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- 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.
- 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.
- ItemA characterization of semisimple plane polynomial automorphisms(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Maubach, Stephan; Furter, Jean-PhilippeIt is well-known that an element of the linear group GLn(C) is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms.
- ItemVector bundles on degenerations of elliptic curves and Yang-Baxter equations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Burban, Igor; Kreussler, BerndIn this paper we introduce the notion of a gemetric associative r-matrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstraß cubic curves.
- ItemUpper tails for intersection local times of random walks in supercritical dimensions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Chen, Xia; Mörters, PeterWe determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of p independent random walks in Zd under the assumption p(d−2)>d. Our approach allows a direct treatment of the infinite time horizon.
- ItemStratifying modular representations of finite groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Benson, Dave; Iyengar, Srikanth B.; Krause, HenningWe classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.
- ItemHölder-differentiability of Gibbs distribution functions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Kesseböhmer, Marc; Stratmann, Bernd O.In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil’s staircases) supported on limit sets of finitely generated conformal iterated function systems in R. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not a-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism.
- ItemNoncommutative topological entropy of endomorphisms of Cuntz algebras(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Skalski, Adam; Zacharias, JoachimNoncommutative topological entropy estimates are obtained for ‘finite range’ endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values are computed for a class of polynomial endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura.
- ItemGeometric flows and 3-manifolds(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Huisken, GerhardThe current article arose from a lecture1 given by the author in October 2005 on the work of R. Hamilton and G. Perelman on Ricci-flow and explains central analytical ingredients in geometric parabolic evolution equations that allow the application of these flows to geometric problems including the Uniformisation Theorem and the proof of the Poincare conjecture. Parabolic geometric evolution equations of second order are nonlinear extensions of the ordinary heat equation to a geometric setting, so we begin by reminding the reader of the linear heat equation and its properties. We will then introduce key ideas in the simpler equations of curve shortening and 2-d Ricci-flow before discussing aspects of three-dimensional Ricci-flow.