isnÕt complete?Does it always contain countable Hamel basis?f course not. For instance, given a maximal orthonormal et in l_2 and a line which passes through no member of the et, there exist linear complements of the line containing very member of the set. Obviously the complement canÕt be losed (hence is not complete under the induced norm), but he complement has uncountable linear dimension. odd Trimble==ubject: How to visualize limits in category theory came late to this discussion and, for some reason, Google no longerllows me to post follow-ups, so I have to start a new thread. Butet me add my 2c.he original limits were actually colimits along directed sets. Inhis, they very much resembled limits along nets and were doubtlessamed by analogy. So suppose D is a directed set (a partially orderedet in which for every x and y, there is a z with x < z and y < z (forur purposes, < will stand for less than or equal)). Assume for each in D, you are given a group (say, other models of finitary firstrder theories, such as fields, will do) G_d and whenever d < e aomomorphism f_{ed}: G_d --> G_e such that G_{dd} is the identity andhenever c < d < e, f_{ed}.f_{dc} = f_{ec}. This is called a directedystem of groups. Then its (co)limit is constructed by taking thenion of the G_d, identifying each element of G_d with its image under_{ed} in G_e and defining the group operations on that set. It isasy and I leave the details to the reader. The dual notion waseveloped and it was shown that the Pontrjagin dual of a colimit ofbelian groups was the limit of the dual groups (carried out in thecategory-the concept was very new--of compact abelian groups). nyway, calling the limits was very natural. Why limits and colimitsere named as they were is not clear to me, but I remember Eilenbergaying once that roductwas rarely misused (but cf. free product). ourbaki generalized it to filtered limits and colimits and eventuallyilenberg and others realized that even that was unnecessary and thatny functor could be said to have a limit or colimit. Actually, youonÕt even need a functor; it is quite natural to use a graph and araph morphism to a category. By this time, the original connectiono Moore-Smith convergence of nets had been left behind, like theheshire cat.hen I think about limits, I think about subobjects of products.==ubject: Re: Implicit Function Theorem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGMqWo17652;> Let (x*,y*) in S be a vector such h(x*,y*) = 0 and the matrix nabla_y > h(x*,y*) is nonsingular, the implicit function states that > there exist (x,y) in the neigborhood of (x*,y*) such that y = f(x) and > h(x,f(x)) = 0.YouÕve badly mangled the IFT, omitting hypotheses, misstating the conclusion, ...orry..... h is C^1, so because det[dh/dy](x*,y*) is nonzero, det[dh/dy]is nonzero in a neighborhood of (x*,y*) by continuity; yourconclusion follows.ny easy way to show this? 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[0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0  [0 / [0 5 [0 F [0 l [0 r [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 & [0 t [0 ~ [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 [0 OT27097 ==ubject: Re: Poisson process, paradox For a constant rate Poisson process, every specific(upto a finite resolution) sequence of N spikes occuring over a given time interval is equally likely. This seems paradoxical because we certainly do not expect to see all N spikes apprearing within the first 1% of time interval. Resolve this paradox.magine a million monkeys using phones randomly so thatn average a given monkey will reach somepecific number once every million minutes.uppose the phone number is reached 5 timesy the monkeys between 6:00pm and 6:01pm. i.e. 5 spikes in one minute ]f there was a ring or busy signal at:00:01pm, that has nothing to do withhe arrival times of the 4 other calls.avid Bernier==ubject: Re: Non-atomistic continuum: Peirce, Weyl, and? posting-account=jEbKGQ0AAADZF1UpkDsHa5gkWBqABnUE While traditional analysis considers the continuum as the set of itshich has parts and Weyl explained in ÔOn the New Foundational Crisisf Mathematicsin 1921 that the concept of a continuous manifoldemained mathematically sterile. At least within IR+, cosine integral transform translates anyiscrete number into a non-atomistic continuous function and vice versa. May I ask for some elucidating hints? Eckard Blumscheinh, show me the way to the next whiskey barh, donÕt ask why, oh, donÕt ask whyhow me the way to the next whiskey barh, donÕt ask why, oh, donÕt ask whyor if we donÕt find the next whiskey bar tell you we must die, I tell you we must die tell you, I tell you, I tell you we must diehatÕs some more Weill for you Blumschein.==ubject: Re: Non-atomistic continuum: Peirce, Weyl, and? ... Weill...ot Kurt Weillot even Andr Weilut Hermann Weyl==ubject: Re: Non-atomistic continuum: Peirce, Weyl, and? posting-account=jEbKGQ0AAADZF1UpkDsHa5gkWBqABnUE While traditional analysis considers the continuum as the set of itshich has parts and Weyl explained in ÔOn the New Foundational Crisisf Mathematicsin 1921 that the concept of a continuous manifoldemained mathematically sterile. At least within IR+, cosine integral transform translates anyiscrete number into a non-atomistic continuous function and vice versa. May I ask for some elucidating hints? Eckard Blumscheinh, show me the way to the next whiskey barh, donÕt ask why, oh, donÕt ask whyhow me the way to the next whiskey barh, donÕt ask why, oh, donÕt ask whyor if we donÕt find the next whiskey bar tell you we must die, I tell you we must die tell you, I tell you, I tell you we must diehatÕs some more Weill for you Blumschein.==ubject: FLTMA: Cross-checking triples for which S.N=0 (long)t does seem S.1 = c....hy not try a few for yourself? a^n mod c (a^1 mod c = a) b^n mod c (b^1 mod c = b) c^n mod b a^n mod b c^n mod a b^n mod a (a^n + b^n) mod c (c^n - a^n) mod b (c^n - b^n) mod a S 1 1 1 1 1 1 2 0 0 2 3 4 1 3 2 1 2 2 1 5 4 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 7 9 1 7 3 2 6 3 1 10 9 1 1 4 2 4 0 6 5 11 3 9 1 1 6 1 2 0 5 7 1 1 1 7 4 2 2 3 2 7 7 9 1 4 5 4 6 6 1 13 9 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 7 9 4 7 6 2 3 6 4 13 10 3 7 4 1 4 0 3 4 7 5 1 1 1 6 1 6 0 5 11 9 9 4 7 1 2 5 6 6 17 11 3 7 4 6 4 1 3 2 6 12 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 8 9 4 8 5 1 4 5 4 13 12 3 7 1 1 1 2 6 0 8 5 1 1 8 5 1 6 2 4 12 1 9 4 1 1 1 10 3 0 13 8 3 7 8 5 1 11 8 4 23 12 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 7 10 3 7 6 3 4 6 3 13 10 9 9 9 1 2 6 0 6 12 5 12 7 3 6 6 4 4 0 8 9 3 1 1 1 4 12 0 4 16 11 4 3 7 6 5 2 6 1 9 12 1 9 9 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 3 11 2 3 1 2 1 10 2 13 9 4 4 9 1 1 0 6 0 6 1 5 8 5 1 2 6 3 2 11 3 3 5 4 1 1 6 1 0 7 9 7 10 1 1 2 3 9 2 14 1 12 9 3 1 1 0 6 0 6 3 2 7 9 1 2 5 9 2 16 9 9 3 5 1 1 5 9 0 14 1 8 6 4 1 2 9 2 2 130 3 10 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 4 11 2 4 1 3 2 9 2 13 3 4 4 5 1 1 7 10 0 17 12 5 8 9 1 3 4 10 2 16 9 3 5 3 1 1 12 2 0 14 10 7 10 1 1 3 4 9 2 15 1 12 9 4 1 1 0 5 0 5 4 2 7 5 1 3 6 2 2 10 3 9 3 9 1 1 12 5 0 17 12 8 6 3 1 3 7 3 2 120 9 10 1 1 1 1 6 0 0 61 10 6 2 4 1 3 3 9 2 142 1 1 4 5 1 1 2 10 0 123 4 11 8 9 1 3 2 10 2 144 3 4 5 3 1 1 7 2 0 95 12 5 10 1 1 3 4 9 2 156 9 3 9 4 1 1 12 5 0 177 10 7 7 5 1 3 4 2 2 88 1 12 3 9 1 1 0 5 0 59 4 2 6 3 1 3 6 3 2 110 3 9 1 1 1 1 12 0 0 121 12 8 2 4 1 3 7 9 2 182 9 10 4 5 1 1 6 10 0 163 10 6 8 9 1 3 3 10 2 154 1 1 5 3 1 1 2 2 0 45 4 11 10 1 1 3 2 9 2 136 3 4 9 4 1 1 7 5 0 127 12 5 7 5 1 3 4 2 2 88 9 3 3 9 1 1 12 5 0 179 10 7 6 3 1 3 4 3 2 90 1 12 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 9 11 2 9 4 2 7 4 2 13 3 4 4 4 7 4 7 0 3 10 1 5 8 3 1 8 6 5 2 13 9 3 5 5 4 7 12 0 6 18 3 7 10 1 7 5 10 9 2 21 1 12 9 9 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 5 12 1 5 3 2 4 8 1 13 12 1 1 1 4 4 0 0 0 0 1 1 1 1 1 1 2 0 0 2 7 12 1 7 6 5 6 6 1 13 10 1 1 1 1 4 11 0 4 15 5 12 1 7 6 6 4 6 0 10 9 1 1 1 1 2 10 0 6 16 11 12 1 7 6 3 10 6 3 19 12 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 11 12 1 11 2 1 10 2 1 13 4 1 1 1 4 1 5 0 3 8 5 12 1 11 8 1 4 2 7 13 3 1 1 1 5 1 4 0 4 8 7 12 1 11 10 1 6 2 9 17 12 1 1 1 9 1 0 0 8 8 2 12 1 11 7 1 1 2 6 9 9 1 1 1 3 1 10 0 2 12 8 12 1 11 6 1 7 2 5 140 10 1 1 1 1 1 11 0 0 111 6 12 1 11 2 1 5 2 1 82 1 1 1 1 4 1 2 0 3 53 11 12 1 11 8 1 10 2 7 194 4 1 1 1 5 1 5 0 4 95 5 12 1 11 10 1 4 2 9 156 3 1 1 1 9 1 4 0 8 127 7 12 1 11 7 1 6 2 6 148 12 1 1 1 3 1 0 0 2 29 2 12 1 11 6 1 1 2 5 80 9 1 1 1 1 1 10 0 0 101 8 12 1 11 2 1 7 2 1 102 10 1 1 1 4 1 11 0 3 143 6 12 1 11 8 1 5 2 7 144 1 1 1 1 5 1 2 0 4 65 11 12 1 11 10 1 10 2 9 216 4 1 1 1 9 1 5 0 8 137 5 12 1 11 7 1 4 2 6 128 3 1 1 1 3 1 4 0 2 69 7 12 1 11 6 1 6 2 5 130 12 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 9 10 7 9 8 1 2 8 7 17 13 15 9 1 1 1 11 8 0 19 15 14 3 9 8 1 12 4 7 23 16 4 1 1 1 1 3 0 0 3 8 6 7 9 8 1 14 8 7 29 4 9 9 1 1 1 13 8 0 21 2 5 3 9 8 1 7 4 7 18 1 16 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 0 0 2 7 11 6 7 3 4 1 10 6 17 15 2 3 5 2 2 0 9 0 9 3 5 7 2 6 1 8 5 5 18 4 4 9 3 4 4 8 6 0 14 11 10 10 10 5 2 4 0 3 7 9 8 5 4 1 1 0 1 0 1 12 3 8 6 3 4 15 2 6 23 16 16 4 9 2 2 15 6 0 21 10 6 2 8 6 1 16 5 5 260 2 15 1 1 4 4 0 0 0 0 1 1 1 1 1 1 2 0 0 2 8 11 6 8 1 3 2 9 6 17 13 2 3 9 1 1 15 5 0 20 2 5 7 6 1 3 7 1 6 14 16 4 9 4 1 1 3 5 0 8 9 10 10 10 1 3 2 0 6 8 4 8 5 3 1 1 12 2 0 14 15 3 8 2 1 3 1 6 6 13 1 16 4 5 1 1 0 10 0 10 8 6 2 7 1 3 14 6 6 260 13 15 1 1 1 1 11 0 0 111 2 12 6 8 1 3 14 9 6 292 16 13 3 9 1 1 12 5 0 173 9 7 7 6 1 3 16 1 6 234 4 9 9 4 1 1 13 5 0 185 15 14 10 10 1 3 12 0 6 186 1 1 5 3 1 1 2 2 0 47 8 11 8 2 1 3 2 6 6 148 13 2 4 5 1 1 15 10 0 259 2 5 2 7 1 3 7 6 6 190 16 4 1 1 1 1 3 0 0 31 9 10 6 8 1 3 2 9 6 172 4 8 3 9 1 1 12 5 0 173 15 3 7 6 1 3 1 1 6 84 1 16 9 4 1 1 0 5 0 55 8 6 10 10 1 3 14 0 6 206 13 15 5 3 1 1 11 2 0 137 2 12 8 2 1 3 14 6 6 268 16 13 4 5 1 1 12 10 0 229 9 7 2 7 1 3 16 6 6 280 4 9 1 1 1 1 13 0 0 131 15 14 6 8 1 3 12 9 6 272 1 1 3 9 1 1 2 5 0 73 8 11 7 6 1 3 2 1 6 94 13 2 9 4 1 1 15 5 0 205 2 5 10 10 1 3 7 0 6 136 16 4 5 3 1 1 3 2 0 57 9 10 8 2 1 3 2 6 6 148 4 8 4 5 1 1 12 10 0 229 15 3 2 7 1 3 1 6 6 130 1 16 1 1 1 1 0 0 0 0ours,oug Gonczeplikon Researcheven Corners, VA 22044-0394==ubject: Numb3rs posting-account=K4N35Q0AAAADkuI33MRIKmRbIrs0Xffuas anyone heard anything about this new show on CBS called Numb3rs?Õve seen a couple of commercials for it (starts in January) and itooks pretty cool--like they use numbers and patterns to solve crimes.==ubject: Re: Numb3rsmartamaven@hotmail.com> Has anyone heard anything about this new show on CBS called Numb3rs? IÕve seen a couple of commercials for it (starts in January) and it looks pretty cool--like they use numbers and patterns to solve crimes.hereÕs a bit here:ttp://www.cbs.com/primetime/numb3rs/==ubject: Re: Numb3rs Has anyone heard anything about this new show on CBS called Numb3rs? > IÕve seen a couple of commercials for it (starts in January) and it > looks pretty cool--like they use numbers and patterns to solve crimes. ThereÕs a bit here: http://www.cbs.com/primetime/numb3rs/aybe theyÕll unearth the dark secret defect at the foundation of abstractlgebra and send all the athematiciansoff to jail where they belong foreaching fraud with government money.==ubject: Re: === Numb3rs==Subject: Re: Numb3rsMessage-id: > > Has anyone heard anything about this new show on CBS called Numb3rs?> IÕve seen a couple of commercials for it (starts in January) and it> looks pretty cool--like they use numbers and patterns to solve crimes.> ThereÕs a bit here:> http://www.cbs.com/primetime/numb3rs/Maybe theyÕll unearth the dark secret defect at the foundation of abstractalgebra and send all the athematiciansoff to jail where they belong forteaching fraud with government money.hoÕs starring in it, M.C. Hammer?- ensanatorce of Clubs==ubject: Re: Numb3rsHas anyone heard anything about this new show on CBS called Numb3rs?IÕve seen a couple of commercials for it (starts in January) and itlooks pretty cool--like they use numbers and patterns to solve crimes.ounds like they stole the idea from the athnetsegment of the old PBS idsshow quare One See e.g. http://www.mathnews.uwaterloo.ca/Issues/mn7805/s1mathnet.php> .Mathematicians... Freeze! obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: How much lossless compression is possible in images?>But I will look what your gif>file amounts to. I suspect it will be a form of>>LZW.>-->dik t. winter> http://www.justsaying.com/rle.gif> http://www.justsaying.com/lzw.gif> hereÕs 2 gifs with the same original image, simple checkers.ot so very simple, but I have written a program to analyse gifÕs.oth are very definitely lzw compressed, no rle in sight. Thenly difference is that the leversion has an LZW clear codehenever the colour changes. Here the first parts of the imageata of the rle and lzw version (as decoded by an LZW decoder): rle lzw 0100 clear 0100 clear 0002 color 0002 color 0102 0102 0103 0103 0104 0104 0105 0105 0106 0106 0107 0107 0104 0104olor changes, here is the first difference, lehas a clear code, soodes differe from here: 0100 clear 0000 color 0000 color 0102 010a 0103 010b 0104 010c 0105 010d 0106 010e 0107 010f 0104 010cecond color change. LZW does not need either a clear or a color code,ust a back pointer. 0100 clear 0002 color 0102 0108 back pointer to first white part 0103 0112 back pointer to first black part 0104 0113 back pointer to first line 0105n all, what is presented as RLE is a horribly inefficient way to do LZW.87 bytes of data for full LZW and 1478 bytes of data for LE- ik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131ome: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/==ubject: Re: Deep Thoughts # 37: I just realized whatÕs wrong with the Continuum Hypothesis posting-account=WZMvOwwAAAC_B1TaayrVN99BJgDWQkUc The cardinality of a set is not well-defined.bviously.bvious at first-order, anyway.very consistent first-order theoryincluding ZFC) has a denumerable model.ll these allegedly uncountable sets are,n THAT model, obviously, countable.ou can get different opinions about he same etÕs cardinality either by looking at it fromutside the theory or by adding new axioms. What is the cardinality of: 1. A = { x | x = N v (eY)YeA ^ x=P(Y) }he fact that A occurs on the right means thiss simply not well-formed as a definition of A.n ZFC, this A is simply not a set. It would stillot be a set even if you removed N. A having anylement in it at all would suffice. The following,or example, is also too big to be a set:={ x | x=O v Ey[yeA & x=p(y)] } 2. A = { x | x = N v x = P(A) }n ZFC, this etsimply does not exist. set cannot contain its own powerset; thatiolates foundation/regularity. 3. A = { x | ~(x e A) }oes not exist in ZF. 4. A = { x | x e A }his is tautologously/syntactically/vacuouslyefinitionally true OF ALL sets and thereforeoes not define a particular set A. 5. A = { x | x e (A u {0}) }his is false of all sets that donÕt contain as an element, and true of all that do, i.e.,t is true only in the case where A u {O} = Aecause O was already in A. So again, this doesot define any particular set. 6. A = { x | x e x }n ZF, this is true of no set whatever, so itoes not define any particular set A. 7. A = { x | ~ (x e x) }n ZF, this A is the class of ALL setsNO set belongs to itself), so its cardinality isust he cardinality of a proper class which is noteally mentionable inside the theory (since the theorynly talks about things that are small enough to be sets). Which have less cardinality than which?ince none of the above can exist at all, thats an entirely meaningless question. Also include P(A) , A u B , A n B etc. for A and B from this list. To tell if A For example, in Q(sqrt(-15)), the norm of (1/4) + (1/4)*sqrt(-15) is> 1, an integer, yet it is clearly not an algebraic integer.> Yes, in some cases such a norm (as Hardy & Wright call it) will not> result in algebraic integers.> IsnÕt this the norm (pun intended)?> If f is an automorphism of the extension K/k> then x/f(x) will have norm 1,> but will not normally be an algebraic integer.es, I was horribly wrong. I think a combination of bad memory and lackf sleep, or whatever. Hardy and Wright do develop Q(sqrt(-1)) and(sqrt(-3)) without regard to algebraic integers, but when they geterious with quadratic fields they seriously prove how to get allintegersin the field and show that they are lgebraic integersection 14.2 is about this stuff. They first show that each algebraicnteger has a minimal polynomial, and next show that algebraic integersave a specific form in quadratic fields.n the other hand, they do *not* show that the algebraic integers form ring. The simplest proof of that I have found in Van der Waerden,ut that requires quite a bit more.- ik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131ome: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/==ubject: Re: Algebraic integersOn the other hand, they do *not* show that the algebraic integers forma ring. The simplest proof of that I have found in Van der Waerden,but that requires quite a bit more.ime to dust off my favourite four-line proof: complex number is an algebraic integer iff it is an eigenvalue of a quare matrix of integers. If x and y are eigenvalues of A and B espectively, then x+y is an eigenvalue of (A @ I) + (I @ B) andy is an eigenvalue of A @ B, where @ denotes tensor product.ave Boyd tells me he heard this from Olga Taussky in the 60Õs.obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: Algebraic integers> On the other hand, they do not show that the algebraic integers form> a ring. The simplest proof of that I have found in Van der Waerden,> but that requires quite a bit more. Time to dust off my favourite four-line proof: A complex number is an algebraic integer iff it is an eigenvalue of a square matrix of integers. If x and y are eigenvalues of A and B respectively, then x+y is an eigenvalue of (A @ I) + (I @ B) and xy is an eigenvalue of A @ B, where @ denotes tensor product. Dave Boyd tells me he heard this from Olga Taussky in the 60Õs.ctually tÕs already in Dedekindas Noether often said. owever Taussky did do much pioneering work in the matrixepresentational approach to algebraic number theory and shelayed a major role in popularizing this viewpoint, so itÕsot too surprising to see this mistakenly attributed to her.n introduction to some of TausskyÕs work here may be found n her two appendices to Harvey CohnÕs excellent textbook: classical invitation to algebraic numbers and class fields.f course one doesnÕt need to explicitly introduce tensor products.nstead, as is well-known, one may proceed more simply as followsfor convenience this is based upon Robin ChapmanÕs notes [1] but,s I mentioned, it already occurs in Dedekind over 100 years ago)HEOREM 1. If b and c are algebraic integers so too are b+c and bc.ROOF. Suppose b and c are algebraic integers. The idea of the proofs to find a nonzero vector v and two integer matrices B and C withv = bv, Cv = cv. Then b+c, bc are eigenvalues of B+C, BC resp.ince (B+C)v = Bv + Cv = bv + cv = (b+c)v and (B C)v = B(Cv) = B(cv) = cBv = bc v ince B+C and BC have integer coefficients, we conclude by Lemma 1hat b+c and bc are both algebraic integers. It remains to show that uch matrices B and C exist. Suppose b^n + b_1 b^(n-1) +...+ b_n = 0 nd c^m + c_1 c^(m-1) +...+ c_m = 0, where the b_j, c_k are integers. Tet v be the mn by 1 vector (b^j c^k : 0 <= j < n, 0 <= k < m)onsider bv. The entries in this vector all have the form b^(j+1)c^khere 0 <= j < n, 0 <= k < m. If j+1 < n this is already an entry in v;f j+1 = n it equals k n k n-1 n-2 c b = c (- b b - b b - ... - b b - b ) 1 2 n-1 nn all cases b^(j+1)c^k is a linear combination, with integer coefficients,f entries in v. Therefore there exists an integer matrix B with Bv = bv. imilarly there exists an integer matrix C with Cv = cv. QEDf course nowadays this is all expressed much more concisely andlegantly in the language of (finitely-generated) modules, theheory of which was constructed by Dedekind as he iterativelyevised and polished his manuscripts on algebraic number theory.-Bill Dubuque1] Robin Chapman, Notes on Algebraic Numbersttp://www.maths.ex.ac.uk/~rjc/notes/algn.dvittp:// www.maths.ex.ac.uk/~rjc/notes/algn.psttp://www.maths.ex.ac.uk /~rjc/notes/algn.pdfttp://www.maths.ex.ac.uk/~rjc/rjc.html== ubject: Re: Algebraic integers>On the other hand, they do *not* show that the algebraic integers form>a ring. The simplest proof of that I have found in Van der Waerden,>but that requires quite a bit more. A complex number is an algebraic integer iff it is an eigenvalue of a square matrix of integers. If x and y are eigenvalues of A and B respectively, then x+y is an eigenvalue of (A @ I) + (I @ B) and xy is an eigenvalue of A @ B, where @ denotes tensor product.hile this is quite a slick proof,t doesnÕt really throw much light on _why_ the result holds.Also the tensor product of 2 matrices is quite a difficult concept.ow does one show that the eigenvalues of A@Bre the products of the eigenvalues?)o my mind, the best proof - even though it uses a bit of algebra,nd is slightly longer - depends on the Lemma that in C is an algebraic integer iff there is a finitely-generated group A < Cuch that cA < A.his fits in rather well with the Lemma that in C is algebraic iff there is a finite-dimensional vector subspace V < Cuch that cV < V.- imothy Murphy -mail (<80k only): tim /at/ birdsnest.maths.tcd.ieel: +353-86-2336090, +353-1-2842366-mail: School of Mathematics, Trinity College, Dublin 2, Ireland==ubject: Re: Algebraic integers> A complex number is an algebraic integer iff it is an eigenvalue of a> square matrix of integers. If x and y are eigenvalues of A and B> respectively, then x+y is an eigenvalue of (A @ I) + (I @ B) and> xy is an eigenvalue of A @ B, where @ denotes tensor product.While this is quite a slick proof,it doesnÕt really throw much light on _why_ the result holds.(Also the tensor product of 2 matrices is quite a difficult concept.How does one show that the eigenvalues of A@Bare the products of the eigenvalues?)f v is an eigenvector of A for eigenvalue x and w is an eigenvector of for eigenvalue y, then (A @ B) (v @ w) = (Av) @ (Bw) = x y (v @ w).hereÕs nothing really difficult, to my mind, about the tensor product of atrices as a concept. You can do it quite concretely as the Kroneckerroduct. Indices for A@B correspond to pairs of indices for A and Bespectively, and (A@B)_{(ij),(kl)} = A_{ik} B_{jl}. All you need toerify is that (A@B)(v@w)=(Av)@(Bw).obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: Algebraic integers>On the other hand, they do *not* show that the algebraic integers form>a ring. The simplest proof of that I have found in Van der Waerden,>but that requires quite a bit more.Time to dust off my favourite four-line proof:A complex number is an algebraic integer iff it is an eigenvalue of a square matrix of integers. mm. One direction is clear even to me, knowing no algebra: theharacteristic polynomial of a matrix is monic, fine. For thether direction: Is it obvious and/or true that any (irreducible?)onic polynomial with integer coefficients is (a factor of?) theharacteristic polynomial of some integer-valued matrix?If x and y are eigenvalues of A and B respectively, then x+y is an eigenvalue of (A @ I) + (I @ B) andxy is an eigenvalue of A @ B, where @ denotes tensor product.Dave Boyd tells me he heard this from Olga Taussky in the 60Õs.