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- ItemA new counting function for the zeros of holomorphic curves(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Anderson, J.M.; Hinkkanen, AimoLet f1, . . . , fp be entire functions that do not all vanish at any point, so that (f1, . . . , fp) is a holomorphic curve in CPp−1. We introduce a new and more careful notion of counting the order of the zero of a linear combination of the functions f1, . . . , fp at any point where such a linear combination vanishes, and, if all the f1, . . . , fp are polynomials, also at infinity. This enables us to formulate an inequality, which sometimes holds as an identity, that sharpens the classical results of Cartan and others.
- ItemThe contact polytope of the Leech lattice(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Dutour Sikiri´c, Mathieu; Schürmann, Achill; Vallentin, FrankThe contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1, 197, 362, 269, 604, 214, 277, 200 many facets in 232 orbits.
- ItemOn test sets for nonlinear integer maximization(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Lee, Jon; Onn, Shmuel; Weismantel, RobertA finite test set for an integer maximization problem enables us to verify whether a feasible point attains the global maximum. We estabish in the paper several general results that apply to integer maximization problems wthe monlinear objective functions.
- ItemClassification of idempotent states on the compact quantum groups Uq(2), SUq(2), and SOq(3)(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Franz, Uwe; Skalski, Adam; Tomatsu, ReijiWe give a simple characterisation of those idempotent states on compact quantum groups which arise as Haar states on quantum subgroups, show that all idempotent states on quantum groups Uq(2), SUq(2), and SOq(3) (q 2 (−1, 0) [ (0, 1]) arise in this manner and list the idempotent states on compact quantum semigroups U0(2), SU0(2), and SO0(3). In the Appendix we provide a simple proof of coamenability of the deformations of classical compact Lie groups.
- ItemNon-standard behavior of density estimators for sums of squared observations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Schick, Anton; Wefelmeyer, WolfgangIt has been shown recently that, under an appropriate integrability condition, densities of functions of independent and identically distributed random variables can be estimated at the parametric rate by a local U-statistic, and a functional central limit theorem holds. For the sum of two squared random variables, the integrability condition typically fails. We show that then the estimator behaves differently for different arguments. At points in the support of the squared random variable, the rate of the estimator slows down by a logarithmic factor and is independent of the bandwidth, but the asymptotic variance depends on the rate of the bandwidth, and otherwise only on the density of the squared random variable at this point and at zero. A functional central limit theorem cannot hold. Of course, for bounded random variables, the sum of squares is more spread out than a single square. At points outside the support of the squared random variable, the estimator behaves classically. Now the rate is again parametric, the asymptotic variance has a different form and does not depend on the bandwidth, and a functional central limit theorem holds.
- ItemOn the [delta] δ=const collisions of singularities of complex plane curves(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Kerner, DmitryWe study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total ± invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A new invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in collisions. We consider in details the ± = const deformations of ordinary multiple point, the deformation of a singularity into the collection of ordinary multiple points and the deformation of the type xp + ypk into a collection of Ak's.
- ItemStein’s method for dependent random variables occuring in statistical mechanics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Eichelsbacher, Peter; Löwe, MatthiasWe obtain rates of convergence in limit theorems of partial sums Sn for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the Curie-Weiss models. Under appropriate assumptions there exists a real number α, a positive number μ, and a positive integer k such that (Sn−nα)/n1−1/2k converges weakly to a random variable with density proportional to exp(−μ|x|2k/(2k)!). We develop Stein's method for exchangeable pairs for a rich class of distributional approximations including the Gaussian distributions as well as the non-Gaussian limit distributions with density proportional to exp(−μ|x|2k/(2k)!). Our results include the optimal Berry-Esseen rate in the Central Limit Theorem for the total magnetization in the classical Curie-Weiss model, for high temperatures as well as at the critical temperature βc=1, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature 1/βn converges to one and obtain a threshold for the speed of this convergence. Single spin distributions satisfying the Griffiths-Hurst-Sherman (GHS)inequality like models of liquid helium or continuous Curie-Weiss models are considered.
- ItemMinimal Riesz energy on the sphere for axis-supported external fields(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Brauchart, J.S.; Dragnev, P.D.; Saff, E.B.We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|−s with d−2 s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s = d − 2 yields interesting phenomena, which we investigate in detail. A weak∗ asymptotic analysis is provided as s ! (d − 2)+.
- ItemWeighted Fourier inequalities for radial functions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Gorbachev, D.; Liflyand, E.; Tikhonov, S.Weighted Lp(Rn) ! Lq(Rn) Fourier inequalities are studied. We prove Pitt-Boas type results on integrability with power weights of the Fourier transform of a radial function.
- ItemA study on gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Kawohl, Bernd; Kutev, NikolaiWe investigate sharp conditions for boundary and interior gradient es- timates of continuous viscosity solutions to fully nonlinear, uniformly ellip- tic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain.