mm-439 Subject: Re: Plotting a 7 vertice graph in which every vertex has degree 4 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21:> I am trying to determine the number of isomorphism classes of simple> 7-vertex graphs in which every vertex has degree 4.It is much easier if you consider their complements instead: 7-vertex 2-regular graphs. A 2-regular graph is just a disjoint union of cycles, and there are only two ways of doing this with seven vertices: a single 7-cycle, or the disjoint union of a 3-cycle and a 4-cycle.-- David Eppstein http://www.ics.uci.edu/~eppstein/Univ. of California, Irvine, School of Information & Computer Science === Subject: Re: Plotting a 7 vertice graph in which every vertex has degree 4David,I came up with three sub-graphs, which I have uploaded the images of withMathematica.The first of the four images is just the complete graph for K_7, and doesn'trelate.Do I have the right idea?http://home.earthlink.net/~diana53/mathematica/1-1-17. htmlDiana> I am trying to determine the number of isomorphism classes of simple> 7-vertex graphs in which every vertex has degree 4.> It is much easier if you consider their complements instead: 7-vertex> 2-regular graphs. A 2-regular graph is just a disjoint union of cycles,> and there are only two ways of doing this with seven vertices: a single> 7-cycle, or the disjoint union of a 3-cycle and a 4-cycle.> -- > David Eppstein http://www.ics.uci.edu/~eppstein/> Univ. of California, Irvine, School of Information & Computer Science === Subject: Re: Plotting a 7 vertice graph in which every vertex has degree 4 <7GtVb.16201$F23.14567@newsread2.news.pas.earthlink.net David,> I came up with three sub-graphs, which I have uploaded the images of with> Mathematica.> The first of the four images is just the complete graph for K_7, and doesn't> relate.> Do I have the right idea?> http://home.earthlink.net/~diana53/mathematica/1-1-17.html I'm not sure why you have the complete graph on 7 vertices, since its not 4-regular. The rest of the graphs look like different drawings of the same graph (the complement of the 7 cycle), as Professor Eppstein suggested. His other suggestion is the complement of a 4 cycle and 3 cycle; i.e. you will have a graph whose vertices can be partitioned into two sets, one set with 3 independent vertices and another set with 4 vertices and two disjoin edges, and then form all edges between these two sets.J === Subject: Re: Plotting a 7 vertice graph in which every vertex has degree 4David,> I am trying to determine the number of isomorphism classes of simple> 7-vertex graphs in which every vertex has degree 4.> It is much easier if you consider their complements instead: 7-vertex> 2-regular graphs. A 2-regular graph is just a disjoint union of cycles,> and there are only two ways of doing this with seven vertices: a single> 7-cycle, or the disjoint union of a 3-cycle and a 4-cycle.> -- > David Eppstein http://www.ics.uci.edu/~eppstein/> Univ. of California, Irvine, School of Information & Computer Science === Subject: greek numeralsHi i require information wrt the above-im writting an essay on how thegreek system came into being-and what impact it had on mathematics-cananyone direct me to good sites etc === Subject: Re: greek numeralsMark im sorry for being vague-im exploring the early Greek system ofusing acrophonics and then the alphanumerical usage-in general imlooking at exploring how the greeks acquired their systems-how theywere modified and how they were supersededneil> .> Perhaps you have a specific thing in mind by the 'Greek system', > not just ancient Greek mathematics in general, Neil? Sounds intriguing. Can you say more? MarkHi i require information wrt the above-im writting an essay on how the>> greek system came into being-and what impact it had on mathematics-can>> anyone direct me to good sites etc http:// www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=numerals+greek +mathematics+notation&btnG=Google+Search === Subject: need help!!!> Mark im sorry for being vague-im exploring the early Greek system of> using acrophonics and then the alphanumerical usage-in general im> looking at exploring how the greeks acquired their systems-how they> were modified and how they were superseded> neil> .> Perhaps you have a specific thing in mind by the 'Greek system', > not just ancient Greek mathematics in general, Neil? Sounds intriguing. Can you say more? Mark >> Hi i require information wrt the above-im writting an essay on how the>> greek system came into being-and what impact it had on mathematics-can>> anyone direct me to good sites etc> > http:// www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=numerals+greek +mathematics+notation&btnG=Google+Searchhi every body can plot this function please say me withwww,hupo19@yahoo.com it is y=arcsin^-1(3/cosx),thank you === Subject: Re: greek numerals> Hi i require information wrt the above-im writting an essay on how the> greek system came into being-and what impact it had on mathematics-can> anyone direct me to good sites etc http://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q= numerals+greek+mathematics+notation&btnG=Google+Search === Subject: Re: apexart randome2> that was *your* claim; they're the same:> 720 degrees (total) divided by twelve vertices (since> the vertices are identical on either of the shapes ...> the math is not Beyond Grade-school ...> taking 3rd, over ?-)You never got back to me about your claim that the icosahedron and> cubeoctahedron had different total angulat deficits.Okay. I misunderstood you. What I said was- the cubeoctahedron hasless curvature per unit area than the icosahedron. It has the sameangular deficit but has a greater area. Do you okay that statement? === Subject: Trigiometrysin(3/2a)/sin(1/2a)=2cosa+1Can anyone see what i should do next?Thx,Roy. === Subject: Re: Trigiometry ETAsAhRxfRHHTY63dHT5jKcpjAV8MwMXCwIUQqcCTJubSOCkdVwjCqJY9H3kTRY = Hint: sin 3x = 4 sin^3 x + 3 sin x, therefore sin 3x / sin x = 4 sin^2x + 3.--OL === Subject: Thx all> Hint: sin 3x = 4 sin^3 x + 3 sin x, therefore sin 3x / sin x = 4 sin^2> x + 3.> --OL === Subject: Re: TrigiometryContent-transfer-encoding: 8bit> sin(3/2a)/sin(1/2a)=2cosa+1Can anyone see what i should do next?Assuming you mean (3/2)*a and (1/2)*a (otherwise, see Virgil's reply):Let u = (1/2)*a to get sin(3*u) / sin(u) = 2*cos(2*u) + 1.Now, sin(3*u) = 3*(cos(u))^2*sin(u) - (sin(u))^3 and cos(2*u) = (cos(u))^2 - (sin(u))^2.So, (3*(cos(u))^2*sin(u) - (sin(u))^3) / sin(u) = 3*(cos(u))^2 - (sin(u))^2 = 2*(cos(u))^2 + (cos(u))^2 - (sin(u))^2 =2*(cos(u))^2 + cos(2*u) =(cos(u))^2 + (cos(u))^2 + cos(2*u) =(cos(u))^2 + 1 - (sin(u))^2 + cos(2*u) =2*cos(2*u) + 1.In other words, the equation is true for _any_ a providing sin((1/2)*a) <> 0.-- Paul SperryColumbia, SC (USA) === Subject: Re: Trigiometry> sin(3/2a)/sin(1/2a)=2cosa+1Can anyone see what i should do next?> Thx,> Roy.Given that the arguments of the sine functions involve reciprocals of the agrument of the cosine function, I suspect that there will be no exact form of solution, at least in terms of any elementary functions that I am aware of, and numerical solutions by approximation methods are the only way to go.Whenever a is a solution, -a will also be a solution, and 0 cannot be a solution, so you need only look at positive solutions. === Subject: applying RiemannHypothesis modification to Kepler Packing ProblemThe below is an old post talking about the Kepler Packing Problem asregards to Kissing points. === Subject: Re: PROOF OF THE KEPLER PACKINGPROBLEM<1993Aug19.021638.256@rp.CSIRO.AU>Radiophysics/ Australia Telescope National Facility <24qasr$moq@news.u.washington.edu><24qasr$moq@ news.u.washington.edu>, tzs@hardy.u.washington.edu (Tim Smith) >Ludwig Plutonium >>(see Figures 2) in Euclidean3-dimensional space. The face->>centered-cubic pack has 12points of tangency for each identical >>sphere. > >And from that, he went straight to this: >>If there is a more dense pack in3-dimensional Euclidean space, >>implies there existsat least one sphere which has 13 points of >>tangency. > >Whoa! You can't make a leap like thatwithout giving proof. This >reminds Unfortunately, Mr Plutonium DEFINEDdensity of a lattice as the numberoriginal post : Define density of identical circles asto the number of points of tangency. I say this is unfortunate because itignores convention. Indeed the number of points of tangency(conventionally, the kissing number) and the sphere packing density are twoquite separate problems in the conventional literature. Ignoringconvention is a great way to ensure ones ideas are enshrined in perpetuity ashaving never been discounted by the experts. However, since I'm not an expert, I thinkit should at least be pointed out that the sphere packing densityproblem and the kissing number problem have separately motivated somevery interesting work over many years. That they are not the same isdemonstrated for example in 9 dimensions, where the densest knownsphere packing is that produced by the Lamda_9 lattice (packing density0.14577, kissing number 272), whilst the greatest known kissing numberis that achieved by the non-lattice packing P_{9a} (packingdensity 0.12885, max kissing number 306). For more details, an excellent bookon this topic is that of Conway and Sloane, Sphere Packings,Lattices and Groups (Springer-Verlag). On the differencebetween the two problems, I quote from this book : . . . in general we should expect thelattice and nonlattice versions of the packing and kissing problems tohave four different answers. The moral is that the kissing number questionis a local problem, while the sphere packing question is a globalproblem! It is interesting to note that fordimensions less than 9, the packings which give the greatest density are alsothose that achieve the greatest kissing number (perhaps MrPlutonium was implicitly assuming this, and therefore mixing histerminology, but that, too, is unconventional. . .)Back in 1993 I had given a proof of the KPP and saying that it was theKissing points. But then Karl Heuer and others such as Andrew Woolfsaid the KPP was not equivalent to the kissing points and they citeddetails such as above.Yesterday I posted another look at the Poincare Conjecture withapplying the RH modification. If we accept as true that NaturalNumbersare the P-adics and accept as true my 2 proofs of RH then the P-adicsall lie on the 1/2 Real Line. But since the p-adics do not form astraightline but are curved implies that many changes in Geometry mustthen take place. That Euclidean straightlines are imaginary. That alllines curve.So then, calling it the RH modification to geometry. Today I want toreexamine the KPP proof. I offered a proof in early 1990s saying theKPP was equal to the Kissing problem. Others said no because higherdimensions distort KPP away from kissing. But, if we accept as truethe RH with its P-adic solution then I suspect it also evaporates theobjections by Karl Heuer and Andrew Woolf and many others with theirhigher dimensions. The RH modification forces the KPP to be equal tothe Kissing. And just as straightlines out to infinity were a fictionand imagination, so also is higher dimensions just imagination.Can someone state the NP problem. I am not sure whether it had somegeometrical elements involved. What I am looking for are other oldoutstanding conjectures in geometry that are unsolved and which havethe assumption that straightlines out to infinity remain straight. Yousee, I would like to apply the P-adic modification of RH.Archimedes Plutoniumwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are galaxies === Subject: NaturalNumbers are the P-adics and why Kissing density jumps in KPP Re: applying RiemannHypothesis modification to Kepler Packing Problem> The below is an old post talking about the Kepler Packing Problem as> regards to Kissing points.> === > Subject: Re: PROOF OF THE KEPLER PACKING> PROBLEM> <1993Aug19.021638.256@rp.CSIRO.AU Radiophysics/Australia> Telescope> National Facility> (snipped to save space)> example in 9> dimensions, where the densest known> sphere packing is> that produced by> the Lamda_9 lattice (packing density> 0.14577, kissing> number 272),> whilst the greatest known kissing number> is that> achieved by the> non-lattice packing P_{9a} (packing> density 0.12885,> max kissing number> 306). For more details, an excellent bookOkay, well, if the NaturalNumbers are the P-adics, and if RH impliesthat there are no straightlines at infinity because the P-adicscompose the 1/2 Realline.Then what an application of RH would do to the Kepler Packing Problemis to first ask the question of does a p-adic dimensional space makemuch sense. Is there a ...99999 dimensional space in 10-adics? Isthere a ....11111 dimensional space in 2-adics?Then further, a RH application of p-adics to the KPP of kissing pointsversus densest-nonkissing plan of attack to prove would then ask theVery Important Question:Question: does the above quoting suggest that the divergence of 9thdimension becomes even more divergent when in the 10th dimension. Thenthe 11th dimension, how much of a divergence if any from the previousdimensions.You see, if NaturalNumbers are really the P-adics, then in KPP thereshould be a linear increase in divergence as we go higher indimensions with the density of packing.For example: the writer above noted that in 9th dimension the kissingdiverges from regular KPP, then the kissing should also diverge in10th dimension, and also in 11th dimension and so forth. But, if theKPP does not diverge in 10th dimension from that of kissing in 10thdimension Suggests or Implies that the P-adics are involved.If Straightlines exist out to infinity and if NaturalNumbers are theFiniteIntegers then the KPP should not be a pockmarked gapping ofkissing points divergence as we increase in dimensions.On the other hand, if NaturalNumbers are the P-adics and that allstraightlines curve as they approach infinity (i.e. straightlines donot exist), then the divergence of the KPP from that of kissing pointsversus densest pack would not be a smooth linear relationship as weincrease in dimensions, and instead have gaps where in say dimension22 the kissing points is the densest pack and where dimension 23 thekissing points are not the densest.Demonstration: If we take oranges to pack and we had a square box(Euclidean Space) and a similar volumed sphere and asked to pack thoseoranges in which container could we get the densest packing? The cubeor the sphere? So that in the KPP, applying the RH would suggest thatthe divergence of kissing is because of the fundamental reason thatNaturalNumbers are really the P-adics.Because if space is Euclidean and that straightlines remain straightout to infinity and that NaturalNumbers are FiniteIntegers then as youincrease in dimensions from say 9 to 10 to 11 to 12 etc etc, that thedivergence from packing should also be a Smooth and linearprogression. But it is not. It is gap ridden and swinging back andforth between kissing as the densest and kissing not the densest.Archimedes Plutoniumwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are galaxies === Subject: Re: JSH: Research question answered Discussion, linux)> I think it interesting as an advanced question to figure out how to> construct one of the imperfect factorizations from a given> tautological space, though it seems that would require a tautological> space only valid in the complex plane.> Hmmm...therefore, it seems logical that an imperfect factorization> cannot be constructed from a tautological space.> The formalism necessary to prove that quick deduction must be rather> impressive.This *does* sound like an interesting and advanced question, if only Iknew what it means.So, following James's lead, I delved into advanced mathematicalresearch into these issues. That is, I googled for tautologicalspace. I got three hits: two regarding some performance art hoohahand one on Postethnic Narrative Criticism. Surprisingly, neither ofthese topics have any mention of imperfect factorization, so it doesappear that James's proposed research is groundbreaking.I eagerly await development of the formalism that relates James'sadvanced polynomial factorization to postethnic performance art.Should be fascinating stuff.-- And the logical extension of free and open-source software in therealm of sex would certainly include publicly shared sex at a sexparty,... queer sexuality and... non-proprietary sexual affection. Annalee Newitz writing in Salon.com === Subject: Re: JSH: Research question answeredI think it interesting as an advanced question to figure out how to> construct one of the imperfect factorizations from a given> tautological space, though it seems that would require a tautological> space only valid in the complex plane. Hmmm...therefore, it seems logical that an imperfect factorization> cannot be constructed from a tautological space. The formalism necessary to prove that quick deduction must be rather> impressive.This *does* sound like an interesting and advanced question, if only I> knew what it means.So, following James's lead, I delved into advanced mathematical> research into these issues. That is, I googled for tautological> space. I got three hits: two regarding some performance art hoohah> and one on Postethnic Narrative Criticism. Surprisingly, neither of> these topics have any mention of imperfect factorization, so it does> appear that James's proposed research is groundbreaking.I eagerly await development of the formalism that relates James's> advanced polynomial factorization to postethnic performance art.> Should be fascinating stuff.The phrase tautological space is one I came up with from my othermath research.It refers to regions other thanx=x, which is x = 0(mod x),as that's the basic tautological space from which mathematicianstraditionally operate, and it's used for most research, while I founduse forx^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)which is the tautological space from which I get *my* examples ofnon-polynomial factorization, which are cubics at a minimum.That's kind of why I like to use Decker's example, as it's aquadratic, and fiddling with it has revealed a LOT of interestingmathematics apparently because it's not derivable in a tautologicalspace using only ring operations.It's fascinating, but the full formalism will have to wait while Ihandle this social stuff.James Harris === Subject: Re: JSH: Research question answered> I think it interesting as an advanced question to figure out how to> construct one of the imperfect factorizations from a given> tautological space, though it seems that would require a tautological> space only valid in the complex plane.> > Hmmm...therefore, it seems logical that an imperfect factorization> cannot be constructed from a tautological space.> > The formalism necessary to prove that quick deduction must be rather> impressive.This *does* sound like an interesting and advanced question, if only I> knew what it means.So, following James's lead, I delved into advanced mathematical> research into these issues. That is, I googled for tautological> space. I got three hits: two regarding some performance art hoohah> and one on Postethnic Narrative Criticism. Surprisingly, neither of> these topics have any mention of imperfect factorization, so it does> appear that James's proposed research is groundbreaking.I eagerly await development of the formalism that relates James's> advanced polynomial factorization to postethnic performance art.> Should be fascinating stuff.The phrase tautological space is one I came up with from my other> math research.It refers to regions other thanx=x, which is x = 0(mod x),as that's the basic tautological space from which mathematicians> traditionally operate, and it's used for most research, Correction, that should be 1=1, which is 1 = 0(mod 1), and yes, I knowit's trivial but it's the base space or explicit space thatmathematicians usually use.James Harris === Subject: Re: JSH: Research question answered>> >> I think it interesting as an advanced question to figure out how to>> construct one of the imperfect factorizations from a given>> tautological space, though it seems that would require a tautological>> space only valid in the complex plane.> Hmmm...therefore, it seems logical that an imperfect factorization>> cannot be constructed from a tautological space.> [...]>> >> The phrase tautological space is one I came up with from my other>> math research.>> >> It refers to regions other than>> >> x=x, which is x = 0(mod x),>> >> as that's the basic tautological space from which mathematicians>> traditionally operate, and it's used for most research, >Correction, that should be 1=1, which is 1 = 0(mod 1), and yes, I know>it's trivial but it's the base space or explicit space that>mathematicians usually use.Correction, calling 1 = 0(mod 1) a space, tautological or otherwise,is just meaningless nonsense.>James Harris === Subject: Re: JSH: Research question answeredThe phrase tautological space is one I came up with from my other> math research.It refers to regions other thanx=x, which is x = 0(mod x),as that's the basic tautological space from which mathematicians> traditionally operate, and it's used for most research, Correction, that should be 1=1, which is 1 = 0(mod 1), and yes, I know> it's trivial but it's the base space or explicit space that> mathematicians usually use.I can imagine that JSH might use it, as he is capable of any sort of idiocy, but unless there is more to it than meets the eye, no one else will ever bother with it.In fact, this seems like a sterling example of a case where there is less to it than meets the eye. === Subject: Re: JSH: Research question answered> I think it interesting as an advanced question to figure out how to> construct one of the imperfect factorizations from a given> tautological space, though it seems that would require a tautological> space only valid in the complex plane.> > Hmmm...therefore, it seems logical that an imperfect factorization> cannot be constructed from a tautological space.> > The formalism necessary to prove that quick deduction must be rather> impressive.This *does* sound like an interesting and advanced question, if only I> knew what it means.So, following James's lead, I delved into advanced mathematical> research into these issues. That is, I googled for tautological> space. I got three hits: two regarding some performance art hoohah> and one on Postethnic Narrative Criticism. Surprisingly, neither of> these topics have any mention of imperfect factorization, so it does> appear that James's proposed research is groundbreaking.I eagerly await development of the formalism that relates James's> advanced polynomial factorization to postethnic performance art.> Should be fascinating stuff.The phrase tautological space is one I came up with from my other> math research.Since there is nowhere on the web where the phrase tautological space occurs in any mathematical context, perhaps JSH will be so kind as to give us a reference to the math research which led him to that phrase.It refers to regions other thanx=x, which is x = 0(mod x),as that's the basic tautological space from which mathematicians> traditionally operate, In all my life, I have never operated from a tautological space, nor have I known of anyone whho did.> and it's used for most research, while I found> use forx^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)which is the tautological space from which I get *my* examples of> non-polynomial factorization, which are cubics at a minimum.Oh! If it is where you get your stuff from, I can understaand that it is totally outside the world of mathematics.That's kind of why I like to use Decker's example, as it's a> quadratic, and fiddling with it has revealed a LOT of interesting> mathematics apparently because it's not derivable in a tautological> space using only ring operations.It's fascinating, but the full formalism will have to wait while I> handle this social stuff.If you handle the social stuff anything like you do with mathematical stuff, that formalism will never havew an opportunity to come into being. Not that any of us were prepared to hold our breaths, anyway. === Subject: Re: JSH: Research question answered> That's kind of why I like to use Decker's example, as it's a> quadratic, and fiddling with it has revealed a LOT of interesting> mathematics apparently because it's not derivable in a tautological> space using only ring operations.> It's fascinating, but the full formalism will have to wait while I> handle this social stuff.Not necessarily. Instead you could handle the full formalism and put your social stuff asidein favor of the mathematics. (I thought you said that's what you were really interested in.)> James Often in error, but never in doubt. Harris--There are two things you must never attempt to prove: the unprovable -- and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com === Subject: Re: JSH: Research question answered>>I think it interesting as an advanced question to figure out how to>>construct one of the imperfect factorizations from a given>>tautological space, though it seems that would require a tautological>>space only valid in the complex plane.>>Hmmm...therefore, it seems logical that an imperfect factorization>>cannot be constructed from a tautological space.>>The formalism necessary to prove that quick deduction must be rather>>impressive.> This *does* sound like an interesting and advanced question, if only I> knew what it means.So, following James's lead, I delved into advanced mathematical> research into these issues. That is, I googled for tautological> space. I got three hits: two regarding some performance art hoohah> and one on Postethnic Narrative Criticism. Surprisingly, neither of> these topics have any mention of imperfect factorization, so it does> appear that James's proposed research is groundbreaking.I eagerly await development of the formalism that relates James's> advanced polynomial factorization to postethnic performance art.> Should be fascinating stuff.I think the deconstructionists (Lacan's group) have done some good work on this, maybe Google didn't find it because it is in French.Gib === Subject: Re: JSH: Research question answered <8765eivets.fsf@phiwumbda.org> Discussion, linux)>> This *does* sound like an interesting and advanced question, if only I>> knew what it means.>> >> So, following James's lead, I delved into advanced mathematical>> research into these issues. That is, I googled for tautological>> space. I got three hits: two regarding some performance art hoohah>> and one on Postethnic Narrative Criticism. Surprisingly, neither of>> these topics have any mention of imperfect factorization, so it does>> appear that James's proposed research is groundbreaking.>> >> I eagerly await development of the formalism that relates James's>> advanced polynomial factorization to postethnic performance art.>> Should be fascinating stuff.> I think the deconstructionists (Lacan's group) have done some good work > on this, maybe Google didn't find it because it is in French.It's not clear to me whether James is thinking deconstructionist orconstructivist.-- The sole cause of all human misery is the inability of peopleto sit quietly in their rooms. -- Blaise Pascal === Subject: Re: JSH: Research question answeredIt's not clear to me whether James is thinking deconstructionist or> constructivist.It's not clear to me whether James is thinking.-- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Sequence of testingIs there a particular sequence to follow in a model when correcting forheteroschedasticity,autocorrelation, stationarity, etc.? What should be detected first andcorrected? Does it make a difference? === Subject: Re: epsilon numbers : a problem of choice> >[...]>>without choice, it is consistent that w1 is not>>regular.>> >What???? I can't believe it (I barely understand how it could be>possible, as the denumerable union of denumerable sets can be not>denumerable, but then I would have sworn it would have a cardinal>uncomparable with w). Any reference?> >Kunen makes this statement without citation, and he goes on to say>>that it is unknown whether one can prove in ZF that there exists a>>cardinal with cofinality >w. (p. 33).>> >A few more thoughts on it>1) every denumerable limit ordinal is of cofinality w (in ZF)>Proof: if n->x_n is an enumeration of x, the sequence f(n+1)=the>smallest p such that x_p>x_(f(n)) gives a set {x_f(n)} cofinal to>x., of order type w.>2) if (x_1exists a injection of w_1 in IR (actually in Q)>First, we inject x_i-> i, then we use the classical (not choice->defined) injection of {x/ x >>What is this classical injection for an arbirtrary countable ordinal?>In fact , *any* countable ordering is a subordering of Q : if we note <* the>ordering, then we define f(x_(n+1)) as 1+sup f((x_i)) if x_(n+1)>* x_i for>all i(f(x_i)+f(x_j))/2 if x_i=max(x_k) for all k such that x_k<*x_(n+1) , and>x_j=min(x_k) for all k such that x_k>*x_(n+1)So how would this work when mapping s(w) (= w U {w}) to Q? Note that in your scheme in (2) above, you are essentially mapping countable order types x_(n+1) x_n to Q, so you need a non-choice scheme that works for all countable ordinals.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: epsilon numbers : a problem of choice[...]>> What is this classical injection for an arbirtrary countable> ordinal?>> In fact , *any* countable ordering is a subordering of Q : if we>> note <* the ordering, then we define f(x_(n+1)) as 1+sup f((x_i)) if>> x_(n+1)>* x_i for all i> all i> that x_k<*x_(n+1) , and x_j=min(x_k) for all k such that x_k>*x_(n+1)> So how would this work when mapping s(w) (= w U {w}) to Q? Note> that in your scheme in (2) above, you are essentially mapping> countable order types x_(n+1) x_n to Q, so you need a non-choice> scheme that works for all countable ordinals.Well, now, who said w U{w} is denumerable? In other words, what (in a non ACworld) do you call a denumerable ordinal? Of course, if the bijection comeswith the ordinal , it is easy to answer your question : if i define w u{w}as (0,1,2,...) in order (1,2,3,...,0), then my map is 0->0,1->-1,2->-1/2,...n->-1/2^(n-1)...).But I agree your argument expose a weakness in mine. If the information isonly there existes a bijection to w, I must admit i need a (very weak)form of AC to map the sequence of ordinals (cofinal to w_1)x_1,x_2,...x_n,... to segments [n,n+1], and thus to inject w_1 in Q. But Istill find the existence of such a sequence hard to believe ;-) === Subject: Moon Unit Radio by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i17KlF611975;I read some web pages about it, and moon unit radio is about situational awarenesss.http://www.darpa.mil/ato/programs/suosas.htmhttp:// citeseer.nj.nec.com/517679.htmlhttp://citeseer.nj.nec.com/ sukthankar94situational.htmlThose are three results from google for searching for situational awareness.I''m concerned about google becoming a verb.By the way, the machine terminator is probably on my side.Results 111, 112, 113:http://www.aviation.unsw.edu.au/readings/avia2100/Week4. ppthttp://www.science9.com/Controlling_Pilot_Error_Situational _Awareness_0071373217.htmlhttp://www.ngnavsys.com/Automated/ Everybody sold everything to Google.Google owns nothing.How many googles is it? What the hell is wrong with you?Ross F.--Ross A. Finlayson === Subject: Re: Moon Unit RadioRoss A. Finlayson> I read some web pages about it, and moon unit radio is about situationalawarenesss.> http://www.darpa.mil/ato/programs/suosas.htm> http://citeseer.nj.nec.com/517679.html> http://citeseer.nj.nec.com/sukthankar94situational.html> Those are three results from google for searching for situationalawareness.> I''m concerned about google becoming a verb.> By the way, the machine terminator is probably on my side.> Results 111, 112, 113:> http://www.aviation.unsw.edu.au/readings/avia2100/Week4. ppthttp://www.science9.com/Controlling_Pilot_Error_Situational _Awareness_0071373217.html> http://www.ngnavsys.com/Automated/:) I don't mind google as a verb, but situational awareness? Bound toget hits on self-help books, new age religions, education for retards, ...darpa-dot-mil is quite a worthwhile site for anyone watching the frontier ofmilitary technology. I'm glad they came in first :)LH === Subject: Re: Moon Unit RadioIn sci.math, Ross A. Finlayson is about situational awarenesss.More likely Moon Unit Radio is what happens whenone of Frank Zappa's daughters finds Mr. Right is reallynamed Mr. Radio, and marries him anyway.:-)[rest snipped]-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: olympiad math contest for students age < 20 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i17KlBY11823;actually it's www.mathlinks.ro :) === Subject: Re: the song you rewrite re: James by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i17KlCG11886;.Sorry Dave, I just came over all pious there. Silly of me not to guess the rewrite of G & S was already current. At least I can try to stop my posts going on for ages off the righthand edge of the screen. Sigh. Mark>> Shame these lists can't be a bit more like that - humour and>> professionalism surviving personality clashes, eh?>Ouch, that hurt! No sign of either humour nor professionalism here?>*sniff*>dave>Member, Society of Professional Humourists === Subject: Re: JSH: Pattern argument by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i17KlDj11906;I went on a trip to the nearby beach to find rocks that have approximately your IQ. It wasn't easy but I think I have found a sufficiently stupid one. === Subject: Re: JSH: Pattern argument===>Subject: Re: JSH: Pattern argument>I went on a trip to the nearby beach to find rocks that have approximately>your IQ. It wasn't easy but I think I have found a sufficiently stupid one.But does it have a BS in Physics and is it a veteran?--MensanatorAce of Clubs === Subject: Re: :: towards a constructive education :: (news server friendly): > I have _rarely_ been the smartest over those around me. When I come to:: well, you certainly spare no effort to convey the idea. For those who: really are't convinced by this posting alone, just google this gal up.I have no problem at all with my newsgroup record. I've posted questionsconcerning my hypergeometric identities, research into the relationshipbetween Bohmian interpretations of quantum mechanics and geometricquantisation, an occasional poem, some help on the newsgroups withmathematical or physical references, several replies to programmingquestions, one proposal for the inclusion of a sizeof mechanism forfunctions into the next c++ standardisation, one argument which extended forsome time concerning a mathematical error in a paper by a physicist namedGhose, and maybe a few political commentaries. I have also spent a lot oftime on codeguru.com, where I usually just answer how do I do this typequestions for people learning how to program or just chat about nonsensewith other regulars. My nastiest moments were in the argument, but I alsotried to keep the focus on the mathematics and do not have any problems withwhat I posted there.: > these groups and read, I find many, many people who are more intelligent: > than I. I seek out intelligent people, just as I seek out creativepeople.: > I don't think there is anything wrong with that, and I do understandquite: > clearly that the criterion are my own definitions. I have been on: > anti-depressants in my life, but I don't see how that has anything to do: > with my post.:: It doesn't in terms of causality. But it shows. Your post is a very: strong indication that you have big difficulties in coping with: reality.What does my post have to do with a grip on reality??? Its aboutmathematical structures that occur in the sciences.And please do not characterise those who have taken anti-depressants as nothaving a grip on reality. The serotonergic system can become damaged due tolong periods of trauma and elevated cortisol, and sometimes requiressupplement in such people. Otherwise, these people often show sign of anexhaustion which can effect mood. Often the trauma was not of their choice,like guns to their head or other violence or abuse. But most patients dohave as good, if not better than average, capabilities to recognise facts,and it is even found that these patients often have less of an avoidancereaction to negative stimuli than the nontreated.It is good to hear, though, that your life has been good.: > It looks like you are just using it as show of negative: > feelings towards me.:: I wasn't using anything. I didn't know, I just made it up along the: way. Jeezes Kryst, I hope you don't start thinking I'm someone you: know personally, this is getting scary.Why would I think you know me? What exactly is scaring you?: > I'm insecure. Its a personal observation that I have found myself drawnto:: Nobody here gives a about that. Nobody wants to know. But most: important, that is no excuse for handing out your idiotic judgement on: the greater part of humanity. It is no excuse for being a blatantly: imbecile arrogant bigot.First you wonder as to my reasoning, and then tell me no one cares about it?I'm honest to you about my emotions, and you scream back at me in anger?There are a few people in this world who care about what I think. There area few people in this world that care about what you think. I am sorry if myone comment about monotheism offended you. There are many monotheists whomI love dearly. I do have opinions about the belief, but they were not theconcern of this post. You have an opposing opinion. Why do _you_ believethat Heyting algebras are not more well known? Do _you_ believe they evenshould?: > Heyting forms say nothing against objectivity of reality. They describethe: > logic we model reality with.:: It seems however they help a good deal in rationalising its denial.: How very comfty it must be not having to choose between I succeeded: and I failed.You are in error in your understanding of Heyting algebras. Boolean algebrais Heyting, for example. However, there are questions like will thisprogram halt? where there are more than two possible answers. You may beable to prove it does (or run it and observe it halting), and you may beable to even show with a proof that it must halt (algorithmics can even giveyou maximum orders of execution time). But there just happen to be programswhere you cannot determine whether they will halt without running them, andwill never know in a finite time whether that proposition is true or false.You need a non-Boolean logic to deal with these.: > If you want more blatant self-criticism, see my postings to the poetry: > newsgroups. You do seem to want very much to think about me in negative: > ways.:: You call that blatant? Holy subtle candlesticks batman! No I don't: want any more of it at all! And no, I don't want to think of you in: any particular way, it's just you rub it on one's face with such: vehemence.I am sorry for the ambiguous referrent. The more was referring toblatant, not want. In other words, I was saying that if you wantedself-criticism that was more blatant than anything in my post, head over tothe poetry groups.I did not believe my post was about myself much at all, except for my onecomment incidental to the focus of the post. It may have been indirectlyabout my approach to the mathematical sciences and concerns of education,but I wanted other voices heard about these issues, and did not see anyvehemence to self aggrandisement.[...]: * So you want to talk about things relevant to Heyting forms and are: cashing in a lot of hating? (should I post that to the poetry groups: ?).: Let's see, you told more than half of humanity that they hold their: religious belief because they are insecure. The same thing for all: those unsuspecting cowards that hold to the more classical views.: Then you told the crowds that you were confused because they hadn't: caught up with you. It seems to me you should have managed to: antagonize everybody by now, but let's asume just for the heck of it,: that there are still some good willed people left after that. Now,: which of the titles you apply to yourself:: prankster, fablist, magician, liar: do you think is the most inviting one to a person wanting to have a: serious conversation about formal systems and education?: moron.I did not tell half of humanity anything. I posted to the usenet, and wouldbe surprised if anywhere near a hundred people actually read the post. AndI already mentioned that I do not have this negativity associated to the useof the term insecure as you do. I personally believe that most humanbeings period have insecurities, and I openly include myself in that group,but I am sorry that angers you. I personally believe that there are manyoutward expressions of insecurities, especially of myself. I use theevidence of my own observations and my readings of others to order thesebeliefs relative to others. If I am presented data that contradicts thesebeliefs, I would eagerly study it and revise my hierarchy as needed. Idon't look to antagonise. Sometimes, I look to provoke, and sometimes Imake mistakes in the process, but I try to learn.If there is anything about my topic you wish to teach me, I am waiting...-- === -=-=-=-=-===Subject: Re: :: towards a constructive education :: (news server friendly)> : > I have _rarely_ been the smartest over those around me. When I come to> :> : well, you certainly spare no effort to convey the idea. For those who> : really are't convinced by this posting alone, just google this gal up.I have no problem at all with my newsgroup record. I've posted questions> concerning my hypergeometric identities, research into the relationship> between Bohmian interpretations of quantum mechanics and geometric[list of newsgroups credentials]Can we cut this part out? I said you were probably brighter than thosearound you. You said no. I said well, you do make an effort to conveyanother impression. Now you post a self-portrait on ngs and otherforums.Can we please agree on your being smarter than average? I will alsoconcede that you are smarter than I. No irony, no subtleties, just getthis part settled.> : > clearly that the criterion are my own definitions. I have been on> : > anti-depressants in my life, but I don't see how that has anything to do> : > with my post.> :> : It doesn't in terms of causality. But it shows. Your post is a very> : strong indication that you have big difficulties in coping with> : reality.What does my post have to do with a grip on reality??? Its about> mathematical structures that occur in the sciences.Not grip, coping. I'm not talking of what it is about but of what itreflects. I will try to resume what I think is wrong with your post ina single segmentfurther down.> And please do not characterise those who have taken anti-depressants as not> having a grip on reality. The serotonergic system can become damaged due to> long periods of trauma and elevated cortisol, and sometimes requires> supplement in such people. Otherwise, these people often show sign of an> exhaustion which can effect mood. Often the trauma was not of their choice,> like guns to their head or other violence or abuse. But most patients do> have as good, if not better than average, capabilities to recognise facts,> and it is even found that these patients often have less of an avoidance> reaction to negative stimuli than the nontreated.Can we leave this part behind too? I at no point characterized thosewho take any kind of drugs in any particular way. I was using theprejudices attached to anti-depressants as way of mocking you, period.> It is good to hear, though, that your life has been good.You haven't heard anything like that. > : > It looks like you are just using it as show of negative> : > feelings towards me.> :> : I wasn't using anything. I didn't know, I just made it up along the> : way. Jeezes Kryst, I hope you don't start thinking I'm someone you> : know personally, this is getting scary.Why would I think you know me? What exactly is scaring you?No, the other way around. I said I hoped you wouldn't think I was someYou knew.> : > I'm insecure. Its a personal observation that I have found myself drawn> to> :> : Nobody here gives a about that. Nobody wants to know. But most> : important, that is no excuse for handing out your idiotic judgement on> : the greater part of humanity. It is no excuse for being a blatantly> : imbecile arrogant bigot.First you wonder as to my reasoning, and then tell me no one cares about it?> I'm honest to you about my emotions, and you scream back at me in anger?There are a few people in this world who care about what I think. There are> a few people in this world that care about what you think. I am sorry if my> one comment about monotheism offended you. There are many monotheists whom> I love dearly. I do have opinions about the belief, but they were not theI wasn't upset about your one remark on monotheism. I find yourattitude in general arrogant and patronizing. I insist that you cannotjustify your judgement of others by including yourself among thejudgement. You cannot make up for this with your attempts to make mefeel more simpathy for you by sharing your emotions and feelings withme.> concern of this post. You have an opposing opinion. Why do _you_ believe> that Heyting algebras are not more well known? Do _you_ believe they even> should?