mm-1042 Subject: Re: Hilbertıs 16th Originator: rusin@vesuvius [George Szpiro] | | can someone tell me the status of the proof of the 2nd part of | Hilbertıs 16th problem? According to news reports a 22-year old | student from Sweden (Elin Oxenhielm) has managed to prove | it. However, Grigori Rozenblum (Chalmers Univ. of Techn, Goeteborg, | Sweden) has apparently found an error. So has her thesis advisor, Yishao Zhou: http://www.math.su.se/~yishao/16thproblem.shtml SA -- Stein Arild Strżmme +47 55584825, +47 95801887 Universitetet i Bergen Fax: +47 55589672 Matematisk institutt www.mi.uib.no/stromme/ Johs Brunsg 12, N-5008 BERGEN stromme@mi.uib.no === Subject: Testing Primality (Slowly!) Content-Length: 201 Originator: rusin@vesuvius Suppose p is a positive integer. Let n[0]=1 and compute n[i]=p-1-((p*i-1) mod n[i-1]). Are there any composite p with n[i]=1 for some i>0? Are there any primes p>7 with n[i]<>1 for all i>0? rich === Subject: orthogonal polynomials and creation/annihilation operators Originator: baez@math-ws-n09.math.ucr.edu (John Baez) Content-Length: 2540 Originator: rusin@vesuvius Iım interested in generalizations of the usual annihilation and creation operators a, a* such that [a,a*] = 1 to Hamiltonians other than the harmonic oscillator Hamiltonian. Lots of different measures w(x)^2 dx on the real line, or half-line, or interval give rise to famous orthogonal polynomials f_0, f_1, f_2, .... like Hermite polynomials, Laguerre polynomials and so on. You get these polynomials just by applying Gram-Schmidt orthonormalization to the functions 1, x, x^2,... Moreover, the functions wf_0, wf_1, wf_2 tend to be eigenfunctions of nice 2nd-order differential operators. For example, if we work on the real line and take w(x) = exp(-x^2/2) the polynomials f_0, f_1, f_2, etc. are called Hermite polynomials, and the Hermite functions wf_0, wf_1, wf_2, etc. are eigenfunctions of the harmonic oscillator Hamiltonian. But if we work on [0,inŜnity) and take w(x) = exp(-x) we get polynomials called Laguerre polynomials, and the functions wf_0, wf_1, wf_2 are eigenfunctions of some *other* 2nd-order differential operator, which Iıll call the Laguerre operator. Now, we can write the harmonic oscillator Hamiltonian as H = aa* where the creation operator a* maps each Hermite function to a multiple of the next one, and the annihilation operator a maps each one to a multiple of the previous one. Even better, the annihilation and creation operators satisfy a cool commutation relation: [a,a*] = 1 In some sense this formula is all you need to know to recover the whole theory of the harmonic oscillator. My question is whether something like this works for other cases... for example the Laguerre case! Can we write the Laguerre operator in terms of some operators vaguely analogous to creation and annihilation operators, that map each function wf_i to some simple linear combination of other ones? And do these operators satisfy some interesting commutation relation? The answer to my question is probably buried in some textbook on orthogonal polynomials. I can almost see it lurking in chapter 22 of Abramowitz and Stegunıs Handbook of Mathematical Functions. But Iım hoping someone can help me out. In particular, Iıd love to just *see* the relevant commutation relations, if they exist. Iım only interested in the Laguerre case because Iım hoping thatıs the simplest example other than the Hermite one. If some other example is nicer, thatıs Ŝne too. === Subject: Re: orthogonal polynomials and creation/annihilation operators Content-Length: 356 Originator: rusin@vesuvius A good resource for such generalizations is The Factorization Method, Rev. Mod. Phys. 23, 21-68 (1951). Best, Bruce > Iım interested in generalizations of the usual annihilation > and creation operators > a, a* such that [a,a*] = 1 > to Hamiltonians other than the harmonic oscillator Hamiltonian. === Subject: Re: orthogonal polynomials and creation/annihilation operators Epigone-thread: strimcongfeld Content-Length: 641 Originator: rusin@vesuvius There has been research in the direction take some commutation relations and Ŝnd the remaining stuff (representation of creation and anihilation operators, orthogonal polynomials, etc.) I am not an expert in this Ŝeld, but I remember that one of the main contributors to this research is Marek Bo.zejko (thatıs z with a dot). The commutation relations that have been investigated are usually something like a_i a_j^* - q_{ij} a_j^* a_i = delta_{ij} I with some matrix (q_ij) of scalars. Another name I remember in connection with this topic is Ilona Krıolak (again o with accent) - a student of Bozejko. Piotr Soltan === Subject: Re: orthogonal polynomials and creation/annihilation operators Content-Length: 3305 Originator: rusin@vesuvius Hi John, Yes I think there you can always deŜne ladder operators as mapping one eigenfunction to the next or previous one (with an appropriate multiplicative constant). The commutation relation would involve the difference between two successive eigenvalues, which turns out to be 1 for the harmonic oscillator. The following paper provides the solution for other potentials, the extension to Laguerre should be easy : Temporally stable coherent states for inŜnite well and Poschl-Teller potentials, by Antoine, Gazeau, Monceau, Klauder and Penson, in JMP June 2001. cheers - L > Iım interested in generalizations of the usual annihilation > and creation operators > a, a* such that [a,a*] = 1 > to Hamiltonians other than the harmonic oscillator Hamiltonian. > Lots of different measures > w(x)^2 dx > on the real line, or half-line, or interval give rise to > famous orthogonal polynomials > f_0, f_1, f_2, .... > like Hermite polynomials, Laguerre polynomials and so on. > You get these polynomials just by applying Gram-Schmidt > orthonormalization to the functions > 1, x, x^2,... > Moreover, the functions > wf_0, wf_1, wf_2 > tend to be eigenfunctions of nice 2nd-order differential > operators. > For example, if we work on the real line and take > w(x) = exp(-x^2/2) > the polynomials f_0, f_1, f_2, etc. are called Hermite polynomials, > and the Hermite functions wf_0, wf_1, wf_2, etc. are eigenfunctions > of the harmonic oscillator Hamiltonian. > But if we work on [0,inŜnity) and take > w(x) = exp(-x) > we get polynomials called Laguerre polynomials, and the functions > wf_0, wf_1, wf_2 are eigenfunctions of some *other* 2nd-order > differential operator, which Iıll call the Laguerre operator. > Now, we can write the harmonic oscillator Hamiltonian as > H = aa* > where the creation operator a* maps each Hermite function > to a multiple of the next one, and the annihilation operator a > maps each one to a multiple of the previous one. Even better, > the annihilation and creation operators satisfy a cool commutation > relation: > [a,a*] = 1 > In some sense this formula is all you need to know to recover > the whole theory of the harmonic oscillator. > My question is whether something like this works for other > cases... for example the Laguerre case! Can we write the > Laguerre operator in terms of some operators vaguely analogous > to creation and annihilation operators, that map each function > wf_i to some simple linear combination of other ones? And do > these operators satisfy some interesting commutation relation? > The answer to my question is probably buried in some textbook > on orthogonal polynomials. I can almost see it lurking in > chapter 22 of Abramowitz and Stegunıs Handbook of Mathematical > Functions. But Iım hoping someone can help me out. In particular, > Iıd love to just *see* the relevant commutation relations, if they > exist. > Iım only interested in the Laguerre case because Iım hoping > thatıs the simplest example other than the Hermite one. If > some other example is nicer, thatıs Ŝne too. === Subject: Conference Announcement Content-Length: 1916 Originator: rusin@vesuvius I would like to draw your attention of the following conference, Crystallography at the start of the 21st century: Mathematical and Symmetry Aspects as a satellite of the XXII European Crystallographic Meeting (ECM-22) This satellite conference will consist of six thematic sessions (half a day per session). Registered participants are encouraged to submit poster presentations, which will be on display during the whole satellite. Full papers are also welcome, and will be printed, after peer-review, in Zeitschrift fur Krystallography, together with the Proceedings, which will consist of the texts