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada***********************avid C. Ullrich==ubject: Re: Algebraic integers>>On the other hand, they do *not* show that the algebraic integers form>>a ring. The simplest proof of that I have found in Van der Waerden,>>but that requires quite a bit more.>Time to dust off my favourite four-line proof:>A complex number is an algebraic integer iff it is an eigenvalue of a>square matrix of integers. Hmm. One direction is clear even to me, knowing no algebra: the characteristic polynomial of a matrix is monic, fine. For the other direction: Is it obvious and/or true that any (irreducible?) monic polynomial with integer coefficients is (a factor of?) the characteristic polynomial of some integer-valued matrix?s Dik said, the maigic words are ompanion matrix I learnt thisroof from Peter Cameron in c. 1982. In my otes on algebraic numbers see webpage) I give an even more simple-minded version of this, avoidingven companion matrices :-)- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: Re: Algebraic integers..>A complex number is an algebraic integer iff it is an eigenvalue of a >square matrix of integers. > Hmm. One direction is clear even to me, knowing no algebra: the> characteristic polynomial of a matrix is monic, fine. For the> other direction: Is it obvious and/or true that any (irreducible?)> monic polynomial with integer coefficients is (a factor of?) the> characteristic polynomial of some integer-valued matrix?up. You can construct the companion matrix of the polynomial. Thisonsists of 1 along the subdiagonal and the negatives of the coefficientsf the polynomial (minus the leading 1) along the top row and 0 elsewhere.he characteristic polynomial is the original polynomial.n the other hand, the remainder is less trivial if you do know nolgebra:>If x and y are eigenvalues of A and B >respectively, then x+y is an eigenvalue of (A @ I) + (I @ B) and>xy is an eigenvalue of A @ B, where @ denotes tensor product.TW, this is similar to the proof in Van der Waerden, but he doesot require knowledge of algebra.- ik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131ome: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/==ubject: Re: Starting Calculus in January...DonÕt worry about getting leg up on the course. Just review youralgebra, geometry, and especially trigonometry. You wonÕt believe howimportant trigonometry is in calculus.es and no on that point. You can take usiness calc which we teach here ithout so much as a mention of any trig function. ItÕs perfectly fine course or its audience. (Well, one would think they would want to work with theccasional _business cycle_ but thatÕs not essential for a short course.)TOH, for the science-and-engineering calculus sequence, itÕs trueouÕll need the trig functions, especially when oingall thosentegrals in the second semester.etting aside the way the generic calculus courses are taught, though,ne has to wonder why the trig functions are so critical for calculus.hat is calculus anyway? Is it really the oingof integrals? Or ist instead the study of limits? Or more informatively, the study ofifferentiation, integration, and series?aking the big-picture view, then, thereÕs no reason to think the trigunctions are any more or less important that the rational functions,xponential functions, hypergeometric functions, or any otherlass. ItÕs very handy to have the trig functions around, so that thealculus teacher can make some examples; and there are some goodpplications which require using trigonometry and calculus inandem. But I wouldnÕt say that calculus really depends on trignometry.f we were plopped into al-KhwarizmiÕs parlor and wanted to guide him toiscover the main ideas of the calculus, we could get very far with justs much math as he knew. At some point he would ask us about things likentegrating 1/x or solving y= y, and so we would add the log andxponential functions as the first functions other than the rationalunctions he could describe. It wouldnÕt be until the comparatively moreomplicated problems of integrating 1/(1+x2) or solving yÕ= - y rose that we would need the trig functions. (IÕm not sure what woulde next -- perhaps integrating exp(-x^2) ? )ave==ubject: Re: Starting Calculus in January... Setting aside the way the generic calculus courses are taught, though, one has to wonder why the trig functions are so critical for calculus. What is calculus anyway? Is it really the oingof integrals? Or is it instead the study of limits? Or more informatively, the study of differentiation, integration, and series?ltimately it is the solution of differential equations. Most of physics an be expressed in terms of differential equations. The trig functions ome in when you have to do intergals of the formNTEGRAL (dx/sqrt(1+-x^2) and such like.ob Kolker==ubject: Re: lim question>Is (1) true? (if not, why?)Yes. But note that the notation in (1) does not imply that thereis such a thing as 1-. The -> 1-means tends to 1 frombelow _not_ that there is a thing named -that n tends to.(The minus sign there is really an adjective modifying whatthe arrow means.)o, itÕs an adverb; youÕre rightly stressing the fact that theresnÕt such a thing as - which notation would imply is aoun and an adjective attached a la mode francaise .ave (Does nal-retentivehave a hyphen?)The last time I used that line, I got fussed at for forgetting theuotation marks. Well! That same day I noticed the sentence While the sign said ÔFish and ChipsÕ, I would have put hyphens between fish and and and and and chips. nd realized it would be clearer if one were to insert quotation marksefore fish and after chips, and pairs of quotation marks between fishnd and and and and and and and and and and and and and and and and andnd and and chips.]==ubject: Re: lim question>>Is (1) true? (if not, why?)>Yes. But note that the notation in (1) does not imply that there>is such a thing as 1-. The -> 1-means tends to 1 from>below _not_ that there is a thing named -that n tends to.>(The minus sign there is really an adjective modifying what>the arrow means.)No, itÕs an adverb; ight. I lied.youÕre rightly stressing the fact that thereisnÕt such a thing as - which notation would imply is anoun and an adjective attached a la mode francaise .dave (Does nal-retentivehave a hyphen?)ell of course it does! Are you blind or what?[The last time I used that line, I got fussed at for forgetting thequotation marks. nd now you get fussed at for not realizing that when youhrase the question as you did the answer is es, bynspection just as itÕs clear that oty*yrcontainsn asterisk, two yÕs and no eÕs.Um. Two s and no s. Or is it and ?)Well! That same day I noticed the sentence While the sign said ÔFish and ChipsÕ, I would have put hyphens between fish and and and and and chips. and realized it would be clearer if one were to insert quotation marksbefore fish and after chips, and pairs of quotation marks between fishand and and and and and and and and and and and and and and and and andand and and chips.]Õm glad you pointed that out. Really.***********************avid C. Ullrich==ubject: Disappointed Differentialhy would anyone care about the foundations of logic?hatÕs a facetious rhetorical question. Grelling:eterological means nonautoheterological.n researching ifferential there are a tremendous numberf named differential equations. for example, if you go toathWorld and rsearch for the term ifferential there areundreds of results.he reference to Erdelyi and them, Ullrich, is to when Iffably asked you to please list everything you had heard ofhat is called a ransform some years ago. You probablyemember that, vicariously by rereading the post, otherserhaps do not, having no part nor parcel in thatiscussion.ifferential equations beyond univariate calculus form aarge blind spot for me, retinal damage. So, I hope toepair that deficiency by generating a broad overview of thetudy of differential equations, necessarily vague as it is complex and vibrant field, so that even if I am ignorantnd worthless in the mechanics of manipulating differentialquations, the vocabulary could be used within variousontexts.hen, I need or rather care to understand to some usableegree the integration with multiple variables, multivariatentegration.his is particularly so where I say things like Int dydx =. ThatÕs mostly silliness. Int 1 dx is x+C. Somewhatess sillily, I have some notions about the continuous andbout how discrete sets on the continuous line behave. return to MathWorld and type ifferential. amn, thatÕs a lot of information. Under the entry forOrdinary Differential Equation an Ordinary Differentialquation has a parameter of its order, similarly to therder of a polynomial, its order is marked in theescription there by the variable letter n. The ODE ofrder n is defined as F(x, yÕ, yÕÕ, ..., y^(n)) = 0.tÕs linear when, similarly to a polynomial,_n x y^(n) + a_n-1 x y^(n-1) + ... + a_1 x y+ a_0 x y =(x)f Q(x) = 0 then the function is called homogenous. HereÕsne thing I donÕt understand, I think there are probablyurther conditions on Q(x) than it just being a functionith domain x, because a function with domain x exists forny function y differentiable n many times as the look-upable.eing linear in this way opens a lot of computational oralculatory methods as would be appropriate witholynomials, eg a polynomial is_n x^n + a_n-1 x^n-1 + ... + a_1 x + a_0 =f linear equations in general, besides the linearifferential equation. Learn something about Viete, orietje, or Vieta, however theyÕre spelling it these days.ndeed, for a function where fÕ(x) = x^2, and fÕx) = x^3,e, f^n(x)= x^n, that linear differential equation would be polynomial with a_n = x^n or so. That never occurs as viahe chain rule fÕ(x) = 1, fÕ(x^2) = 2x, fÕ(x^3) = 3x,tcetera. Do you know any terms for which that is not thease? That is to say, the maximal order of the x term inhe differand in fÕis less than that of fÕ. While that iso, itÕs as well an interesting difference.hatÕs next, frogs from the sky?oss F.==ubject: Re: Disappointed Differential While that is so, itÕs as well an interesting difference.hile that is so, there is another way to consider the linearifferential equation.t is in the case where the variable x of the linearifferential equation is only considered at the value x=1.hen, this is only for a simple function of the order n like^n. Then, consider the coefficients reversed in order, from_n to a_1, with a_0 irrelevant. Then, you can consider theoefficient a_n to be divided by n._n x fÕ^n + a_n-1 x fÕ^n-1 + ... + a_1 x f+ a_0 f_n x^n+1 + a_n-1 n x^n + ... a_1 n! x^2 + a_0 xhen, if you get a root of that polynomial, what is that?ou might notice that some of the coefficients are described inFactorial/Exponential Identity, Infinity. a ha ha!oss F.==ubject: Re: Disappointed Differential > IÕd appreciate that, IÕd like to learn about this haos theorysomeday. OK, but first show some discipline. Learning about chaos theory is different from being a chaotic person. Han de Bruijnour implication is that I am chaotic. ItÕs worse, Han, IÕmirective. appreciate your offer of help. What is haos theory Is ithe blanket definition athematical stuff that seems chaotic? haos theory is basically differential analysis, including partialnd single differential analysis, and linear and non-linearrdinary differential analysis.s that wrong? IÕm asking you because you might know differently.onsider the linear and non-linear differential analysis, aarallel between those two is the polynomial and power series. Ishat not so in an obvious way? Where that is so, is not itarticularly explicit, and rarely made explicit?o, would you please present a one-page formula sheet covering allnown differential analysis? That might seem a ridiculousequest, there can be way too much to cover on a single page. Thedea of the one-page formula sheet is that in academia in someases on exams the student, or rather, test-taker, has the optiono prepare their own page or card containing information that theyould choose to use on the exam. ItÕs very convenient, and leadso very concise expressions of the important technical conceptseeded to correctly answer the questions and solve the problems ofhe examination. The students write in very small type, I waseminded of this concept from coming across a 3x5 index cardovered with statistics formulae for an exam I had in college somewelve or so years ago. I got AÕs, those are good marks, intatistics, that was my minor. Another use of the formula sheets to be laminated, placed on the desk or table, and readilyeferenced, for example the menu in a restaurant. appreciate your input, I know thereÕs a lot I could learn fromou. I know in person people my age who have been studyingifferential equations for fifteen years, publishing post-docifferentialists, several of them. I can and perhaps will writend ask them my questions. I happen to have sci.math as an easyutlet, I also post to sci.logic and sci.physics from time toime. IÕm glad that IÕve written to you, these newsgroups areull of people with all kinds of knowledge.o, what is differential analysis? What good is it? How is itaught, how is it learned? What are the major subfields? Arehere any controversial subfields, and if so, how so and why?hatÕs the bleeding edge of research in the field?ayne Staley once sang something along the lines of: ne timehe teacher hit the ruler on my hand, it bled itÕs a good linetaleyÕs a tragic case. I like the EP and November Hotel, annstrumental. I was listening to Sugar Ray the other day, iteally got me pumped.oss F.==ubject: Re: Disappointed Differential I appreciate your offer of help. What is haos theory Is it the blanket definition athematical stuff that seems chaotic? tÕs a kind of mathematical culture where certain subjects and habitsre prevalent. Anyway, different when compared with e.g. calculus orumerical analysis. And no, IÕm not an expert in it, though IÕve doneome work that uses ttractors epellersand the like. A mixturef virtually everything (numerical, calculus, chaos) can be found in:ttp://huizen.dto.tudelft.nl/deBruijn/grondig/calculus.htm Chaos theory is basically differential analysis, including partial and single differential analysis, and linear and non-linear ordinary differential analysis. Is that wrong? IÕm asking you because you might know differently.ou have already done a better survey than this by yourself, if IÕvenderstood your next poster in this thread. Consider the linear and non-linear differential analysis, a parallel between those two is the polynomial and power series. Is that not so in an obvious way? Where that is so, is not it particularly explicit, and rarely made explicit?ith Operator Calculus, ordinary (linear) differential equations can beolved, indeed, as if you are factoring a polynomial: So, would you please present a one-page formula sheet covering all known differential analysis? That might seem a ridiculous request, there can be way too much to cover on a single page. ... snip ... ]s such, this seems to be a mighty good idea! But I donÕt think IÕmoing to undertake it, because it might turn out to be too much a hellf a job. covered with statistics formulae for an exam I had in college some twelve or so years ago. I got AÕs, those are good marks, in statistics, that was my minor. [ ... snip ... ]hat was your major, Ross? (Mine has been theoretical physics.) So, what is differential analysis? What good is it? How is it taught, how is it learned? What are the major subfields? Are there any controversial subfields, and if so, how so and why?or physicists, differential analysis - as you call it - is perhapshe most important field in mathematics. You can only master it byctually solving a lot of problems, in class or in practice. Therere still remaining some controversial subfields as well - such ashe infamous diverging integrals in Quantum ElectroDynamics(?). WhatÕs the bleeding edge of research in the field? [ .. snip .. ] donÕt think there is just _one_ bleeding edge of research.an de Bruijn==ubject: Re: Disappointed Differential <41BFB75A.8E80A14F@tiki-lounge.com> <41C004B8.B8258CAC@tiki-lounge.com> posting-account=7ryOqgsAAABSV_46k1efyFxO01THH4J8i Han,h, great, the Operator Calculus. I think that has to do with Odlyzkond Ito.here appears to be descriptions of the discrete, and continuous,perator calculi. studied economics. The predictor of growth in the economy isnvestment in research and development, and economic data in the U.S.eßects that statistical correlation to more than 95% confidence. So,ooks are better for the economy than bombs. Learning takes forever.ouÕre a physicist. Your webpage contains a variety of numericaloftwares, and you have expressed interest in things musical, and youall yourself a Dijkstrian humble programmer. ThatÕs great.bout the chaos theory, there appears to be some application of theellular automata, indeed the content of the perator Calculusentryt WolframÕs MathWorld is a reference to WolframÕs book. There mighte more utility in sinc, and various combinations of sin( a 2pi x) / b, than some numbered limited edition game, ie there is more utility inhe understanding than the generator, viz fish and fishing.hen, with the operator calculus, in deeming the linear ODE in someases in a way the polynomial, and solving for roots, then there is theuestion of the solution for the general case of the polynomial ofrbitrary order. There was some discussion of the arbitrary polynomialolutions some years ago, how is that done?ack to differential equations, theyÕre difficult, but with a varietyhain rule to integration by parts and substitution, to piecewisentegration, .... This is beyond my understanding, besides very simplenivariate cases, even surface integrals boggle my mind, with del,abla. Then, there is partial differentiation, a tool of multivariatenalysis.f I think something is unintuitive IÕll often find another way to dot.ne problem with that is to get the intuitive understanding of whathose things mean. The integral of the univariate function gives therea between the funciton plot and x axis, loosely, as an intuitiveescription of what it is.ne problem with the differential equations, or indeed otherathematical things that have to do with your study area ofathematical physics, is that the computations readily explode in termsf the variables so that you can start with a simple equation and noto many steps later have several pages overßowing with one equation.ertain applications such as the use of the Clifford algebras vis-a-vishe tensor representations can see a decrease in the number ofariables with no loss of information, and indeed in some casesncreased power.n some few paragraphs we can see the introduction, if only in context,f wide applications of the differentials, differential methods,ifferential operators, differential sysems, differential analyis, andther divers things differential and integral.o IÕm hoping you have that brief survey that you could send to me aopy.oss F.==ubject: Re: Disappointed Differentialhis Google is amazing. If it werenÕt for Google and other searchngines, sci.math would be full of many looking for information.ttp://www.imho.com/grae/chaos/chaos.htmlo, chaos theory is not just differential analysis. It also includesractal behavior. ItÕs like the chemical reaction that cycleshrough seven colors, but that is just about buffering, criticaloints, and I guess thatÕs probably an endothermic cycle. I forget.s it is, itÕs full of many looking for information.hat, and many that have it.o, chaos theory is not just differential analysis. ItÕs the studynd theories of explanations of systems that exhibit varying behaviorased upon input conditions, with extrema. So, learning about chaosheory, besides finding a damned definition of chaos theory in theirst place, involves crunchy differential analysis, fractalnalysis, and asymptotic and otherwise mathematically tractablenterpretations of many problems for which those expansive methodsffer solutions. Also, back in the 80Õs, it meant understoodroblems for which the solutions hadnÕt been developed yet, eg, annknown integral form. A lot has changed in differential analysisnd as well fractal analysis in the past twenty-five years or so, asompared to geometry, unless you consider for example new models ofeometry.oss F.==ubject: Re: The State-of-the-Art in Mathematics-Spam-This: SpamCopies@YahooGroups.Com The complex number i is nonsense since i = sqrt(-1) = sqrt(1/-1) = 1/i ...efore anyone can make any sense out of such equations, you need toefine precisely what you mean by the sqrt function. Please provide usith such a definition before expecting us to respond to anything youost.he normal definition of sqrt as a single-valued function is as follows:or positive real argument r, itÕs the positive solution to x^2 = r.or negative real argument r, itÕs i times the positive solution to x^2 = -r.or complex argument a + b*i, each a,b real, where b is nonzero,tÕs one of the two solutions to (x + y*i)^2 = (a + b*i) where y>0.ote that with that definition of sqrt,he rule sqrt(a*b) = sqrt(a) * sqrt(b) is not generally true.ikewise itÕs not generally true that sqrt(a/b) = sqrt(a) / sqrt(b),hich you seem to have mistakenly assumed in your equation chain.o either that wasnÕt the definition of sqrt you were using (hence myuestion what definition you are using), or you made a stupid mistakessuming those rules are generally true when in fact they arenÕt.==ubject: Re: SVD Algorithmm 15.12.04 00:23 schrieb Luigi Napolitano:>An alternative would be to create an indexing array and sort that. Fill >the indexing array like this:>The values w[idx[1 ...n]] are then in decending order. The original w[1 >... n] values are unchanged, and there is no need to re-arrange the U and >V matricies. - W with elements in descending order; - U re-arrange; - V re-arrange. So: U (re-arrange) * W (descending) * V transpose (re arrange) = A. (I will take the first columns of U, the first rows of V transpose, the first elements of W.) Any idea?ust do an ndirectsort, which provides an index for your array,long which you can access the elements of the array in sortedannerindex:=indirectSort(diagonal(W))hen access indirectly:U[index[i],j], W[index[i],index[i]], V[j,index[i]]our singular-values-matrix and the other matricesia the same index..Helms==ubject: Re: SVD Algorithm-RFC2646: Format=Flowed; OriginalGottfried Helms ha scritto nel messaggio Just do an ndirectsort, which provides an index for your array, along which you can access the elements of the array in sorted manner index:=indirectSort(diagonal(W)) then access indirectly: U[index[i],j], W[index[i],index[i]], V[j,index[i]] your singular-values-matrix and the other matrices via the same index.erfect!tÕs a good idea, really.uigi Napolitano ==ubject: Re: SVD Algorithm> ik T. Winter ha scritto nel messaggio > What is the problem with that? Just sort them and make appropriate> interchanges to the U and V matrices as well.