[lots of stuff, please refer to the original]Now let's get to what is wrong with your post. You've already worndown my rage with your stamina, (I never expected you to reply, Ipresumed it would bebelow you) and it's Sunday, so my tone is softened down a bit. Youhave had more negative responses than positive, and you really haven'tstarted any constructive discussion anywhere. Some have openlydeclared that they didn't even bother to read the whole lot ofnonsense. I actually printed it out andread it through a few times. It really angered me a lot. Why? Because of the hocus pocus with the W at the beginning.Because of the endless list of assertions for which you neitherprovide proofs, examples, or clues as to why they should be ofinterdisciplinary interest.Because of your explanation in terms of insecurity, (You choose not tointerpret insecurity as something negative. But that is yoursubjective choice. I think most people would prefer not to beinsecure), which is arrogant and belittling even if your personalopinion is another.Because you keep rubbing under ones nose, how naturally it all comesto you, as opposed to everybody else.Because I couldn't get rid of the feeling that you are biassed againstboolean logic for private reasons.The last one, combined with your failure to provide compelling reasonsfor considering your point of view, and your galathaea: prankster,fablist, magician, liar give me the impression of someone showing ofher stylish chic new theory just to impress others, like a teeny witha newly discovered worldview. There are moral implications in whatyour attitude reflects, and I don't have the impression that I likethem. This of course is my personal subjective interpretation of whatyou exposed of yourself, but I think I'm not that off the mark, sinceI seem to have struck a nerve or two. You don't seem to realise howmuch of yourself comes across your post, and you also don't seem torealise how much effect your words have on others the second you getinto expressing opinion.As for your intended topic, you mentioned the halting problem as anexample where one needs polyvalent logic. When I studied computerscience, there neverwas a mention of the need for polivalent logic in this context. MaybeI missed something. As I see it, for a given algorithm, and a giveninput, you know it's halting behaviour or you don't. That is prettyboolean to me. So I would like you to elaborate more on this example.Then I would like you to establisha relationship based on heyting structures and another area ofknowledge from among the ngs you posted to, and to explain why thisrelationship is importantto both areas. My advice to you, if you want to be more successful in your quest togather a multidisciplinary group of people willing to discuss yourtopic:Try to establish a path of such examples, relationships andexplanations that goes through all of the disciplines you addressed.Expose this path assuccinctly as you can, without compromising understandability.Leave out all the hocus pocus stuff.Leave out as much of your personal opinion as you can.And consider dropping (this is again very personal, I'm afraid) theprankster, fablist, magician, liar it is no great recommendation ofyourself. Maybe it works wonders among intelectuals of the liberalarts, but I think it's a real turnoff for the more scientific orientedmind.> If there is anything about my topic you wish to teach me, I am waiting...if we ever get to discussing your topic and I can contribute. === Subject: Re: :: towards a constructive education :: (news server friendly)> It doesn't in terms of causality. But it shows. Your post is a very> strong indication that you have big difficulties in coping with> reality.As do yours.:-) === Subject: Re: :: towards a constructive education :: (news server friendly)> It doesn't in terms of causality. But it shows. Your post is a very> strong indication that you have big difficulties in coping with> reality.>As do yours.:-)Hey twerp, get lost. I'm talking to the lady here. We'll let you knowif there is anything we need.p.s.: I don't think you are making a good impression with yoursycophantic droolings. And the girl can handle the discussion withoutyour help. === Subject: Re: :: towards a constructive education :: (news server friendly) > Galathaea, from what I gather from other postings of yours, you are> probably brighter than those around you and those you grew up with.> You probably are acustomed to receiving praise and approval at the> merest trembling of your lips. Before you go on reading I sugest you> eat some chocolate burn some inciense and say a prayer to your> favorite god of no hard feelings. And yes, maybe take some of that> prozac too, you know, the one you take to take the edgess off your> extremely sharp mind.The various systems that have been formed> concerning the standard of right and wrong,> may all be reduced to the principle of sympathy> and antipathy. One account may serve for> all of them. They consist all of them in so many> contrivances for avoiding the obligation of> appealing to any external standard, and for> prevailing upon the reader to accept of the> author's sentiment or opinion as a reason for> itself. The phrases are different, but the principle> the same. --Jeremy Bentham> ????> There are two types that rationalise and argue the denial of objective> right and wrong. Those who have had their faces violently stuck into a> very big pile of reality , and those who sorely need it. The> former sometimes deserve pity the latter always deserve contempt. The> rest of the people just shut up and hope they never really have to> find out.> according to this theory, it would be moral to e.g. torture one person> if this would produce an amount of happiness in other people> outwheighing the unhappiness of the tortured individual.. So y,> how large does the gang need to be in order for a gang rape to be> moraly ok?> dimwit.Just one foul-mouthed Guenther capable of enforcing his own propensity for self-love.:-) === Subject: Re: :: towards a constructive education :: (news server friendly) > Galathaea, from what I gather from other postings of yours, you are> probably brighter than those around you and those you grew up with.> You probably are acustomed to receiving praise and approval at the> merest trembling of your lips. Before you go on reading I sugest you> eat some chocolate burn some inciense and say a prayer to your> favorite god of no hard feelings. And yes, maybe take some of that> prozac too, you know, the one you take to take the edgess off your> extremely sharp mind.> > The various systems that have been formed> concerning the standard of right and wrong,> may all be reduced to the principle of sympathy> and antipathy. One account may serve for> all of them. They consist all of them in so many> contrivances for avoiding the obligation of> appealing to any external standard, and for> prevailing upon the reader to accept of the> author's sentiment or opinion as a reason for> itself. The phrases are different, but the principle> the same.> > --Jeremy Bentham ????> There are two types that rationalise and argue the denial of objective> right and wrong. Those who have had their faces violently stuck into a> very big pile of reality , and those who sorely need it. The> former sometimes deserve pity the latter always deserve contempt. The> rest of the people just shut up and hope they never really have to> find out.> according to this theory, it would be moral to e.g. torture one person> if this would produce an amount of happiness in other people> outwheighing the unhappiness of the tortured individual.. So y,> how large does the gang need to be in order for a gang rape to be> moraly ok?> dimwit.Just one foul-mouthed Guenther capable of enforcing his own propensity for self-love.Thats all you could come up with? Any 8 year would do better...After one single round Michy the Moocha cornered in his own corner andbeaten with his own tricks. What a sorry sight.:-)And stop grinning, until you put up a decent argument. It's just sad. === Subject: Moon Unit RadioI have gotten into a discussion over on sci.space.policy, we aretalking about space exploration.That previous post, that was just drunk-talk.So anyways I wonder what people on sci.math about actually sendingpeople to the moon. My plan is to use an Earth to Orbit Mass Driverto lower the cost of putting cargo in space and on the moon, to enableenough materiel on the moon, Luna, for astronauts to put it togetherand make a space station on the moon. The concept of the coilgun fora mass driver is that an electromagnetic coil is energized and themagnetic field flux draws the projectile, a pod of 2000 to 40000kilograms, through the coil, the coil is deneenergized as the podflies through the coil, it is electrmagnetically levitated, thus notholding back the pod. A sequence of the coils are energized anddeenergized in succession to accelerate the pod at some 300 times theforce of Earth's gravity, G's, for over two seconds the pod toaccelelerate over ten kilometers through Mach 30, 11+km/s, Earthescape velocity, the speed at which a massy object flies directly offthe planet.I am wondering how to calculate the forces on the pod, given so manywatts energizing the coil. (?) I'm hoping somebody would explainsome mathematical techniques for calculating the forces on the coiland pod, what is the magnetic field mathematically? I don't know muchabout electricity and magnetism. I looked at Mathworld web pages andit is describing the use of surface integrals, an area in which I ammathematically lacking. How are surface/contour integrals evaluated,how do they work? How are they solved numerically, Runge-Kutta,Newton-Raphson? How do magnetism and the magnetic characteristics ofthe materials of various materials and components of a pod and coileffect each other?We're talking about it on sci.space.policy, I'm hoping to get somefeedback from sci.math about it.Ross F.--Ross A. Finlayson === Subject: Re: Lagragian, Quantum Mechanics,Path of least actionThe Lord of the Rain( Suresh __NoJunkMail kumar)> interesting thought. I will finish up laterDo us a favour and don't finish it up in this newagroup.Franz === Subject: Re: Lagragian, Quantum Mechanics,Path of least actiongo away idiot. Nobody wants u around. You aint hip.-suresh> The Lord of the Rain( Suresh __NoJunkMail kumar)> interesting thought. I will finish up later> Do us a favour and don't finish it up in this newagroup.> Franz === Subject: one dimensional dynamic iterationI'm having trouble proving something that should be rather simple, and wouldappreciate help (feel free to reply or email me at nageeb@stanford.edu).Let g(x)=d*x if x>1/d or g(x)=1+y-d*y*x where y >=0.Now, g has a single fixed point, which is a sink (attracting) if d*y<1 or asource (repelling) if d*y>1.Now we can consider iterating the function, (g^n)(x) = d*(g^(n-1))(x) if(g^(n-1))(x)>1/dor (g^n)(x) = 1 + y - d*y* (g^(n-1))(x).If d*y > 1 but (d^2) * y <1, then it is straightforward to show that (g^2)has one repelling fixed point and two attracting fixed points. I'd like togeneralize this to n. i.e., find how many attracting and repelling fixedpoints there are for arbitrary n.Any suggestions?-- Nageeb === Subject: Re: Elevator problem> >For the record, for the case n = 2 and k = 3, I get the following >fractions of discrete time that there is at least one elevator on the >indicated floor:>1: 2/3>2: 11/30>3: 7/15Here are the simulation results for the same case:1: 0.64999> 2: 0.35906> 3: 0.51598> >I assumed that when the elevators are on floors 1 and 3 and the >call comes from floor 2, then the elevator from floor 3 responds.The code picks randomly either 1 or 3. This could explain the small> variance in the results.Oh, this seems like a good example. Here's what I get using the states{1,1}, {2,2}, {3,3}, {1,2}, {1,3}, and {2,3}. There are four possible(and equally likely) trip requests: 1 -> 2, 1 -> 3, 2 -> 1, and 3 -> 1.Here's how each state responds to each request: 1->2 1->3 2->1 3->1{1,1} -> {1,2}, {1,3}, {1,1}, {1,1}{2,2} -> {2,2}, {2,3}, {1,2}, {1,2}{3,3} -> {2,3}, {3,3}, {1,3}, {1,3}{1,2} -> {2,2}, {2,3}, {1,1}, {1,1}{1,3} -> {2,3}, {3,3}, two!, {1,1}{2,3} -> {2,3}, {3,3}, {1,3}, {1,2}.The state two! is split equally between the states {1,1} and {1,3}.Therefore the transition matrix M is[ 1/2 0 0 1/2 3/8 0 ][ 0 1/4 0 1/4 0 0 ][ 0 0 1/4 0 1/4 1/4 ][ 1/4 1/2 0 0 0 1/4 ][ 1/4 0 1/2 0 1/8 1/4 ][ 0 1/4 1/4 1/4 1/4 1/4 ].I don't have my Mathematica crutch at the moment, so rather than tryingto compute the full eigensystem of M, I'll just compute the eigenvectorcorresponding the the eigenvalue 1. It works out to (63, 10, 27, 30, 44, 37) / 211.If we're interested in the probability of at least one elevator being ona given floor, then we just add up the probabilities for states containingthe floor. For floor 1, this means indices 1, 4, and 5; for floor 2, it's2, 4, and 6; and for floor 3, it's 3, 5, and 6. This yieldsFloor 1: 137/211 = 0.64929Floor 2: 77/211 = 0.36493Floor 3: 108/211 = 0.51185Toni's numbers are pretty close to these.-Jim Ferry === Subject: Re: Elevator problem>>I assumed that when the elevators are on floors 1 and 3 and the >>call comes from floor 2, then the elevator from floor 3 responds.The code picks randomly either 1 or 3. This could explain the small> variance in the results.Picking 3 is better than picking randomly, from a service point of view.-- --Tim Smith === Subject: Re: Elevator problem>>I assumed that when the elevators are on floors 1 and 3 and the >>call comes from floor 2, then the elevator from floor 3 responds.The code picks randomly either 1 or 3. This could explain the small> variance in the results.Picking 3 is better than picking randomly, from a service point of view.Service? Well that complicates things, doesn't it?We could generalize Toni's problem as follows: suppose there are stilln elevators serving k floors, and that the 2k-2 possible elevator requestsstill happen serially, independently, and with equal frequency. Toni gavea protocol elevators would use to respond to requests. We could considerinstead the space of all possible protocols and ask for one that minimizesthe expected wait time.The expected wait time could be modeled simply as the distance between thefloor making the request and the floor of the elevator that gets sent. Inanother post in this thread, I gave the limiting distribution for Toni'soriginal protocol in the n=2, k=3 case. This allows us to compute theexpected wait time E:E = (1/211) ( (1/2) (0*(63 + 30 + 44) + 1*(10 + 37) + 2*27) + (1/4) (0*(10 + 30 + 37) + 1*(63 + 27 + 44) ) + (1/4) (0*(27 + 44 + 37) + 1*(10 + 30) + 2*63) = 251/422 = 0.59479.I believe Tim's assertion that picking 3 is better amounts to theassertion that the expected wait time for that protocol is lower thanthis. Stephen made an exact calculation for that case, so maybe hewill tell us that protocol's expected wait time.Possible protocols would have the following form (using the unorderedstate space S I defined elsewhere in this thread). They are functionsf: S x {1,2,...,k} -> [0,1]^k, where f(j,x) = {p_1,p_2,...,p_k) givesthe probabilities p_y of an elevator being chosen from floor y given thatthe elevators are in state j and that the request is made from floor x.For all j and x, we stipulate that the p_y's must sum to 1, and that if yis not a member of j (j being the set of floors where elevators are), thenp_y = 0. (Note that it would seem more straightforward to make the rangeof f be [0,1]^n, assigning probabilities to elevators rather than floors.This would be appropriate if we were using the ordered state space, butwith the (more efficient) unordered state space elevators are essentiallyunlabeled.)Call a protocol mixed if there exist values of j and x such that atleast two p_y's are positive. My intuition tells me that there is nevera need for a mixed protocol -- that one could always achieve a minimumexpected wait with a pure (i.e., non-mixed) protocol instead. I offerno proof of this, though. Where mixed protocols would become importantis in the bizarre scenario in which the tenants conspire to make tomaximize the expected wait. I doubt anyone will want to take thisthread in that direction, though.-Jim Ferry === Subject: Perspective projection matrix (quick question)Hi all,I am aware that the quantity f in the perspective projection matrix standsfor the normal distance from the 3d scene to the projection plane, but isthis value always positive?E.g. If I want to project onto the plane z=-1, then would f=1 or -1?My textbook doesn't really make this clear.And I take it that if my viewpoint were at z=-a then I would have to add ato all the z-ordinates of my points in 3d space (and subtract in the casez=a)? === Subject: JSH: Tautological spacesI did a post recently where I said the base tautological space thatmathematicians operate in is x = 0(mod x), and I realized later that'swrong as it's 1 = 0(mod 1).That's the base tautological space where by tautological space I meana region of truth.In mathematics it's then a region of mathematical truth.x=2, is a condition in the tautological space 1=0(mod 1).In a different tautological space, like x=0(mod x), everything withinthat space has x as a factor.In a tautological space like x+y+z = 0(mod x+y+z), everything hasx+y+z as a factor, but also you have 3 distinct elements x, y and z,which give form to the space without regard to their values.I've spent a lot of time working the tautological spacex^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)which you'll notice has 4 elements.My thinking is that part of the problem I'm facing is thatmathematicians are used to complex solutions in the simpletautological space 1=0(mod 1), but are not yet ready to move intoconsidering even simple solutions in complex spaces likex^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)so I kind of have infinity as my backyard with no one else playing inmy sandbox.Oh yeah, I don't think the human being has been born yet that canhandle complex solutions in a complex topological space. I call aperson who can handle such a challenge a third generationmathematician.Today's mathematicians are first generation.James Harris === Subject: Re: JSH: Tautological spaces>I did a post recently where I said the base tautological space that>mathematicians operate in is x = 0(mod x), and I realized later that's>wrong as it's 1 = 0(mod 1).>That's the base tautological space where by tautological space I mean>a region of truth.>In mathematics it's then a region of mathematical truth.Making up another silly word is not going to make yourarguments correct, sorry.>[...]>Oh yeah, I don't think the human being has been born yet that can>handle complex solutions in a complex topological space. If this actually meant something it would be ridiculous.>I call a>person who can handle such a challenge a third generation>mathematician.>Today's mathematicians are first generation.You really don't believe it, but it's really true regardless:When you make these comments about things that areBeyond today's mathematicians you sound really reallystupid, even to someone who's not following the math.Honest. >James Harris === Subject: Re: JSH: Tautological spaces> Oh yeah, I don't think the human being has been born yet that can> handle complex solutions in a complex topological space. I call a> person who can handle such a challenge a third generation> mathematician.Today's mathematicians are first generation.> James HarrisVery eloquently stated. But I'll say this: The mathematicians of todaythink they are living in Star Trek the Next Generation, when inreality they have not even gotten to the point of Star Trek the OldSeries with Capt. Kirk and Mr. Spock.They say boldly go where no man has gone before but they believe inprimitive notions like infinity, yet no one has ever measured thisnumber. I am searching for a partner that will help me in my quest tofind M, the largest natural number. I believe that I have almostsucceeded, but I need someone to confirm my findings.Could you possibly help me? I believe there is much profit if I canfind M.Dr. Ben Zona === Subject: Re: JSH: Tautological spaces> I did a post recently where I said the base tautological space that> mathematicians operate in is x = 0(mod x), and I realized later that's> wrong as it's 1 = 0(mod 1).Tautological space. That is very funny. I have noted this before. JSH is a humorist. === Subject: Re: JSH: Tautological spaces> I did a post recently where I said the base tautological space that> mathematicians operate in is x = 0(mod x), and I realized later that's> wrong as it's 1 = 0(mod 1).That's the base tautological space where by tautological space I mean> a region of truth.In mathematics it's then a region of mathematical truth.x=2, is a condition in the tautological space 1=0(mod 1).In a different tautological space, like x=0(mod x), everything within> that space has x as a factor.In a tautological space like x+y+z = 0(mod x+y+z), everything has> x+y+z as a factor, but also you have 3 distinct elements x, y and z,> which give form to the space without regard to their values.I've spent a lot of time working the tautological spacex^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)which you'll notice has 4 elements.No doubt they are earth, air, fire and water.My thinking is that part of the problem I'm facing is that> mathematicians are used to complex solutions in the simple> tautological space 1=0(mod 1), but are not yet ready to move into> considering even simple solutions in complex spaces likex^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)so I kind of have infinity as my backyard with no one else playing in> my sandbox.Oh yeah, I don't think the human being has been born yet that can> handle complex solutions in a complex topological space. I call a> person who can handle such a challenge a third generation> mathematician.Today's mathematicians are first generation.Judging by the mathematical content of the above, JSH is, at best, of the zeroeth generation, but more probably of a negative generation of large order. === Subject: Re: JSH: Tautological spaces> >> >> I've spent a lot of time working the tautological space>> >> x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)>> >> which you'll notice has 4 elements.No doubt they are earth, air, fire and water.Given the breadth and versatility of James' imagination and intellect,they're probably more like earth, dirt, dust and soil.-- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: JSH: Tautological spaces === >Subject: JSH: Tautological spaces>Message-id: <3c65f87.0402071646.2f988ec0@posting.google.comI did a post recently where I said the base tautological space that>mathematicians operate in is x = 0(mod x), and I realized later that's>wrong as it's 1 = 0(mod 1).>That's the base tautological space where by tautological space I mean>a region of truth.So now you're an astronaut?>In mathematics it's then a region of mathematical truth.>x=2, is a condition in the tautological space 1=0(mod 1).>In a different tautological space, like x=0(mod x), everything within>that space has x as a factor.>In a tautological space like x+y+z = 0(mod x+y+z), everything has>x+y+z as a factor, but also you have 3 distinct elements x, y and z,>which give form to the space without regard to their values.>I've spent a lot of time working the tautological space>x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)>which you'll notice has 4 elements.>My thinking is that part of the problem I'm facing is that>mathematicians are used to complex solutions in the simple>tautological space 1=0(mod 1), but are not yet ready to move into>considering even simple solutions in complex spaces like>x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)>so I kind of have infinity as my backyard with no one else playing in>my sandbox.>Oh yeah, I don't think the human being has been born yet that can>handle complex solutions in a complex topological space. I call a>person who can handle such a challenge a third generation>mathematician.>Today's mathematicians are first generation.>James Harris--MensanatorAce of Clubs === Subject: Re: Tautological spaces> so I kind of have infinity as my backyard with no one else playing in> my sandbox.Great! Let us know when you solve FLT in Harris-world. It won't have anyrelevancy to actual real-world mathematics, but I'm sure it will beentertaining nonetheless.Doug === Subject: Re: JSH: Tautological spaces> My thinking is that part of the problem I'm facing is that> mathematicians are used to complex solutions in the simple> tautological space 1=0(mod 1), but are not yet ready to move into> considering even simple solutions in complex spaces like> x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)> so I kind of have infinity as my backyard with no one else playing in> my sandbox.Why not just stay in your sandbox and stay out of the newsgroups? Seemslike that's the best choice for you and the newsgroups, too.> James HarrisJSH Motto: I'm happiest when I'm making love to myself!--There are two things you must never attempt to prove: the unprovable --and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com === Subject: Re: Measure theory / Rudin 1.5 (b)> In order to show that the set {x in X; (f_n(x)) has a finite limit when> n->+oo} is measurable (where (f_n) is a sequence of measurable functions> with values in R), can I say that this set equals:> limsup(limsup(abs(f_n-f_m),n),m)^(-1)({0}) ? (using Cauchy's criterion)It might be easier to consider {x in X: limsupf_n(x) = liminff_n(x)}. === Subject: Re: Measure theory / Rudin 1.5 (b)> In order to show that the set {x in X; (f_n(x)) has a finite limit when> n->+oo} is measurable (where (f_n) is a sequence of measurable functions> with values in R), can I say that this set equals:> limsup(limsup(abs(f_n-f_m),n),m)^(-1)({0}) ? (using Cauchy's criterion)> It might be easier to consider {x in X: limsupf_n(x) = liminff_n(x)}.Indeed, thank you. In that case, i'll separate the cases where limsup(f_n)is finite or infinite, but that's still easier.--Julien Santini === Subject: Re: 1 + 1/3 + 1/5 + ... + 1/(2*N - 1) ETAtAhRJj8dkPqdy2vp++1kZazHF7HXs7AIVALjBnuCdsI2t5ofrbh+ ZDQQTXGMV Consider the sum:H(n) = 1 + 1/2 + 1/3 + ... + 1/nThis is approximatd by ln(n) + gamma where gamma is Euler's Constant.With that in mind, render your sum as H(2n)-(1/2) H(n).--OL === Subject: Re: Help: Mathematical Equation to Rotate a Linehey thanks manI have a straight line and I want to rotate it a certain number of> degrees and find the points of its new coordinates. The line has to be> rotated about its center.What math equation can I use, assuming a variable 'd' for degrees and> coordinates of the lines endpoints and center point> ..alright, alright. Was a little tough on ya. Here it goes.Your problem is this: you have two points, you want to rotate these> two points around a third point. Well, then, the problem is actually> just to rotate a *point* around another point. As simple as that.I don't know how guys with projective geometry knowledge do it, but> here is my newbie plan.1) Translate the point around which you want to move to the origin. If> your center is and the point you want to rotate is