of the six sessions. Details, including the program of the six sessions of the satellite, can be found at the following address: http://www.lcm3b.uhp-nancy.fr/mathcryst/satellite.htm We have done all the efforts to keep the costs at the minimal possible level, and thus the registration fees as low as possible. Even if you are not directly interested, I take the liberty of asking you to forward this message to those colleagues of yours who may be interested. Also, if you run a website with schedule of conferences and events, please add a link to this satellite. or better to the address in the signature below. ///////////////////////////////////////////////////////////// Dr. Massimo Nespolo Associate Professor of Mineralogy and Crystallography Laboratoire de Cristallographie et de Modelisation des Materiaux Mineraux et Biologiques (LCM3B) UMR - CNRS 7036 Universiteı Henri Poincareı Nancy1 BP 239 Boulevard des Aiguillettes F54506 Vandoeuvre-les-Nancy cedex France mailto:mathcryst.satellite@lcm3b.uhp-nancy.fr http://www.lcm3b.uhp-nancy.fr/mathcryst/satellite.htm ///////////////////////////////////////////////////////////// === Subject: Relations in groups/algebras Content-Length: 367 Originator: rusin@vesuvius DeŜnition: a k-relation of a group G is deŜned as an element of the intersection of the kernels of all homomorphisms from the free group on k generators to G. Question: If a group satisŜes a non-trivial 3-relation, then does it always satisfy a non-trivial 2-relation? I can ask the analogous question in the context of Lie algebras, or of associative algebras. === Subject: Re: Relations in groups/algebras Content-Length: 644 Originator: rusin@vesuvius > DeŜnition: a k-relation of a group G is deŜned as an element of the > intersection of the kernels of all homomorphisms from the free group > on k generators to G. > Question: > If a group satisŜes a non-trivial 3-relation, then does it always > satisfy a non-trivial 2-relation? I think so. A free group on two generators contains a free subgroup on 3 generators. Hence if you take a non-trivial 3-relation and substitute into it three elements of a free subgroup on 2 generators that generate a free subgroup. This substitution is therefore also non-trivial and it is a 2-relation for the group in question. === Subject: Commutators Content-Length: 789 Originator: rusin@vesuvius Hi! Suppose you have a tripartite hilbert space H_A x H_B x H_C where x is the tensor product. dim(H_A)= dim(H_B)=dim(H_C). Consider all the operations acting on H_A x H_B that commutes with all the elements of a non abelian group G acting on H_A x H_B . Letıs the set of those operations them O_AB. Consider all the operations acting on H_A x H_C that commutes with all the elements of a non abelian group G acting on H_A x H_B. Letıs the set of those operations them O_AC. We consider the trivial extesion of the operators O_AB and O_AC in H_A x H_B x H_C just tensoring with the identity acting rispectively on H_C and H_B. Is it true that all the operators commuting with the group G acting on H_A x H_B x H_C can be contructed using the extensions of O_AB and O_AC ? Fabio. === Subject: Uniform distribution Epigone-thread: proyyanwen Content-Length: 375 Originator: rusin@vesuvius To autumn: Why donĞt you check existing textbooks on uniform distribution? There is one by E. Hlawka, originally in German but having been translated into English; there is another one by Kuipers and Niederreiter. By the way, Hlawka invented uniform distribution on compact sets, and that topic is treated in his book. Hoping this will help, Peter Flor (Graz, Austria). === Subject: Re: Ŝnite group Content-Length: 985 Originator: rusin@vesuvius This is not true. DeŜne the following twist: if t >0, x in R^2 g(t,x)=(t, (e^2pi/t)x). Consider the set A={(t,x)| x=0 or t natural}. Then the group {g^n} on A has the desired property, but is not Ŝnite. Maybe this is true for manifolds. Simeon > Let G a sub-group of the group of homeomorphisms of a metric space E, > Hom.8eo(E), such that each orbit of G is Ŝnite (the cardinality of > orbits is not uniformly limited i.e for all n in IN there exists xin > E such that card(G(x)) > n (G(x) is the orbit of x by G). > A point x in E is regular if there exists an open neighboorhod