> I could sort the Singular Value Matrix (W is a vector).> But, How to interchange the columns of U and the rows of V?> Is there an algorithm?uring sorting you have to interchange singular values. If, when doinguch an interchange, you also interchange the appropriate rows of U and. But this is more a programming question than a mathematical question.n most cases the matrices one of the orders (n or m) is pretty small.n that case a simple sorting algorithm is appropriate. Find the largestingular value (say K) and integerchange with W[1]. Also interchangeows 1 and K of U and V. Next go on with the largest singular valuef W[2..N]. This is much less inefficient than it appears. Thenterchanges in U and V are mostly needed anyhow, and they are theulk of the process.- ik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131ome: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/==ubject: Re: New number ETAsAhRYJxJtY8d4ZxeaXu3/0gctBIZ+ rQIUHXYHpQ7QudAEuILJu22wZo7UfKk= snÕt there an upside down A symbol used ofr llin logicalxpressions? Example: Ax in R, x^2 >= 0 with that A really being anpside-down A.-OL==ubject: Re: New number IsnÕt there an upside down A symbol used ofr llin logical expressions? Example: Ax in R, x^2 >= 0 with that A really being an upside-down A. --OLi Oscar,hey call that the universal quantifier.ormally itÕs read as or anyor or all ThereÕs some differencen those two interpretations, and thereÕs some vagueness in theireanings. For example, when you say or any you might think thateans or one or more or, or any single one.Similarly with orll that might be interpreted as or eachor or all at once thiss all about the universal quantifier, or as some say, the generaluantifier.hose vague interpretations, while sometimes applicable and useful, areot in the general case. For example, the set of all singletons is nottself a singleton. For each element, that element is a singleton, forll of them together as a set, it is not.hile that is so, there are some considerations where or allmeans asell or each and everyand also or the collection of each andvery. ThatÕs most often so in the particularly set-theoretic.t is normally the case that or anymeans or each and everyarticular single one and NOT or all at once together. hen, the backwards E is the existential quantifier. That basicallyeads or at least oneor here exists ItÕs not generallytherwise or at most oneoss F.==ubject: Re: New number > Here is an interesting concept: > This is a new number representing the ppositeof zero, but not > exactly infinity. I call the number lland it is written as a > filled in zero (or a zero with a dot in the middle for convenience). > It exists on the basis that in an infinite universe, all possibilities > that can exist must exists, therefore when all the possibilities do > exist the number is All Wow! That is such an amazing discovery that I have to divulge my discovery of a new number that is the ppositeof three, but itÕs not exactly infinity! I call it eerht, and it is written using the numeral 3, only setting it on its side atop a zero, and drawing a smiley face inside the zero portion of eerht. Any time I have a problem where the solution isnÕt really three, and it isnÕt exactly infinity, I just yell out EERHT! and draw the numeral for eerht on something, or IÕll just draw it in the air! This math research is so much fun! > . I am probably not the first to think of this, but I am sure some fun > could be had with the concept. IÕm going to invent a number that is written by throwing a handful of confetti! And another that uses one of those blowy things where a paper thing rolls out and makes a squawking sound! DaleAn obscure planet of the S-K system, your majesty, itÕs inhabitants refero it as the planet ÔÕEarthÕÕ. ey hatchet man, you should research HegelÕs eingand othing. WhatouÕre talking about there is the ig infinity. ood luck,oss Finlayson==ubject: Re: New number posting-account=IgDbbA0AAACoESZw-tVujzasm3CdWTai see it is the time to invent something opposit infinity, like...mmmm... finity!!! :))))) not funny==ubject: Re: New number i see it is the time to invent something opposit infinity, like... hmmmm... finity!!! :))))) not funnys too!==ubject: Re: Why did you choose math?-CompuServe-Customer: Yes-Coriate: interspeed.co.nz-Ecrate: tanandtanlawyers.com-Pose: George Cox-Punge: Micro$oft-Sanguinate: The MVS Guy-Terminate: SPA(GIS)-Tinguish: Mark Griffith-Treme: C&C,DWS at 02:12 PM, delta01211@yahoo.com (delta01211) said:I would like to know the reason you chose mathematics.ecause itÕs fun.I am pondering about whether I choose physics or mathematics. hat are your interests?To me, doing math can really make my personality more unique.But a problem with concentrating on math is that I might not learn anything by reading a book independently.hat would be a problem in any academic discipline.I tried Principles of Mathematical Analysis, which, according to a review in Amazon com, seems very famous. hich? Apostol? Rudin?I somewhat understood it, but I can not remember much.id you do the exercises?Another problem is that if I become a complete math geek, my social life will be gone forever.hat says more about you than about Mathematics.As for physics, if I were to study physics independently, my planwould be to read a calculus-based textbook and get a sense ofphysics.hat would be only the beginning of the beginning.A problem with this is that this is not too progressive.ou need to walk before you can run. The basic Physics is needed inrder for you to understand the more advanced Physics.- hmuel (Seymour J.) Metz, SysProg and JOAT nsolicited bulk E-mail subject to legal action. I reserve theight to publicly post or ridicule any abusive E-mail. Reply toomain Patriot dot net user shmuel+news to contact me. Do noteply to math? at 02:12 PM, delta01211@yahoo.com (delta01211) said:>I would like to know the reason you chose mathematics.ecuase IÕm not good enough at anything else (or perhaps it chose me)-irascible since 1957==ubject: Re: Santa Claus is the god of math !!!!n sci.math, Thomas Nordhausthnord2002@yahoo.de>lpsur01fiklo5j3hct2bioegnsfun8ts68 @4ax.com>:>YouÕre just a bunch of wining idiots who only make complaints and ^^^^^^ I think what you mean is hining- very few people that are wining start whining. That usually happens if youÕre beering.ersonally, IÕd prefer winning. :-)rest snipped]- 191, ewill3@earthlink.nettÕs still legal to go .sigless.==ubject: Re: Santa Claus is the god of math !!!! <3297phF3jelotU1@individual.net> <329a96F3ikobsU1@individual.net> posting-account=_-j7cgwAAADnQK9-r68QgRsgfV-jhA3Anly an idiot or a philosopher would take an opposing point of view onomething trivial like CantorÕs proof of the uncountability of theeals. Meanwhile, the mathematicians out there are *creating*athematics. Nobody taught Wiles how to prove FLT.cid Ôooh==ubject: Re: Santa Claus is the god of math !!!! Only an idiot or a philosopher would take an opposing point of view on something trivial like CantorÕs proof of the uncountability of the reals. Meanwhile, the mathematicians out there are *creating* mathematics. Nobody taught Wiles how to prove FLT.ou completely lose the preceding argument, so why not demonstratehat your actual method is aswell, in a completely unrelated argument.antor and Wiles are both accepted because they were academics andany people got the fruit of the labour. There is no scientific method inaccepting Xand isallowing Yow the algebra is there its *trivial*.. 1. L(a,a) = L(a,a) (obvious) 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a) 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2) 4a. not(exists a, forall b, L(a,b) != L(b,b)) (negation of 3) 4b. not(exists a, forall b, L(a,b) = !L(b,b)) (! is some suitable digit change function) 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r(b)= !L(b,b,))efore it was a rugged foundation of elaborate formal systems pages and pages long.nfinite people each toss coins infintie times> all sequences to infinite length have been tossed.nly an idiot would claim otherwise. 10 year olds can see that.erc==ubject: Re: Santa Claus is the god of math !!!! <3297phF3jelotU1@individual.net> <329a96F3ikobsU1@individual.net> <329gvrF234pg2U1@individual.net> posting-account=_-j7cgwAAADnQK9-r68QgRsgfV-jhA3A > Only an idiot or a philosopher would take an opposing point of viewn > something trivial like CantorÕs proof of the uncountability of the > reals. Meanwhile, the mathematicians out there are *creating* > mathematics. Nobody taught Wiles how to prove FLT. you completely lose the preceding argument, so why not demonstrate what your actual method is aswell, in a completely unrelatedrgument.h my, that was quite arbitrary. But itÕs clear you missed the point.K -- who taught Galois how to prove his correspondence theorem? (Hint- nobody did.) ThatÕs a pretty gosh darned constructive proof. Whoaught Godel how to prove the incompleteness theorem? (Hint -- nobodyid.) ThatÕs a pretty gosh darned constructive proof. Cantor and Wiles are both accepted because they were academics and many people got the fruit of the labour. There is no scientificethod in ccepting Xand isallowing Yamn straight there isnÕt. Mathematics isnÕt open to empiricalerification in any meaningful sense. What does matter is consistencynd interestingness. Now the algebra is there its *trivial*.. 1. L(a,a) = L(a,a) (obvious) 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a) 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2) 4a. not(exists a, forall b, L(a,b) != L(b,b)) (negation of 3) 4b. not(exists a, forall b, L(a,b) = !L(b,b)) (! is some suitableigit change function) 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r(b)=L(b,b,))his notation is horrible. What is L? A function on two variables?here do a and b live? Why use ! and not as negation? (Keep in mindhat I donÕt follow your posts religiously -- heh heh) Before it was a rugged foundation of elaborate formal systems pagesnd pages long. Infinite people each toss coins infintie times => all sequences to infinite length have been tossed.ot really. If the coins are fair, every single person could tosseads every single time. Only an idiot would claim otherwise. 10 year olds can see that.elow me. cid Ôooh==ubject: Re: Santa Claus is the god of math !!!! > Only an idiot or a philosopher would take an opposing point of view on > something trivial like CantorÕs proof of the uncountability of the > reals. Meanwhile, the mathematicians out there are *creating* > mathematics. Nobody taught Wiles how to prove FLT. > you completely lose the preceding argument, so why not demonstrate > what your actual method is aswell, in a completely unrelated argument. Oh my, that was quite arbitrary. But itÕs clear you missed the point. OK -- who taught Galois how to prove his correspondence theorem? (Hint -- nobody did.) ThatÕs a pretty gosh darned constructive proof. Who taught Godel how to prove the incompleteness theorem? (Hint -- nobody did.) ThatÕs a pretty gosh darned constructive proof.he topic is accepting peoples proof, not constructing them, namelyaranormal proofs and use of scientific method. > Cantor and Wiles are both accepted because they were academics and > many people got the fruit of the labour. There is no scientific method in > ccepting Xand isallowing Y Damn straight there isnÕt. Mathematics isnÕt open to empirical verification in any meaningful sense. What does matter is consistency and interestingness.o, what matters to you is the label attached to the proof, and some rudimentaryine by line es that looks ok > Now the algebra is there its *trivial*.. > 1. L(a,a) = L(a,a) (obvious) > 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a) > 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2) > 4a. not(exists a, forall b, L(a,b) != L(b,b)) (negation of 3) > 4b. not(exists a, forall b, L(a,b) = !L(b,b)) (! is some suitable digit change function) > 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r(b)= !L(b,b,)) This notation is horrible. What is L? A function on two variables? Where do a and b live? Why use ! and not as negation? (Keep in mind that I donÕt follow your posts religiously -- heh heh)he 1st 3 lines are DarylÕs, since people complained about my notation.gain you judge by the person, not the substance. > Before it was a rugged foundation of elaborate formal systems pages and pages long. > Infinite people each toss coins infintie times > => all sequences to infinite length have been tossed. Not really. If the coins are fair, every single person could toss heads every single time.his is avoiding the issue which is only simplified so you can follow it.f a finite length sequence is not present given infinite resources itÕs not random. > Only an idiot would claim otherwise. 10 year olds can see that. Below me.ndergrads are stupid, the net will be a swampland when 100,000 versions ofou come here blithering your parrot talk.erc==ubject: Re: Santa Claus is the god of math !!!! <3297phF3jelotU1@individual.net> <329a96F3ikobsU1@individual.net> <329gvrF234pg2U1@individual.net> <32a1bkF3jql54U1@individual.net> posting-account=_-j7cgwAAADnQK9-r68QgRsgfV-jhA3A > Only an idiot or a philosopher would take an opposing point ofiew > on > something trivial like CantorÕs proof of the uncountability ofhe > reals. Meanwhile, the mathematicians out there are *creating* > mathematics. Nobody taught Wiles how to prove FLT. > you completely lose the preceding argument, so why notemonstrate > what your actual method is aswell, in a completely unrelated > argument. > Oh my, that was quite arbitrary. But itÕs clear you missed theoint. > OK -- who taught Galois how to prove his correspondence theorem?Hint > -- nobody did.) ThatÕs a pretty gosh darned constructive proof.ho > taught Godel how to prove the incompleteness theorem? (Hint --obody > did.) ThatÕs a pretty gosh darned constructive proof. The topic is accepting peoples proof, not constructing them, namely paranormal proofs and use of scientific method.aranormal proof? I think youÕve just coined a new phrase in theogical lexicon. What, praytell, is a aranormal proofut since you feel that you can set my agenda, let me remind you thatou accused us math guys of being parrots. A parrot canÕt come up with new proof, now can he? That is exactly what my examples illustrate.ou miss the point yet again. > Cantor and Wiles are both accepted because they were academicsnd > many people got the fruit of the labour. There is no scientific > method in > ccepting Xand isallowing Y > Damn straight there isnÕt. Mathematics isnÕt open to empirical > verification in any meaningful sense. What does matter isonsistency > and interestingness. No, what matters to you is the label attached to the proof, and someudimentary line by line es that looks ok > Now the algebra is there its *trivial*.. > 1. L(a,a) = L(a,a) (obvious) > 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a) > 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2) > 4a. not(exists a, forall b, L(a,b) != L(b,b)) (negation of 3) > 4b. not(exists a, forall b, L(a,b) = !L(b,b)) (! is someuitable > digit change function) > 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r(b)= > !L(b,b,)) > This notation is horrible. What is L? A function on twoariables? > Where do a and b live? Why use ! and not as negation? (Keep inind > that I donÕt follow your posts religiously -- heh heh) the 1st 3 lines are DarylÕs, since people complained about myotation. again you judge by the person, not the substance.f you quoted Daryl verbatim, his notation is horrible. You need toexplain_ what these symbols are supposed to be, otherwise itÕs just atring of gobbledygook. > Before it was a rugged foundation of elaborate formal systemsages > and pages long. > Infinite people each toss coins infintie times > => all sequences to infinite length have been tossed. > Not really. If the coins are fair, every single person could toss > heads every single time. this is avoiding the issue which is only simplified so you can followt. if a finite length sequence is not present given infinite resourcestÕs not random.hatÕs bull. Like I said, every single person could toss headsvery single time *using fair coins.* > Only an idiot would claim otherwise. 10 year olds can see that. > Below me. undergrads are stupid, the net will be a swampland when 100,000ersions of you come here blithering your parrot talk.Õm 100% unique, baby. DonÕt act like you know anything about me, youutjob.cid Ôooh==ubject: Re: Santa Claus is the god of math !!!! Paranormal proof? I think youÕve just coined a new phrase in the logical lexicon. What, praytell, is a aranormal proof > There is NOTHING that Santa cannot prove !!!! > Wish for a proof for Christmas and youÕll get it! We already had a thread discussing acceptance of GodÕs proofs in this group recently. I vainly refer tohatÕs what I joined the thread to discuss, you turned it into the 225thantor VS Idiot thread. But since you feel that you can set my agenda, let me remind you that you accused us math guys of being parrots. A parrot canÕt come up with a new proof, now can he? That is exactly what my examples illustrate. You miss the point yet again.he point is will a new proof be missed, not will an old proof be accepted. > 1. L(a,a) = L(a,a) (obvious) > 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a) > 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2) > 4a. not(exists a, forall b, L(a,b) != L(b,b)) (negation of 3) > 4b. not(exists a, forall b, L(a,b) = !L(b,b)) (! is some If you quoted Daryl verbatim, his notation is horrible. You need to _explain_ what these symbols are supposed to be, otherwise itÕs just a string of gobbledygook.o its machine parsable actually, youÕre just ingnorant and like to abuse peoplend complain ad infinitum > Before it was a rugged foundation of elaborate formal systems pages > and pages long. > Infinite people each toss coins infintie times > => all sequences to infinite length have been tossed. > Not really. If the coins are fair, every single person could toss > heads every single time. > this is avoiding the issue which is only simplified so you can follow it. > if a finite length sequence is not present given infinite resources itÕs not random. ThatÕs bull. Like I said, every single person could toss heads every single time *using fair coins.*hen rephrase the question, can you *always* find a new sequence,ork on the worst case.gain : a missing finite length string is improbable GET IT?herefore, you CANT find a new sequence at all, inverting ßips is completely useless. > Only an idiot would claim otherwise. 10 year olds can see that. > Below me. > undergrads are stupid, the net will be a swampland when 100,000 versions of > you come here blithering your parrot talk. IÕm 100% unique, baby. DonÕt act like you know anything about me, you nutjob.eople with 18 months of reading text under their belt trying to run the newsgroups.ou contribute boring rubbish thatÕs a direct paraphrase of a text book, thinking youan knock new approaches and new theories which is what the dynamic usenet mediums for... to get away from likes of you. newsgroups should be banned from undergrads.erc==ubject: Re: Santa Claus is the god of math !!!! <3297phF3jelotU1@individual.net> <329a96F3ikobsU1@individual.net> <329gvrF234pg2U1@individual.net> <32a1bkF3jql54U1@individual.net> <32ak57F3jh09pU1@individual.net> posting-account=_-j7cgwAAADnQK9-r68QgRsgfV-jhA3A > Paranormal proof? I think youÕve just coined a new phrase in the > logical lexicon. What, praytell, is a aranormal proof > There is NOTHING that Santa cannot prove !!!! > Wish for a proof for Christmas and youÕll get it! We already had a thread discussing acceptance of GodÕs proofs inhis group recently. I vainly refer to thatÕs what I joined the thread to discuss, you turned it into the25th Cantor VS Idiot thread. > But since you feel that you can set my agenda, let me remind youhat > you accused us math guys of being parrots. A parrot canÕt come upith > a new proof, now can he? That is exactly what my examplesllustrate. > You miss the point yet again. The point is will a new proof be missed, not will an old proof beccepted. > 1. L(a,a) = L(a,a) (obvious) > 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a) > 3. forall a, exists b, L(a,b) = L(b,b) (generalization of) > 4a. not(exists a, forall b, L(a,b) != L(b,b)) (negationf 3) > 4b. not(exists a, forall b, L(a,b) = !L(b,b)) (! is some > If you quoted Daryl verbatim, his notation is horrible. You needo > _explain_ what these symbols are supposed to be, otherwise itÕsust a > string of gobbledygook. no its machine parsable actually, youÕre just ingnorant and like tobuse people and complain ad infinitumt just might be machine parsable, but only because the machinenderstands the syntax well enough to parse it. Until I know what L isupposed to be, I canÕt. > Before it was a rugged foundation of elaborate formal systems > pages > and pages long. > Infinite people each toss coins infintie times > => all sequences to infinite length have been tossed. > Not really. If the coins are fair, every single person couldoss > heads every single time. > this is avoiding the issue which is only simplified so you canollow > it. > if a finite length sequence is not present given infiniteesources > itÕs not random. > ThatÕs bull. Like I said, every single person could toss heads > every single time *using fair coins.* Then rephrase the question, can you *always* find a new sequence, work on the worst case. Again : a missing finite length string is improbable GET IT?mprobable and impossible are distinct concepts. Therefore, you CANT find a new sequence at all, inverting ßips isompletely useless.mprobable and impossible are distinct concepts. A *lot* of improbablehings happen every day. GET IT? > Only an idiot would claim otherwise. 10 year olds can seehat. > Below me. > undergrads are stupid, the net will be a swampland when 100,000 > versions of > you come here blithering your parrot talk. > IÕm 100% unique, baby. DonÕt act like you know anything about me,ou > nutjob. people with 18 months of reading text under their belt trying to runhe newsgroups. you contribute boring rubbish thatÕs a direct paraphrase of a textook, thinking you can knock new approaches and new theories which is what the dynamicsenet medium is for... to get away from likes of you. newsgroups should beanned from undergrads. Hercesus christ herc! Some might say the same about the mentally ill.cid Ôooh==ubject: Re: Santa Claus is the god of math !!!! > Again : a missing finite length string is improbable GET IT? Improbable and impossible are distinct concepts. > Therefore, you CANT find a new sequence at all, inverting ßips is completely useless. Improbable and impossible are distinct concepts. A *lot* of improbable things happen every day. GET IT?o, you havenÕt made a point.ctually IÕm pretty sure you donÕt have the faculties to comprehend what youre commenting on, hence the syntactic feedback.erc==ubject: Re: Santa Claus is the god of math !!!! <3297phF3jelotU1@individual.net> <329a96F3ikobsU1@individual.net> <329gvrF234pg2U1@individual.net> <32a1bkF3jql54U1@individual.net> <32ak57F3jh09pU1@individual.net> <32c22kF3jsi6rU1@individual.net> posting-account=_-j7cgwAAADnQK9-r68QgRsgfV-jhA3AouldnÕt god already know the answer to that? Can you make a post soudicrous even you canÕt read it?cid Ôooh==ubject: Re: Santa Claus is the god of math !!!! undergrads are stupid, the net will be a swampland when 100,000 versions of you come here blithering your parrot talk. Hercf course youÕre OK A.P., IÕm just planning for the onslaughterc==ubject: Quick Orthagonal matrix questions | 0 0 1 1 | = | 1 1 0 0 | | 1 -1 0 0 | | 0 0 (1 - sqrt(5))/2 (1 + sqrt(5))/2|n orthagonal matrix?---------------------------------------------* Binary Usenet Leeching Made Easy* http://www.newsleecher.com/?usenet--------------------------- ------------------==ubject: Re: Quick Orthagonal matrix question Is | 0 0 1 1 | U = | 1 1 0 0 | | 1 -1 0 0 | | 0 0 (1 - sqrt(5))/2 (1 + sqrt(5))/2| an orthagonal matrix? ---------------------------------------------- * Binary Usenet Leeching Made Easy * http://www.newsleecher.com/?usenet ----------------------------------------------o you know what the definition of an orthogonal matrix is? Can you heck if this matrix satisfies it?==ubject: Re: Quick Orthagonal matrix question Is | 0 0 1 1 | U = | 1 1 0 0 | | 1 -1 0 0 | | 0 0 (1 - sqrt(5))/2 (1 + sqrt(5))/2| an orthagonal matrix? ---------------------------------------------- * Binary Usenet Leeching Made Easy * http://www.newsleecher.com/?usenet ----------------------------------------------Do you know what the definition of an orthogonal matrix is? Can you check if this matrix satisfies it?hoops on last reply wasnÕt thinking..... :ah I know the definition, and IÕm pretty confident that it is a rthogonal set, but not a orthonormal set. I just wanted to ouble check because itÕs an important assignment.---------------------------------------------* Binary Usenet Leeching Made Easy* http://www.newsleecher.com/?usenet--------------------------- ------------------==ubject: Re: Quick Orthagonal matrix question> Is> | 0 0 1 1 |> U = | 1 1 0 0 |> | 1 -1 0 0 |> | 0 0 (1 - sqrt(5))/2 (1 + sqrt(5))/2|> an orthagonal matrix?> ----------------------------------------------> * Binary Usenet Leeching Made Easy> * http://www.newsleecher.com/?usenet> ---------------------------------------------->Do you know what the definition of an orthogonal matrix is? Can>you>check if this matrix satisfies it? Whoops on last reply wasnÕt thinking..... : Yah I know the definition, and IÕm pretty confident that it is a orthogonal set, but not a orthonormal set. I just wanted to double check because itÕs an important assignment.o, itÕs not orthogonal.- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: Re: Quick Orthagonal matrix question> Is> | 0 0 1 1 |> U = | 1 1 0 0 |> | 1 -1 0 0 |> | 0 0 (1 - sqrt(5))/2 (1 + sqrt(5))/2|> an orthagonal matrix?> ----------------------------------------------> * Binary Usenet Leeching Made Easy> * http://www.newsleecher.com/?usenet> ---------------------------------------------->Do you know what the definition of an orthogonal matrix is? Can>you>check if this matrix satisfies it? Whoops on last reply wasnÕt thinking..... : Yah I know the definition, and IÕm pretty confident that it is a orthogonal set, but not a orthonormal set. I just wanted to double check because itÕs an important assignment.es, it is orthogonal.- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: Re: Quick Orthagonal matrix questionah I know the deffinition, I donÕt think it is, so I have to ivide by its length and redo some steps... but I wanted to double y interpretation of the deffinition before I do a lot of algebra.---------------------------------------------* Binary Usenet Leeching Made Easy* http://www.newsleecher.com/?usenet--------------------------- ------------------==ubject: Ring Isomorphism am having trouble with what I thought would be an easy exercise ( I amreparing for my final)onsider the map M : Z_2 + Z_3--> Z_6 given by M ( ( [a]_2 , [b]_3) ) = [3a 4b] _6 am having trouble showing that M is one to one.I started off with supposing that M ( ( [a]_2 , [b]_3) ) = M ( ( [a]_2 ,bÕ ]_3) ).hen [3a + 4b] _6 = [3a+ 4bÕ] _6parently I am not sure what to do from here.lso donÕt I have to show that M is well defined?ny help would be appreciated.kajuma==ubject: Re: Ring Isomorphism I am having trouble with what I thought would be an easy exercise ( I am preparing for my final) Consider the map M : Z_2 + Z_3--> Z_6 given by M ( ( [a]_2 , [b]_3) ) = [3a + 4b] _6 I am having trouble showing that M is one to one. I started off with supposing that M ( ( [a]_2 , [b]_3) ) = M ( ( [a]_2 , [b]_3) ). Then [3a + 4b] _6 = [3a+ 4bÕ] _6 Aparently I am not sure what to do from here. Also donÕt I have to show that M is well defined? Any help would be appreciated. MkajumaereÕs a simple proof that Z_a X Z_b is (ring) isomorphic to Z_abf gcd(a,b) = 1.. Consider the mapping f: Z->Z_a X Z_b given by f(n) =([n]_a,[n]_b).. Show f is a homomorphism with kernel (ab).. Hence Z_ab = Z/(ab) is isomorphic to the image of f.. But Z_ab and Z_a X Z_b contain the same number of elements,o the isomorphism in 3 is actually onto.==ubject: Re: Ring Isomorphism Consider the map M : Z_2 + Z_3--> Z_6 given by M ( ( [a]_2 , [b]_3) ) = [3a + 4b] _6 I am having trouble showing that M is one to one.his may not be what the solution is looking for, but for maps of verymall groups, surjectivity (onto-ness) and injectivity (1-1-ness) areasily checked and proven simply by listing the elements and what they mapo one by one.his will also make well-definedness pretty clear.ustin==ubject: Re: Ring Isomorphism I am having trouble with what I thought would be an easy exercise ( I am preparing for my final) Consider the map M : Z_2 + Z_3--> Z_6 given by M ( ( [a]_2 , [b]_3) ) = [3a + 4b] _6 I am having trouble showing that M is one to one. I started off with supposing that M ( ( [a]_2 , [b]_3) ) = M ( ( [a]_2 , [b]_3) ). Then [3a + 4b] _6 = [3a+ 4bÕ] _6 Aparently I am not sure what to do from here. Also donÕt I have to show that M is well defined? Any help would be appreciated. Mkajumaint 1 : when is a *group* *homomorphism* one to one? (This assumes thatour course has already covered some elementary group theory, which may oray not be the case.)int 2 : it is actually enough to show that the map is onto (do you seehy?).D==ubject: Re: Ring Isomorphism (This assumes that your course has already covered some elementary group theory, which may or may not be the case.)illy me: of *course* it must be the case, given the title of your post.D==ubject: Re: Ring Isomorphism> (This assumes that> your course has already covered some elementary group theory, which may or> may not be the case.) Silly me: of *course* it must be the case, given the title of your post.onÕt some algebra courses cover rings before groups?- ave Seamanudge YohnÕs mistakes revealed in Mumia Abu-Jamal ruling.http://www.commoncouragepress.com/index.cfm?action= book&bookid=228>==ubject: Re: Ring Isomorphism>> (This assumes that>> your course has already covered some elementary group theory, which may>> or may not be the case.)> Silly me: of *course* it must be the case, given the title of your post. DonÕt some algebra courses cover rings before groups?es, but then my hint remains valid in a modified form: when is a *ring*omomorphism one to one?D==ubject: Re: eXtreme Proving? (math research processes) posting-account= y3wZYhMAAABYsCtaDBjCWE5oFd14ElQZbfvQjxC1czdFUKdrfKUl4g suspect mathematicians were air provingin the sense of using onenother as sounding boards to find gaps in proposed proof attempts longefore computers needed programming.==ubject: Re: eXtreme Proving? (math research processes)...about how mathematicians work]As to working mathematicians trying out alternatives, sure thereÕs lots of alternatives: long walks, chemical enhancement, conferences and workshops, email, napkins at coffeehouses, computer experimentation.Most mathematicians are highly computer literatehatÕs not my experience. Take enough long walks and write on enough napkinsnd you wonÕt have a lot of time left to practice writing ello worldave==ubject: Cheaters Definition of Trigonometric Functions. posting-account=y_8zEgwAAADgSV7JOUCWfc4H-PfoU2q_ quote an Appendix of a calculus textbook have.Let U be a unit circle, that is, a circlef radius 1, with center at the origin of rectangular coordinate system, and Let be the point with coordinates (1,0).f t is any number between 0 and 2PI, thens illustrated in Figure III.1 there isrecisely one point P(x,y) on U such thathe length of arc AP measured in theounterclockwise direction from A is t. think your textbooks have also similarescriptions to start the definition ofrigonometric functions. Or worse, noefinition is given at all.hen I was a college student, the definitionf trigonometric functions based on a similarlaim without proof was a stumbling block.ell me if you have more rigorous but alsontuitive introduction (not using power seriesut using an improper definite integral)n your textbook? I found and bought onen Japanese after great efforts, but do not have it now. akao==ubject: Re: CheatersDefinition of Trigonometric Functions. I quote an Appendix of a calculus textbook I have. et U be a unit circle, that is, a circle of radius 1, with center at the origin of a rectangular coordinate system, and Let A be the point with coordinates (1,0). If t is any number between 0 and 2PI, then as illustrated in Figure III.1 there is precisely one point P(x,y) on U such that the length of arc AP measured in the counterclockwise direction from A is t. hen first introducing trig to students, I start with the definition ased on triangles and then show how the circle definition (as started bove) is a natural extension of it. I think your textbooks have also similar descriptions to start the definition of trigonometric functions. Or worse, no definition is given at all. When I was a college student, the definition of trigonometric functions based on a similar claim without proof was a stumbling block.efinitions do not have proofs. They simply are. Perhaps you are xpecting too much from the defining aspects of these functions. Tell me if you have more rigorous but also intuitive introduction (not using power series but using an improper definite integral) in your textbook? I found and bought one in Japanese after great efforts, but I do not have it now. ower series and improper integrals are the things you need to prove are quivalent to the definition you started to quote. As far as rigor is oncerned, where do you find a lack of rigor in the complete definition which was not included)?- ill Twentymanmail: wtwentyman at copper dot net==ubject: Re: Cheaters Definition of Trigonometric Functions. I quote an Appendix of a calculus textbook I have. et U be a unit circle, that is, a circle of radius 1, with center at the origin of a rectangular coordinate system, and Let A be the point with coordinates (1,0). If t is any number between 0 and 2PI, then as illustrated in Figure III.1 there is precisely one point P(x,y) on U such that the length of arc AP measured in the counterclockwise direction from A is t. I think your textbooks have also similar descriptions to start the definition of trigonometric functions. Or worse, no definition is given at all. When I was a college student, the definition of trigonometric functions based on a similar claim without proof was a stumbling block. Tell me if you have more rigorous but also intuitive introduction (not using power series but using an improper definite integral) in your textbook? I found and bought one in Japanese after great efforts, but I do not have it now. TakaoWhat you quoted is apparently the beginning of a longer story,eading to the definition of sin(t) as the x-coordinate of theoint P(x,y) and so on.)ow: what is intuitive to some may be not intuitive to others.he kinematic approach goes like this:A point starts traveling from (1,0) in such a way that its velocityector is obtained from its position vector by rotating counterclockwisey a right angle. (Remember Euclid? All right angles are equal.)We sort of anticipate that the motion will be circular about (0,0) atonstant speed, but we need not assume it. It will pop out of theathematics of the situation.Now put it into equations: let (x(t), y(t)) be the position at time t,henxÕ(t) = -y(t)yÕ(t) = x(t)ith initial conditions x(0)=1, y(0)=0.Before you protest that I am making an unwarranted excursion into systemsf differential equations, let me obtain some elementary conclusions,ssuming that the system does have a solution:F(t) = (x(t))^2 + (y(t))^2 must be constant, namely 1.hy? Write down FÕ(t) and simplify, using the defining equations. Youill find FÕ(t)=0 everywhere, F(0)=1.We do have circular motion: center (0,0), radius 1.Now get the square of the speed: (xÕ(t))^2 + (yÕ(t))^2it will also be 1).hen give x(t), y(t) their traditional names, cos(t) and sin(t),espectively.hat is still hanging in the air: the existence of those x(t), y(t). Thateed not have anything to do with intuition; power series or some integralalculus trick can establish that (first get arcsine by finding out thatt/dy = 1/sqrt(1-y^2) in a small enough interval,...) bonus: the addition formulas follow from the differential properties:or example, cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b) will follow from(d/dt) (cos(a+b-t)*cos(t) - sin(a+b-t)*sin(t)) = 0,o the inside is a constant, and has the same value for t=0 as for t=b.We know that sin(0)=0 and cos(0)=1.)nd similarly for sin(a+b).==ubject: Re: CheatersDefinition of Trigonometric Functions. posting-account=y_8zEgwAAADgSV7JOUCWfc4H-PfoU2q_ > I quote an Appendix of a calculus textbook > I have. > et U be a unit circle, that is, a circle > of radius 1, with center at the origin of > a rectangular coordinate system, and Let > A be the point with coordinates (1,0). > If t is any number between 0 and 2PI, then > as illustrated in Figure III.1 there is > precisely one point P(x,y) on U such that > the length of arc AP measured in the > counterclockwise direction from A is t. > I think your textbooks have also similar > descriptions to start the definition of > trigonometric functions. Or worse, no > definition is given at all. > When I was a college student, the definition > of trigonometric functions based on a similar > claim without proof was a stumbling block. > Tell me if you have more rigorous but also > intuitive introduction (not using power series > but using an improper definite integral) > in your textbook? I found and bought one > in Japanese after great efforts, but > I do not have it now. > Takao (What you quoted is apparently the beginning of a longer story, leading to the definition of sin(t) as the depends on this unproven assumption. Now: what is intuitive to some may be not intuitive to others. The kinematic approach goes like this: A point starts traveling from (1,0) in such a way that its velocity vector is obtained from its position vector by rotatingounterclockwise by a right angle. (Remember Euclid? All right angles are equal.) We sort of anticipate that the motion will be circular about (0,0)t constant speed, but we need not assume it. It will pop out of the mathematics of the situation. Now put it into equations: let (x(t), y(t)) be the position at time, then xÕ(t) = -y(t) yÕ(t) = x(t) with initial conditions x(0)=1, y(0)=0. Before you protest that I am making an unwarranted excursion intoystems of differential equations, let me obtain some elementary conclusions, assuming that the system does have a solution: F(t) = (x(t))^2 + (y(t))^2 must be constant, namely 1. Why? Write down FÕ(t) and simplify, using the defining equations.ou will find FÕ(t)=0 everywhere, F(0)=1. We do have circular motion: center (0,0), radius 1. Now get the square of the speed: (xÕ(t))^2 + (yÕ(t))^2 (it will also be 1). Then give x(t), y(t) their traditional names, cos(t) and sin(t), respectively. What is still hanging in the air: the existence of those x(t), y(t).hat need not have anything to do with intuition; power series or somentegral calculus trick can establish that (first get arcsine by finding outhat dt/dy = 1/sqrt(1-y^2) in a small enough interval,...) A bonus: the addition formulas follow from the differentialroperties: for example, cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b) will followrom (d/dt) (cos(a+b-t)*cos(t) - sin(a+b-t)*sin(t)) = 0, so the inside is a constant, and has the same value for t=0 as for=b. (We know that sin(0)=0 and cos(0)=1.) And similarly for sin(a+b).o do you agree on the deficiency of the common practicen our textbooks? And do you propose that a trigonometricunction should be defined as a solution of a system ofifferential equations? For a textbook we should satisfyoth the logical rigor and intuitive understanding.s long as you satisfy these conditions I am not againstour proposal. think we should start the definition of trigonometricunctions, for example, fromos^{-1}(x) Integral_{t}^{1} 1/sqrt(1-t^2)dt.his is an improper definite integral, butt is consistent with the commonly practiced definitionf the logarithmic function ln asn(x) = Integral_{1}^{x}(1/t)dtnd then the exponential function exp as the inversef ln.akao==ubject: Re: Cheaters Definition of Trigonometric Functions.>>I quote an Appendix of a calculus textbook>>I have.>>et U be a unit circle, that is, a circle>>of radius 1, with center at the origin of>>a rectangular coordinate system, and Let>>A be the point with coordinates (1,0).>>If t is any number between 0 and 2PI, then>>as illustrated in Figure III.1 there is>>precisely one point P(x,y) on U such that>>the length of arc AP measured in the>>counterclockwise direction from A is t. >>I think your textbooks have also similar>>descriptions to start the definition of>>trigonometric functions. Or worse, no>>definition is given at all.>>When I was a college student, the definition>>of trigonometric functions based on a similar>>claim without proof was a stumbling block.>>Tell me if you have more rigorous but also>>intuitive introduction (not using power series>>but using an improper definite integral)>>in your textbook? I found and bought one>>in Japanese after great efforts, but>>I do not have it now.>>Takao>(What you quoted is apparently the beginning of a longer story,>leading to the definition of sin(t) as the depends on this unproven assumption.his is where you are confused. It is not an assumption, but a definition*. It may not be one that you find intuitive or satisfying, ut it is a definition.>Now: what is intuitive to some may be not intuitive to others.>The kinematic approach goes like this:> A point starts traveling from (1,0) in such a way that its velocity>vector is obtained from its position vector by rotating counterclockwise>by a right angle. (Remember Euclid? All right angles are equal.)> We sort of anticipate that the motion will be circular about (0,0) at>constant speed, but we need not assume it. It will pop out of the>mathematics of the situation.> Now put it into equations: let (x(t), y(t)) be the position at time t,>then> xÕ(t) = -y(t)> yÕ(t) = x(t)>with initial conditions x(0)=1, y(0)=0.> Before you protest that I am making an unwarranted excursion into systems>of differential equations, let me obtain some elementary conclusions,>assuming that the system does have a solution:> F(t) = (x(t))^2 + (y(t))^2 must be constant, namely 1.>Why? Write down FÕ(t) and simplify, using the defining equations. You>will find FÕ(t)=0 everywhere, F(0)=1.> We do have circular motion: center (0,0), radius 1.> Now get the square of the speed: (xÕ(t))^2 + (yÕ(t))^2>(it will also be 1).>Then give x(t), y(t) their traditional names, cos(t) and sin(t),>respectively.>What is still hanging in the air: the existence of those x(t), y(t). That>need not have anything to do with intuition; power series or some integral>calculus trick can establish that (first get arcsine by finding out that>dt/dy = 1/sqrt(1-y^2) in a small enough interval,...)>A bonus: the addition formulas follow from the differential properties:>for example, cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b) will follow from> (d/dt) (cos(a+b-t)*cos(t) - sin(a+b-t)*sin(t)) = 0,>so the inside is a constant, and has the same value for t=0 as for t=b.>(We know that sin(0)=0 and cos(0)=1.)>And similarly for sin(a+b). So do you agree on the deficiency of the common practice on our textbooks? And do you propose that a trigonometric function should be defined as a solution of a system of differential equations? For a textbook we should satisfy both the logical rigor and intuitive understanding. As long as you satisfy these conditions I am not against your proposal.our proposal of using differential equations to define the trig unctions sounds ludicrous. How would you teach a pre-calculus student hat they mean? I think we should start the definition of trigonometric functions, for example, from cos^{-1}(x) = Integral_{t}^{1} 1/sqrt(1-t^2)dt. This is an improper definite integral, but it is consistent with the commonly practiced definition of the logarithmic function ln as ln(x) = Integral_{1}^{x}(1/t)dt and then the exponential function exp as the inverse of ln. usually define ln(x) as the inverse function of e^x. This is enerally quite sufficient for many purposes.- ill Twentymanmail: wtwentyman at copper dot net==ubject: WAR IMMINENT betweeen Mathematicians and Physicists!he other day, I was doing a web search for information aboutathematical typography. happened to come across a forum on the web site of a noted author,amed for his work on the effective graphical presentation ofnformation. In the discussion there, he showed an example of how oneould, using todayÕs technology, typeset a mathematical equation inembo instead of one of the fonts more typically used for that purpose,uch as Modern Series 7, Times New Roman, or Computer Modern.As everyone knows, the font Bembo is based on the font used by Aldusius Manutilus to typeset the book e Aetnain 1495, written by Pietroembo. Some feel this font came closer to perfection as slightly revisedor the later ypnerotomachia Poliphiliof 1499, and there is also aont named Poliphilus as well. And then thereÕs Aldine Roman, which mayr may not have been used by a Colonel in the U.S. Coast Guard for hisemoranda... but I digress.)n any event, he also decided, for purposes of clarity, to use Romanather than italic type for the letter e when it represented 2.71828...,he base of the natural logarithms.ne comment objecting to this as being in defiance of time-honoredraditions of mathematical typography was recorded in the forum.or myself, I thought this was a good idea, but the distinction betweenoman and italic, applied to single letters in isolation, didnÕt seem toe enough to make a clear distinction. And Roman is already in use forunctions like cos and sin. I had some vague memory of seeing one bookn which cos and sin were typeset in a bolder face; that might help withart fo the confusion. But boldface is already used to indicate vectoruantities.he next day, I happened across a PDF document which outlined new ISOtandards and new IPU recommendations concerning the typesetting ofathematics. It turns out that the Roman e, rather than the italic e, isctually now the preferred standard.he document is available at:ttp://www.tug.org/TUGboat/Articles/tb18-1/ tb54becc.pdfowever, when I read further in the document, a disturbing patternmerged.lthough e, the base of the natural logarithms, and i, the principalalue of the square root of minus one, were to be set in Roman type, h,lanckÕs constant, would remain in italics.fter all, there are some cosmological theories in which the fundamentalhysical constants change over time.t first, this seemed to be a commendable exercise in humility on theart of the Physics community. It certainly is true that the values oßanckÕs constant, the speed of light, epsilon-nought, and theine-structure constant are all contingent facts, as opposed to thealues of e, pi, CatalanÕs constant, or EulerÕs constant (also known ashe Euler-Mascheroni constant). (Is that e, pi, C, or gamma, or e, pi,, or C? Ah, well.)owever, on further reßection, it occurred to me that such an abstrusehilosophical distinction would hardly be a consideration for day-to-daysers of mathematical formulae, and distinguishing only half of theonstants in equations from the variables in them by typographic meansould be more likely to cause confusion than to relieve it.nd then it struck my notice that the activities of the Internationaltandards Organization are far more likely to involve engineers thanathematicians.as this it? Was this attempt to reform mathematical notation to be donet the expense of mathematicians, by changing e from the italic formhey knew and loved, while the physicists refused to surrender their owneloved italic h for the same noble cause of clearer notation?learly, this is the sort of thing that could introduce a note ofcrimony into the otherwise blissfully harmonious relations between theathematics and physics communities.an disaster be averted?fter giving some thought to the matter, I believed I had come up with aay to achieve the goal of distinguishing variables from constants inquations, while allowing everyone to retain the constants of concern toechnology, mathematical notation need not be constrained by the compassf the matrix-case.f we use a sans-serif typeface, like Helvetica, or Univers, or Futuraor the variables in our equations, they will be nicely distinguishedrom all the constants appearing therein, be they italic or Roman.ut thereÕs just one problem with that. While some people are fond of pithe constant, that is; no one wants to spend a whole afternoon sorting font of type somebody dropped on the ßoor), and some are fond of e,nd some are fond of -bar(h/(2*pi))... others have come to know andove the symbols representing _variables_ as well. Such as that mostommon of variables, x.f x does not remain in the italic of a modern typeface, such as Moderneries 7, or Century Expanded, or Bodoni... were it to change to thatateful icepick form found in the oldstyle and transitional faces, orven, as proposed here, to the plain form associated with a sans-seriface, then, even if peace were to reign between Mathematics and Physics,here would be strife with Computing Science. Because Dr. Donald E.nuth would surely rally the Computing Scientists against us were we tonßict such an indignity on his favorite variable!ohn Savardttp://home.ecn.ab.ca/~jsavard/index.html==ubject: Re: WAR IMMINENT betweeen Mathematicians and Physicists!-CompuServe-Customer: Yes-Coriate: interspeed.co.nz-Ecrate: tanandtanlawyers.com-Pose: George Cox-Punge: Micro$oft-Sanguinate: The MVS Guy-Terminate: SPA(GIS)-Tinguish: Mark Griffith-Treme: C&C,DWSAlthough e, the base of the natural logarithms, and i, the principalvalue of the square root of minus one, were to be set in Roman type,h, PlanckÕs constant, would remain in italics.ust be glad that the EEÕs arenÕt running the show, or it would be jnstead of i.- hmuel (Seymour J.) Metz, SysProg and JOAT nsolicited bulk E-mail subject to legal action. I reserve theight to publicly post or ridicule any abusive E-mail. Reply toomain Patriot dot net user shmuel+news to contact me. Do noteply to betweeen Mathematicians and Physicists!The document is available at:http://www.tug.org/TUGboat/Articles/tb18-1/tb54becc.pdft first, I thought the following may be a parent URL for that:ttp://www.tug.org/TUGboat/Contents/contents8-1.htmlut I see that the actual parent URL is:ttp://www.tug.org/TUGboat/Contents/contents18-1.htmlohn Savardttp://home.ecn.ab.ca/~jsavard/index.html==ubject: Re: Please help quickly! posting-account=W9drAQ0AAAAwnUCtDicSi-THtOHRPCY8mm... letÕs see here.or the first one, I get sin(arcsec4), and sin(2arcsec4).or the second one, I get -Pi/6or the third one, I get 3x(x - a)(x - b)(x - c) = y, where a, b, and cre some constants.f you needed to, you can find a, b, and c by solving the followingystem of equations: + b + c = -5/3b + ac + bc = 1bc = -1/3ope that helps..==ubject: Re: Please help quickly! please help me with the following problems ...by showing me the work and how to do them..i am tryin to check my answers and work if sec x=4, find sin x and sin 2x solve for x: -cos x _______ = tan x 1- sinx factor: y= 3x^4- 5x^3+ 3x^2 - xec(x) = 1/cos(x)an(x) = sin(x)/cos(x)hatÕs the common factor?- ill Twentymanmail: wtwentyman at copper dot net==ubject: SchrodingerÕs equation project posting-account=MBqP4A0AAADmTXsiNvaGzTW4Fx5uIecd am an undergraduate and I just finished a mathematical physicsourse, where we spent about a week on SchrodingerÕs equation. We allave final projects to do, and mine is on just this topic. Basically Iave to give a 45-minute long presentation on something interestingnvolving SchrodingerÕs equation, and it can literally just be meopying something out of a book and presenting it. I donÕt really knowny neat 45-minute long vignettes on such a topic, so I was wonderingf anybody could recommend anything like that? Like just a book thatalks about this in a readable format is great.B==ubject: Re: SchrodingerÕs equation projectI am an undergraduate and I just finished a mathematical physicscourse, where we spent about a week on SchrodingerÕs equation. We allhave final projects to do, and mine is on just this topic. Basically Ihave to give a 45-minute long presentation on something interestinginvolving SchrodingerÕs equation, and it can literally just be mecopying something out of a book and presenting it. I donÕt really knowany neat 45-minute long vignettes on such a topic, so I was wonderingif anybody could recommend anything like that? Like just a book thattalks about this in a readable format is great.he book Feyneman and Computation, chapter 20 recounts a delightfultory: Feynman, Barton and the Reversible Schrodinger Differencequation. Very readible, easily summarized. ttp://www-users.cs.york.ac.uk/~susan/bib/nf/h/anthnyjg.htm# 10240ee also:http://ice.prohosting.com/~jwah/sch.htmldapt their difference equation to Excel and you will have anxtremely original talk.ohn Baileyttp://home.rochester.rr.com/jbxroads/mailto.html== ubject: some material needed need some material concerned with wave equation and fourierransformation (i.e. FT applied to solve WE in arbitrary dimension).an you suggest me where to find some intenet interesting materiallease?aem Tuarg==ubject: Re: Disappointed As I said, itÕs a very standard argument in some circles.aybe. But itÕs not very standard that your vicious circles intersectith mine. ;-)an de Bruijn==ubject: Re: Disappointed Differential IÕd appreciate that, IÕd like to learn about this haos theory someday.K, but first show some discipline.earning about chaos theory is different from being a chaotic person.an de Bruijn==ubject: Re: Presentations of D_n when n = 2^m >The number of generators of a group G obviously depends on the >presentation, and can be as many as |G|. ...or more! E.g. Z/2Z = K. :-) Does that kind of thing actually come up much inractice? Maybe in research on the word problem in presentationheory?- im Heckman==ubject: Re: Presentations of D_n when n = 2^m>The number of generators of a group G obviously depends on the>presentation, and can be as many as |G|.> ...or more! E.g. Z/2Z = OK. :-) Does that kind of thing actually come up much inpractice? Maybe in research on the word problem in presentationtheory?ure thing! If you want to know the group of a knot (i.e. theundamental group of its complement in S^3) you just write down aunch of generators and relations from a drawing of it. You have noay of knowing a priori how fficientthat presentation is, andertainly the group can even be trivial. (Just toss a big loopaphazardly on the table to get an example.)s IÕm sure you know (since you mention the Word Problem), therean be no algorithm which could take any finite presentation andecide whether the group is trivial or not. And thatÕs just theost basic thing you might want to know from a presentation; uestions about the number of generators are harder (i.e. impossible).ave==ubject: Re: Maxwell laws are spherical because atoms are spherical Re: Lorentz transformations, Maxwell Eq., and Special Relativity imply the Cosmos is a huge sphere such as the inside of an atom What you say?lready you are confused but that is partly my fault for not making mine clear. YouÕre confusing the (necessary) spaces with the fields. Both can have curvature, but they are different kinds of animals. Faraday is field equations, presume 3D + time. am doing something deeper than your comment. I am on a different level than you.et me try to explain it with an example in mathematics and physics. We start from the beginningf mathematics and create numbers. We create 0 and 1 and than mark off 2, 3 etc Then we createational numbers of numbers between the Counting numbers. We create more numbers of Reals butet bored and want to create Geometry. Now it gets exciting. We see create circles and then weealize there is a special number of how many diameters fit a circumference of a circle.e call this special number pi. We work out its value and find it is 3.14... and irrational. Soar in all of this creating we do not bring into the picture physics. But now we do so.e say that we can not create any mathematics in a vacuum but that it is created with aackground of physics. We ask why should pi have a value of 3.14... and why not purely 3 or 4?f the Universe is created and destroyed periodically would pi in every one of those universese the same pi of 3.14... In other words is the value of pi independent of physics?f we realize that the Universe itself is an atom and since an atom has curvature of a spherehat outlines the electron dot cloud and the number of subshells for plutonium is 22 inside of 7hells we see that this number of 22/7 is a Rational approximation of pi.o the value of pi is created because of the physical nature of the Cosmos itself.f intelligent life were to appear in a Helium Atom Totality then the value of pi in thatniverse would be different from the pi in a Plutonium Atom Totality because there are differentumbers of subshells and shells.o, Tom, what I am saying about Maxwell Equations, about Special Relativity, and about Lorentzransformations is that they have the mathematical form of a sphere/cylinder is because of theosmos or Universe itself is spherical/cylindrical and not ßat plane Euclidean geometry.f the Cosmos were ßat plane Euclidean geometry, then intelligent life would spring forth andiscover physics and the laws of physics and the Maxwell Equations in that Universe would beifferent from the Maxwell Equations we know.he Maxwell Equations are what they are because the Cosmos is spherical/cylindrical.he Maxwell Equations are Lorentz transformation invariant because the Cosmos ispherical/cylindrical. If the Cosmos were ßat Euclidean then the Maxwell Equations would not benvariant to Lorentz transformation.nd Special Relativity has the speed of light as invariant. Why invariant? Because the Cosmos ispherical and when you shot a beam of light that has infinite speed in a spherical/cylindricalosmos, it no longer is infinite but becomes finite and bounded because a sphere is finite andounded. So the reason light speed is a constant c is because the Cosmos is finite andpherical.ummary: pi has a value and that value is determined by the physical constraints of the Cosmosith its 22 subshells inside of 7 shells and for e where 19 subshells are occupied inside of 7hells. The speed of light is constrainted to a finite constant because the Cosmos physically ispherical. The laws of electricity and magnetism as codified by the Maxwell Equations will have mathematical form because the Cosmos is a sphere/cylinder of a big atom. GRT links energy tensor to curvature (as an axiom). 4D.om you are doing what 20th century physicists and mathematicians have been doing for a centuryn that of pushing around math models to make sense of physics.orget the model pushing.tart at a different level and begin by asking ow does physics create the math that is Maxwellquations, that is the Lorentz transformation invariance, that is Special Relativity. How doeshysics determine or set the value of what pi and e are.ou see, I do not push around models to try to see how Maxwell Equations behave in a higherimension.nstead I link and join and connect physics with mathematics and ask deeper questions as to howhysics creates the mathematics. If you add another 2D you can have ßat, with a nicely curved sub-manifold AND consistency. Push up to 10D and you can seem to get Lorentz for free (but I have been suspicious about such freedom for years). Tomasso.his is more model pushing which in the end leads to no solving of any problems and instead justore confusion and lack of understanding and lack of knowledge. Higher dimensions reminds me oftolemy adding on more and more epicycles.f the Cosmos were ßat plane Euclidean geometry then the Maxwell Equations would not be Lorentzransformation invariant and the speed of light would be infinite, or variable which includes anfinite speed, and in fact Special Relativity would not hold.o in a sense, the summary is that the Cosmos is spherical/cylindrical because the Maxwellquations and Lorentz transform invariance and Special Relativity are spherical/cylindrical.ust as pi and e are in Rational form of 22 subshells in 7 shells and 19 occupied subshells in 7hells because the Cosmos is a big atom with those subshells and shells.o you see, Tom, I am connecting physics with math itself and explaining why math is what it isecause the Physics is what it is. The Physical Nature of the Cosmos creates the mathematicalorm.rchimedes Plutoniumww.iw.net/~a_plutoniumhole entire Universe is just one big atom where dotsf the electron-dot-cloud are galaxies==ubject: Re: Maxwell laws are spherical because atoms are spherical Re: Lorentz transformations, Maxwell Eq., and Special Relativity imply the Cosmos is a huge sphere such as the inside of an atom> What you say? Already you are confused but that is partly my fault for not making mine clear.ou write long. You make connections between physical observations and instrinsicroperties of mathematical spaces. That is not silly, per se. But your geometricalroperties (and interpretations) arenÕt necessary. You want Lorentz invariance?K, in 3D + time, curve the space. The SPACE basically means your metric.his is the intimate property mirrored in constancy of [in both Maxwell andR]. In SR, the 3D space can be ßat - but that is an eventless space, where talkingbout change, information, observation, etc is meaningless. No problem.ut the 3D + time WITH Lorentz you have the metric and measure distances theay physicists do (as you know). Let me try to explain it with an example in mathematics and physics. We start from the beginning of mathematics and create numbers. We create 0 and 1 and than mark off 2, 3 etc Then we create Rational numbers of numbers between the Counting numbers. We create more numbers of Reals but get bored and want to create Geometry. Now it gets exciting. We see create circles and then we realize there is a special number of how many diameters fit a circumference of a circle. We call this special number pi. We work out its value and find it is 3.14... and irrational. So far in all of this creating we do not bring into the picture physics. But now we do so.orry, you donÕt get circles until you invent dimensions and have 2D. 1D geometry is called Intervallgebra and is pretty dull. iis a property of circles in ßat spaces. We say that we can not create any mathematics in a vacuum but that it is created with a background of physics. We ask why should pi have a value of 3.14... and why not purely 3 or 4?tÕs tied to distance, which is tied to how you measure (as I would tell a physicist). The sums whichield ican be reduced to geometric inprepretations of distance measurement. If the Universe is created and destroyed periodically would pi in every one of those universes be the same pi of 3.14... In other words is the value of pi independent of physics?aively I would say yes, but that depends on the notions of dimension and Euclidean distanceaving a place. Maybe this would not apply in some topological universes which associateoints differently than the way we think (well, many of us). Eg, people who like strings, likeo have ackdoordimensions that on a very small scale change things, providing a kind ofocal esilienceof interations between oints If we realize that the Universe itself is an atom and since an atom has curvature of a sphere that outlines the electron dot cloud and the number of subshells for plutonium is 22 inside of 7 shells we see that this number of 22/7 is a Rational approximation of pi. am too naive to make this connection without more explanation. So the value of pi is created because of the physical nature of the Cosmos itself.itto. If intelligent life were to appear in a Helium Atom Totality then the value of pi in that universe would be different from the pi in a Plutonium Atom Totality because there are different numbers of subshells and shells. So, Tom, what I am saying about Maxwell Equations, about Special Relativity, and about Lorentz transformations is that they have the mathematical form of a sphere/cylinder is because of the Cosmos or Universe itself is spherical/cylindrical and not ßat plane Euclidean geometry.his notion of athematical formcauses me problems. If it characterising a surface, then thatust be IN a space. If it is defined by a metric/curvature/etc, then OK. GR provides a theoryo link that to energy content. If the Cosmos were ßat plane Euclidean geometry, then intelligent life would spring forth and discover physics and the laws of physics and the Maxwell Equations in that Universe would be different from the Maxwell Equations we know. miss that necessity. The Maxwell Equations are what they are because the Cosmos is spherical/cylindrical. The Maxwell Equations are Lorentz transformation invariant because the Cosmos is spherical/cylindrical. If the Cosmos were ßat Euclidean then the Maxwell Equations would not be invariant to Lorentz transformation. And Special Relativity has the speed of light as invariant. Why invariant? Because the Cosmos is spherical and when you shot a beam of light that has infinite speed in a spherical/cylindrical Cosmos, it no longer is infinite but becomes finite and bounded because a sphere is finite and bounded. So the reason light speed is a constant c is because the Cosmos is finite and spherical.inkowski and Lorentz provided the geometry before SR. Summary: pi has a value and that value is determined by the physical constraints of the Cosmos with its 22 subshells inside of 7 shells and for e where 19 subshells are occupied inside of 7 shells. The speed of light is constrainted to a finite constant because the Cosmos physically is spherical. The laws of electricity and magnetism as codified by the Maxwell Equations will have a mathematical form because the Cosmos is a sphere/cylinder of a big atom.he 22 and 7 seems accidental (and approximate), to me.> GRT links energy tensor to curvature (as an axiom). 4D. Tom you are doing what 20th century physicists and mathematicians have been doing for a century in that of pushing around math models to make sense of physics.rue. Forget the model pushing.hysical models are ushed(and tested) against observations, for consistency. Noticing thatome ratios of numbers approximate pi has less grounding. Start at a different level and begin by asking ow does physics create the math that is Maxwell Equations, that is the Lorentz transformation invariance, that is Special Relativity. How does physics determine or set the value of what pi and e are. You see, I do not push around models to try to see how Maxwell Equations behave in a higher dimension.ut your spheres and cylinders are similar kinds of constructs.omasso.==ubject: Re: Restriction on Longitudinal Waves Given monotony, what you claim in appears to me to be true. .. snip .. ] However, in physical media, are you assuming infinite compressiblity? That is, is the spacing between points allowed to become infinitely small? In gas media, there is a characteristic exponent greater than one producing heating on compression....ho knows? But the restriction amplidute < wavelength / 2.pi could beerived from monotony as the sole requirement. ThatÕs why I crosspostedo Ôsci.mathas well. Under conditions of turbulence, could monotony be violated?r even more general: is 2-D / 3-D different from 1-D?an de Bruijn I fight terrorism by: Using less gasoline.==ubject: Re: Restriction on Longitudinal Waves ... snip ... ]> Theorem: the amplitude of a longitudinal wave cannot be larger> than its wavelength, divided by 2.pi . ... snip ... ] See: coustic Fields and Waves in Solids, Vol I & IIby B.A. Auld, 2nd ed (February 1990), Krieger Publishing Company; ISBN: 089874783X found that they are in our UT library. So maybe I could borrow them.