, this means> you work with =

- , then after the rotation...2) Rotate using the formula (rotation about the origin)...x' = x cos theta - y sin theta> y' = x sin theta + y cos thetatheta must be radian, mind you.3) Translate the point back. This means finally you have + That's about it. === Subject: Re: Help: Mathematical Equation to Rotate a Line> I have a straight line and I want to rotate it a certain number of> degrees and find the points of its new coordinates. The line has to be> rotated about its center.> What math equation can I use, assuming a variable 'd' for degrees and> coordinates of the lines endpoints and center pointx' = (x - x_0) cos(d) - (y - y_0) sin(d) + x_0y' = (x - x_0) sin(d) + (y - y_0) cos(d) + y_0 === Subject: A web site that teaches Discrete math ?I need a website that teaches Discrete Math.I have visited Mathworld.com but it didn't look good.Do you know a good one ? Omid === Subject: Re: Need Help !!!> Let p be in (A')*. If n is a natural number then the open disk D with> center p and radius 1/n contains some point q of A' (because p is in> its closure). But since the disk D is an open set and since q is in A',> there is some element a_n of A that belongs to D.Why ? Because A in R^2 ??To me, the fact thatD is an open set and q is in A'=> D has at least one point a_1 in A distinct from q.=> now, for obtaining a_2 in A distinct from a_1 (in your post), what isneeded ??That is, I'm not sure when x is in A', A contains infinitely many points ofA. ---(*)(i,e, As it stands, I can only accept that when x is in A', A contains atleast one point of A)If somebody know the reason for (*), please post reply. === Subject: Re: Need Help !!!sorry i make a mistake some spelling.please read this instead of above my post.> Let p be in (A')*. If n is a natural number then the open disk D with> center p and radius 1/n contains some point q of A' (because p is in> its closure). But since the disk D is an open set and since q is in A',> there is some element a_n of A that belongs to D.Why ? Because A in R^2 ??To me, the fact thatD is an open set and q is in A'=> D has at least one point a_1 in A distinct from q.=> now, for knowing existence of a_2 in A&D distinct from a_1 (in yourpost), what isneeded ??That is, I'm not sure when x is in A' & D is an open set containing x, D contains infinitely many points of A. ---(*)(i,e, As it stands, I can only accept that when x is in A', D contains atleast one point of A)If somebody know the reason for (*), please post reply. === Subject: Re: Need Help !!!> => now, for knowing existence of a_2 in A&D distinct from a_1 (in your> post), what is needed ??Why do you think you need to have a_2 distinct from a_1?Best regards,Jose Carlos Santos === Subject: Re: Need Help !!!>>Let p be in (A')*. If n is a natural number then the open disk D with>>center p and radius 1/n contains some point q of A' (because p is in>>its closure). But since the disk D is an open set and since q is in A',>>there is some element a_n of A that belongs to D.> Why ? Because A in R^2 ??By definition of derived set, any disc centered at any element q ofA' contains some element of A distinct from q. So, since q is in Dand D is an open set, D contains some element of A. The onlyobjection that you can raise here is that I have not proved thatevery a_n is different from p; I need that in order to deducethat p, being the limit of (a_n)_n, belongs to A'. But that'seasy: by the same argument used to prove that D contains an elementof A (which used only the fact that D is an open set), you canassert that D{p} contains an element of A.Best regards,Jose Carlos Santos === Subject: re:Mathematical ScreensaversThe screensaver I use, straight out of Windows 98, is called 3D flowerbox. It is a series of shapes segueing into one another.----== Posted via Newsfeed.Com - Unlimited-Uncensored-Secure Usenet News==----http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: T1 spaceIn T1 space A,x is a limit point of a set A => every neighborhood of xcontains infinitely many points of A.How can I prove this??If someone know this, please post reply. === Subject: Re: T1 spaceBut, I have one question about your post.> Why do you use the expression at most?> Since [quote:aeca3a415a]the intersection of {U,U_i; i=1..n}[/quote:aeca3a415a] is the subset of U, it can only intersects A, if possible, at a_1,...,a_n, and since itis the subset of U_i; i =1..n, a_1,...,a_n are therefore excluded from theintersection with A.Is that must needed ?> If that expression is omitted, Is you're proof wrong ??> necessary and precise----== Posted via Newsfeed.Com - Unlimited-Uncensored-Secure Usenet News==----http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: Re: T1 spaceHello> In T1 space A,> x is a limit point of a set A => every neighborhood of x> contains infinitely many points of A.Suppose there exists an open set U containing {x} and such that U containsonly a finite number of points of A, say a_1,...,a_n (and we may assume a_i<> x for all i).By T1-axiom, for all i, there is an open set U_i such that x belongs to U_ibut a_i doesn't.Now take the intersection of {U,U_i; i =1..n}. This is an open setcontaining {x}, and whose intersection with A is at most {x}. Contradiction(x cannot be a limit point).--Julien Santini === Subject: Re: T1 spaceYou're reply is very helpful to me.But, I have one question about your post.Why do you use the expression at most?Is that must needed ?If that expression is omitted, Is you're proof wrong ??> Hello> In T1 space A,> x is a limit point of a set A => every neighborhood of x> contains infinitely many points of A. Suppose there exists an open set U containing {x} and such that U contains> only a finite number of points of A, say a_1,...,a_n (and we may assumea_i> <> x for all i).> By T1-axiom, for all i, there is an open set U_i such that x belongs toU_i> but a_i doesn't.> Now take the intersection of {U,U_i; i =1..n}. This is an open set> containing {x}, and whose intersection with A is at most {x}.Contradiction> (x cannot be a limit point).> --> Julien Santini === Subject: Re: T1 space> But, I have one question about your post.> Why do you use the expression at most?> Is that must needed ?> If that expression is omitted, Is you're proof wrong ??A limit point of A (where A is a subset of X) is defined to be a point x ofX such that any open neighborhood of x in X intersects A in at least oneother point which is different from x. As a result, x may not belongs to A,and the intersection of {A,U,U_i; i =1..n} may be empty . That's why I saidit's at most {x} (actually either it is {x} or the empty set).--Julien Santini === Subject: Re: James>>To be sung to the 'Pirates Of Penzance' tune 'Modern Major General' (author>>unknown):the original tune is Arthur Sullivan; lyrics William[?] Gilbert.I *think* (but I'm not sure) that this version might be by Jim Ferry, from> before he saw the light. If so, you'll probably make him very unhappy by> reposting it, as I'm sure he doesn't want to be reminded of his former> Disturbing Lack of Faith.You flatter me. I wish I could parodize like that.> [snip most of it]>>I always have the last word; so, with utmost finality,>>That's all from me, the model of a Newsgroup Personality.I think the last line is missing a two-syllable word near the end. Apart> from that the whole thing scanned beautifully.Yes, it would sound better, for example, with have the final word.But wait! Compare it to how the original ends: But still in matters vegetable, animal, and mineral, He is the very model of a modern Major-General.Reading this, it is clear that there is a caesura between vegetableand animal, which corresponds to pausing after so, in the parody.Reading the parody this way makes it flow much better (as well as morefaithfully!) than not pausing at all (one's first impulse) or afterthe semicolon (one's second, perhaps).But anyway, this is all rather off-topic. We would be wise to considerthe words of James himself and stick to the math. This newsgroup isno place for ridicule, sarcasm, and general tomfoolery, particularly ifit is aimed at a great man like James S. Harris.-Jim Ferry === Subject: Re: JamesI always have the last word; so, with utmost finality,>That's all from me, the model of a Newsgroup Personality.>> >> I think the last line is missing a two-syllable word near the end. Apart>> from that the whole thing scanned beautifully.Yes, it would sound better, for example, with have the final word.> But wait! Compare it to how the original ends: But still in matters vegetable, animal, and mineral,> He is the very model of a modern Major-General.Reading this, it is clear that there is a caesura between vegetable> and animal,[You correctly guessed that I meant *second*-last line]But actually I've always heard the original second-last line performed with 4 very clear syllables to the word ve-ge-ta-ble, so that the line has the same constant rhythm as all the other lines.This is called a patter song by the way, because it just patters alongat a constant rhythm with no let-up. Gilbert and Sullivan included one inmost of their shows. === Subject: Re: JamesContent-transfer-encoding: 8bit>>To be sung to the 'Pirates Of Penzance' tune 'Modern Major General' (author>>unknown):> [snip most of it]>>I always have the last word; so, with utmost finality,>>That's all from me, the model of a Newsgroup Personality.I think the last line is missing a two-syllable word near the end. Apart> from that the whole thing scanned beautifully.Eh? The second last line seems to me to trip along better if last isreplaced by final, but the last line looks just fine; I hear stresseson all, me, mod, of, News, Per, and al. Maybe there's abit of a stress on the final y. But the whole art of prosody lies inknowing when to stop refining your distinctions.-- Chris HenrichThe total lack of evidence is the surest sign that the conspiracy is working. === Subject: Re: James> >To be sung to the 'Pirates Of Penzance' tune 'Modern Major General' (author>unknown):>> [snip most of it]I always have the last word; so, with utmost finality,>That's all from me, the model of a Newsgroup Personality.>> >> I think the last line is missing a two-syllable word near the end. Apart>> from that the whole thing scanned beautifully.> Eh? The second last line seems to me to trip along better if last is> replaced by final, but the last line looks just fine; Sorry, I meant the second last line. Replacing last with final is astart, but you still need another syllable. Adding and after thesemi-colon completes the fix. (The rest of the lyric scans just as well asGilbert's, who was an unparallelled master of scansion, no joke about it.) === Subject: Re: Derivative of a sum = sum of derivative?>if you are sure that you have a valid derivative of the whole sum, and>you can differentiate each member of the sum, then derivative of the>sum will be equal to the sum of derivatives of the members of your>sum.Not necessarily. For example, consider the telescoping series f(x) = sum_{n=1}^infinity f_n(x) = sin(x), wheref_n(x) = 1/n sin(n x) - 1/(n+1) sin((n+1) x)Each (f_n)'(0) = 0, but f'(0) = 1.Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: MathML runtime.|>C++ to a more open standard - MathML to be more specific. Is anyone |>aware of a piece of software that can read in MathML and create code (C, |>C++, C# etc.. ) from it?Maple can read Content MathML and create C code. For example:> S := x2x:> MathML[Import](S); 2 x + sqrt(x)> CodeGeneration[C](%);cg = x * x + sqrt(x);Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Question for logarithm experts>> I have heard the following remarks about this problem: it's an>> unfair question. Its an equation that's not an equation. It is a>> single equation with two variables.>The second is right, it's not an equation. There is no value of X that can>make this true.Nonsense. An equation with no solutions is still an equation.Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Question for logarithm experts>> I am looking for a step-by-step method (proof) on the solution of>> this particular equation - (Solve for x) and equations like it:>> 7^X = 4*X.>Using a graphing calculator it's easy to show that there is *no* solution>for this equation. I plugged both into mine, and then zoomed into where>they're at their closest, and it's very obvious that there's no solution.Oh, so your graphing calculator thinks for you, does it? I happen to likethe function f(x) = 40 + log(x) - x/100 . Does your calculator tellyou anything about the solutions to, say f(x) = 1 ?Calculators are fine tools, but they're not a replacement for thinking!dave === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i185QZR17811;Is confusion evil, padre? Did the Good Lord not say that it was easier for aninfinitely large camel to pass through a zero than for a rich man to enter the kingdom of heaven? >root/administrator, and I thusly replied: >and I thusly replied: > >> >> Zero divided by zero is infinity.>> >> I say :>> >> Wharrrrrrrff>> Return To School !!ROTFL!You are obviously holding on to some genius proof that the mathematical> community has missed for centuries. What a clever person you are!Tell us all! Don't go all coy on us now.> What is your answer for zero divided by zero?>> >> When I took an advanced math course, we had such a discussion and the>> answer is UNDETERMIN. However, I raised an issue, i.e. 1=1 (as well as>> 0=0). If 0=0, then the numerator and denominator of 0s are cancelling>> out. Therefore, the correct answer of 0/0 is 1. Go figure!>This is a shocking development.>-- >The Reverend Parson Peter Parsnip>Smiting Sinful Usenet Users Since 1874>A bastard shall not enter into the congregation of the Lord; even to his>tenth generation shall he not enter into the congregation of the Lord. >- Deuteronomy 23:2 === Subject: a infinite integration problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i185Qc717862;HI,Can someone tell me the results of the following integration?int_{-infty+j*epsilon}^{infty+j*epsilon}frac{1}{v( v-j*a)(v+j*b)}dvwhere v is complex variable,a,b,epsilon are real varialbe and >0, j=sqrt{-1}. Thank you!Tao === Subject: Re: a infinite integration problem>Can someone tell me the results of the following integration?>int_{-infty+j*epsilon}^{infty+j*epsilon}frac{1}{v (v-j*a)(v+j*b)}dv>where v is complex variable,a,b,epsilon are real varialbe and >0, j=sqrt{-1}. The three poles of the integrand (0, ja, and -jb) all lie on theimaginary axis. Consider the rectangular contour with the corners (inorder) -x+je, x+je, x+jy, -x+jy, where y > max(a,e). The integral alongthe vertical pieces of the contour are less than 2(y-e)/x^3. If we letx->oo, this goes to 0 and leaves us with the horizontal pieces of thecontour. The integral along the top horizontal piece of the contour isless than |oo dt 2 | ------------------- = ------- |-oo (t^2+(y-a)^2)^{3/2} (y-a)^2If we let y->oo, this goes to 0 and leaves us with the bottom horizontalpiece, which is the integral in question. If e > a, then none of thepoles are inside the contour (i.e. above the bottom horizontal piece).Thus, if e > a, |oo+je dv | ------------- = 0 |-oo+je v(v-ja)(v+jb)If e < a, then only the pole at ja is inside the contour. The pole atja has residue 1/(ja(ja+jb)) = -1/(a(a+b)). Thus, if e < a, |oo+je dv 2 pi j | ------------- = - ------ |-oo+je v(v-ja)(v+jb) a(a+b)Rob Johnson take out the trash before replying === Subject: Silly question for someone with a big calculator. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i185QbB17846;Suppose x, y, e (unit) are elements of a 2-group F. Fortunately, there is no need to go into proving that F is a group nor to discuss properties of F extensively, here (I will have a lot more to say about F once I get my homepage up and running- hopefully soon.).Let R(F) be the ring generated by F. Choose 2 x + 2 y + ( -3) e Now, the following is just a particular observation I made and has little to do with my previous work. In fact, there`s probablynothing to it, just that it caught my eye this evening.For any given m in naturals which is a power a 2, we would like to find the coefficient (i.e., a whole number by definition) of the unit e in (2 x + 2 y + (-3) e) ^ m raised to the powerof m, assuming that the associative/distibutive laws hold. Thetask is to find for what m this coefficient is not a prime number.The rules of the game are: x*x = y*y = e*e = e and x*y = -y*xFor example (2 x + 2 y + (-3) e) ^ 2 = = 4x*x + 4 x*y - 6x +4 y*x + 4y*y - 6y - 6x - 6y + 9 = = 17e - 12x - 12yProceeding, I get (2 x + 2 y + (-3) e) ^ 4 = -408x - 408y + 577e (2 x + 2 y + (-3) e) ^ 8 = -470832x -470832y +665857ewhich I quickly checked to be prime from http://www.numbertheory.org/php/prime_generator.html:665801, 665803, 665813, 665843, 665857, 665897, 665921, 665923, 665947, 665953, 665981, 665983, 665993the number of primes in the range 665800 to 666000 is 13(but... I didn't check any further)C. Dement === Subject: Re: Silly question for someone with a big calculator. to find the coefficient (i.e., a whole number by definition) > of the unit e in (2 x + 2 y + (-3) e) ^ m raised to the power> of m, assuming that the associative/distibutive laws hold. The> task is to find for what m this coefficient is not a prime number.The rules of the game are: x*x = y*y = e*e = e and > x*y = -y*x> For example (2 x + 2 y + (-3) e) ^ 2 = > = 4x*x + 4 x*y - 6x +4 y*x + 4y*y - 6y - 6x - 6y + 9 => = 17e - 12x - 12yProceeding, I get (2 x + 2 y + (-3) e) ^ 4 = -408x - 408y + 577e> > (2 x + 2 y + (-3) e) ^ 8 = -470832x -470832y +665857ewhich I quickly checked to be prime from > http://www.numbertheory.org/php/prime_generator.html:Well if I've done it right, for 2^4 the coefficient of e is 886731088897 = 257 * 1409 * 2448769. === Subject: Re: polynomials that produce only primes>There was a recent post asking if all numbers of the form n^2 + n + 41 are>prime.>This set me thinking ...>No polynomial ax^n + bx^(n-1) ... + fx + g can produce only primes, because>if we set x=k*g (for any k) then its divisible by g. The polynomial can be>zero for only at most n of these cases. For all other values of k, the>polynomial must be a non-zero number divisible by g.>The only cases this doesn't work for are g=0 and g=1. The g=0 case is also>trivial, as x is a factor of the polynomial. So the only case left where its>not trivial is g=1.Hint: 1 is not a prime.You might also note that f(n + q) = f(n) mod q. Try it with q = f(n).Conclude that any prime that divides one value of f will divide infinitelymany of them.Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: polynomials that produce only primes>>There was a recent post asking if all numbers of the form n^2 + n + 41 are>>prime.>>This set me thinking ...>>No polynomial ax^n + bx^(n-1) ... + fx + g can produce only primes, because>>if we set x=k*g (for any k) then its divisible by g. The polynomial can be>>zero for only at most n of these cases. For all other values of k, the>>polynomial must be a non-zero number divisible by g.>>The only cases this doesn't work for are g=0 and g=1. The g=0 case is also>>trivial, as x is a factor of the polynomial. So the only case left where its>>not trivial is g=1.>Hint: 1 is not a prime.>You might also note that f(n + q) = f(n) mod q. Try it with q = f(n).>Conclude that any prime that divides one value of f will divide infinitely>many of them.Another way to say this is that f(a) | f(f(a)k+a) when f is a polynomialwith integer coefficients and k and a are integers. When k = 1 and a is0 or -1, we get the two most common refutations to x^2+x+41 being prime,x = 41 and x = 40.Rob Johnson take out the trash before replying === Subject: Re: polynomials that produce only primesOn the other hand...There is a polynomial in several variables, P(a,b,c,...,z) such thatwhen you plug in natural numbers, the primes (all the primes and only the primes) are the positive values. This is from the work on Hilbert's 10th problem.-- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Space>A simplified view of the surface of a torus, was implemented in the game>asteroids two decades or so ago.>Your spacecraft sits on a 2D screen and moving off screen at any point>brought you back from the opposite side. Wrapping once against the two>sides, one gets a cylinder. Wrapping twice, a torus.It's a torus if you mean topologically (or if you view it in R^4); thatis, you're ignoring metric data and any other additional structure (e.g.as a complex curve). That's fine -- it makes the next paragraph easier!>What does a space look like, if, when I sit in it (as a 3D object now) it>has the property that if I shoot a gun against ANY direction, the bullet>eventually hits me coming from the (spherically [*]) opposite direction>where I shot at?FIre a bullet in any direction and follow it halfway through its trip.The set of points you traverse is homeomorphic to the closed ball. Ifyou collect together the second halves of the bullets' journeys, youget the same ball. So the whole space you are envisioning is simply thisclosed ball, but with the points P and -P on the boundary sphereidentified (collapsed into one).