U_x and > an integer k_x such that for all yin U_x we have card(G(y)) < k_x. > if E is connected and G is Ŝnitely generated and if each point x in > E is regular is it true that G is Ŝnite? > In particular we have if E is compact (or each orbit meets a compact K > subset E) then G is Ŝnite. > cordially. === Subject: Principal bundles / K-theory / Gerbe references? Content-Length: 1273 Originator: rusin@vesuvius Hello all, Iım looking for some texts which discuss principal bundles with a view toward (topological) K-theory. Iıve found plenty of (and have some) differential geometry books with nice discussions of principal bundles in that context, and some algebraic topology books talk about them too, with an eye towards stable homotopy theory (e.g. Switzer, which does go through some K-theory too), but are there any places to look on the relations between principal bundles and K-theory? Iıve had some introduction to K-theory already (mainly from Atiyahıs book), and Iıve seen principal bundles pretty extensively in differential geometry/gauge theory, and while there are (some) other K-theory books Iım looking at, Iım not sure where to look if I want to see some of the interrelations between the subjects. Any suggestions? Also, btw, anyone have any good references for studying gerbes, from a topological viewpoint? I know thereıs the stack picture, and the higher category picture, but Iım not an algebraic geometer nor nice introduction to gerbes and their topology and geometry from a Cech or differential geometry standpoint would be appreciated as well. -Jake Mannix === Subject: Re: Principal bundles / K-theory / Gerbe references? Content-Length: 412 Originator: rusin@vesuvius Hi Jake, you might be interested in bundle gerbes, they play the same roles as gerbes but involve bundles rather than sheaves etc. They were introduced in the following paper: M.K. Murray Bundle Gerbes dg-ga/9407015 further bundle gerbe theory: math.DG/9908135 math.DG/0106018 math.DG/0004117 applications to physics: math.DG/0106179 hep-th/0204199 K-Theory: hep-th/0106194 I hope this is helpful, Stuart === Subject: Re: Principal bundles / K-theory / Gerbe references? Content-Length: 1485 Originator: rusin@vesuvius > Hello all, > Iım looking for some texts which discuss principal bundles with a > view toward (topological) K-theory. Iıve found plenty of (and have > some) differential geometry books with nice discussions of principal > bundles in that context, and some algebraic topology books talk about > them too, with an eye towards stable homotopy theory (e.g. Switzer, > which does go through some K-theory too), but are there any places to > look on the relations between principal bundles and K-theory? > Iıve had some introduction to K-theory already (mainly from > Atiyahıs book), and Iıve seen principal bundles pretty extensively in > differential geometry/gauge theory, and while there are (some) other > K-theory books Iım looking at, Iım not sure where to look if I want to > see some of the interrelations between the subjects. > Any suggestions? Husemoller? > Also, btw, anyone have any good references for studying gerbes, > from a topological viewpoint? I know thereıs the stack picture, and > the higher category picture, but Iım not an algebraic geometer nor > nice introduction to gerbes and their topology and geometry from a > Cech or differential geometry standpoint would be appreciated as well. Hitchin, Lectures on Special Lagrangian Submanifolds math.DG/9907034 Aaron === Subject: Re: characterization of cumulants Content-Length: 1099 Originator: rusin@vesuvius > The cumulants k_n of a probability distribution are given by > E(exp(tX)) = exp( SUM_{n=1}^inŜnity k_n * t^n / n! ) > where X is a random variable having the distribution in > question. BTW, I just learned a combinatorial interpretation of the coefŜcients in the polynomials that express the moments as functions of the cumulants; perhaps this will edify the audience of this forum. (I am reliably told this is universally known, in a context that makes it appear that that means that maybe only combinatorialists know it.) The nth moment m_n is a polynomial in the Ŝrst n cumulants. Each monomial is a product such as, for example, k_3 k_2^2 k_1^2. Think of this as