Õd prefer a shortcut, though. What do you mean, Sam? Is that theoremell konwn? And is it true or not?an de Bruijn==ubject: Re: Restriction on Longitudinal Waves [ ... snip ... ]>> Theorem: the amplitude of a longitudinal wave cannot be larger>> than its wavelength, divided by 2.pi . [ ... snip ... ]> See:> coustic Fields and Waves in Solids, Vol I & IIby B.A. Auld, 2nd ed> (February 1990), Krieger Publishing Company; ISBN: 089874783X I found that they are in our UT library. So maybe I could borrow them. IÕd prefer a shortcut, though. What do you mean, Sam? Is that theorem well konwn? And is it true or not? Han de Bruijn Shortcuts? You owe it to yourself to see what others have done.==ubject: Re: Restriction on Longitudinal Waves> I found that they are in our UT library. So maybe I could borrow them.> IÕd prefer a shortcut, though. What do you mean, Sam? Is that theorem> well konwn? And is it true or not? Shortcuts? You owe it to yourself to see what others have done.Õll go to the library, if I donÕt get a sensible answer right here.o thatÕs settled on my part (though it will cost me an hour or so).ow on your part. Instead of this well-known strategy of iding behindhe fat books why donÕt you simply admit that _you donÕt know_?an de Bruijn==ubject: Re: Restriction on Longitudinal Waves posting-account=JUztfAwAAADv1PUBBOcJTiECyhwXix6x IÕll go to the library, if I donÕt get a sensible answer right here. So thatÕs settled on my part (though it will cost me an hour or so). Now on your part. Instead of this well-known strategy of idingehind the fat books why donÕt you simply admit that _you donÕt know_?ell I do -- at least now that IÕve read and thought about youruestion.o cut right to the chase (since I just unintentionally erased myeautiful ßorid draft), your result is based on the assumption that aedium cannot overrun itself -- that is, in a longitudinal displacementave, a point A in the medium originally to the left of a point Bannot wind up to the right of B. (In fairness you mention somethingike this).he sharpest way of expressing this seems to be the requirement thaty/dx > -1, where x is a coordinate position in the undisplaced medium,(x) the displacement of that point; this at a particular snapshot t.nyway, assuming a waveform y = Asin(kx), this gives us the requirement < wavelength/2pi, as you said. So subject to (1) the assumption thathe medium cannot interpenetrate itself and (2) the caveat that thelope condition is more fundamental than its specialization to a sineave, and (3) the emphasis that this applies only to the displacementn a longitudinal displacement wave -- yes, you are correct.e could ask what kind of medium might violate assumption (1). Anudsen gas comes to mind, or something that looks like a Knudsen gas.or example, a loosely spaced platoon of marching soldiers. If routedrom behind, the rear line might well overrun the front line!ne question: when you write ransversalwave, did you mean to writetransversewave, or is that some other term? I canÕt see whatonnection a transverse wave has with any of this. And may I say, youeem to have taken a simple idea and converted it into an impenetrablef impressively typeset line of math! One shudders to think what mightappen if you exposited something truly difficult. ;-)==ubject: Re: Restriction on Longitudinal Waves>> I found that they are in our UT library. So maybe I could borrow them.>> IÕd prefer a shortcut, though. What do you mean, Sam? Is that theorem>> well konwn? And is it true or not?> Shortcuts? You owe it to yourself to see what others have done. IÕll go to the library, if I donÕt get a sensible answer right here. So thatÕs settled on my part (though it will cost me an hour or so). Now on your part. Instead of this well-known strategy of iding behind the fat books why donÕt you simply admit that _you donÕt know_? Han de Bruijn It is well known that wave propagation modes spawn other wave propagation modes... I thought you might be interested in the mathematics of the same. To that end I suggested an excellent textbook written by one of the experts in the field and you insult me, insisting I do your work for you. Bye==ubject: Re: Restriction on Longitudinal Waves posting-account=JUztfAwAAADv1PUBBOcJTiECyhwXix6x conjecture he means the Windows variety. ;-)==ubject: Re: A vector calc related question> Back in the day, a visiting professor challenged my class to find an> exception to the following general statement:>> F sub xy = F sub yx (Partial to x then y is equal to the partial of> then x).>> This is used, as yall may know, to find relative extrema where d => Fxx*Fyy - (Fxy)^2. The reason it came up was the lecturer said that> they are one in the same. The visitor challenged us to find an> exception. Later, when my lecturer came back, he admited there were> cases where it isnÕt equal. Anyone have an idea?> A standard example is> F(x,y) = xy(x^2-y^2)/(x^2+y^2) if x =/=0 or y=/=0,> F(x,y) = 0 if x = y = 0.> The two mixed partial derivatives at (0,0) are different.> --> Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html> acan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 > Francis Wheen, _How Mumbo-Jumbo Conquered the World_ I submitted the result. He was so impressed I pursued it that he asked my professor to grant me a load of extra credit. :) YeehaawhatÕs the only reason to pursue mathematics? For rofessors o give redit :-(- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: Re: A vector calc related question posting-account=2OYlAwwAAAAzuGZHzY8fB1XqLzeo4Yd5ead what I posted again.==ubject: Re: A vector calc related question posting-account=2OYlAwwAAAAzuGZHzY8fB1XqLzeo4Yd5ead what I posted again.==ubject: Re: A vector calc related questionhe fact remains, that if(x,y) = xy(x^2 - y^2)/(x^2 + y^2) if (x,y) =/= (0,0)(x,y) = 0 if (x,y) = (0,0)hen F_{x,y}(0,0) =/= F_{y,x}(0,0).hether or not a student riesthis and concludesFxy does equal Fyxis irrelevant.- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: Re: A vector calc related question posting-account=2OYlAwwAAAAzuGZHzY8fB1XqLzeo4Yd5==ubject: Re: A vector calc related question posting-account=2OYlAwwAAAAzuGZHzY8fB1XqLzeo4Yd5==ubject: Re: A vector calc related question posting-account=2OYlAwwAAAAzuGZHzY8fB1XqLzeo4Yd5==ubject: all equivalence relations on a finite sete have lattice of all equivalence relations on the finite set.et n be number of elements in this finite set. How many elements hashe smallest set of equivalence relations from which the whole latticean be generated using the lattice operations meet and join.asy to achieve is n. WhatÕs the best result and how can be achieved?aciek==ubject: Re: all equivalence relations on a finite set==ubject: all equivalence relations on a finite set We have lattice of all equivalence relations on the finite set. Let n be number of elements in this finite set. How many elements has the smallest set of equivalence relations from which the whole lattice can be generated using the lattice operations meet and join. Easy to achieve is n. WhatÕs the best result and how can be achieved?ouÕre not going to do any better.he partitions of S = { a,b,c } { {a}, {b}, {c} } { {a,b}, {c} }, { {b,c}, {a} }, { {c,a}, {b} } { {a,b,c} }re generated by the equivalence relations { (a,a), (b,b), (c,c) } (a,a), (b,b), (c,c), (a,b), (b,a) } (a,a), (b,b), (c,c), (b,c), (c,b) } (a,a), (b,b), (c,c), (c,a), (a,c) } (a,a), (b,b), (c,c), (a,b), (b,a), (b,c), (c,b), (c,a), (a,c) } */ | * * | / *---==ubject: The Possible Cure For AIDShe Possible Cure For AIDS-------------------------------------------------- believe this may be the cure for AIDS.See reference LINKS)ttp://www.prweb.com/releases/?00000032492ttp:// www.keelynet.com/interact/arc_1_99-4_99/00000071.htmttp:// www.tetrasil.com/science.html# I also did a bible code analysis and the bible matrix showed many repeatedccurrances in the matrix showing the cure for HIV is AG4O4 ( Tetrasilveredroxide). It showed up in the matrix something like this.G4O4/CURE/HIV These words are found right next to each other and crossing each other inhe bible matirx. Is this just a conincidence? I donÕt think so because therere repeated occurances of this in different locations of the matrix, andcientific research is showing that 8 out of 10 patients treated with IVnjected tetrasil was cured with a one time treatment. The other 2 didnÕt getured because they were too far gone from the disease. IÕm not sure of the dosage they used, but it seems to have other usefulures too for external use on the skin.martÕs Alt. Physics News Groupttp://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1. Enterprize (Science Journal)ttp://smart1234.s-enterprize.com/==ubject: Re: The Possible Cure For AIDS I also did a bible code analysis and the bible matrix showed many repeated occurrances in the matrix showing the cure for HIV is AG4O4 ( Tetrasilver Tedroxide). It showed up in the matrix something like this. AG4O4/CURE/HIVo you decode the hebrew of Gen 1:17-19 by the ancient rabinical ashbet ubstutition. The english translation of the decrypt is: S Enterprise is putz/ob Kolker==ubject: Re: The Possible Cure For AIDS> I also did a bible code analysis and the bible matrix showed manyrepeated> occurrances in the matrix showing the cure for HIV is AG4O4 ( Tetrasilver> Tedroxide). It showed up in the matrix something like this.> AG4O4/CURE/HIVDo you decode the hebrew of Gen 1:17-19 by the ancient rabinical ashbet substutition. The english translation of the decrypt is: S Enterprise is a putz/Bob Kolker I used software that analyzes the torah in hebrew in matrix form. And by theay, mart/1234s in the bible code matrix.martÕs Alt. Physics News Groupttp://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1. Enterprize (Science Journal)ttp://smart1234.s-enterprize.com/==ubject: Re: The Possible Cure For AIDS posting-account=W9drAQ0AAAAwnUCtDicSi-THtOHRPCY8s IÕve mentioned in my other post, that is why the Bible code isurely combinatorical and offers no significant results. The fact thatmart/1234 appears in the code is no more surprising than AG404,ilter, Jesus, apocalypse, George Bush, etc. other than the fact thatome have more letters than others..==ubject: Re: The Possible Cure For AIDSDiscussion, linux) The Possible Cure For AIDS --------------------------------------------------- I believe this may be the cure for AIDS. (See reference LINKS) http://www.prweb.com/releases/?00000032492 http://www.keelynet.com/interact/arc_1_99-4_99/00000071.htm http://www.tetrasil.com/science.html# I also did a bible code analysis and the bible matrix showed many repeated occurrances in the matrix showing the cure for HIV is AG4O4 ( Tetrasilver Tedroxide). It showed up in the matrix something like this. AG4O4/CURE/HIVhe outcome of the query. 04is the standard code for age notoundhus, you have found the Bible Code equivalent of the Magic 8 ballÕsAsk Again Lateretter luck next time.- And yes, for those who think that just maybe I did find a short prooff FermatÕs Last Theorem, and THE prime counting function, if Iucceed at what IÕm working on now world economy as you know it wille gone.-- James Harris branches out.==ubject: Re: The Possible Cure For AIDS> The Possible Cure For AIDS> ---------------------------------------------------> I believe this may be the cure for AIDS.> (See reference LINKS)> http://www.prweb.com/releases/?00000032492> http://www.keelynet.com/interact/arc_1_99-4_99/00000071.htm> http://www.tetrasil.com/science.html#> I also did a bible code analysis and the bible matrix showed manyrepeated> occurrances in the matrix showing the cure for HIV is AG4O4 ( Tetrasilver> Tedroxide). It showed up in the matrix something like this.> AG4O4/CURE/HIVthe outcome of the query. 04is the standard code for age notfoundThus, you have found the Bible Code equivalent of the Magic 8 ballÕssk Again Later Most people will agree that AG4O4 is the chemical structure for tetrasilveredroxide. ThatÕs 4 atoms of silver with 4 atoms of oxygen. What are thehances of finding this repeatedly in the bible matrix associated with HIV andIDS and the words CURE and HEAL? Very slim chance. And then you look up the scientific research done that has already provenhat AG4O4 does cure AIDS with one injection treatment. This doesnÕt lookike magic 8 balls to me. It looks like their is something in the bible matrixodes. Other things found in the bible matrix codes, shows the date of JFKÕsssassination and more. I think there is something in this matrix that could beignificant enough to study. The bible code matrix I used was directly from the torah in hebrew. Therere intelligent logic patterns of words next to other words, that appears toelate to world events. These patterns are based on equivalent equal spacesetween the letters of words that can be diagonal, vertical or horizontal.ike for example the word cure,123u123r123e ^ |spacesr it could be like this c 1 2 u 1 2 r 1 2 e And these patterns of different words are right nest to each other.ne search I found was,IV/AG4O4/CURE/HEAL What are the chances of this happening in any book set up like a matrix?ery slim.Better luck next time.-- nd yes, for those who think that just maybe I did find a short proofof FermatÕs Last Theorem, and THE prime counting function, if Isucceed at what IÕm working on now world economy as you know it willbe gone.-- James Harris branches out. martÕs Alt. Physics News Groupttp://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1. Enterprize (Science Journal)ttp://smart1234.s-enterprize.com/==ubject: Re: The Possible Cure For AIDS Most people will agree that AG4O4 is the chemical structure for tetrasilvertedroxide. ThatÕs 4 atoms of silver with 4 atoms of oxygen. What are thechances of finding this repeatedly in the bible matrix associated with HIV andAIDS and the words CURE and HEAL? Very slim chance. The bible code matrix I used was directly from the torah in hebrew. Thereare intelligent logic patterns of words next to other words, that appears torelate to world events. These patterns are based on equivalent equal spacesbetween the letters of words that can be diagonal, vertical or horizontal.here are no AÕs, GÕs, 4Õs, OÕs, etc. in Hebrew. Did you transliteratehese according to some arbitrary scheme, or did you search for Hebrewetter strings?obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: The Possible Cure For AIDS There are no AÕs, GÕs, 4Õs, OÕs,he G is gimmel the 3 rd letter of the Hebrew alphabet.he vowels (mplicit in old Hebrew but made explicit by the Masorites) nclude an ah sound and a oh sound.ob Kolker==ubject: Re: The Possible Cure For AIDS> There are no AÕs, GÕs, 4Õs, OÕs,The G is gimmel the 3 rd letter of the Hebrew alphabet.The vowels (mplicit in old Hebrew but made explicit by the Masorites) include an ah sound and a oh sound.FAIK the Torah codes are never done with the vowels.obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: The Possible Cure For AIDS AFAIK the Torah codes are never done with the vowels.hat is because there are no vowels in Torah scrolls. The vowels are nderstood or implicit. Like thr rm frnsh aprtmnt (three room furnished partment). However this gives rise to some real ambiguities, general esolved by traditional authority.ob Kolker==ubject: Re: The Possible Cure For AIDSThe Possible Cure For AIDS---------------------------------------------------I believe this may be the cure for AIDS.(See reference LINKS)http://www.prweb.com/releases/?00000032492http:// www.keelynet.com/interact/arc_1_99-4_99/00000071.htmhttp:// www.tetrasil.com/science.html# I also did a bible code analysis and the bible matrix showed many repeatedoccurrances in the matrix showing the cure for HIV is AG4O4 ( TetrasilverTedroxide). It showed up in the matrix something like this.AG4O4/CURE/HIV These words are found right next to each other and crossing each other inthe bible matirx. Is this just a conincidence? I donÕt think so because thereare repeated occurances of this in different locations of the matrix, andscientific research is showing that 8 out of 10 patients treated with IVinjected tetrasil was cured with a one time treatment. The other 2 didnÕt getcured because they were too far gone from the disease. IÕm not sure of the dosage they used, but it seems to have other usefulcures too for external use on the skin. Other info showing bible matrix code occurrences I found about the cure forIDS:IV/AG4O4/CURE/HEALIV/AG4O4/HEALEAL/AG4O4/CURE/HIVEAL/ AG4O4/CUREURE/AG4O4/HEALIDS/AG4O4G4O4/CURE/HEALG4O4/VIRUSG4O4 /AIDSIDS/CUREIDS/HIVEAL/AIDS/CUREEAL/HIVIV/CURE/HEAL These are some of the words found related to the cure for AIDS-HIV. Theyppear in different locations in the bible matrix and they show up next to eachther in different crossing patterns next to each other like this: Patterns something like this of words near each other in the matrix are foundhowing the possible cure for HIV. / / / / --- -- / --- --- / | / / | / | / | martÕs Alt. Physics News Groupttp://pub39.bravenet.com/forum/show.php?usernum= 3320272813&cpv=1. Enterprize (Science Journal)ttp://smart1234.s-enterprize.com/==ubject: Re: The Possible Cure For AIDS The Possible Cure For AIDS --------------------------------------------------- I believe this may be the cure for AIDS. (See reference LINKS) http://www.prweb.com/releases/?00000032492 http://www.keelynet.com/interact/arc_1_99-4_99/00000071.htm http://www.tetrasil.com/science.html# I also did a bible code analysis and the bible matrix showed manyepeated occurrances in the matrix showing the cure for HIV is AG4O4 ( Tetrasilver Tedroxide). It showed up in the matrix something like this. AG4O4/CURE/HIV==ubject: access to sciencedirecto you have free access to the journalhysics Reports, Volume 195, Issues 4-5,ovember 1990, Pages 127-293rom http://www.sciencedirect.com/ ?ean-Philippe BouchaudAntoine Georgesnomalous diffusion in disordered media:tatistical mechanisms, models and physical applicationsttp://www.sciencedirect.com/science?_ob= GatewayURL&_method=citationSearch&_ oikey=B6TVP-46SXPMN-7F&_origin=SDEMFRASCII&_version=1&md5= 4160e0122f1e3b829 214ec713582d26 will grateful for you.aldekaldemar_wolski@vp.pl==ubject: Re: Moments over a Simplex What do you mean by hevariance over a multi-dimensional simplex? hm, should I rather say he moments of a multi-dimensional simplexspecially: integral x dV / integral dV = first order moment andintegral x^2 dV - (integral x dV )^2 ) / integral dV = second orderhere dV = infinitesimal simplex volume . Do I make myself clear now?an de Bruijn==ubject: Re: Moments over a Simplex> What do you mean by hevariance over a multi-dimensional simplex? Uhm, should I rather say he moments of a multi-dimensional simplex Especially: integral x dV / integral dV = first order moment and (integral x^2 dV - (integral x dV )^2 ) / integral dV = second order Where dV = infinitesimal simplex volume . Do I make myself clear now?ecause you are squaring a vector.- tephen J. Herschkorn sjherschko@netscape.net==ubject: Re: Moments over a Simplex> Especially: integral x dV / integral dV = first order moment and> (integral x^2 dV - (integral x dV )^2 ) / integral dV = second order Because you are squaring a vector.o. Apologies again for causing so much misunderstanding. What I meanto ask is the outcome with ÔxÕ as a scalar. Because IÕm actually afterhe moments of a simplex which is stretched along the x-axis. It makeso difference for the algebra, I hope (does it?).an de Bruijn==ubject: Re: Moments over a Simplex>> Especially: integral x dV / integral dV = first order moment and>> (integral x^2 dV - (integral x dV )^2 ) / integral dV = second order> Because you are squaring a vector. No. Apologies again for causing so much misunderstanding. What I meant to ask is the outcome with Ôxas a scalar. Because IÕm actually after the moments of a simplex which is stretched along the x-axis. It makes no difference for the algebra, I hope (does it?).et me see if I understand you now: Let (x1, x2,..., x_n) vary over our n-1 dimensional simplex. Is the integral you want (integral x_j)^2 dV - (integral x_j dV )^2 ) / integral dV? In that case, refer o an earlier posting of mine (and the followup with the correction), here I showed that the variance of the jth component is[(n-1) sum(i=1..n, [(r_i)_j]^2) - 2 sum( (i,k): i < k, (r_i)_j (r_k)_j ] / [n^2 (n+1)] [n sum(i=1..n, [(r_i)_j]^2) - sum(i=1..n, (r_i)_j)^2] / [n^2 (n+1)] v_j / (n+1),here r1, r2,..., r_n are the vertices of the simplex and v_j is the ariance of a uniform distribution on {r1_j, r2_j,..., (r_n)_j}. (I hink - I just noticed that last way of expressing it.)hat we all have been deriving previously is the the average value of x - m|^2 = the average value of |x|^2 - |m|^2, where | .| indicates he Eucliean norm and m is the average value of the simplex (i.e., the irst moment).r are you refering to the projection of the simplex onto one of its dges? That is something different. For this, you may use the same inear transformation technique I described in my previous post.- tephen J. Herschkorn sjherschko@netscape.net==ubject: Re: Moments over a Simplex posting-account=uMDgiw0AAAANoAxJOs_DnrtdjhRMBFah > What do you mean by hevariance over a multi-dimensional > simplex? Uhm, should I rather say he moments of a multi-dimensional simplex Especially: integral x dV / integral dV = first order moment and (integral x^2 dV - (integral x dV )^2 ) / integral dV = second order Where dV = infinitesimal simplex volume . Do I make myself clear now? Han de Bruijnhat Stephen is saying, I think, is that the most useful generalizationf variance to multiple dimensions is the covariance matrix. What Hanants is the trace of this matrix, which is what Robert Israel derived.ereÕs my own, longwinded derivation of the covariance matrix V, whichllustrates a general procedure for deriving moments of any order:irst, we define the following:0,x1,...,xn are n-vectors,k = xk-x0 for k = 1 to n, = [y1,y2,...,yn] (assume for convenience Det(A)>0), = the simplex with vertices at 0 and each yk,k = the k^th unit n-vector, and = the unit simplex (vertices at 0 and each zk).e can compute integrals over S by converting them tontegrals over T using the transformation y = Az:ntg_S dy = Det(A) Intg_T dz,ntg_S y_i dx = Det(A) Intg_T (Az)_i dz,ntg_S y_i y_j dx = Det(A) Intg_T (Az)_i (Az)_j dz,here y_i, for example, is the i^th component of y.imilarly,_ik = yk_iill denote the (i,k) component of A, which is the^th component of the vector yk.tÕs easy to integrate a monomial over T:emma:ntg_T z_1^r1 z_2^r2 ... z_n^rn dz =1! r2! ... rn! / (r1 + r2 + ... + rn + n)!roof:Left to reader)herefore:ntg_S dy = Det(A)/n!ntg_S y_i dy = Det(A) (Sum_k A_ik Intg_T z_k dz) Det(A) (Sum_k A_ik) / (n+1)!ntg_S y_i y_j dy Det(A) ( Sum_kl A_ik A_jl Intg_T z_k z_l dz) Det(A) ( (Sum_k!=l A_ik A_jl 1! 1!/(n+2)!) +Sum_k==l A_ik A_jl 2! 0!/(n+2)!) ) Det(A) ( (Sum_k A_ik) (Sum_l A_jl) +Sum_k A_ik A_jk) )/(n+2)!ere Sum_k is the sum from k = 1 to n, and, e.g.,um_k!=l is the sum from k = 1 to n and l = 1 to n with not equal to l.e can now express the moments in terms of the yk:et M(y_i) = Intg_S y_i dy / Intg_S dy, and(y_i y_j) = Intg_S y_i y_j dy / Intg_S dy.hen M(y_i) = (Sum_k yk_i) / (n+1), and(y_i y_j) = ( (Sum_k yk_i) (Sum_k yk_j) +Sum_k yk_i yk_j) ) / ((n+1)(n+2)).hese are necessarily identical in form when we rewritehem in terms of the xk:et M(x_i) = M(y_i) + x0_i, and(x_i x_j) = M(y_i y_j) + x0_i M(y_j) +(y_i) x0_j + x0_i x0_jhen M(x_i) = (Sum0_k xk_i)/(n+1), and(x_i x_j) = ( (Sum0_k xk_i) (Sum0_k xk_j) +Sum0_k xk_i xk_j) ) / ((n+1)(n+2)),here Sum0 is the sum from k = 0 to n.inally, weÕd like to shift the second-order moment to bebout the mean:_ij = M((x_i - M(x_i)) (x_j - M(x_j))) M(x_i x_j) - M(x_i) M(x_j),hich yields this formula for the covariance matrix V,_ij = ((Sum0_k xk_i xk_j)/(n+1) - M(x_i) M(x_j))/(n+2) ( Sum0_k (xk_i-M(x_i)) (xk_j-M(x_j)) )/((n+1)(n+2)).he trace of this matrix isr(V) = Sum_i V_ii ((Sum0_k |xk|^2)/(n+1) - |M(x)|^2)/(n+2). (Sum0_k |xk-M(x)|^2)/((n+1)(n+2)).o cast this in terms of Sum0_kl |xk-xl|^2, we writeum0_kl |xk-xl|^2 2(Sum0_k |xk|^2)(Sum0_l 1) - 2(Sum0_k xk).(Sum0_l xl) 2(n+1) Sum0_k |xk|^2 - 2 (n+1)^2 |M(x)|^2 Tr(V) * 2 (n+1)^2 (n+2),o, restricting the sum to k>l,r(V) = (Sum0_k>l |xk-xl|^2) / ((n+1)^2 (n+2)),hich agrees with Robert IsraelÕs result.-Jim FerryMetron, Inc.f rr @m tsc .c m e y e i o==ubject: Re: Moments over a Simplex>>What do you mean by hevariance over a multi-dimensional>>simplex?>Uhm, should I rather say he moments of a multi-dimensional>simplex Especially: integral x dV / integral dV = first order>moment and (integral x^2 dV - (integral x dV )^2 ) / integral dV>= second order Where dV = infinitesimal simplex volume . Do I make>myself clear now? What Stephen is saying, I think, is that the most useful generalization of variance to multiple dimensions is the covariance matrix. What Han wants is the trace of this matrix, which is what Robert Israel derived.o. IÕm sorry that IÕve caused so much confusion. By ÔxI simply mean acalar variable, just One of the second order moments, just One elementf the inertial tensor in N dimensions (now IÕm talking as a physicist).he secret behind it is the following. It is seen that the numerator andhe denominator in the expressions for the moments of a simplex _both_ontain the volume of the arallelogramassociated with the simplex,s a factor (apart from 1/N!). Therefore these moments are _independent_f the question whether that simplex is degenerate or not. The point ishat IÕm actually unsing a _degenerated_ simplex, which is wrapped alonghe x-axis alone. Therefore I am only interested in the moments with ÔxÕs the (bounded) unknown. (Oh well, I didnÕt say nly interested...)an de Bruijn==ubject: Re: Moments over a Simplex posting-account=tyq2IQ0AAAAuNErVkOWXc866QgLV9Cn9 > The midpoint of a line segment (x1,x2) is (x1+x2)/2 > The midpoint of a triangle (r1,r2,r3) is (r1+r2+r3)/3 > The midpoint of a tetrahedron (r0,r1,r2,r3) is (r0+r1+r2+r3)/4 > The midpoint of a simplex in N dimensions is sum(k<=N) r_k/N ? > The variance in x over a line segment in 1-D is: (x2-x1)^2/12 > The variance in x over a triangle in 2-D is, if I made no errors: > ((x2-x1)^2 + (x3-x1)^2 + (x3-x2)^2)/36 > Question: What is the variance in x over a simplex in N dimensions? > Is there a sensible way to make these calculations (multiplentegrals) > more amenable to treatment? Letting m be the vector average, sum(i=1..n, r_i). You are apparently trying to calculate the average over the simplex of the squared distance from m; it is incorrect to call this scalar the variance. The desired average is the trace of the variance-covariance matrix. The uniform distribution over a simplex is the Dirichlet distribution with parameters 1, 1, 1,..., 1 (n times for the (n-1)-dimensional simplex). If random vector X has D(a1, a2,..., a_n)istribution, X has n nonnegative components which sum to unity. Letting s = sum(i=1..n, a_i), EX = a / s Var(Xi) = a_i (s - a_i) / [s^2 (s+1)] Cov(Xi, Xj) = -a_i a_j / [s^2 (s+1)] for i != j, If random vector Y is uniformly distributed over the (n-1)-dimensional simplex with vertices r1, r2,..., r_n, then Y =r1 r2 r3 ... r_n] X, where the vectors are written in columns and Xas D(1, 1,..., 1) distribution.orm. Let R = [r1 r2 ... r_n]. Then = R Xar(Y) = R Var(X) R^There Var indicates here the covariance matrix. Thus, the averagequared distance from the mean isr(R C R^T) / (n^2 (n+1))here C_(i,j) = n d(i,j) - 1nd d(i,j) is the Kronecker delta (i.e., 1 if i = j, 0 otherwise).==ubject: Re: Fibonacci optimized, aliquot parts were used previouslyow!!! IÕm a junior in highschool, and I am doing a term paper on how egyptian ways and traditions inßuenced greek philosophy which resulted in civilization today. I found your information to be very helpful. Can you tell me more about the egyptians math systems? THANX CAT==ubject: Re: The State-of-the-Art in Mathematics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2Dj30426;> The complex number i is nonsense since i = sqrt(-1) = sqrt(1/-1) = 1/i ...Before anyone can make any sense out of such equations, you need todefine precisely what you mean by the sqrt function. Please provide uswith such a definition before expecting us to respond to anything youpost.The normal definition of sqrt as a single-valued function is as follows:For positive real argument r, itÕs the positive solution to x^2 = r.For negative real argument r, itÕs i times the positive solution to x^2 = -r.For complex argument a + b*i, each a,b real, where b is nonzero,itÕs one of the two solutions to (x + y*i)^2 = (a + b*i) where y>0.Note that with that definition of sqrt,the rule sqrt(a*b) = sqrt(a) * sqrt(b) is not generally true.Likewise itÕs not generally true that sqrt(a/b) = sqrt(a) / sqrt(b),which you seem to have mistakenly assumed in your equation chain.So either that wasnÕt the definition of sqrt you were using (hence myquestion what definition you are using), or you made a stupid mistakeassuming those rules are generally true when in fact they arenÕt.he point IÕm raising is that this contradiction arising from the concept i stems from the fact that i is ill-defined. The mapping sqrt is well-defined only on perfect square such as 1, 4, 100, etc. . E. Escultura==ubject: Re: Please please help with log(z). Practical/Practice by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2GC30629;|Let G = {z : z is not 0, and 0 < arg(z) < 2pi} and let g(re^(it)) =|log(r) + it where 0 < t < 2pi. I do not understand why g is a branch of the |logarithm (by your definition) but if 0 < t <= 2pi, then g is not a branch |of the logarithm. I am grateful for your explanation.You appear to be describing a second function g (letÕs call it g_2)that we could define thus: g_2(z) is the w satisfying 00 from above of g_2(1+ui) is 0.The kind of confusion youÕre describing is normally well dispelled bypaying more careful attention to definitions, and that often seemsto be about the only way to treat it.The fact that the definition of ranchis chosen so as to excludeg_2 is somewhat of the nature of a technical detail, but donÕtunderestimate the value of getting such technical details straight.You will find that there are many situations in mathematics wherethere are multiple possible choices of definition that differ onlyin some detail, and it may be that none seems obviously better thanany other, but where the theory develops much more elegantly lateron if you make certain choices of definition. You will also findthat when youÕre reading a well-written exposition, the author willhave taken the time to make good choices of definition for you, andthat it will typically be worth taking seriously that they mean bytheir definitions *exactly* what they say.Caratheodory defines the rincipal valueof the log function at zto be the w such that e^w=z and -pi0}. Note, however, that I donÕt see him referto anything but ANALYTIC branches of log f from then on. This isconsistent with other textbooks that use phrases like nalytic branchof log The most general concept of nverse functionjust isnÕt allthat nice unless one assumes some additional property likecontinuity or analyticity.However the definitions are set up, there are key lemmas that come uplater, telling you that there exist analytic functions under suitableconditions, provided that the domain is a simply connected region.For instance, if f is analytic on R and z_0 is some point in R,then there exists a function g that is also analytic on R satisfyingg(z_0)=0 and gÕ(z)=f(z) on R. Applying this to f(z)=1/z gives us abranch of the log on R, provided 0 is not in R. ThereÕs no direct wayof applying this lemma to get functions like the one you alluded to(that I called g_2).If you want, later, to consider how a function g analytic on aregion behaves in limits approaching the boundary of R, fine, butget the easy part first.Now that I think of it, it occurs to me that I have myself atopological question related to this. Suppose R is a maximal simply-connected region (i.e. open set) in the complex plane, notcontaining 0, where by maximal I mean simply that there is noregion Rsatisfying the same conditions such that R is a propersubset of RÕ. Is R necessarily the complement of a curve runningthat he would provide on request.) ;-)Keith Ramsay I wonder if in practice these logarithms are actually calculated, is there an actual way of calculating the log of the image of, say, |z-2|=1 under z^2 , or, say the log of some other function? If so, do we calculate it with the formula Log(z)=Integral dz/z along any path ( because of path independence ) from some z_0 to z ? ==ubject: Has anybody encountered the following strange object .. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2B730287;i everybody,as anybody seen an object like this f(x,y)f(u,v)> =c d(x-v) d(y-u) here c is a number, d stands for the Dirac-Deltafunction, and he <> is a functional average over the function f. Does anyone know name for this? Has anyone done some work using/involving this?e are doing research in physics and do not know how to deal with his object and wonder if any of you mathematicians have ideas.ny help will be appreciated!hoshi==ubject: Re: (open?) questions related to basic linear algebra.QUESTIONS: Let M be a module over a commutative ring R. 1) does there always exist minimal generating sets in M ? 2) suppose M has a minimal generating set G; does the finitnessof G imply that of the maximal free sets. Or more weakly, suppose Mhave a base B; does the finitness of B imply that of maximal freesets? 3) suppose M have a base B; does the finitness of B imply that ofminimal generating sets? 4) do always free sets have cardinality less than generating sets?or more weakly, do always free sets have cardinality less than bases? 5) are two maximal free sets always of the same cardinality? Of course, if one suppose R to be an integral domain, many of thesequestions can be answered easily. But I ask for general commutativerings. If anyone knows about a paper that have dealt with thesequestions, please let me know; and else, I would be happy to knowwhat you think about.Michael.nswer to question 3: if M is finitely generated, then everyinimal set of generators is finite.xpress all members of the finite set of generators as linearombinations of the elements of the minimal set of generators.nly finitely many elements of the minimal set will be involved.hese elements form a set of generators of M. The minimalityhus implies the finiteness of the minimal set.==ubject: Re: Recent resolution of issues by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2Ev30456;>This is called the Perron paradox, the use of necessary>condition without existence principle or theory. It involves >assuming the existence of something that is not established>yet and then proving that it is equal to something. Using >this method you can prove that the greatest integer is 1.>Please see, Exact solutions of FermatÕs equation, Nonlinear>Studies, Vol. 5, No. 2, pp. 227 - 254.So inearis a synonym of ensical WhoÕd have thought it?Lee Rudolpho, they are different categories. I really donÕt understand your question.. E. Escultura==ubject: Re: A continuous iteration of f(x,y) r>0 ; f(x,y)^[r] . by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2CP30393;> f:R*R->R .Let us start with a simple case:> g(x,g(x,y))= f(x,y) f given (1) > or f(x,y)^[1/2]=g(x,y) > Example: f(x,y)=y/(1+x*y) , g(x,y)=2*y/(2+x*y) verifies (1).> By the same way will have:f(x,y)^[1/p]=p*y/(p+x*y)=g(x,y) p integer> verifying g(x,g(x....)))n times =f(x,y)=y/(1+x*y).> We may generalize:> if f(x,y)=phi^-1(phi(y)+n(x)) or m^[n(x)](y) ,> phi(m(y))=phi(y)+1 ;> then f(x,y)^[r]=phi^-1(phi(y)+r.n(x)) or m^[r.n(x)](y) ,> here r is a positive real number.> Please your comments and ideas,alain.Alain, my tetration.org web site is dedicated to continuous iteration.See for resources on continuous iteration and for an outline of my combinatorial approach to the subject.For ideas consider Stephen WolframÕs question: > How can one extend recursive function definitions to continuous > numbers? What is the continuous analog of the Ackermann function? The > symbolic forms of the Ackermann function with a fixed first argument > seem to have obvious interpretations for arbitrary real or complex > values of the second argument. But is there a general way to extend > these kinds of recursive definitions to continuous cases? Given a way > to do this, how does it apply to recursive definitions like those on > page 130? What happens to all the irregularities when one is between > integer values? Or is it only possible to find simple continuous > generalizations to functions that show fundamentally simple behavior? > Can this be used as a characterization of when the behavior is simple?For my non-peer-reviewed response to WolframÕs question on the NKS Forum see HereÕs why IÕm interested in the subject. A theory of continuously iterated smooth matrix functions would probably encompass all of dynamics in physics. A theory of continuously iterated smooth complex functions would be powerful enough to extend the Ackermann function to complex numbers.Daniel Geislert seems your email : daniel@finestplanet.comm doesnÕt work !. would like exchanging on some themes...riendly,Alain.==ubject: More on the State-of-the-Art in Physics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2Fv30547;ORE ON THE STATE-OF-THE-ART IN PHYSICShe most significant element in the new physics is the introduction of dynamic modeling upon which all the other discoveries rest. Dynamic modeling explains nature in terms of its laws in contrast to conventional modeling that simply describes nature mathematically. This shifts the task of the physicists from solving mathematical equations to discovering the laws of nature. The discoveries that stem from dynamic modeling are as follows:) The existence of the two fundamental states of matter: dark and visible matter. They are both familiar to physicists now and they even calculated that 95% of our universe is dark matter. There are also conversion laws governing conversion of matter from one fundamental state to another.) The laws of nature anchor the ßux theory of gravitation which is a TOE. It says that gravity is turbulence and is part of the dynamics of vortex ßuxes of superstrings. To-date, there are 42 laws of nature discovered so far that anchor the theory and its subtheories of quantum gravity, macro gravity, turbulence, enriched turbulence, intelligence and learning. It provides a definitive cosmology of our universe. In particular, its destiny is a cluster of black holes in dark matter. A black hole is massive concentration of non-agitated superstrings. or details including list of references visit my websites:ttp://www.users.bigpond.com/pidro/home.htmttp:// home.iprimus.com.au/pidro/. E. Escultura==ubject: Re: More on the State-of-the-Art in Physics MORE ON THE STATE-OF-THE-ART IN PHYSICSORE OFF-TOPIC, MORE LIKE!- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: MORE INFORMATION FROM THE NEW PHYSICS by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2DL30416;ORE INFORMATION FROM THE NEW PHYSICS) The big bang was explosion of a black hole, the destiny of the core of a previous universe.) Our universe is a supersuper galaxy10^10 billion light years across.) The accumulated mass around its eye is a tightly-packed, cocoon-shaped galaxy cluster 650 million light years across discovered by French astronomers in 1994.) As a vortex ßux of superstrings our universe spins and imparts centrifugal force on its galaxies and catapults them outward at the rate of 10^20 km/sec. This is the present rate of its radial expansion. Moreover, this expansion is accelerated at the rate of 1/10^10 km/secsec. This means that our universe is still rising in power of spin towards its peak when it begins to decline towards zero. Then our universe dies as supersuper galaxy and its galaxies will be deprived of the inßuence of its core spin and remain in disarray. The Sagittarius cluster is the remnant of a dead galaxy. Parts of it have been gobbled up by the gravitational action of the Milky Way. ) Contrary to popular belief a black hole does not suck matter, dark or visible, because it is dark matter, a massive concentration of non-agitated superstring. It is the eye of the cosmological vortex that nurtures and builds it up that does. ttp://www.users.bigpond.com/pidro/home.htmttp:// home.iprimus.com.au/pidro/. E. Esculturaniversity of the Philippinesnternational Federation of Nonlinear Analysts==ubject: Re: MORE INFORMATION FROM THE NEW PHYSICS MORE INFORMATION FROM THE NEW PHYSICSou know this is sci.math, not sci.physics?- obin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 rancis Wheen, _How Mumbo-Jumbo Conquered the World_==ubject: Re: The State-of-the-Art in Mathematics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBFD2DG30421;>>In sci.math, S. Enterprize Company>>> MATHEMATICAL UPDATE AND CLARIFICATION> THIS IS part of my occasional update of mathematics and physics. It> refers to the resolution of mathematical issues that have appeared in> several journals recently. (1) The complex number i is nonsense since> i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i.> -i = 1/i>> -i = 1/sqrt(-1)>typo; i =/= -i> i =/= -1>> i = sqrt(-1)> i = sqrt( -1)> i^2 = 1> i^3 = -i>>Be *VERY* careful here. sqrt(4) = +2 or -2. sqrt(-1) is>>traditionally called i, but again sqrt(-1) = +i or -i.>>(ThereÕs at least one oke proofrunning around that>>deliberately confuses the issue by inappropriately taking>>the square root and proving a nonsense result.)>> LetÕs see if the reciprocal of 1/i really is -i.>>Well, if b is the reciprocal of a, then a * b = +1.>>Since i^2 = -1, i * i * (-1) = +1, and one can take>>i * (-1) = -i to be the reciprocal of i. QED.>> It looks like to me, the reciprocal of i is,> i^-1.>> But, letÕs check,>> sqrt(-1)/sqrt(-1) = 1>> i^1 * i^-1 = i^(1-1) = i^0 = 1>> If the reciprocal of i was -1 then>>Did you mean -i here?>> i^1 * (-i^1) = -i^(1+1) = -i^2 = 1 >> It looks like it gives the same answer in this case.>> But what about this ? >> (i^-1) * (i^-1) = i^(-2) = 1/i^2 = 1>>Eh? 1/i^2 = 1/(-1) = -1.>> compare,>> (-i^1) * (-i^1) = i^2 = 1>> It looks like to me, the reciprocal of i can equal -i or i^-1,> because you get the same results for either way you express it.>>And the two are in fact equal.>> But does,>> i^3 = i^-1> i^3 = -i>> It was shown that,>> -i = i^-1>> So it does.>> And,>> i =/= -i>> But when you raise this to the power of 2, two negatives turns into> positive, THEN they are equal to 1.>> i^2 = 1>> -i^2 = 1>>I think you mean (-i)^2 = 1, which is also true.>>-(i^2) = 1 is also true. However, i^2 = -1.Nope. (-1^.5)*(-1^.5) = 1^1 = 1negative * negative = positive >>[.sigsnip]>>-- >>#191, ewill3@earthlink.net>>ItÕs still legal to go .sigless.>the problem starts with the definition that i = sqrt(-1). The mapping sqrt is>well-defined only on perfect square. It is nonsense otherwise. Nope. Perfect squares doesnÕt matter with the sqrt. Example,sqrt(2) = 1.41421...(1.41421...)^2 = 2A perfect square would be,sqrt(4) = 2sqrt(-1) = i <--- This is correct. This accounts for the imaginary side ofreality.To make sqrt(-1) real, you square it,(sqrt(-1))^2 = 1>E. E. E. Esculturahe problem here is not the choice of values of sqrt(-1) but the concept i itself. It is not well-defined because the mapping sqrt is well-defined only on perfect square such as 1, 4, etc. . E EsculturaSmartÕs Alt. Physics News GroupS. Enterprize (Science Journal)==ubject: Re: ext. field-Enigmail-Version: 0.89.5.0-Enigmail-Supports: pgp-inline, pgp-mime Hi I was wondering if someone could help me out. IÕm trying to figure out how to construct an extention field for lets say g(x) = x^3 + 2x + 1 and I have to create the extention field GF(27) = Z3[x]ell since 27 = 3^3, the extension filed will be a polynomial of the ind a x^2 + b x + c where a, b and c are each in the range 0..2. Using our field polynomial we know that x^3 == -2 x - 1 == x + 2 mod 3 x^3 == -4 x - 2 == 2 x + 1 mod 3ow if we denote a x^2 + b x + c by (a b c) we can start with unity (0 0 ) and multiply by x to get x (0 1 0), x by x to get x^2 (1 0 0) and x^2 y x to get X^3. But we know that x^3 is the same as x + 2 (above) so e can make this substitution to get (0 1 2). We can then multiply this y x to get x^4 as (1 2 0) since x just moves the coefficients left by 1 lace. Everytime we get an overßow into x^3 we can use one of the bove equalities to reduce the result to a polynomial with x^2 as it ighest term.f we do this we get: 0 0 1 0 1 0 1 0 0 0 1 2 1 2 0 2 1 2 1 1 1 1 2 2 2 0 2 0 1 10 1 1 01 1 1 22 1 0 23 0 0 24 0 2 05 2 0 06 0 2 17 2 1 08 1 2 19 2 2 20 2 1 11 1 0 12 0 2 23 2 2 04 2 2 15 2 0 16 0 0 1here the last result shows that we have returned to unity. So we have 0 0 0) and elements 0..25 above as the 27 elements of GF(27) Brian Gladman==ubject: Re: ext. field I was wondering if someone could help me out. IÕm trying to figure out how to construct an extention field for lets say g(x) = x^3 + 2x + 1 and I have to create the extention field GF(27) = Z3[x]he elements of this field are the polynomials ax^2 + bx + c.hey are multiplied modulo g(x).ut youÕd have to say what exactly you want to do in the field.