>Can such a space be visualized in 3D?You can't even embed that boundary sphere into R^3 ! (It's the projectiveplane RP^2. The smallest dimension into which it embeds topologically is R^4.)dave === Subject: Re: there is no such thing as infinity>can use their computer time. The program in FORTRAN is simple:00001 n=1>00002 1 n=n+1>00003 print(3,4)n>00004 if(n.eq.M) then print(3,4)M>00005 else go to 1>00006 end if>00007 endWhat bastard version of FORTRAN is this?Last time I used unit 3, it was a card READER on an ICL1906A.Where's your format definition for label 4?Where's the STOP ?Why are the sequence numbers in columns 1 to 5 instead of 72 to 80?Why is GO TO spelt as two words?Why is ENDIF spelt as two words?Why doesn't the compiler bork when it spots the use of an undefined variable in the IF?Jokes are supposed to be internally consistent, you know.-- Looking for a 23 === Subject: Re: there is no such thing as infinity>>can use their computer time. The program in FORTRAN is simple:>>00001 n=1>>00002 1 n=n+1>>00003 print(3,4)n>>00004 if(n.eq.M) then print(3,4)M>>00005 else go to 1>>00006 end if>>00007 end> What bastard version of FORTRAN is this?> Last time I used unit 3, it was a card READER on an ICL1906A.Lots of programs use unit 3. Some programs work with dozens of files and needa unit number for each one. If unit 3 is not mentioned in an OPEN statement,then it will probably be connected with a file named 'fort.3' or somethingsimilar.> Where's your format definition for label 4?That is a problem, but there also is no definition of M.> Where's the STOP ?Fortran has not required a STOP statement for the past quarter of acentury.> Why are the sequence numbers in columns 1 to 5 instead of 72 to 80?That would be 73 to 80, but it's probably because those sequence numbersare not actually part of the file and are not seen by the Fortrancompiler. Some text editors can be told to display line numbers on thescreen while editing.> Why is GO TO spelt as two words?> Why is ENDIF spelt as two words?Why not? Both are perfectly legal, even if you are using modernfree-format Fortran in which blanks are significant.> Why doesn't the compiler bork when it spots the use of an undefined > variable in the IF?Because it's not a Swedish chef?Seems like that's the one really important point, since the otherproblems can be easily fixed. He can't very well define M, since that'sthe unknown value that the program is supposed to be looking for.> Jokes are supposed to be internally consistent, you know.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: there is no such thing as infinityDave Seaman wibbled:> Last time I used unit 3, it was a card READER on an ICL1906A.Lots of programs use unit 3. Some programs work with dozens of files and need> a unit number for each one. If unit 3 is not mentioned in an OPEN statement,> then it will probably be connected with a file named 'fort.3' or something> similar.When I learnt FORTRAN, we were advised that anything up to 9 was likely to be reserved by the compiler for various h/w devices.Where's your format definition for label 4?That is a problem, but there also is no definition of M.Where's the STOP ?Fortran has not required a STOP statement for the past quarter of a> century.Well, it's a long time since I learnt it, it's true. I always put one in, just in case.Why are the sequence numbers in columns 1 to 5 instead of 72 to 80?That would be 73 to 80, but it's probably because those sequence numbers oops! quite right.> are not actually part of the file and are not seen by the Fortran> compiler. Some text editors can be told to display line numbers on the> screen while editing. === Subject: Re: there is no such thing as infinity> Dave Seaman wibbled:>> Last time I used unit 3, it was a card READER on an ICL1906A.>> >> Lots of programs use unit 3. Some programs work with dozens of files and need>> a unit number for each one. If unit 3 is not mentioned in an OPEN statement,>> then it will probably be connected with a file named 'fort.3' or something>> similar.> When I learnt FORTRAN, we were advised that anything up to 9 was likely > to be reserved by the compiler for various h/w devices.Not any more. There is a standard input (unit=*) and a standardoutput (also unit=*). Card readers and card punches are gone, andprinting and plotting are done by sending files to the appropriate devicein a postprocessing step.Even if unit 3 had a predefined association in some implementation (whichit doesn't in any modern Fortran that I am aware of), you can still usean OPEN statement to connect unit 3 to a file.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: there is no such thing as infinityDave Seaman wibbled:> Dave Seaman wibbled:>> Last time I used unit 3, it was a card READER on an ICL1906A.>> >> Lots of programs use unit 3. Some programs work with dozens of files and need>> a unit number for each one. If unit 3 is not mentioned in an OPEN statement,>> then it will probably be connected with a file named 'fort.3' or something>> similar.When I learnt FORTRAN, we were advised that anything up to 9 was likely > to be reserved by the compiler for various h/w devices.Not any more. There is a standard input (unit=*) and a standard> output (also unit=*). Card readers and card punches are gone, and> printing and plotting are done by sending files to the appropriate device> in a postprocessing step.* was default input or output1 and 2 were input and output, which might, depending on whether it was the ICL, CDC, or Minnesota compiler, have been the same as 5 and 6, which were always the tty in and out. iirc 3 and 4 were card reader/punch, and 7 and 8 were tape reader/punch. Might have been the other way around. We didn't have a direct connection to the plotters afair.Even if unit 3 had a predefined association in some implementation (which> it doesn't in any modern Fortran that I am aware of), you can still use> an OPEN statement to connect unit 3 to a file.I knew that really, I was just being silly. I mean, the whole 'no such thing as infinity' thing was a joke anyway. I hope.-- Wanted: 24 === Subject: Re: there is no such thing as infinityI didn't know retarded people posted to newsgroups. In fact, I didn't> know that retarded people could get phds. What mail-order diploma factory> did you get your degree from again?I am not retarded and have an above average IQ, thank you. I got myPhD from Univerisity of San Moritz, a non-acredited but well-respecteduniversity in England. Just because it is non-acredited does not meanthat it is a diploma factory. It is better than the average stateuniversity in the USA and its curriculum is much more flexible. Inscientific thought.As I said, infinity does not exist and no one has ever observed it. Ifsomeone here could prove its existence then I would gladly concede,but no one has. My FORTRAN program will eventually find M and I willrevolutionize mathematics. Everyone will discard the notion ofinfinity from all math books and replace it with M=max N, the largestnumber possible.People think they contradicted me by giving numbers like 5.0 x 10^100and saying that it contradicts my predicted (but not yet verifiedvalue of M). But this misses the point - Has anyone ever counted tothis number? If you can prove to me that you have counted that high orhave at least programmed a computer to do such, then I will concede.But no one has, so it is doubtful that this number 5.0 x 10^100exists and therefore is doubtful that infinity exists.Ben Zona, PhD === Subject: Re: there is no such thing as infinity>> >> I didn't know retarded people posted to newsgroups. In fact, I didn't>> know that retarded people could get phds. What mail-order diploma>> factory did you get your degree from again?I am not retarded and have an above average IQ, thank you. I got my> PhD from Univerisity of San Moritz, a non-acredited but well-respected> university in England. ? What part of England is San Moritz in?-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Partridge, _Bouncing Back_ (14 times) === Subject: Re: there is no such thing as infinityDear Lord, it's easy to troll some of you people.Clue #1: Blatant Simpsons reference. > Ben Zona, PhDClue #2: Yiddish === Subject: Re: there is no such thing as infinity@reader2.panix.com:> Dear Lord, it's easy to troll some of you people.Hush. We're enjoying this. === Subject: Re: there is no such thing as infinity> It has currently reached about 2.0 x 10^18. Just as Einstein proved> that there is no aether, I am convinced that I will prove that there> is no infinity and then write a book or two.Hey, you could be on to something! Just one question. What happens when you> multiply the highest number by 2?;-PeterThe Great Computer in the sky gives an overflow error, of course.And to answer other people's questions about the foundations ofCalculus, dx=1/M, so in fact, we can do away with derivatives andreplace them with difference equations. lim_{n goes toinfinity}f(n)=f(M).So there you have it. Good wholesome math with no paradoxes. I feelsorry for you people as you believe in stuff that doesn't even exist.No one has ever measured anything beyond Asimov's constant, A, that Igave before! My hypothesis is that M=A*c^2.Dr. Ben Zona === Subject: Re: there is no such thing as infinityX-SessionID: C5fVb-14222-_4-24352@news.uchicago.eduX-Hash-Info: post-filter,v:1.4X-Hash: f5d229e2 6436bebe 66bd34f5 51d64e5c 71449827>> It has currently reached about 2.0 x 10^18. Just as Einstein proved>> that there is no aether, I am convinced that I will prove that there>> is no infinity and then write a book or two.>> >> Hey, you could be on to something! Just one question. What happens when you>> multiply the highest number by 2?>> >> ;-Peter>The Great Computer in the sky gives an overflow error, of course.>And to answer other people's questions about the foundations of>Calculus, dx=1/M, so in fact, we can do away with derivatives and>replace them with difference equations. lim_{n goes to>infinity}f(n)=f(M).>So there you have it. Good wholesome math with no paradoxes. I feel>sorry for you people as you believe in stuff that doesn't even exist.>No one has ever measured anything beyond Asimov's constant, A, that I>gave before! My hypothesis is that M=A*c^2.>Dr. Ben ZonaDoes you Mom know how you sign your posts?Mati Meron | When you argue with a fool,meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: there is no such thing as infinityDoes you Mom know how you sign your posts?Mati Meron | When you argue with a fool,> meron@cars.uchicago.edu | chances are he is doing just the sameMom is proud of her son the doctor. Dr. Ben Zona === Subject: Re: there is no such thing as infinityX-SessionID: AQjVb-14876-_4-25462@news.uchicago.eduX-Hash-Info: post-filter,v:1.4X-Hash: cfbfed18 b61da1e3 62492b52 4eafd49d d77e6b0d>> >> Does you Mom know how you sign your posts?>> >> Mati Meron | When you argue with a fool,>> meron@cars.uchicago.edu | chances are he is doing just the same>Mom is proud of her son the doctor. >Dr. Ben ZonaIt is not the Dr. part that I had in mind.Mati Meron | When you argue with a fool,meron@cars.uchicago.edu | chances are he is doing just the same === Subject: Re: there is no such thing as infinity>I've thought really hard about this one and came to the conclusion>that there is no scientific evidence of infinity existing. The highest>number that anyone has ever measured to according to Isaac Asimov in>his book Science and Human Thought is only about 5.0 x 10^48. No one>has ever gotten past that number. Doesn't this sound weird?You are trying to relate infinity to a quantity. Infinity is not aquantity. Nor is it the absence of quantity. If it could be sorelated, then it would be a number that is not a number, and hencewould have no identity. But infinity IS NOT A NUMBER.Your program cannot find the largest number because it derives eachnumber from the previous one. This is basic Set Theory. We say thereexists the number n = 0 (or 1), and that there exists the number n+1.By this definition, we have 5.0 x 10^48. Thus we have (5.0 x 10^48)+1,along with (5.0 x 10^48)^48^48^48^48^48 . . .I think you're grappling with the fact that infinity is defined to bethat which has no bound, no limit. You can't quantify infinity,because inherent in its definition is that fact that it is not anumber, and as such cannot be quantified.Xevious === Subject: Re: there is no such thing as infinity> >I've thought really hard about this one and came to the conclusion>that there is no scientific evidence of infinity existing. The highest>number that anyone has ever measured to according to Isaac Asimov in>his book Science and Human Thought is only about 5.0 x 10^48. No one>has ever gotten past that number. Doesn't this sound weird?You are trying to relate infinity to a quantity. Infinity is not a> quantity. Nor is it the absence of quantity. If it could be so> related, then it would be a number that is not a number, and hence> would have no identity. But infinity IS NOT A NUMBER.You should take a look at Cantor's theory of sets. In it, Cantor doestreat infinities as numbers, cardinals and ordinals. Try a Googlesearch on it.One of the brilliant things Cantor did was to define an infinite setas a set whose elements can be put into 1-1 correspondence with aproper subset of itself (obviously something one cannot do with afinite set). For instance, you can put the set of positive integersinto 1-1 correspondence with the set of positive even integers by thecorrespondence n -> 2nIt seems to go against common sense to say that these two sets havethe same cardinalities. Then again, we don't have any commonexperience working with infinities.Patrick === Subject: Re: there is no such thing as infinity> The highest>number that anyone has ever measured to according to Isaac Asimov in>his book Science and Human Thought is only about 5.0 x 10^48.I don't understand what this means. You don't measure numbers, you use numbers to measure things.-- Looking for a 23 === Subject: Re: there is no such thing as infinity00001 n=100002 1 n=n+100003 print(3,4)n00004 if(n.eq.M) then print(3,4)M00005 else go to 100006 end if00007 endYour program can't possibly work.It has to start with 1, and no one has ever observed a 1. So 1 probably doesn'texist.Also, what happens if 1 is bigger than the number M that you are looking for?You will never find it. Perhaps you should consider trying n=n-1 somewhere inyour program.And what if M is irrational, which seems likely? Or imaginary, which seems evenmore likely.Also, what if you find M and try to add 1 to it? What do you get?Perhaps you could program it in Smalltalk? === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)In sci.math, root/administrator:>> and I thusly replied:>> Zero divided by zero is infinity.I say :WharrrrrrrffReturn To School !!>> ROTFL!>> >> You are obviously holding on to some genius proof that the mathematical>> community has missed for centuries. What a clever person you are!>> >> Tell us all! Don't go all coy on us now.>> What is your answer for zero divided by zero?>When I took an advanced math course, we had such a discussion and the answer> is UNDETERMIN. However, I raised an issue, i.e. 1=1 (as well as 0=0). If> 0=0, then the numerator and denominator of 0s are cancelling out.> Therefore, the correct answer of 0/0 is 1. Go figure!Not quite, of course. Consider the following limits:lim (x->0) x/x = 1.Naturally. But...lim (x->0) 2*x/x = 2lim (x->0) x/(2*x) = 1/2lim (x->0) x^2/x = 0lim (x->0) x/(x^2) = oolim (x->0) (x^2 - x)/x = -1lim (x->0) number of tea breaks + (miles to London * x) / (speed * x) = time to Londonand so on.:-)-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)>According to standard analysis, infinity is not a number.>However, depending on the particular case, 0/0 could>have a +-Inf limit, of course.Assuming that zero is a real number, it would be considered thesmallest concievable number. So, that is the inverse of Zero? Thelargest conceivable number, or Infinity.>According to standard analysis, infinity is not a number.>However, depending on the particular case, 0/0 could>have a +-Inf limit, of course. === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)>According to standard analysis, infinity is not a number.>However, depending on the particular case, 0/0 could>have a +-Inf limit, of course.It is a bit misleading to consider infinity as a number. There are certainmathematical operations which yield infinity as a result, 1/0 being themost obvious. But the paradoxes which occur quickly show that normalmathematics is inadequate to deal with infinity. Throughout the history ofmathematics, notational conventions have been adopted to cope with conceptswhich cannot adequately be described within existing systems. Theseconventions are flexible enough to deal with those concepts but, in turn,have thrown up their own paradoxes.> Assuming that zero is a real number, it would be considered the> smallest concievable number. So, that is the inverse of Zero? The> largest conceivable number, or Infinity.I wish it were that simple. The simplest version of infinity is the ideaof a set of things which can be put into one to one correspondence with theordinals, or counting numbers - integers, if you prefer. Hence, the evennumbers are countably infinite by the simple mapping x -> x/2. By a morecomplex analysis involving the x-y plane, the rationals can also be counted(think of x and y both going from -infinity to +infinity: the intersectionsof these coordinates in the x-y plane define all the rational numbers.Starting from the point 0,0 and taking a spiral path outwards, all therationals can be assigned a unique ordinal which is unlimited, yetcountable. Duplicates, yes, but all countable.But when we come to irrational numbers, we, er, lose count. Cantor put thefollowing (simplified) argument:1. By arithmetic transforms, any interval of rational numbers can betransformed into the interval (0, 1) by algebraic transformation - in thecase of the countable infinity, this is done by the reciprocal function suchthat 1/infinity -> 0, and 1/1 ->1.2. Consider this range expressed as binary fractions, so 0 = 0.000000....and 1 = 0.111111... (if the latter seems problematic, multiply the equationby two and subtract 1. It works.)3. If we write these expressions down, we can also insert between them anyarbitrary collection of zeros and 1's such that our new value differs fromboth zero and 1. Not only that, because the binary expansions are infinite,but countably infinite, we can do this an infinite number of times such thatany of our binary fractions differs from its neighbours in at least onebinary place. And there is no way this process can terminate, hence the setof binary fractions were get is not only infinite, but uncountably infinite.Simply, we can always insert a new fraction.It actually gets worse (I prefer to call it more interesting) when yourealise that we've only done this in one dimension! David Hilbert realisedthis, and if you're really interested, I suggest you Google for the HilbertHotel - it explains these ideas very well.Cheers === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)>> Assuming that zero is a real number, it would be considered the>> smallest concievable number. So, that is the inverse of Zero? The>> largest conceivable number, or Infinity.>I wish it were that simple....and I just notice that we spanned into the sci.math group, whichmeans we're talming to some really big heads here. === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)>> Assuming that zero is a real number, it would be considered the>> smallest concievable number. So, that is the inverse of Zero? The>> largest conceivable number, or Infinity.>I wish it were that simple.Boy. lots of Doctors here. Doc, it hurst when I do this!!! Well,DONT DO THAT!!!. Anyway...Aside from all that brain-boggling blather which you dunped on us, Iassume that any number divided by Zero is Infinity. Looks right. Takefer instance...1/.00000000001=10000000000Number divided by Really Small Number = Really Big Number.THEREFORE...1/0= Really, Most Sincerely The Biggest Freakin' Number You CanImagine, or INFINITAY. Ja? Goot! Undt So... Ve haf zum zimplemathematics, as least that's what I leant in school. === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)>> Assuming that zero is a real number, it would be considered the>> smallest concievable number. So, that is the inverse of Zero? The>> largest conceivable number, or Infinity.I wish it were that simple.> 1/.00000000001=10000000000This is OK as far as it goes. But, it has limits.> Number divided by Really Small Number = Really Big Number.> THEREFORE...> 1/0= Really, Most Sincerely The Biggest Freakin' Number You Can> Imagine, or INFINITAY. Ja? Goot! Undt So... Ve haf zum zimple> mathematics, as least that's what I leant in school.That's right, it's simple - but it quickly throws up issues which HAFF tobe dealt with!Und ziss ist der weg von vitch ve begin to unterstandt der Mathematik, und,spater, die Welt! Als Wittgenstein sagt, Die Welt ist alle vitch ist dercase. I don't really thing he meant cheese, but my German is poor. So poor,in fact, that I'm setting up a trust fund for him. === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)>Und ziss ist der weg von vitch ve begin to unterstandt der Mathematik, und,>spater, die Welt! Als Wittgenstein sagt, Die Welt ist alle vitch ist der>case. I don't really thing he meant cheese, but my German is poor. So poor,>in fact, that I'm setting up a trust fund for him.Yawole, Mine Hair!Confidentially, Doc, I really don't like posting into intellectualnewsgroups. That means I have to use my brain, and that makes my headhurt. When I'm on the internet, I don't want to use my brain al all,which is why I read groups like alt.free.newsserver, where I can chewthe fat with other brainless clods like myself.By Einstien's Ghost, I don't how the heck this ever flowed over tosci.math, but if you'll excuse me, I'll be going back tofreenewsservers and forget this silly thread ever happened.So, thanks for all the fish, and see you back on campus. === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)> So, thanks for all the fish, and see you back on campus.no probs there. === Subject: Re: When 0 divided by 0 isn't infinity... (was: Re: I lost an account...)root/administrator, and I thusly replied: > >> and I thusly replied: >> Zero divided by zero is infinity.I say :WharrrrrrrffReturn To School !!>> >> ROTFL!>> >> You are obviously holding on to some genius proof that the mathematical>> community has missed for centuries. What a clever person you are!>> >> Tell us all! Don't go all coy on us now.>> What is your answer for zero divided by zero?When I took an advanced math course, we had such a discussion and the> answer is UNDETERMIN. However, I raised an issue, i.e. 1=1 (as well as> 0=0). If 0=0, then the numerator and denominator of 0s are cancelling> out. Therefore, the correct answer of 0/0 is 1. Go figure!This is a shocking development.-- The Reverend Parson Peter ParsnipSmiting Sinful Usenet Users Since 1874A bastard shall not enter into the congregation of the Lord; even to histenth generation shall he not enter into the congregation of the Lord. - Deuteronomy 23:2 === Subject: discrete mathematics fat setsGiven a set S {1,2,3,...,n}A subset P is said to be a fat set if every element in it is >= thecardinality of the subset P.The problem is to find the number of such fat sets. Also to come upwith a recurrsion logic. === Subject: Re: discrete mathematics fat sets> Given a set S {1,2,3,...,n}> A subset P is said to be a fat set if every element in it is >= the> cardinality of the subset P.