corresponding to the partition 9 = 3 + 2 + 2 + 1 + 1, and generalize the obvious pattern. How many partitions of a set of 9 things break it into a set of three, two sets of two, and two sets of one? The answer is the coefŜcient of that monomial in the expansion of the 9th moment (and again, generalize the obvious pattern). -- Mike Hardy === Subject: Re: characterization of cumulants Content-Length: 1535 Originator: rusin@vesuvius > The cumulants k_n of a probability distribution are given by > E(exp(tX)) = exp( SUM_{n=1}^inŜnity k_n * t^n / n! ) > where X is a random variable having the distribution in > question. > BTW, I just learned a combinatorial interpretation of the > coefŜcients in the polynomials that express the moments as > functions of the cumulants; perhaps this will edify the audience > of this forum. (I am reliably told this is universally known, > in a context that makes it appear that that means that maybe only > combinatorialists know it.) > The nth moment m_n is a polynomial in the Ŝrst n cumulants. > Each monomial is a product such as, for example, > k_3 k_2^2 k_1^2. > Think of this as corresponding to the partition > 9 = 3 + 2 + 2 + 1 + 1, > and generalize the obvious pattern. How many partitions of a > set of 9 things break it into a set of three, two sets of two, > and two sets of one? The answer is the coefŜcient of that > monomial in the expansion of the 9th moment (and again, generalize > the obvious pattern). -- Mike Hardy This and related results are well-known in statistical mechanics and quantum Ŝeld theory, where it is the basis of diagram calculus. A general mathematical treatment in terms of the Œpolymer expansionı is given in J Glimm and A Jaffe Quantum Physics: A Functional Integral Point of View Springer Verlag; 2nd edition, 1996 Arnold Neumaier === Subject: Re: characterization of cumulants Content-Length: 850 Originator: rusin@vesuvius > k_1 (X + c) = k_1 (X) + c > k_n (X + c) = k_n (X) for n > 1. > > Are there theorems saying the cumulants are the only > well-behaved quantities with these invariance and equivariance > properties, for suitable interpretations of well-behaved? > Of course there are. > k_1(X) = > k_n(X) = (X - )^n + P_n(X - ), n = 2, 3, 4, ... > where P_n(z) is an arbitrary degree n-1 polynomial for n = 2, 3, 4, ... Sorry about the delay in answering this. I assume that, like physicists, by you mean the expected value of X. But your expression (X - )^n + P_n(X - ) appears to be a random variable, whereas k_n(X) should be a constant -- the nth cumulant of X. Can you clarify what you mean? -- Mike Hardy === Subject: inscriptable quadrilateral Epigone-thread: climpcunblay Content-Length: 109 Originator: rusin@vesuvius I posted a simple proof at CTK Exchange. See http://www.cut-the-knot.org/htdocs/dcforum/DCForumID6/406. shtml [Mod. note: this is a follow-up to the query posted several weeks ago with === Subject: tangential quadrilateral --djr] === Subject: Braid groups: order of an element Content-Length: 414 Originator: rusin@vesuvius Does anyone know, if there exists an algebraic proof for the fact that every element of a braid group is of inŜnite order? Fox and Neuwirth proved this fact in 1962 (Fox, Neuwirth: The braid groups, Math. Scand. 10 (1962)) and claimed that the result was not known previously. However, they used a topological argument, and I am interested in hearing of an algebraic proof for the result. -- === Subject: Question about Linear Operators on the outer product Content-Length: 447 Originator: rusin@vesuvius Given a linear operator A on a Hilbert space H it is often natural to extend the deŜnition of A to the outer product H x H : A (a,b) = (Aa,b) + (a,Ab) [*] But suppose we have some other way to deŜne A as a linear operator on the outer product, so that condition [*] is NOT satisŜed. What language should be used to describe this? What is the correct name for condition [*]? -- Charles Francis === Subject: Re: Question about Linear Operators on the outer product Epigone-thread: tryrneuprerm If by outer product you mean the Cartesian product or, in the terminology of Hilbert spaces, the dierct sum then the way to look at this is to consider 2x2 martices. More precisely any operator on H(+)H can be written as a 2x2 matrix of operators from H to H. The way to act with a given A:H-->H on H(+)H you proposed is to act with |A 0 | |I 0| |A+I 0 | |0 I | + |0 A| = | 0 A+I| (these are upposed to be matrices) where I is the identity map on H. Now a very common way to amplify an operator onto H(+)H is to use a diagonal matrix with both entries equal to A. Then in your notation NewA(a,b)=(Aa,Ab) I guess this is more natural as this has much better algebraic properties. If you write H(+)H as the tensor product of H with C^2 then the diagonal martix is just A(x)I_2 where I_2 is the identity on the two dimensional space and (x) is the tensor product. The advantage is, for example, that (assume that A is normal) for functions on the spectrum of A you have f(A(x)I_2)=f(a)(x)I_2. Piotr === Subject: Special forms of Hillıs Differential Equation Consider the special type of Hillıs equations rıı[t] + a(1 + q cos[2t])^K r[t] = 0, K = 1, 2, 3, ... It appears that the eigenvalue structure for odd K is sharply different than that for even K. In particular, except for the trivial case, q = 0, when K is even the odd and even periodic solutions for a given order never coexist, i.e. have the same values of a and q. Conversely, in the case of odd K the eigenfunctions for orders greater than one have a Ŝnite number of nontrivial coexistence points. The plot of the eigenvalue curves in (a,q) space appears braided. Are there any good references for this property. Particularly, a qualitative explanation of whatıs going on. Iıve got Magnus and Winklerıs book but their results on coexistence are limited to Inceıs (K = 2) and Lameıs differential equations. Further, their results seem, to me at least, more mechanical than explanatory. === Subject: Regulators of elliptic curves Content-Length: 489 Originator: rusin@vesuvius Hi - (1) What does the regulator of an elliptic curve tell us about the number of rational points of low or moderate height on it? Say the elliptic curve in question has no rational points of (canonical) height below X. What does the regulator imply about the number of rational points of height between X and X*(1+epsilon)? (2) What do we know about the regulator of an elliptic curve? Is there anything besides the Swinnerton-Dyer conjecture? How can one get at the regulator? Harald === Subject: Hill-type equation I would like some info on the equation yıı(t) = w(t) * y(t) (1) where w is a smooth C^infty function and t is in R. In particular I would like to know under what conditions (on w) the solution of (1) posses at most one zero. Any pointers in the literature are welcome. I assume only C^infty not analyticity, therefore C^infty-ŝatness is allowed. If w>=0 for all t, and t0=0 implies that y must have only one root. I want to know if the same holds when w take negative values. Obviously w canıt be always negative, since, if for instance, w = negative constant y would oscillate around t-axis. Thus, qualitatively speaking, if w assumes negative values for some subset I of R, the measure m(I) of I must be comparatively small (if not zero). How small must m(I) must be? Since, at least for me, this problem seems difŜcult, it is convenient to have a physical picture for it. So, (1) can be viewed as a Scrondinger equation with w being U(x)-E and inside a 1-dimensional potential U(x). Observe that now t The previous paragraph describes the common sense result that if U>E and the wave function has two nodes, then the wave function is zero between them. Thus, physically speaking, Iım looking for 1-dimensional potentials which are realistic in the sense to exist), and the wave function must have at most one node, i.e., at most one point of zero appearance probability. Of course, unlike quantum mechanics, in this context y may or may not be square integrable. In fact y may de(in)crease indeŜnitely. I donıt know if, piecewise constant approximation of w and solution of (1) in each interval of constancy of w will be of any use. If possible, passing to the limit may miss some solutions with two or more zeros. === Subject: Re: Finding cycles/circuits in directed graph -- Johnson Paper, Mateti Paper Epigone-thread: prinvangquald I have been trying to locate a source/copy of the papers you mention that far. Do you know how I can gain access to them?