- imothy Murphy -mail (<80k only): tim /at/ birdsnest.maths.tcd.ieel: +353-86-2336090, +353-1-2842366-mail: School of Mathematics, Trinity College, Dublin 2, Ireland==ubject: Re: An interesting matrixour are correct in that the algorithm is almost (but not quite) identicalo the Sieve of Eratosthenes. The algorithm obviously builds on the Sievend the result is this beautiful little matrix that has fractal-likeualities.avid > Here is an interesting matrix at www.guffy.net/matrix.htm. > It is constructed from a very simple algorithm with the result beinghat > the matrix is upper triangular with the appearance of diagonals thatave > the same pattern as the rows from which the matrix is constructed. > I have no conclusions about the matrix but simply wanted to share itith > others. > David Will you call it the Matrix of Eratosthenes? -- John jmatthews at wright dot edu www dot wright dot edu/~john.matthews/==ubject: What is the name of this curve? posting-account=9Lp_BA0AAADb80yh5G0s_940Xs_4VvWd was recently looking at my closet and asked myself...What does theurve look like that my closet doors sweep out on the ßoor? So I didome calculations.irst off, My door is not a simple one pane door sweeping out a circle.tÕs a 2-pane door where one end pivots, one end slides on a track andhereÕs a hinge in the middle. What this boils down to is determininghe shape swept out by a line of constant length with one endonstrained to a circle and the other stuck to the x-axis.Õll step through my methodology and let me know if I made any mistakesItÕs been a long time since I did a lot of trig and calculus). IÕveoticed by doing some reading that a lot of curves formed by moving onebject along another are hyperbolic functions where this turns out toe trigonometric.y terms are L is each doors length, theta is the angle that the firsts open (0=closed pi/2 is open). First I looked at the positions ofhe doors(ItÕs piecewise):(theta,x)={0=3;(1)=F(2)=1 hank you.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?ays...How to prove that Fibonacci sequence is primitive recursive?F(n)=F(n-1)+F(n-2) n>=3;F(1)=F(2)=1ereÕs a hint: Suppose instead of F(n) you tried to define a function(n) that returned the ordered pair of numbers . (Therere many ways to code a pair of numbers as a single number.)-aryl McCulloughthaca, NY==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? HereÕs a hint: Suppose instead of F(n) you tried to define a function g(n) that returned the ordered pair of numbers . (There are many ways to code a pair of numbers as a single number.) Now thatÕs a real hint.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? How to prove that Fibonacci sequence is primitive recursive? F(n)=F(n-1)+F(n-2) n>=3; F(1)=F(2)=1 thank you.s addition primitive recursive?s subtracting 1 primitive recursive?s subtracting 2 primitive recursive?s 1 primitive recursive?s 2 primitive recursive?- ill Twentymanmail: wtwentyman at copper dot net==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?ill Twentyman says...> How to prove that Fibonacci sequence is primitive recursive?> F(n)=F(n-1)+F(n-2) n>=3;> F(1)=F(2)=1 > thank you.Is addition primitive recursive?Is subtracting 1 primitive recursive?Is subtracting 2 primitive recursive?Is 1 primitive recursive?Is 2 primitive recursive? think youÕre missing the point: F(n) is defined in termsf F(n-1) and F(n-2), while a primitive recursive definitionives F(n) in terms of F(n-1).-aryl McCulloughthaca, NY==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? Will Twentyman says... >> How to prove that Fibonacci sequence is primitive recursive? >> F(n)=F(n-1)+F(n-2) n>=3; >> F(1)=F(2)=1 > thank you. >>Is addition primitive recursive? ...... I think youÕre missing the point: F(n) is defined in terms of F(n-1) and F(n-2), while a primitive recursive definition gives F(n) in terms of F(n-1).his definition is an instance of ourse of values ecursion, and Rozsa Peter showed (Recursive Functions,cademic Press, 1967) that this scheme does not lead outsidehe primitive recursive functions. In fact her example ishe F sequence. Dennis==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? How to prove that Fibonacci sequence is primitive recursive? F(n)=F(n-1)+F(n-2) n>=3; F(1)=F(2)=1 thank you.irst step: * find the definition of rimitive recursive econd step: * see if the above recursion fits that definitionone.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? First step: * find the definition of rimitive recursive Second step: * see if the above recursion fits that definition itshow?==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? posting-account=uJhfTw0AAACZ85X1hg4ZQuYw9kXQVPPG > First step: > * find the definition of rimitive recursive > Second step: > * see if the above recursion fits that definition itshow?heesh. terminological puritan...or its that definition substitute an be defined in terms ofombinations of 0, succ, proj, comp, and prim recne of those will be especially relevant...==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?herifffruitßy@gmail.com says...> First step:> * find the definition of rimitive recursive > Second step:> * see if the above recursion fits that definition> itshow?sheesh. terminological puritan...For its that definition substitute an be defined in terms ofcombinations of 0, succ, proj, comp, and prim recOne of those will be especially relevant... think you missed the point. The fact that F(n) (theth Fibonacci number) is primitive recursive does *not*ollow by inspection from the definition of rimitiveecursive So what does ee if the above recursionits that definitionmean? Does it mean heck all possiblerimitive recursive definitions, and see if any of themomputes the same function as F(n)? -aryl McCulloughthaca, NY==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? I think you missed the point. The fact that F(n) (the nth Fibonacci number) is primitive recursive does *not* follow by inspection from the definition of rimitive recursive hether it does or not depends on the particular definitionf rimitive recursivethat is intended. This is the reasonor asking that the definition be provided to us, beforee can help further.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? posting-account=S4DtCQwAAABnGcRBFZz3eigcfwscMK8D > I think you missed the point. The fact that F(n) (the > nth Fibonacci number) is primitive recursive does *not* > follow by inspection from the definition of rimitive > recursive Whether it does or not depends on the particular definition of rimitive recursivethat is intended. This is the reason for asking that the definition be provided to us, before we can help further.efinition of primitive recursive function is:)f(0) is primitive recursive;)f(x)=f(x)+1 is primitive recursive;)fn_m(u1, u2, ... un)=um is primitive recursive;)Composition: Given f(x) and g(x) primitive recursive functions, the(g(x)) function is primitive recursive;)Primitive recursion: Given g(x) a primitive recursive function,e form a new primitive recursive function f(x) defined by:(0)=x0;(x+1)=g(x, f(x));y the way, my speciality is computer application and I am onlynterested in computability theory.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?Discussion, linux)> I think you missed the point. The fact that F(n) (the> nth Fibonacci number) is primitive recursive does *not*> follow by inspection from the definition of rimitive> recursive> Whether it does or not depends on the particular definition> of rimitive recursivethat is intended. This is the reason> for asking that the definition be provided to us, before> we can help further. Definition of primitive recursive function is: 1)f(0) is primitive recursive;uh? 2)f(x)=f(x)+1 is primitive recursive;ha? think something ainÕt quite right here.- esse F. HughesLike the ski resort full of girls hunting for husbands nd husbands hunting for girls, the situation is not asymmetrical as it might seem. -- Alan MacKay==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? N Niel says...> I think you missed the point. The fact that F(n) (the> nth Fibonacci number) is primitive recursive does *not*> follow by inspection from the definition of rimitive> recursive Whether it does or not depends on the particular definitionof rimitive recursivethat is intended. This is the reasonfor asking that the definition be provided to us, beforewe can help further.Õm sorry. I thought that was pretty standard.he primitive recursive functions is the smallest collectionf functions including the primitive functions zero(x) = 0 successor(x) = x+1 proj^i_n(x_1,x_2,...,x_n) = x_ilosed under the operations of composition and primitive recursion,here primitive recursion takes a function g(x_1,...,x_n) and aunction h(x_1,...,x_n,y) and returns a function f(x_1,...,x_n,y)uch that f(x_1,...,x_n,0) = g(x_1,...,x_n) f(g(x_1,...,x_n,y+1) = h(x_1,...,x_n,f(x_1,...,x_n,y))ow, what you can prove (with a bit of work) is that anyotal recursive function that is bounded by a primitive recursiveunction is primitive recursive. So once youÕve proved that,t is pretty simple to show that something is primitive recursiveithout getting into the details of how rimitiveecursionis defined.-aryl McCulloughthaca, NY ==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? Whether it does or not depends on the particular definition of rimitive recursivethat is intended. I think you will find that there is no significant variation in theefinition of rimitive recursive==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?K, letÕs use the NIST definition:rimitive recursiveefinition: A total function which can be written using only nestedonditional (if-then-else) statements and fixed iteration (for) loops.--------National Institute of Standards and Technology...ttp://www.nist.gov/dads/HTML/primitivrecr.html== ubject: Re: How to prove that Fibonacci sequence is primitive recursive? N Niel says...OK, letÕs use the NIST definition:primitive recursiveDefinition: A total function which can be written using only nestedconditional (if-then-else) statements and fixed iteration (for) loops.o, letÕs not. ThatÕs not the usual definition. (Although it giveshe same set of functions---but there is a proof involved in showinghat it is the same set of functions as the usual definition.)-aryl McCulloughthaca, NY==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? OK, letÕs use the NIST definition: primitive recursive Definition: A total function which can be written using only nested conditional (if-then-else) statements and fixed iteration (for) loops. This isnÕt actually a definition of anything. However, somebody whonows a bit about programming will be able to specify a suitableanguage and make it into a definition: a function which is definabley a program in language P.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? posting-account=uJhfTw0AAACZ85X1hg4ZQuYw9kXQVPPGouÕre right - damn godel numbers....==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? For its that definition substitute an be defined in terms of combinations of 0, succ, proj, comp, and prim rec One of those will be especially relevant... In other words, you are just answering econd step: see if theunction is in fact primitive recursive. ==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? posting-account=uJhfTw0AAACZ85X1hg4ZQuYw9kXQVPPGup.ome people like to vary the vocabulary used to describe one and theame thing... quirky, I know...==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? Some people like to vary the vocabulary used to describe one and the same thing... quirky, I know... Quirky? No, just pointless.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?> Some people like to vary the vocabulary used to describe one and the> same thing... quirky, I know... Quirky? No, just pointless.he point is that the OP should put some of his/her own effort into oing this perfectly straightforward homework question.obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?obert Israel says...The point is that the OP should put some of his/her own effort into doing this perfectly straightforward homework question.tÕs not perfectly straightforward. The recursive definition of(n) (the nth Fibonacci number) is not a primitive recursiveefinition, since F(n) involves both F(n-1) and F(n-2), while primitive recursive definition can only give F(n) in termsf F(n-1). To solve the problem, you have to do something thats *not* straight-forward, which is show that course-of-valuesecursion can be defined in terms of primitive recursion. Thatnvolves showing that a list of numbers can be coded as a singleumber, and that both the coding and the decoding is primitiveecursive. That might be obvious once youÕve seen itone once, but it isnÕt obvious otherwise.-aryl McCulloughthaca, NY==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?Robert Israel says...>The point is that the OP should put some of his/her own effort into >doing this perfectly straightforward homework question.ItÕs not perfectly straightforward. The recursive definition ofF(n) (the nth Fibonacci number) is not a primitive recursivedefinition, since F(n) involves both F(n-1) and F(n-2), whilea primitive recursive definition can only give F(n) in termsof F(n-1). To solve the problem, you have to do something thatis *not* straight-forward, which is show that course-of-valuesrecursion can be defined in terms of primitive recursion. Thatinvolves showing that a list of numbers can be coded as a singlenumber, and that both the coding and the decoding is primitiverecursive. That might be obvious once youÕve seen itdone once, but it isnÕt obvious otherwise.n the Fibonacci case, you can encode F(n) and F(n-1) as ^n*F(n)+F(n-1). You do need to know that multiplication andivision are primitive recursive.obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? The point is that the OP should put some of his/her own effort into doing this perfectly straightforward homework question. The point of what?==ubject: Re: How to prove that Fibonacci sequence is primitive recursive?> The point is that the OP should put some of his/her own effort into > doing this perfectly straightforward homework question. The point of what?he point of providing a not-too-explicit hint.obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? The point of providing a not-too-explicit hint. No hint whatever has been posted.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? posting-account=uJhfTw0AAACZ85X1hg4ZQuYw9kXQVPPGt conceivable that you would enjoy checking out what some pointlessude named Frege had to say about he morning starand he eveningtar People like him are of the opinion that have different ways oficking the same thing out is quite the *opposite* of pointless...==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? It conceivable that you would enjoy checking out what some pointless dude named Frege had to say about he morning starand he evening star People like him are of the opinion that have different ways of picking the same thing out is quite the *opposite* of pointless... I doubt that Frege would have thought it very interesting tonswer the question ow do I prove that the Fibonacci sequence isrimitive recursive?by ry showing that it is primitive recursive==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? posting-account=uJhfTw0AAACZ85X1hg4ZQuYw9kXQVPPG > It conceivable that you would enjoy checking out what someointless > dude named Frege had to say about he morning starand hevening > star People like him are of the opinion that have different waysf > picking the same thing out is quite the *opposite* of pointless... I doubt that Frege would have thought it very interesting to answer the question ow do I prove that the Fibonacci sequence is primitive recursive?by ry showing that it is primitiveecursiveshrugs> ok.==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? > First step: > * find the definition of rimitive recursive > Second step: > * see if the above recursion fits that definition itshow?hat is the definition of rimitive recursive- . A. Edgar http://www.math.ohio-state.edu/~edgar/==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? What is the definition of rimitive recursiveoogle .27,000 hitsob Kolker==ubject: Re: How to prove that Fibonacci sequence is primitive recursive? What is the definition of rimitive recursive You can look it up. So what does it mean to say that the aboveecursion itsthe definition? And what conclusion can we drawrom the observation that it does or that it does not ittheefinition?==ubject: Need help with boolean problem. Figure 10(a) shows the intersection of a main highway with aecondary access road.ehicle detection sensors are placed along lanes C and D (main road)nd lanes A and Baccess) road. These sensor outputs are LOW (0) when no vehicle isresent and HIGH (1)hen a vehicle is present. The intersection traffic light is to beontrolled according to theollowing logic:) The east west (E-W) traffic light will be green whenever both lanes and D are occupied.) The E-W light will be green whenever either C or D is occupied butanes A and B are nototh occupied.) The north -south (N-S) light will be green whenever both lanes And B are occupied but and D are not both occupied.) The N-S light will also be green when either A or B is occupiedhile C and D are bothacant.) The E-W light will also be green when no vehicles are present.sing the sensor outputs A, B, C and D as inputs, design a logicircuit to control the trafficight. There should be two outputs, N-S and E-W, that go HIGH when theorrespondingight is green. Simplify the circuit as much as possible and show allteps!. For the above question.a) Make 2 Truth tables (one for the E-W light, one for the N-Sight).b) Write out the Boolean equation for each Truth Table.c) Simplify the equations to the least # of ICÕs.d) Draw the logic circuits to control both lights.e) Connect the east west circuit on your breadboard and demonstrateo the instructorhat your circuits are functioning properly.==ubject: Re: Need help with boolean problem1. Figure 10(a) shows the intersection of a main highway with asecondary access road.Vehicle detection sensors are placed along lanes C and D (main road)and lanes A and B(access) road. ou forgot to copy Figure 10(a). I presume C and D represent theriverÕs lanes on the main highway and A and B represent the driverÕsanes on the access (perpendicular?) road.o what have you done so far on the problem? Have you studied Karnaughaps? Have you set up a Karnaugh map with inputs A, B, C, and D? What,xactly are you stuck on?-Lynn==ubject: help on a recursive functionÕm interested on the behaviour on a H-length circular queue, empty before i=0, equal to P(0)=a and P(i)= b Sum(j=i-1, i-H [P(j)]), that is:(i) = a(1+b)^i i=0, ..., H-1(i) = (1+b)P(i-1) - b P(i-H) i>= H got just the following results:(i) = a (b(i+1) - 1)/(b - 1) *if H=1*ndim(i->Inf [ P(i) ]) = 1 / (1-b) for each H, if | b | <1y question is: is it possible to express this function in closed form for each H?hank you in advance,moll-est==ubject: Re: help on a recursive function IÕm interested on the behaviour on a H-length circular queue, empty before i=0, equal to P(0)=a and P(i)= b Sum(j=i-1, i-H [P(j)]), that is: P(i) = a(1+b)^i i=0, ..., H-1 P(i) = (1+b)P(i-1) - b P(i-H) i>= H I got just the following results: P(i) = a (b(i+1) - 1)/(b - 1) *if H=1*h? If H = 1, I get, for i > 0,(i) = (1+b)P(i-1) - bP(i-1) = P(i-1) = ... = a and lim(i->Inf [ P(i) ]) = 1 / (1-b) for each H, if | b | <1 my question is: is it possible to express this function in closed form for each H?obert has already given you one answer. HereÕs anotherf k = mH + n, with 0 <= n < H, m >= 0, then(k) = (1+b)^n sum{i=0}^{m} (-1)^i comb{(m-i)H + n + i}{i} b^i (1+b)^{(m-i)H}dmittedly, thatÕs not in a very losedform. ick==ubject: Re: help on a recursive function> IÕm interested on the behaviour on a H-length circular queue, empty > before i=0, equal to P(0)=a and P(i)= b Sum(j=i-1, i-H [P(j)]), that is:> P(i) = a(1+b)^i i=0, ..., H-1> P(i) = (1+b)P(i-1) - b P(i-H) i>= H> I got just the following results:> P(i) = a (b(i+1) - 1)/(b - 1) *if H=1* Oh? If H = 1, I get, for i > 0, P(i) = (1+b)P(i-1) - bP(i-1) = P(i-1) = ... = aops... excuse me Rick, I meant with:(i) = a (b^(i+1) - 1)/(b - 1)I forgot the Robert has already given you one answer. HereÕs another If k = mH + n, with 0 <= n < H, m >= 0, then P(k) = (1+b)^n sum{i=0}^{m} (-1)^i comb{(m-i)H + n + i}{i} b^i (1+b)^{(m-i)H} Admittedly, thatÕs not in a very losedform. h, itÕs perferct :-) tried by Hwith a few results...ould you help me understanding *how* have you got in this?moll Est==ubject: Re: help on a recursive functionsnip>> If k = mH + n, with 0 <= n < H, m >= 0, then> P(k) => (1+b)^n sum{i=0}^{m} (-1)^i comb{(m-i)H + n + i}{i} b^i (1+b)^{(m-i)H}> Admittedly, thatÕs not in a very losedform. oh, itÕs perferct :-) I tried by Hwith a few results... Could you help me understanding *how* have you got in this?ure luck. :-)ctually, all I did was the usual stuff:. Write the values for some P(i). Identify a pattern (this is the agicstep). Prove that your guess is right (often by induction).ut IÕm sure that IÕm not telling you anything you didnÕtlready know.ick==ubject: Re: help on a recursive functionother. Multiply the expression below by a. I left it off whileorking and forgot to put it back in when posting. If k = mH + n, with 0 <= n < H, m >= 0, then P(k) = (1+b)^n sum{i=0}^{m} (-1)^i comb{(m-i)H + n + i}{i} b^i (1+b)^{(m-i)H} Admittedly, thatÕs not in a very losedform. Rick==ubject: Re: help on a recursive functionP(i) = a(1+b)^i i=0, ..., H-1P(i) = (1+b)P(i-1) - b P(i-H) i>= Hmy question is: is it possible to express this function in closed formfor each H?t looks to me like the answer is (i) = a/(1-(H-1)b) (1 - sum_r r^(-i) (1-r^H -(1-r)H)/(r^H (H+r-rH) - r))here the sum is over the roots r of the polynomial (z) = 1 - b sum_{j=1}^{H-1} z^j = (1 - (1+b) z + b z^H)/(1 - z)assuming that polynomial has all its roots distinct, and b <> 1/(H-1)).obert Israel israel@math.ubc.caepartment of Mathematics http://www.math.ubc.ca/~israel niversity of British Columbia Vancouver, BC, Canada= set. It isasy and I leave the details to the reader. The dual notion waseveloped and it was shown that the Pontrjagin dual of a colimit ofbelian groups was the limit of the dual groups (carried out in thecategory-the concept was very new--of compact abelian groups). nyway, calling the limits was very natural. Why limits and colimitsere named as they were is not clear to me, but I remember Eilenbergaying once that roductwas rarely misused (but cf. free product). ourbaki generalized it to filtered limits and colimits and eventuallyilenberg and others realized that even that was unnecessary and thatny functor could be said to have a limit or colimit. Actually, youonÕt even need a functor; it is quite natural to use a graph and araph morphism to a category. By this time, the original connectiono Moore-Smith convergence of nets had been left behind, like theheshire cat.hen I think about limits, I think about subobjects of products.==ubject: Re: Implicit Function Theorem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGMqWo17652;> Let (x*,y*) in S be a vector such h(x*,y*) = 0 and the matrix nabla_y > h(x*,y*) is nonsingular, the implicit function states that > there exist (x,y) in the neigborhood of (x*,y*) such that y = f(x) and > h(x,f(x)) = 0.YouÕve badly mangled the IFT, omitting hypotheses, misstating the conclusion, ...orry..... h is C^1, so because det[dh/dy](x*,y*) is nonzero, det[dh/dy]is nonzero in a neighborhood of (x*,y*) by continuity; yourconclusion follows.ny easy way to show this?