> The problem is to find the number of such fat sets.Infinitely Countablenulset{n}, n >= 1{n,m}, min n,m >= 2...{n1,.. nj }, min (n1,.. nj) >= j... === Subject: Re: discrete mathematics fat sets>> Given a set S {1,2,3,...,n}>> A subset P is said to be a fat set if every element in it is >= the>> cardinality of the subset P.>> The problem is to find the number of such fat sets.>Infinitely CountableNo. He _started_ with Given a set S {1,2,3,...,n};this set has only finitely many subsets.>nulset>{n}, n >= 1>{n,m}, min n,m >= 2>...>{n1,.. nj }, min (n1,.. nj) >= j>... === Subject: Re: discrete mathematics fat sets> Given a set S {1,2,3,...,n}A subset P is said to be a fat set if every element in it is >= the>> cardinality of the subset P.The problem is to find the number of such fat sets.>Infinitely Countable> No. He _started_ with Given a set S {1,2,3,...,n};> this set has only finitely many subsets.nulset{j}, j in S{j,k}, distinct j,k in S, min j,k >= 2...S - {j,k}, distinct j,k in S, min S-{j,k} >= n-2Sj, j in S, min Sj >= n-1S === Subject: Re: Solving linear inhomogenous recursion>f_(n) = f_(n-1) + f_(n-2) + n for n>=2 and f_(0) = f_(1) = 1Double this equation and compare it to the sum of> f_(n-1)=...> f_(n+1)=...> This gets you to the level of a linear recursion with constant coefficients.> Do you know how to solve those (without generating functions, if you prefer> to have one hand tied behind your back) ?daveThe orginal equation can solved using the combination of thehomogeneous solution and the particular solution (which I suppose isthe method the OP wanted). However, I am curious whether there a wayto solve this equation with generating functions directly, withoutfirst having to do the algebraic manipulation that you mention to getrid of the n term? Without, the n term, it would have been prettyeasy, of course, coz that part is just the Fibonacci sequence.Also, there is a slight technicality that is nagging me. When you aretakingf_(n-1) = f_(n-2) + f_(n-3) + n - 1 (in order to add it to f_(n+1) =f_(n) + f_(n-1) + n + 1), isn't the condition that n>=2 gettingviolated?Sabyasachi === Subject: easy....analysis problem........continuous function f : [0,1]->R is f(x) >= 0 andint f(x) dx = 00~1show that f(x) is constant function.------------------------um.....i think.......let P_n(x) = a_0 + (a_1)x + (a_2)x^2 +......+(a_n)x^nint f(x) dx = int lim P_n(x) dx (n->00) (because, Weierstrassapproximation theorem)0~1thusint lim P_n(x) dx = lim int P_n(x) dx (because, {P_n} ->f :uniformlyconvergence)lim int P_n(x) dx = lim {a_0 + (a_1)/2 + (a_2)/3 +.......+(a_n)/(n+1)}it is zero.thusany n=0,1,2,..... , a_n = 0thusf(x) = lim P_n(x) = 0-------------------------but, i am not sure my proof, please let me check my process.thank you ....teacher... === Subject: Re: easy....analysis problem........hot-girl a .8ecrit dans le message de> continuous function f : [0,1]->R is f(x) >= 0 and> int f(x) dx = 0> 0~1> show that f(x) is constant function.Let F(x)=Int(f(t),t=0..x)F(0)=0F(1)=int (f(t) ,t=0..1) = 0F'=f>=0F increasingF(0)=F(1) then F constant , then F'=f=0 === Subject: Re: easy....analysis problem........> continuous function f : [0,1]->R is f(x) >= 0 andint f(x) dx = 0> 0~1show that f(x) is constant function.> f has to be identically zero. If f(x) > 0 for any point x, then, since f is continuous, f is > 0 in some neighborhood of x, which we can take to be a closed interval, hence f has a min value on that interval.Any Riemann partition finer than this interval must have sum > min(f)*length(interval) > 0. This is a contradiction, therefore f is identically 0. === Subject: Re: easy....analysis problem........> continuous function f : [0,1]->R is f(x) >= 0 and> int f(x) dx = 0> 0~1> show that f(x) is constant function.Your proof seems ok to me, but I think you don't need the W. approximationtheorem; if f(a)>0 at one point, then using continuity you show that f(x)>0on [a-e,a+e] (e>0), and in that case int(f) cannot be 0. === Subject: Re: easy....analysis problem........> continuous function f : [0,1]->R is f(x) >= 0 andint f(x) dx = 0> 0~1show that f(x) is constant function.It's mora than being constant; f = 0.Suppose otherwise, that is, suppose that f(a) > 0 for some a.Then, since f is continuous, f(x) > 0 for every x in someinterval [b,c] around a. Consider the partition P = {0, b, c, 1}.Then the lower sum of f with respect to this partition is >= (b - c).min{f(x) : x in [b,c]} > 0. So, you cannot haveinf f(x) = 0.Best regards,Jose Carlos Santos === Subject: Bound of a sumHi all,I've found the following inequality at a book: if a=1/sqrt(2), thena^((n + 1)(n + 2)) + a^((n + 2)(n + 3)) + a^((n + 3)(n + 4)) + ... << a^(n(n + 3)).This is stated as a matter of fact, without any hint of a proof; itis not even suggested that the reader tries to prove it as anexercise. So, my guess is that it should be quite obvious, butthe fact is that I have been unable to do it. Any idea?Best regards,Jose Carlos Santos === Subject: Re: Bound of a sum>Hi all,>I've found the following inequality at a book: if a=1/sqrt(2), then>a^((n + 1)(n + 2)) + a^((n + 2)(n + 3)) + a^((n + 3)(n + 4)) + ... <>< a^(n(n + 3)).>This is stated as a matter of fact, without any hint of a proof; it>is not even suggested that the reader tries to prove it as an>exercise. So, my guess is that it should be quite obvious, but>the fact is that I have been unable to do it. Any idea?>Best regards,>Jose Carlos Santos Dividing both sides by a^(n(n + 3)), you want to prove a^2 + a^(2n + 6) + a^(4n + 12) + a^(6n + 20) + ...If n >= -1, then this is bounded by a^2 + a^4 + a^6 + ... < a^2/(1 - a^2) = 1.-- John Adams served two terms as Vice President and one as President, but lostreelection. Later his son became President despite losing the popular vote.That son lost his reelection attempt badly. Now history is repeating itself.pmontgom@cwi.nl Microsoft Research and CWI Home: San Rafael, California === Subject: Re: Bound of a sum>>I've found the following inequality at a book: if a=1/sqrt(2), then>>a^((n + 1)(n + 2)) + a^((n + 2)(n + 3)) + a^((n + 3)(n + 4)) + ... <>>< a^(n(n + 3)).>>This is stated as a matter of fact, without any hint of a proof; it>>is not even suggested that the reader tries to prove it as an>>exercise. So, my guess is that it should be quite obvious, but>>the fact is that I have been unable to do it. Any idea? Dividing both sides by a^(n(n + 3)), you want to prove a^2 + a^(2n + 6) + a^(4n + 12) + a^(6n + 20) + ...If n >= -1, then this is bounded by a^2 + a^4 + a^6 + ... < a^2/(1 - a^2) = 1.Like I said, it was quite obvious. Unfortunately, not for me. :-)Best regards,Jose Carlos Santos === Subject: 1/0 now allowedIts defined as 1.6367348238383838Dont ask why, but must be used or your calculations will be wrong. === Subject: Re: 1/0 now allowed[...]http://groups.google.com/groups?safe=off&ie=UTF-8& oe=UTF-8&as_uauthors=LV.-- email: lastname at cs utk eduhomepage: cs utk edu tilde lastname === Subject: Re: 1/0 now allowed> Its defined as 1.6367348238383838Dont ask why, but must be used or your calculations will be wrong.As my calculations never require that I divide by zero, I remain content that none of my computation devices can use that value for that operation. === Subject: The Universal SetIf f(x) is a homeomorphism from T onto S, and for every point p in T,f(U(p)) = U(f(p)), and the monad is invariant under standardtopological transformations, with the caveat that the definition alsocomprizes a type of dynamic situation sematics, where concepts, suchas proper set, ordinal and cardinal are relativised to context,taking care of paradox at all levels via symmetry, or an invariantmany-valued logic, and the top[set of all sets], would naturally notexist, of course. since there is nothing outside the universe itbecomes an infinite chain, or composition, of ever more inclusivesituated sets expressing an interesting informational -topologicaldynamic. === Subject: Re: The Universal Set> If f(x) is a homeomorphism from T onto S, and for every point p in T,> f(U(p)) = U(f(p)), and the monad is invariant under standard> topological transformations, with the caveat that the definition also> comprizes a type of dynamic situation sematics, where concepts, such> as proper set, ordinal and cardinal are relativised to context,> taking care of paradox at all levels via symmetry, or an invariant> many-valued logic, and the top[set of all sets], would naturally not> exist, of course. since there is nothing outside the universe it> becomes an infinite chain, or composition, of ever more inclusive> situated sets expressing an interesting informational -topological> dynamic.www.elsewhere.org/cgi-bin/postmodern/ -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Partridge, _Bouncing Back_ (14 times) === Subject: A trig.InequalitySuppose that pi/n =< a_n < pi/2 , n= 3,4, ... .Then sin(n*x) =< sin(x)/(5*n) for all x in [a_n,pi/2] ? === ==========Subject: Re: A trig.Inequality>Suppose that pi/n =< a_n < pi/2 , n= 3,4, ... .>Then> sin(n*x) =< sin(x)/(5*n) for all x in [a_n,pi/2] ?> === ======= How about n = 10, a_n = pi/5, x = pi/4?Then sin(n*x) = sin(5*pi/2) = sin(pi/2) = 1,whereas sin(x)/(5*n) = sqrt(2)/100 < 1.-- John Adams served two terms as Vice President and one as President, but lostreelection. Later his son became President despite losing the popular vote.That son lost his reelection attempt badly. Now history is repeating itself.pmontgom@cwi.nl Microsoft Research and CWI Home: San Rafael, California === Subject: Re: A trig.Inequality>Suppose that pi/n =< a_n < pi/2 , n= 3,4, ... .Then> sin(n*x) =< sin(x)/(5*n) for all x in [a_n,pi/2] ?> === ======= How about n = 10, a_n = pi/5, x = pi/4?> Then sin(n*x) = sin(5*pi/2) = sin(pi/2) = 1,> whereas sin(x)/(5*n) = sqrt(2)/100 < 1.Thank you and Sorry for a Misprint . The corect for of inequality is === ======================================================= sin(n*x) =< (n/5)* sin(x) , for x in [a_n,pi/2] === ================================================= =Alex . === Subject: Re: Generalized Lie Bracket with Some Elementary ExamplesI've argued that the Lie Bracket can be generalized to the> difference between a sequence of elements, steps, procedures,> indeterminates, etc., and its reverse. For example:1) [A, B] = P(A-->B) - P(B-->A) = P(B) - P(A)where the center equality is the definition of the far left-hand-> side and P( ) is probability of, and (A-->B) is the set/event> defined by:2) (A-->B) = (AB')' = A' U Bwith AB' the intersection of A and the complement B' of B.Is there any point to this?Does this Lie bracket of yours satsify the Jacobi identity?If not, then why call it a Lie bracket.> It sometimes happens, but not by any means always, that the> generalized Lie Bracket can be written as f(x, y) - f(y, x) for> f a function and x, y some objects.Can the genuine Lie bracket (say in the Lie algebra sl(2)) beexpressed in this way?> Aside from probability-statistics, the definition seems rather> useful for algebra in general and even number theory.Oooh! Excellent, can it be used to prove Goldbach, orthe Riemann hypothesis?> For example,> it is rather easy to prove that for noncommutative algebras or> rings or modules, we have:3) (x o y) - (y o x) = x - xy + y -(y - yx + x) = yx - xy4) (x o y)' - (y o x)' = x + xy + y - (y + yx + x) = xy - yxwhere x o y is the Jacobson Radical star product x + y - xy which> can alternatively be formulated as what I label (x o y)' = x + y> + xy for (noncommutative) a ring or module with elements x, y.(Oh, you mean the multiplcative formal group).Yup rather easy indeed --- perhaps a bit of an understatement.> Notice also:5) (x^n - y^n) - (y^n - x^n) = 2(x^n - y^n)Brilliant! You're doing better than most of my first-yearstudents :-)> but for a noncommutative algebra we don't have the usual factoriza-> tion of x^n - y^n but rather:6) (x - y)(x^(n-1) + x^(n-2)y + ... + y^(n-1)) => = x^n + x^(n-1)y + ... + xy^(n-1) - [yx^(n-1) + ... + y^n]> = (x^n - y^n) + (x^(n-1)y - yx^(n-1)) + ... + (xy^(n-1) - yx^(n-1))So, for example, the following is wrong:7') (x^2 - y^2) - (y^2 - x^2) = (x + y)(x - y) - (y + x)(y - x)> = x^2 - xy + yx - y^2 - (y^2 - yx + xy - x^2) = 2x^2-2xy + 2yx -2y^2because it claims that 2x^2 - 2y^2 = 2x^2 - 2y^2 + 2yx - 2xy which> is only true iff xy = yx. Or if 2yx = 2xy = 0 (one might be working in characteristic 2).> The error is writing x^2 - y^2 => (x + y)(x - y) in (7') because the latter is really x^2 -xy + yx> -y^2 which isn't x^2 - y^2 in general. However, we do have:You really are doing pretty well!> 7) (x + y)(x - y) = 2x^2 - 2y^2 + 2(yx - xy)which relates things again to the Lie bracket yx - xy.Wow! That's amazing!Now, what happened to the applications to number theory?-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Partridge, _Bouncing Back_ (14 times) === Subject: Re: D.E. problem2 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i15IVpx07169;>>y*y'' - (y')^2 = 0 ; y(0) = 2; y'(0) = 3>.>.>.>> Where does this 3 = C*0 come from?>>3 = 0>y'(0) = 3 was given as an initial condition for the problem. But your formula was y= 3y not 3x. y(0)= 2 so y'(0)= 3= C*y= C*2. C= 3/2, not 0. Now that you know y'= (3/2)y. Now it should be easy to getthe answer below.>The answer is given as y = 2e^[(3/2)*t] === Subject: Re: Fast integer division by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i15IVqU07186;Mersenne number 2^n - 1.Put M = 2^n A + B (i.e. split M into high and low order n bits)now M - A = 2^nA - A + B = A(2^n-1) + BThus M - A mod 2^n - 1 = B. Thus M mod 2^n-1 = A + B.So just add the high order bits to the low order n bits.If you want M mod 2^n + 1, subtract instead of add..... === Subject: appeal to check argument by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i15IVp907173;.Does anyone have a couple of minutes to point out the errors if I post a two-screen-length argument about properties odd perfect numbers need to have? === Subject: Re: appeal to check argument>Does anyone have a couple of minutes to point out the errors if I >post a two-screen-length argument about properties odd perfect >numbers need to have?What makes you think if people haven't got time to do that, then theyhave time to respond to you telling this is the case?Oh, and if you post something to sci.math, someone will point out theerrors. Don't worry about that.-- I'm not interested in mathematics that might have anythingto do with reality. -- Easterly, in sci.math === Subject: Re: D.E. problem2 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i15IVqI07179;>>y*y'' - (y')^2 = 0 ; y(0) = 2; y'(0) = 3>.>.>.>> Where does this 3 = C*0 come from?>>3 = 0>y'(0) = 3 was given as an initial condition for the problem. But your formula was y'= 3y, not 3x. The condition isy'(0)= 3= Cy(0)= C(2) so C= 3/2 and y'= (3/2)y. Now it should be easy to get the result below.>The answer is given as y = 2e^[(3/2)*t] === Subject: Note: Re: Please read my preprint for a proof of Goldbach Conjecture by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i15IVqb07195;>Thus, if our rule (Definition 2.2) was not defined, simply we can find a layering that satisfies the condition (1) (in the paper);>J(0,p) for all p up to an appropriate odd natural number will do.Hoping that this note might help the readers.The quote is misleading. There is NO guarantee that the prime for theupper limit ( p_x_ ) is equal to the largest prime in the firstrow of a matrix of the form (3) at some odd natural number q (, whichis for N(3,q) ).Yes, the method is really combinatorial, making use of a matrixrepresentation (3) and property in the problematic Definition 2.2.It seems that we could construct a covering other than P(2, p_l_) towhich there is no layering that corresponds.Hisanobu Shinya === Subject: Re: Note: Re: Please read my preprint for a proof of Goldbach Conjecture by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i15NafG01582;>It seems that we could construct a covering other than P(2, p_l_)[ actually, P(3, p_l_) ] to>which there is no layering that corresponds.I meant that, under the assumption that the proof were correct,it makes me feel that the proof of the lemma could be applied toother covering as well; I do not feel the necessity that the coveringmust have been P(3, p_l_) for the proof to work. In this direction,perhaps a major mistake might be found.Hisanobu Shinya === Subject: Re: Please read my preprint for a proof of Goldbach Conjecture by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i161smX12673;>If there is a reader out there with extreme patience and insight into>the cluttered mind, then I suspect he would still have trouble with>this paper because it contains genuine errors.Before posting my question to you, honestly I would like to know ifthis discussion has been in any way rude to all of those involved. I am asking this question because in a way I do feel so; I am notretreating although an error was suggested by christian bau.The discussion has continued, whose ultimate reason is that myquestion to christian bau has not been answered at all. All I want toknow is a clear argument againt my paper. Please forgive me forassuming that pursuing a clear, undeniable conterargument beconsidered as nonguilty.Now, let me ask Dr. Blackburn about his comment. I truly, definitely,would like to know the genuine errors in my paper. Just speak to me.I would be quite ready for that. There will be no nervous breakdown ordepression or something like that. In fact, I just got a refusal ofadmission from a university, and yet I am quite energetic toeverything in my life :)The reason I submitted the paper to the Annals of Mathematics is thatI read a posting in Sci.math.research on the proof of Kepler'sConjecture [ or maybe Mathworld ], saying that the Annals does notpuslish a paper whose correctness is ambiguous. This implies thatwhatever pusblished in the Annals must be true; I would like my paperto be guaranteed as absolutely as possible. Even I will visit aneditor to explain what I got, if the editor generously gives me timeto explain. Even if I find a mistake in that occasion, that would beno waste of time since, THEN, I can throw away my paper.Therefore, an opinion of Dr. Blackburn would be very important.I have said really a big thing in one posting. But wouldn't that bethe way it is? Well, partly this might be because of my beingtremendously focused.Hisanobu ShinyaP.S. Human beings are really strange; I just felt that I remembertyping the last three sentences. === Subject: Number Theory Problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i163dfP20645;I found the following problem in an old number theory text by William J. LeVeque. Neither I nor anybody I know can solve this problem. Let N=(a-1)(b-1), where a,b are positive integersand gcd(a,b)=1Show that every integer c>=N is representable in the form c=ax+by with x,y>=0, while c=N-1 is not so representable.Thank you.Stas Sheynkop === Subject: Re: Number Theory Problemwhile c=N-1 is not so representable.> I can only prove this trivial case. Suppose c=N-1 is representable in the form that c=ax+by with x,y>=0 while N-1=(a-1)(b-1)-1=ab-a-b.Then ab-a-b=ax+by and hence the equality ab=(x+1)a+(y+1)b ...(1) holds.Since gcd(a,b)=1 and therefore x+1 is divisible by b while y+1 is divisible by a, which implies that (x+1)a+(y+1)b>=2ab, in contradiction to (1).----== Posted via Newsfeed.Com - Unlimited-Uncensored-Secure Usenet News==----http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: Re: Question for logarithm experts by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1647ww23096;>I am looking for a step-by-step method (proof) on the solution of>this particular equation - (Solve for x) and equations like it:>7^X = 4*X.>This seems to be a relatively simple problem but I've been tortured by>it since the eleventh grade, 28 years ago.> I have heard the following remarks about this problem: it's an>unfair question. Its an equation that's not an equation. It is a>single equation with two variables.>Those observations have not released me from my relative turmoil.>I can be reached at blue_rose01@msn.com>If you graph y = 7^x and y = 4x for real x,y you will see that the two curves do not intersect. For 11th grade math in 1976,that should be good enough to say there is no solution. === Subject: Re: Math of Hydrogen by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i16EFpP03006;>The probability distribution field for a Hydrogen atom>plots where the Hydrogen atom's single electron is most>likely to be found. It is a series of points plotted within>a sphere having a radius the size of the spherical orbital.>Generally the points are more dense the closer to the>center you get.>However, suppose the plot is reversed. Instead of>plotting the most probable places, we plot the least>probable places that the electron can be found.>Beyond the radius of the orbital, all points are least>probable. This is merely a photographic negative of>the initial probability field.>Let>P = position vector pointing to least probable points>r = radius of Hydrogen atom>Turning the atom inside-out through the center of the>proton, the proton now occupies the outer surface of>the orbital as a thin shell, and the outer surface of the>orbital occupies the center. They swap places. This>transformation may fall along the lines,>P' = ( r/|P| - 1)P>where P' points to the new coordinates of the least>probability.>You arrive at something similar to the initial probability>field again. The atom changes states in the process of>successive inversions, and makes up a spherical wave>that inverts on itself in an oscillation.>As the cycle continues,>P = ( r/|P'| - 1)P'>Hydrogen carries with it the resounding frequency of>the singularity preceding the Big Bang, the bandwidth>at which all time and space vibrates, the crux of all>time....> perhaps.>Jon Giffen... perhaps??? What's that supposed to mean? === Subject: Re: Derivative of a sum = sum of derivative? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i16IG7422669;>Does>d/du Sum (from i = 0 to infinity) e^(tu)>=>Sum (from i = 0 to infinity) d/du e^(tu)>??>Does a derivative of a sum always equal a sum of a derivative? I know for>integrals it's not true....Do you mean Sum (from t = 0 to infinity)?FIRST: The derivative of the sum.If u < 0, then Sum (from t = 0 to infinity) e^(tu) is equal to 1/(1 - e^u), because 1 + x + x^2 + ... = 1/(1 - x) when 0 < x < 1.The derivative of this is (e^u)/(1 - e^u)^2SECOND: The sum of the derivatives.The derivative of the terms of the series is e^u + 2 e^(2u) + 3 e^(3u) + ...which is equal to (e^u + e^(2u) + e^(3u) + ...)+ (e^(2u) + e^(3u) + e^(4u) + ...)+ (e^(3u) + e^(4u) + e^(5u) + ...)+ ...which equalse^u (1 + e^u + e^(2u) + e^(3u) + ...)+ e^(2u) (1 + e^u + e^(2u) + e^(3u) + ...)+ e^(3u) (1 + e^u + e^(2u) + e^(3u) + ...)+ e^(4u) (1 + e^u + e^(2u) + e^(3u) + ...)+ ...which equals(1 + e^u + e^(2u) + e^(3u) + ...) * (e^u)((1 + e^u + e^(2u) + e^(3u) + ...)which equals(e^u)/(1-e^u)^2if u < 0HENCE: They are equal in this case. === Subject: Re: Tricky integration - a silly error? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i16LncV07676;>I'm trying to integrate sin3x/(1+cosx) >This is how I've done it>this is the same as integrating (2sinx(cos x)^2)/(1+cosx) + (cos2xsinx)/(1+cosx)>Now the numerator is a derivative of the denominator so the answer is>-2(cosx)^2ln(1+cosx) + -cos2xln(1+cosx)>Is this correct because numerical integration say's it wrong?>Cheers,>SarahHow about differentiating your result? Do you get your originalfunction back? === Subject: Re: Abstracting out the method, non-polynomial factorization by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i16MLMw10647;> Some corrections...Despite all the controvery over my method, basically what I've done is> a bcing act--simple against complex--and here's an abstraction of> the technique:Consider f_1(x) f_2(x) = g F(x), and gab = gc, where(f_1(x) + ga)(f_2(x) + b) = g(F(x) + G(x) + c)where f_1(0) = f_2(0) = F(0) = G(0) = 0.> > then dividing g from both sides *must* giveab = c,Actually that factorization is available but so are an infinity of> other unit factor factorizations like(-a)(-b) = cso I went for a strong condition and got it wrong, when the proper> condition that works well is simply that the factorization is> available in the ring being considered.whereas, if you're in some other ring, like the field of algebraic> numbers, then you have an *infinity* of factorizations on the left,> like(sqrt(g) a)(b/sqrt(g)) = c.Correcting following up from before, the issue isn't the number of> factorizations but availability of a given factorization.Here the proper point is that the factorization shown is NOT available> in the ring of algebraic integers.> >> Let's be direct about this.>> Your methods, if correct, would imply that>> Q(x) = (25*x^2 + 30*x + 2)>>cannot be factored in the form>> (5 a_1 + c_1)*(5 a_2 + c_2),>>when x > 0 and a_1, c_1, a_2, and c_2 are all>>algebraic integers.>> You say it's impossible.>> However, Rick Decker shows that when x = 1,>> >> Q(x) = (5 sqrt(-2) + sqrt(7))*(-5 sqrt(-2) + sqrt(7)).>> All the numbers in sight are algebraic integers.>> What you keep saying is impossible is clearly, unambiguously>>possible. Not just for x = 1, but for any x > 0.>Where for any x > 0 means ___ ? Means x a positive integer - more generally I believe,any integer other than 0 or -1. Nora B.>> It is strange that you keep worrying about Rick's example,>>but you never quote what it actually says. You just keep>>saying it is impossible.>> Anyone who can do the arithmetic can see that you are >>wrong. Now you are taking your wrong conclusion and trying>>to generalize it into a wrong *method*. Even worse, you >>are trying to glorify it into some kind of astounding (but >>wrong!) GREAT DISCOVERY.>> Generalizing bad math still leaves you with just ... bad math.>> You know what counterexamples do. They disprove methods. >>That's what has happened here. Your attempt to deal with it,>>running at top speed in the wrong direction, changes nothing.>>You are still stuck with a hard-core fact: what you keep saying>>is impossible is sitting there, unrefutable, like a chunk of concrete.>> Nora B.>> James Harris>> === Subject: Re: Biomathematics -How Great the Waste![Folks 'Love' WDB2T, even as they'refuse' to see it :-]K. P. Collins> [...]> [...]> [...]> Nobody can usefully model turbulence> through space and over time.> [...] True, but only because of an implicit> Falsehood. No such thing as 'time' exists within> physical reality. What's been referred to as time is> an Erroneous partial conceptualization> of the one-way flow of energy from> order to dis-order that is what's> =described= by 2nd Thermo [WDB2T]. When this Falsehood is eliminated,> turbulence is easy. In the limit of 3-D space -> 0, energy> will flow toward decreasing order. Period.> Whoever gets the Clay Millenium Navier-> Stokes Prize will, necessarily, end with this> same Conclusion :-]> K. P. Collins === Subject: Why the question which I posted is misssing, I post it again! Please help me if element a is one GF(2^m) primitive element, and g(x)=x+a generate allpolymials whose stage is n. i.e. p(x)=g(x)*h(x) and p(x) can be described as: p(x)=p0 + p1x + p2x^2 + .....pn-1x^n-1 and pi is an elemnet inGF(2^m) First we define weight of pi wpi is the number of non-zero's in it. Forexample, if in GF(2^2) , a=(0 1), so wa=1; and b=(1 1), so wb=2 Then we define weight of p(x) Wp is Wp=wp0 + wp1 + wp2 +.....wpn-1 My question is how to calculate the number of polymials whose weight isequal to l.Yang Jun === Subject: Cardinal of Farey sequenceWhat is the Cardinal of a farey sequence ? === Subject: Re: Cardinal of Farey sequence>What is the Cardinal of a farey sequence ?Huh?!?May it be that you mean? Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore Aursand on comp.lang.perl.misc === Subject: help me pleaseI know the notions of pull back bundle, tensor product of two bundles over aI don't know how to construct the notion of tensor product of two differentbundles over two distinct manifolds, say M_1 and M_2 starting (i supposefrom the previous one notions).i ask you kindley to Help me please. === Subject: Re: help me please> I know the notions of pull back bundle, tensor product of two bundles over> I don't know how to construct the notion of tensor product of two> different bundles over two distinct manifolds, say M_1 and M_2 starting (i> suppose from the previous one notions).How about this?Consider M = M_1 x M_2.Let B_1 and B_2 be the bundles on M_1 and M_2.If pi_1: M -> M_1 is the projection consider the pullbackpi_1^*(B_1) from M_1 to M. Similarly considerpi_2^*(B_2) and form the tensor productpi_1^*(B_1) (x) pi_2^*(B_2).-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Partridge, _Bouncing Back_ (14 times) === Subject: Re: Pronunciation>How do you pronounce: schwa. However, this is entirely a guess on my part; he does not come > up in my conversations.I guess by zh, you mean the French j sound. I am not an expect but Ithink that j in Indian words usually represents a sound more likeEnglish j than French.>9. Fermat Two syllables, fair MAHThat is a reasonable approximation to the French pronunciation but Idon't recall hearing it often among English speakers. I thought thatmost pronounce it as if it were English with the first syllablesounding like fur (as on coat) and the second like the thing you wipeyour feet on. Certainly, I don't think that you will surprise anyoneor risk misunderstanding if you pronounce it as if it were English.J === Subject: Re: JSH: Don't talk to me>Back in the 70's when it came out that Nixon had a secret list of > journalists he disliked, it became a badge of honor to be on the list. > In fact, many journalists NOT on the list were embarassed to have been > omitted.I don't find your analogy very convincing. Posts from the cult of JSH >> detractors are often sad documents that should prove embarrassing to their >> authors when and if their heads are ever removed from their posteriors. And >> I don't think many sci.math posters are embarrassed at being ommited from >> JSH's list, do you?>> >> What, you don't find mocking responses to JSH comparable to>> investigative reporting making a corrupt presidency uncomfortable?>> >> Golly, the analogy worked for me. Weirdo.>I don't necessarily mind mocking responses.Of course you don't - on days when you're sober and not screamingobscenities you manage to convince yourself that your posts are_always_ reasonable and to the point, and hence the mockingreplies give you a way to (erroneously) conclude you're betterthan someone. It's the simple and direct _refutations_ you can't stand...>James Harris === Subject: Re: Don't talk to me> That's it. I just want those three people to OFF!!!!!!!!!> James HarrisDiagnostic criteria for 301.81 Narcissistic Personality Disorder (cautionary> statement)Uh-oh. Looks like there's some particularly gratuitous Harris-bashing ahead.> A pervasive pattern of grandiosity (in fantasy or behavior), need for> admiration, and lack of empathy, beginning by early adulthood and present in> a variety of contexts, as indicated by five (or more) of the following:(1) has a grandiose sense of self-importance (e.g., exaggerates achievements> and talents, expects to be recognized as superior without commensurate> achievements)But James *is* extremely important! What would be delusional would be forhim to view himself in any other way *but* extremely important! Yes, heexpects to be recognized as superior, but he *has* commensurate achievements.He's produced, for example, a simple proof of Fermat's Last Theorem! He has*counted* the primes *exactly*!> (2) is preoccupied with fantasies of unlimited success, power, brilliance,> beauty, or ideal loveHis brilliance is unlimited, so why shouldn't his success be? Isn't it onlyjust that he should be hailed as the best mathematician in history? But whathe gets falls woefully short of this. It's despicable.> (3) believes that he or she is special and unique and can only be> understood by, or should associate with, other special or high-status people> (or institutions)Again, James is special and unique. Why he mingles with the scum on sci.mathis a mystery to me.> (4) requires excessive admirationI think James has shown that he could care less what anyone thinks about him,provided they recognize the fantastic caliber of his mathematical genius.> (5) has a sense of entitlement, i.e., unreasonable expectations of> especially favorable treatment or automatic compliance with his or her> expectationshave WMDs and was upset that the President didn't listen to him? You peopleneed to remember that the President works for us, not the other way around.> (6) is interpersonally exploitative, i.e., takes advantage of others to> achieve his or her own endsJames does brag about how he uses sci.math like a tool. But what has sci.mathever done to deserve better?> (7) lacks empathy: is unwilling to recognize or identify with the feelings> and needs of othersWhatever.> (8) is often envious of others or believes that others are envious of him or> herHa! Why should James be envious of anyone else! It is to laugh! It'severyone else who's envious of him! Oh, that's the second part. Fine.> (9) shows arrogant, haughty behaviors or attitudesArrogant? It's mathematicians who are arrogant, not James. Duh.> Reprinted with permission from the Diagnostic and Statistical Manual of> Mental Disorders, fourth Edition. Copyright 1994 American Psychiatric> Association> ***Perhaps instead of spewing your useless nonsense, you should study this> disorder, go see a doctor and then drug and drink yourself to death.> Research into your delusional fits of grandeur would be a more> appropriate vocation for you!You're not supposed to be addressing James directly, you asshole! Whatwould you do in an audience with the Queen of England? Call her tootsand slap her on the ass? There are protocols to be followed, as Jameshimself has deigned to point out!Look, it's very simple: if Napoleon thinks he's Napoleon, does that makehim crazy? Sure, if it were actually *true* that all of James's writingswere not the mathematically revolutionary material they are, but wereinstead some sort of muddled and trivial scrawlings, then of course hewould have to be crazy to think he's some monumental genius. In this case,yes, all the above criteria would apply, and James would be, perhaps, theworld's greatest raving, out-of-control narcissist.But the fact is, James really is everything he claims to be, so your pointis moot. How do I know that James is what he is? It's as simple as this:he claims to be, so I believe him. I can't follow his mathematics, butthat's just because it's too complex for me. But this doesn't matter. Iknow that James is right. Why else would he *claim* to be right, repeatedlyand emphatically, if he weren't? He would, as you point out, have to be*crazy* to do that! And James isn't crazy. If he were, he would tell us,I'm sure.> Think about, eh, let's wack you!Eh?-Jim Ferry === Subject: Re: JSH: Don't talk to me> I don't necessarily mind mocking responses.Exactly. It isn't mockery or abuse that gets to you, it is simple and clear proof that you are wrong. That's the unforgiveable offence, isn't it?Gib === Subject: Re: JSH: Don't talk to meI don't necessarily mind mocking responses.Exactly. It isn't mockery or abuse that gets to you, it is simple and > clear proof that you are wrong. That's the unforgiveable offence, isn't it?GibHell no!!! That's a relief!!! I'm not a mathematician. I'm some guywho decided he'd go looking for something that might have been missedin the great rush of math society to build upon itself.And I found it.Like don't try the bull of saying I don't admit when I'm wrong,when time after time over a period of years I have.And besides there's my prime counting function which any person outthere with the balls to go do a Google search on can see is unique inthat it uses a partial difference equation, and it doesn't take longto find out that no one else in recorded history managed to find sucha gem.I think the problem is that today's mathematicians are pencilpushers--and not in a good way--who do NOT have guts.Freaking cowards are running as fast as they can.So I'm in the process of chasing them down.James Harris === Subject: Re: JSH: Don't talk to me> Freaking cowards are running as fast as they can.> So I'm in the process of chasing them down.> James HarrisHey Jim. What would you do if you caught one? === Subject: Re: JSH: Don't talk to me> I think the problem is that today's mathematicians are pencil> pushers--and not in a good way--who do NOT have guts.Freaking cowards are running as fast as they can.So I'm in the process of chasing them down.Be awfully careful not to catch any of them or you just might get a little of those humongous loads of crap squeezed out of you. Then you might be too tiny to reach the keyboard and amuse us any more. === Subject: Repunits prime factors: a result.I read Paulo Ribenboim's book The book of prime number records and Ifound a chapter about repunits (numbers that are formed only by 1's inbase 10: 1, 11, 111, ....) and its primality. Essentially the Book saysthat there's a little few known about it. And in general, it's not knownwhen a repunit is prime or is composite, or if the number of primerepunits is finite or infinite.I had interest in this class of numbers and I wanted to investigate moreabout these. With some lucky and investigated another thing, I discovered(at least) a surprised result (at least for me). It points out that thenumber of factors of the repunits is, in mean, very bigger. Sorry, but Idon't check it computationally.I offer it with some comments to you. Please, comment any part that youthink that it's good (or bad) or anything you want. I hope that anyonethink that it's useful. Please, if reply me, not by mail if it's possible.1. Notation: I denote as R(n) the n-th repunit, that is a number formed byn 1's. [R(1)=1, R(2)=11, R(3)=111, ...]Now two simply lemmas that are obvious true, and that I only prove thesefor formality. Clearly, these results are more general, but we only areinterested in interval [0,1]2. Lemma 1: Any rational number of [0,1] belongs to any of the followingclasses: a) Periodic numbers (mixed or pure) (in base 10) b) Numbers with finit number of digits (in base 10) Dem: We need a lemma: Lemma 1a: Let be a a number in [0,1] with finite number of decimal digits or a periodic number (mixed or pure). Then, for all b>=1, a/b is a number with finite number of decimal digits or a periodic number (mixed or pure). Dem: Induction on N(b)=number of prime factors of b=n. n=0: Then b=1, and a/b=a and all it's ok. n-->n+1: N(b)=n+1, so b=p_1....p_np_{n+1}. a/b = a/(p_1...p_np_{n+1}) = [a/(p_1...p_n)]/p_{n+1}. The numerator is periodic number or a number with finit number of digits (induction hipotesis) and then (by the case n=1) all the fraction is too. It finishes the dem. Prove of the lemma properly: We prove that a/b is of the two previous classes by induction on N(b). N(b)=0. Then b=1. So, because a/b is in [0,1], a=0 or a=1 that is clearly a number with finite number of digits. N(b)=n-->N(b)=n+1: So b=p_1...p_np_{n+1}. So a/b = a/(p_1...p_np_{n+1})=[a/p_1...p_n]/p_{n+1}. a/p_1...p_n is a periodic number or a number with finit number of digits by induction hipotesis. And so [a/p_1...p_n]/p_{n+1} is a number with finit number of digits or a periodic number applying lemma (1a).3. Lemma 2: Any rational numbers of [0,1] has the form: 1a) a/10^n for some n>=0, a>=1. 1b) a/(9R(n)10^r), for some a>=1, r>=0, n>=1 Dem: Any rational number of [0,1] is a periodic number or a number with finit number of digits. We prove that the first class has the second form, and the second class has the first form. - If we have x rational number in [0,1] that have only a finite number of digits, then x=0.a_1.....a_n, where a_i are the digits of x (in base 10). So x = (a_1...a_n)/10^n. And saying a = a_1...a_n we have that x = a/10^n, as we want. - If we have x rational number in [0,1] that it's a periodic number, then x has the form x=0.b_1...b_ra_1....a_na_1....a_na1_....a_n.... where a_1...a_n is the period (n>=1) and b_1...b_r is the non-peridic part (r>=0. We can have r=0 in the case that x is pure periodic number). So 10^(r+n)x = b_1....b_ra_1....a_n.a_1....a_na_1...a_n.... -10^rx = - b_1....b_r.a_1....a_na_1...a_n.... If we add these, we have: x = (b_1....b_ra_1....a_n - b_1...b_r)/(10^r(10^n-1)) = (b_1....b_ra_1....a_n - b_1...b_r)/(10^r9R(n)) = a/(9R(n)10^r), where a = b_1....b_ra_1....a_n - b_1...b_r as we want prove.4. Theorem: For every prime p>=7, there exist n such that p divides R(n)[it not implies that R(n) could not be prime. In that case (R(n) prime), pwere equal to R(n)] Dem: Let be the fraction 1/p, p prime distinct of 2, 3 and 5 (p>=7). This number is (clearly) a rational number. So, by lemma 2, 1/p is has the form (1a) or (1b). Clearly it could not have the form (1a): If 1/p=a/10^n for some n>=0, then 10^n = pa. So p divides 10^n, that it's impossible because p is not equal to 2,5. So it has the form (1b). So there exist some n>=1, r>=0, a>=1 such that a/(9R(n)10^r) = 1/p So ap = 9R(n)10^r. So p divides 9R(n)10^r. But, because p is not equal to 2,3,5, p does not divide 9 and 10^r. So, because p is prime, p divides R(n). So we proved that if p>=7, then there are some n such that p divides R(n), as we want.5. Notes: a) I believe that it's interesting investigate the sequences of numbers (a_n) with the similar propierty of that result, because a general study of this topic could provide us a information about repunits. Fixed m>0, we could define that (a_n) is m-anything iff for all p >=m, there exist n>=0 such that p divides a_n. Our case is m=7. I don't know if anyone investigated/discovered anything about it. b) If we count the repunits and the primes, obviously there are much more primes than repunits: Let Ro(x)=Card({n repunit <= x}), and Pi(x) is the counting prime function, then it's easy to prove that Ro(x) = [log(9x+1)], where [x] is the integer part of x, that it's clear more more smaller function than Pi(x). So it's reasonably to think that the composite repunits have more prime factors in its descomposition. But I don't know if it's true and how prove it. c) We can generalizate this result, that have a great corollary. Now, it's.6. Lemma: For all b>=1 such that 2,3,5 do not divide b, 1/b has the form(1b) Dem: Suppose that 1/b has the form (1a). Then there were a, n>=0 such that a/10^n = 1/b So ab = 10^n. In particular, b divides 10^n, which is impossible because 2, 5 do not divide b.7. Theorem: For all b>=1 such that 2,3,5 do not divide b, then thereexist n>=0 such that b divides R(n) Dem: Considering 1/b with b such that 2,3,5 do not divide b. Then 1/b has the form (1b) by lemma. So there is n>=0 such that 1/b = a/(9R(n)10^r) So 9R(n)10^r = ab. In particular, b divides 9R(n)10^r. But b does not divide 9 (because 3 does not divide b) nop 10^r (because 2,3 do not divide b). So b divides R(n). Then we prove that for such b there are n such that b divides R(n), as we want.8. Corollary: Let be w(x) the number of distinct prime factors or x. Thenthere are repunits with arbitrary values of w(x), that is, for all m>2,there is some n>=0 such that w(R(n))=m [for m=1 is true too: R(2)=11 thatis prime] Dem: Chossing b=p_1...p_m and applying the previous result.The question is: Are there infinite number of repunits with thispropierty?. That is, for all m>=1, if define W(m)={r repunit such thatw(r)=m}, have we got that card(W(m))=infinity? (Now we know thatcard(W(m))>=1). The case m=1 is if prime repunits are infinit.Well, thank you very much for reading my post.Best regards,Xan. === Subject: Re: Series>>|What are the next ten characters in the following series?>>|>>|1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5>>there aren't any more- it ends right there.>> Nice one. I've sometimes tended to give facetious answers>> to these ill-defined what's-the-next-number questions, but>> I have to stop now, this is better than anything I've ever done.>Hmm... but not half as witty as Rob Johnsons answer, sorry.>I always wonder, why people who don't like this kind of>riddles feel like responding at all. It sounds like>Well I didn't get it. And so it won't be worth it.There _is_ a serious point behind those replies: thatthe questions as posed do not have a unique answer.(It seems likely that some of them are homework,and it's simply not right to mark _any_ answer toone of these problems wrong...)>In German: Dem Fuchs sind die Trauben zu sauer.>Rainer Rosenthal>r.rosenthal@web.de === Subject: Re: Series> There _is_ a serious point behind those replies: that> the questions as posed do not have a unique answer.Hello David,surely there are always infinitely many solutions for thesequestions of the kind tell the next number.But some people, including me, like to find out, what theposer of the question did have in mind, when he or she askedthis question. It's sort of *communication* I'd say. Othersare out or feel outside, because they don't enjoy this sortof communication.I always like people playing that sort of game, even if Idon't have a solution or even don't know what they are talkingabout. It's just nice to have them use their brains in afriendly manner.Think of the great OEIS, maintained by Neil Sloane and hisfriends all over the world (greetings to all SeqFans ...).It's a great plesure to find out that two reasonably definedsequences are identical on the first 100 numbers and thendiverge.You could become a well-respected member in the SeqFan community,I believe, if you could provide a welldefined and nice problem,where a sequence is popping out, which starts 1, 4, 9, etc.until 169 = 13^2 and then continues with an element other than196 = 14^2.Greetings and cheer up :-)Rainer Rosenthalr.rosenthal@web.de === Subject: Re: SeriesIn sci.math, Jones<_jones92057@yahoo.com><962b628d.0402062200.78dfd414@ posting.google.com>:> What are the next ten characters in the following series?> 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5I'm thinking gray. Graycode, that is. 654321 000000 <- start state1 0000012 0000111 0000103 0001101 0001112 0001011 0001004 001100and so on.Therefore, the next few characters are 12131214121312161213121412131215.(Gray's Encoding or Graycode is used occasionally to generate veryclean counters; the next number is 1 bit away from the previous one,as opposed to the more standard binary encoding where a large numberof transitions may occur e.g. from 111111 to 1000000. There'sprobably a better definition somewhere on the Web, of course;I'd have to look.)-- #191, ewill3@earthlink.net -- insert random dirty mind hereIt's still legal to go .sigless. === Subject: Re: Series> What are the next ten characters in the following series?1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5Lookup ID Number A001511 in the On-Line Encyclopedia of Integer Sequences === Subject: Re: Series|What are the next ten characters in the following series?> |> |1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5there aren't any more- it ends right there.If the sequence ended, the last number would be '4' not '5' === Subject: Re: SeriesWhat are the next ten characters in the following series?1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 I think you mean 'sequence.' 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6JProbably 'symbols' vice 'characters' as well.The 32nd 'symbol' can either be '6' or '5' === Subject: Re: Series <962b628d.0402071606.7312d934@posting.google.com > What are the next ten characters in the following series?> 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5> I think you mean 'sequence.'> 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6Probably 'symbols' vice 'characters' as well. Either way, you original use of series is incorrect.> The 32nd 'symbol' can either be '6' or '5' It is 6 if you want to make it at least remotely interesting. But if you meant that one can come up with a rule so that the 32nd symbol can be either '5' or '6', then you might as well say that the 32nd symbol can be any other real number as well. But the nicest extension of the sequence maintains the following property: since '1' is every second character, remove all the '1's. You are left with 2 3 2 4 2 3 2 5 2 3 2 4 2 3 2 6Now '2' is ever second character, so remove it and be left with3 4 3 5 3 4 3 6Now '3' is every second character, so get rid of those:4 5 4 6Now 4 is every second character, so get rid of those: you get 5 6 (5 7...)An alternate generation scheme is to define the function of a sequence f(a1 a2 ... a(n-1) an) = [a1 a2 ... a(n-1) an] [a1 a2 ... a(n-1) an+1](i.e. concatenate the same sequence but with the last symbol incremented by 1.)Start with the sequence '1' and keep applying f().f(1) = [1] [2]f(f(1)) = [1 2] [1 3]f(f(f(1))) = [1 2 1 3] [1 2 1 4]f(f(f(f(1)))) = [1 2 1 3 1 2 1 4] [1 2 1 3 1 2 1 5]This is the sequence you started with. The next symbols are then given by f(f(f(f(f(1))))) .J === Subject: Re: SeriesWhat are the next ten characters in the following series?1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 Donald Duck and Superman.I get it - characters - but why Donald Duck and Superman specifically? === Subject: Re: Series> What are the next ten characters in the following series?> > 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 Donald Duck and Superman.> I get it - characters - but why Donald Duck and Superman specifically?Donald=6 charactersDuck=4 characterstogether: 10 characters :-) === Subject: Combinatoric:4 groups of 4This is a planning problem for scheduling players in teams of 4.Not sure if it is feasible.16 players, 5 days of play, 4 teams each day.Each player teamed with each other player only once over the 5 days.This is a start, but how to proceed ?day 1 abcd efgh ijkl mnop 2 a... b... c... d... 3 a... b... c... d... 4 a... b... c... d... 5 a... b... c... d...I have a feeling, if there is a way, it's related to some cyclicarrangement pattern.Any suggestions or pointers to places of study are much appreciated.Richard === Subject: Re: Dense Subset of Sobolev Space?> Hi Everybody,> I'm interested in finding a dense set in the Sobolev>space of functions on the interval [-1,1], such that>f' is in L2[-1,1], with the inner product> (f,g)=int( f'(x)*g'(x) + f(x)*g(x), x = -1..1).> I think -- but am not sure -- that the set of (finite)>linear combinations of x and exp(i*n*x) (where n runs over all>integers) is dense. I think that the exp(i*n*x) are dense, but that has to dowith nonharmonic Fourier series and the fact that1 < pi; I think, but am not sure, that you simply left outa pi somewhere. I'm going to assume you meantL2[-Pi, Pi] instead of L2[-1,1]; you could instead talkabout [-1,1] and put some Pi's into the exponentials.You also need to note that since you're clearly includingan inner product! You need to use the complexconjugate of g and g'.With those modifications it's clear that the span ofthe exponentials is dense in the space; let's callthe space H. First show this:(i) The elements of H are continuous (so for example it makes sense to talking about f(0) for f in H).Hint: Some inequality shows that |f(x) - f(y)| <= something.(ii) If f is in H and f'(0) = 0 then ||f||_2 <= c1 ||f||_infinity <= c2 ||f'||_2.Now it follows that the trigonometric polynomialsare dense in H: Given f in H, you can find atrig poly Q such that ||f' - Q||_2 < epsilon/c, andthen the above shows that there is a trig polyP such that ||f||_H < epsilon.>I think I must add x, since its derivative >is 1, and I must have 1 to approximate the derivatives in the >L2 norm. The x is a nuisance for various reasons, but I don't>see how I can get rid of it.> So, is my set indeed dense? If not, what do I need>to add to it? === Subject: good analysis book with categorical viewpoints?I am very interested in category theory and found it extremely usefulin understanding concepts in algebra. And I believe that some time agoI read something about an analysis book with a categorical viewpointon this newsgroup. Unfortunately I could not find the thread any more.Can anybody recommend a good book? Topics should possibly includeanalysis on manifolds, a bit measure theory and integration.TIA,Tobias === Subject: grateful for comments: argument by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i18Dgo723440;.Grateful for any corrections. Apologies if any lines go .Outline: If an odd perfect number exists and has three primefactors a,b,c then we can partition the total list offactors, both prime and composite, into those dividingby a [call this 'a.sum'], remaining factors dividingby b [b.sum], remaining factors dividing by c [c.sum],and 1.For perfect number status, what we call 'complements'of each sum must divide by that prime:for example 1 + a.sum + b.sum = c.comp must divide by c. In general x must divide x.comp for each of x = a,b,c.By examining each of a.comp = 1 + b.sum + c.sum,b.comp = 1 + a.sum + c.sum,c.comp = 1 + a.sum + b.sum we find it is never the case that all three divide by they primes they should, and hence there are no odd perfect numbers. Further there are no even perfect numbers dividing by more than two primes. Argument: [1] We note that any gap of size p between a multipleof x and a multiple of y only occurs twice betweeneach pair of consecutive multiples of the lowestcommon multiple of x and y [call this 'LCMxy']. Thismeans that p occurs between kLCMxy and [k+1]LCMxytwice, once as +p [x multiple - y multiple = +p] andonce as -p [x multiple - y multiple = -p]. For suppose p could occur three times, then therewould be either two +p gaps or two -p gaps. Supposetwo +p gaps. Then we have larger x multiple - larger ymultiple = +p = smaller x multiple - smaller ymultiple. But then larger x multiple - smaller xmultiple = z = larger y multiple - smaller y multiple,and both can only = z if z is the lowest commonmultiple of x and y. So p occurs at most twice betweenkLCMxy and [k+1]LCMxy.And if p occurs only once, as +p say, then we havethat [for example] kLCMxy + 4x - 3y = +p, but[k+1]LCMxy - 4x + 3y does not = -p, a contradiction.So p occurs at least twice between kLCMxy and[k+1]LCMxy. placed symmetrically between kLCMxy and [k+1]LCMxy,since when calculating differences between multiplesof x and y, counting downward in increments of x or yfrom one multiple of LCMxy only differs in direction,not size, from counting upward in increments of x or yfrom the previous multiple of LCMxy.The process where we obtained complements frompartitioned sums of factors in an odd perfect numberpreserves differences between the multiples but withsign reversed [a.sum - b.sum = b.comp - a.comp, forexample] we must have, for the complements to divideby the primes they should:jLCMabc - a.sum = z = a.comp - gLCMabc.Where j and g may not be consecutive multiples of LCMabc, but if a.sum < [or >] its closest LCMabc, thena.comp > [or <] its closest LCMabc. But because 1 is always included, the complements and the partitioned sums are never equal distances from nearest multiples of LCMabc, and so do not all divide by the appropriate primes for the number to be perfect. Example: Let a = 3, b = 5, c = 7, and a.sum = 51, b.sum = 65,c.sum = 77, and LCMabc = 105.The complements are a.comp = 1 + 65 + 77 = 143 [not divided by 3]b.comp = 1 + 51 + 77 = 129 [not divided by 5]c.comp = 1 + 51 + 65 = 117 [not divided by 7].With these difference gaps between multiples, the nearest numbers, not complements, which would work:a.comp = 210 [2LCMabc here] - 51 = 159 [3 divides]b.comp = 210 - 65 = 145 [5 divides]c.comp = 210 - 77 = 133 [7 divides]. In a more general example, try to deliberately bce partition sums to obtain complements which could be the same distance from an LCM. Where a,b,c,d are four primes in some perfect number, such that a.sum = LCMabcd - 11b.sum = LCMabcd - 7c.sum = LCMabcd + 5d.sum = LCMabcd + 13,we get the complementsa.comp = 3 LCMabcd + 11 + 1b.comp = 3 LCMabcd + 7 + 1c.comp = 3 LCMabcd - 5 + 1d.comp = 3 LCMabcd - 13 + 1. This clarifies how for potential perfect numbers with threeor more prime divisors, not every complement will divide by the prime it should. Complements have the same difference gaps as the partitionedsums, ensuring by [1] that they can only all divide asrequired if jLCMabc - a.sum = z = a.comp - gLCMabc, which they cannot because the right-hand term above is at least 1 too large. We note that a perfect number can divide by only twoprimes, but only if one of them is 2. For if a.comp = 1 + b.sumthen either a or b must be 2. And since every second number is a multiple of 2, and any odd number + 1 divides by 2 as easily as that odd number - 1, one of the constraints is removed from a number divided by two primes, one of them 2, being perfect. However, this constraint that partitioned sums must 'mirror' complements ensures there are no odd perfect numbers. Mark Griffith.>.>Does anyone have a couple of minutes to point out the errors if I >post a two-screen-length argument about properties odd perfect >numbers need to have? === Subject: Re: Silly question for someone with a big calculator. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i18Dgoe23444;Sorry, wasn't thinking that putting a colon at the end of a link would cause it not to work:http://www.numbertheory.org/php/prime_generator.html.org/ php/prime_generator.html A few more points-1. It was not as if I kept plugging in different number combos until I finally got the element 2 x + 2 y + (-3) e with the properties I wanted. 2 x + 2 y + (-3) e can actually be written as (z^2) itself, where z is nothing other than the sum of certain elements of F.2. (obvious) any such prime p can be written in the form p = 2(q)^2 + p' where p' is prime.3. I had a premenition about this last year when I first began researching F. http://mathforum.org/discuss/sci.math/t/538068?hi_n= 538068,538153 Looking back on this link, now, almost makes me want to laugh- if it hadn't been such a humbling experience at the time. === Subject: Inverse of a Laplace TransformIs the inverse transform of the Laplace transform necessarily linear?More generally, is the inverse operator of a linear operatornecessarily linear? If not, under what condition should it be linear?(Not a HW question, but something that came up while studyingEngineering Process Control.) === Subject: Re: Inverse of a Laplace Transform> Is the inverse transform of the Laplace transform necessarily linear?> More generally, is the inverse operator of a linear operator> necessarily linear? If not, under what condition should it be linear?> (Not a HW question, but something that came up while studying> Engineering Process Control.)If we write F as the inverse f^(-1).then for every A and B there is a and b such that A = f(a) and B = f(b) [1]F( p A + q B ) = F( p f(a) + q f(b) ) because [1] = F( f( p a + q b ) ) because f is linear = p a + q b because F is inverse of f = p F(f(a)) + q F(f(b)) because F is inverse of f = p F(A) + q F(B) because [1] === Subject: Re: Inverse of a Laplace Transform> Is the inverse transform of the Laplace transform necessarily linear?Yes.> More generally, is the inverse operator of a linear operator> necessarily linear?Yes: if L(af + bg) = aL(f) + bL(g) (constant a,b), then it would behelpful if L^(-1)(aL(f) + bL(g)) = aL^(-1)(L(f)) + bL^(-1)(L(g)).-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will find these people and we will bring them to justice. === Subject: Math/Physics ValentinesIt is time to repost this again: MATH/PHYSICS VALENTINESRemember those cheesy valentines you used to get when you were inelementary school? Well I give something similar to my Math Physicsstudents on Valentines day. I tried it last year with good success and Iwas hoping that I could get some more ideas for cards. So it wasrecommended that I tap some of the greatest minds around, which is why I'mhere. I'm not sure if anyone can help me, but if you have any more ideas orMathewThe following is a list I have come up with already:* You're one of the fundamental forces in my life. Be my Valentine.* You're the net force that makes my heart accelerate. Be my Valentine.* There's an attraction between us, I think its gravity. Be my Valentine.* You're so great they should name a constant after you. Be my Valentine.* We're like opposite charges. Be my Valentine.* The lines of force point me towards you. Be my Valentine.* Like resonance tubes, we're in harmony together. Be my Valentine.* We add up to a good team. Be my Valentine.* You're a positive exponent in my life. Be my Valentine.* If they plotted you and I on a scatter-plot, they would find a positive correlation. Be my Valentine.* You're a positive exponent in my life. Be my Valentine.* You're the only variable for me. Be my Valentine.* Like functions, you're the only value for me. Be my Valentine.* Looking for an affractionate girl. Be my Valentine.* Met you at the decimall. Be my Valentine.* Talking about you, I told my best friend I would never lever! .Be my Valentine* I don't care if your breasts pendulum, be my Valentine < not for young students ;-) >* Do you also feel the attraction? Be my Valentine* Do you see the gravity of this situation? You have to be my Valentine* When I first saw you I felt the Big Bang ! Be my Valentine* Ion the other hand, would love for you to be my Valentine* The only predicate: be my Valentine* At absolute zero you would still move me. Be my Valentine* Be my Valentine, even if it's only Faraday* My theorem is : you'd be great as my Valentine* I'm attracted, don't repel me. Be my Valentine* We'd make a nice tuple on Valentine, be mine* I want our relationship to be Ex-Static, be my Valentine* Don't be square, be my Valentine* Love hertz, be my Valentine* Wave if you accept to be my Valentine* The frequency of our dating would amplify if you are my Valentine* Oh, you could try the nerdy approach... My love for you is incalculable.* Or you could try Tweety Bird talk... I'm equate-y for you!* Or you could get a bit racy... Algebras in the world can't measure up to your curvilinear structure. Beta Valentine of my life.* I've finally worked up the courage to ask ... be a joule and Be My Valentine.* Would you be inclined to Be My Valentine?* It would matter to me if you'd agree to Be My Valentine.* End the chaos in my life. Be My Valentine.* I feel an impulse to ask you to Be My Valentine.* I've finally overcome the inertia of shyness to ask you to Be My Valentine.* I love your body with naked singularity. Be My Valentine.* I hope that my boldness in asking you to Be My Valentine won't cause friction in our relationship!* I think of you with more and more frequency. Be My Valentine.* I can't resist asking you to Be My Valentine.* You generate excitement in my life. Be My Valentine.* I hope that from the smile on my face, you can extrapolate that I want you to Be My Valentine.* If you'll agree to Be My Valentine from across the room, please signify by giving me a standing wave.* If I ask you to Be My Valentine on February 15th, will you overlook that relative deviation?* I think our relationship has potential. Be My Valentine.* My heart is sad. You can rectify that by agreeing to Be My Valentine.* My affection for you will never decay. Be My Valentine.* Are you going to Be My Valentine, or watt?* Every second I think of you riding that Schwinn, and it hertz. Be My Valentine.* Be my valen[cy]-tine.* Be my valentine and I square I'll be yours.* You're the root of my affection. I really mean it!* In case you didn't hear, I'll theta gain... Be mine!* Give me a sine... Will you be my Valentine?* My heart and my foot-pounds when you are around.* We could be dynamic together!* I'll give you a moment to decide if you'll be mine.* I need to ask yaw... will you be mine?* If you're looking for synthesis your opportunity, Be my Valentine.* I can't wait to explore your fuzzy boundaries. Be my Valentine.* It was a magnetic moment when we met. Be my valentine.* You're the Great Attractor. Be my valentine.Roses reflect a light frequency at one end of the visibleelectromagnetic spectrum,Violets reflect a light frequency at the other end of the visibleelectromagnetic spectrum,Sugar is C12H22O11,And you release the endorphins in my brain.* I would like to make you an eigenfunction of my Hamiltonian.* May I Lorentz boost into your centre of mass?* I expect your wave function to be degenerate under this operator.I am a positron spiralling in your electric field.Let my electron tunnel trough your barrier acquering negative energy defyingspace-time quantization.There is a force (F = r + 1/(r^4), r=distance) between us that gets largerwith distance.,,,sigma(me)/me = sigma(you)/you -- Be my Valentine!The sum of my divisors equals the sum of yours -- Be my Valentine!You = K_f(Me) -- Be my Valentine! (K_f is the Love transform)We differ by multiplication of a unit -- Be my Valentine!My module is faithful -- Be my Valentine!We are connected by a natural homomorphism -- Be my Valentine!You are a purely inseparable extension of me -- Be my Valentine!My love for you is an invariant under the transformations of others -- Bemy Valentine!You are a primitive element of my life -- Be my Valentine!You are the sum of your divisors -- Be my Valentine!Every polynomial is separable over you -- Be my Valentine!All our loops are contractible to a point -- Be my Valentine!My atoms are attracted to your electrons. -- This post is free post; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. Joachim Verhagen (jcdverha@xs4all.nl)WWW http://www.xs4all.nl/~jcdverha/ (Science Jokes) === Subject: Re: about vector bundleprooftern ha scritto nel messaggio> i ask you kindley to read this paper:> I am persuaded it is correct.> I ask you to correct it. Thank you in every circumstance.> Best regards> Tern> Tp M denotes the tangent space to M at p ;> xi denotes a vector bundle over M ;> I will denote the differential of f at p with d(f)p . More> precisely, d(f)p : Tp M ---> Tf(p) xi> Now, suppose that f : M ---> xi belongs to Z^0_p (xi) = the set ofsection> of xi such that f(p) = 0. Then it follows that f vanishes at p and hence> (using Taylor series in a local coordinate system for M at p and a local> trivialization of xi near p) we can write f near p as a finite sum> f = g_1 f_1 + ... + g_n f_n where the f_i are sections of xi and theg_i> are smooth complex-valued functions(defined on M) THAT VANISH AT p.> Then it follows that d(f)p = d( g_1 )p tensor f_1 (p) +g_1(p) tensor d> (f_1)p + ... + d( g_n )p tensor f_n (p) + g_n(p) tensor d(f_n)p ,> where tensor denotes the tensorial product ,> g_i :M ---> C ,> d( g_i )p is the differential of g_i at p ,> d(g_i)p : Tp M ---> Tg_i (p) C ; C denotes the complex field, Tg_i (p) C> is canonically isomorphic to C, so d(g_i)p is an element of T*M_p ;> f_i : M ---> xi ,> d( f_i )p is the differential of f_i at p> d(f_i)p : Tp M ---> Tf_i (p) xi> Observation: g_i(p) tensor d(f_i)p is trivially g_i(p) * d(f_i)p> where * denotes the product of the scalar g_i(p) by the vectord(f_i)p> .> Since g_i vanish at p , we have d(f)p = d( g_1 )p tensor f_1 (p) + ...+> d( g_n )p tensor f_n (p)> .> Conclusions:> d(f)p : Tp M ---> T0 xi where T0 xi denotes the tangent space to the> vector bundle xi at 0 .> d(f)p : Tp M --->Span( f_1 (p) , ..., f_n (p) ) ;> Span( f_1 (p) , ..., f_n (p) ) is contained in T0 xi_p === Subject: Re: straightlines curve at infinity; Riemann H. connects with Poincare Conjecture Re: when NaturalNumbers = p-adics what alters in the Riemann HypothesisIf we accept as true that the NaturalNumbers are the P-adics, then the1/2 Realline in the RiemannHypothesis must be a curved line and thatno straightlines ever exist but curve as the further we go out.In 1993 or 1994 I claimed this number of p-adics in the 10-adics of....999999 as the largest number that exists. I claimed it wasinfinity itself. I am proud of that claim for it has not diminishedin stature in these intervening years.And today I can put further use to that number .....99999 for in thePoincare Conjecture of a point compactification at infinity where youwant to take the infinite Euclidean Plane and sort of take its 4edge-points and like a sheet of wrapping paper want to join those fouredge points and make a sphere.Well, I am proud to say that Euclidean geometry at infinity is afiction a illusion and purely imaginery just as ghosts and witches areimaginary. That Euclidean Geometry is curved lines at infinity becausethe NaturalNumbers are the P-adics and that no-one needs to pointcompact the Euclidean Plane because it is already forming into asphere and that this number .....999999 is the point that is the4-pointedge of the infinite Euclidean plane.I suppose if you take just the 10-adics then it is a infinite circle.But if you take collectively all the p-adics of 2-adics, 3-adics etcetc they form an infinite sphere.I am guessing that the Collective P-adics is similar (I do not know ifthey are equal) to the geometry formed by the positive-Reals which isRiemannian Geometry.I do not know the relationship between the geometry formed by theCollective P-adics and the geometry formed by the positive Reals asRiemannian geometry. Both have positive curvature. But the positiveReals seem to have numbers such as pi and e which the CollectiveP-adics do not have and vice versa.Archimedes Plutoniumwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are galaxies === Subject: Differential operetor and variable changingI have an important question for my studies. I have a function, for example x=Rsin{theta}cos{phi}, so the differential dx became dx=sin{theta}cos{phi}dR+Rcos{theta}cos{phi}dtheta+(-Rsin{theta }sin{phi}dphi).In other words the differential of an n variables function is the sum of differentials depending each only by a variable. How became the operator frac{partial}{partial x} knowing that x is function of other variables like in the precedent example? Thank you for your attemption and sorry for my not perfect english.-- <<<>Ah, contact me to fedelemail@yahoo.it === Your inanity is truly unparalleled. I humble myself before your greatness.> Are you crazy?I bet, you do not have enough power of imagination to understand magnitude> of my craziness.Physics, this field is of crazy people. If you are sane, better to do 9 to 5> job and never look at this NG.-Abhi.