Oberwolfach Preprints (OWP)

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Nonlinear matroid optimization and experimental design
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Berstein, Yael; Lee, Jon; Maruri-Aguilar, Hugo; Onn, Shmueel; Riccomagno, Eva; Weismantel, Robert; Wynn, Henry
We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.
Secondary heat flow between confocal ellipses : an application of extended thermodynamics
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Barbera, Elvira; Müller, Ingo
Much as non-Newtonian fluids exhibit secondary flows in elliptic pipes, rarefied gases exhibit secondary heat flows between elliptical cylinders. The phenomenon may be called non-Fourierian, because it is not covered by Fourier’s law of heat conduction. The effect is demonstrated in the paper by exploiting the 13-moment theory of gases. Apart from secondary heat flows the theory predicts shear stresses balanced by gradients of the heat flux, and the need for a distinction of the kinetic and the thermodynamic temperature.
A 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Parker, Christopher; Rowley, Peter
In this article we give 3-local characterizations of the alternating and symmetric groups of degree 8 and use these characterizations to recognize the sporadic simple group discovered by McLaughlin from its 3-local subgroups.
Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Trofimchuk, Elena; Tkachenko, Victor; Trofimchuk, Sergei
We study positive bounded wave solutions u(t,x)=ϕ(ν⋅x+ct), ϕ(−∞)=0, of equation ut(t,x)=δu(t,x)−u(t,x)+g(u(t−h,x)), x∈Rm(*). It is supposed that Eq. (∗) has exactly two non-negative equilibria: u1≡0 and u2≡κ>9. The birth function g∈C(R+,R+) satisfies a few mild conditions: it is unimodal and differentiable at 0,κ. Some results also require the positive feedback of g:[g(maxg),maxg]→R+ with respect to κ. If additionally ϕ(+∞)=κ, the above wave solution u(t,x) is called a travelling front. We prove that every wave ϕ(ν⋅x+ct) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate c∗∈R+∪+∞ such that (∗) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity c>c∗. Our results are based on a detailed geometric description of the wave profile ϕ. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass G of ’asymmetric’ tent maps such that given g∈G, there exists exactly one travelling front for each fixed admissible speed.
Vector bundles on degenerations of elliptic curves and Yang-Baxter equations
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Burban, Igor; Kreussler, Bernd
In 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.
On test sets for nonlinear integer maximization
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Lee, Jon; Onn, Shmuel; Weismantel, Robert
A 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.
Hölder-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.
Resolution of symplectic cyclic orbifold singularities
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Niederkrüger, Klaus; Pasquotto, Federica
In this paper we present a method to obtain resolutions of symplectic orbifolds arising as quotients of pre-symplectic semi-free S1–actions. This includes, in particular, all orbifolds arising from symplectic reduction of Hamiltonian S1–manifolds at regular values, as well as all isolated cyclic orbifold singularities. As an application, we show that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold.
The McKay conjecture for exceptional groups and odd primes
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Späth, Britta
Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G ! G and G := GF , such that the root system is of exceptional type or G is a Suzuki-group or Steinberg’s triality group. We show that all irreducible characters of CG(S), the centraliser of S in G, extend to their inertia group in NG(S), where S is any F-stable Sylow torus of (G, F). Together with the work in [17] this implies that the McKay-conjecture is true for G and odd primes ` different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [15].
Drawing large pictures at random : Oberwolfach lecture 2007
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Werner, Wendelin
This lecture is of very introductory nature. The goal will be to describe specific concrete questions, and to use them as a tool to convey some general ideas. Let me therefore skip the general introduction and immediately start with a first simple question: Is there a way to choose at random and uniformly among all possible choices a continuous curve (a d-dimensional curve, say)?
Geometric flows and 3-manifolds
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Huisken, Gerhard
The 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.
Zeta functions of 3-dimensional p-Adic lie algebras
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Klopsch, Benjamin; Voll, Christopher
We give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers Zp. To this end, we associate to such a Lie algebra a ternary quadratic form over Zp. The formula for the zeta function is given in terms of Igusa's local zeta function associated to this form.
Self-dual polygons and self-dual curves
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Fuchs, Dmitry; Tabachnikov, Serge
[no abstract available]
Stratifying modular representations of finite groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Benson, Dave; Iyengar, Srikanth B.; Krause, Henning
We 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.
Upper tails for intersection local times of random walks in supercritical dimensions
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Chen, Xia; Mörters, Peter
We 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.
Epsilon-hypercyclic operators
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Badea, Catalin; Grivaux, Sophie; Müller, Vladimir
For each fixed number " in (0, 1) we construct a bounded linear operator on the Banach space `1 having a certain orbit which intersects every cone of aperture ", but with every orbit avoiding a certain ball of radius d, for every d > 0. This answers a question from [8]. On the other hand, if T is an operator on the Banach space X such that for every " > 0 there is a point in X whose orbit under the action of T meets every cone of aperture ", then T has a dense orbit.
Preconditioning of block tridiagonal matrices
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Axelsson, Owe; Karátson, János
Preconditioning methods via approximate block factorization for block tridiagonal matrices are studied. Bounds for the resulting condition numbers are given, and two methods for the recursive construction of the approximate Schur complements are presented. Illustrations for elliptic problems are also given, including a study of sensitivity to jumps in the coefficients and of a suitably motidied Poincaré-Steklov operator on the continuous level.
Nonlinear optimization over a weighted independence system
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Lee, Jon; Onn, Shmuel; Weismantel, Robert
We consider the problem of optimizing a nonlinear objective function over a weighted independence system presented by a linear-optimization oracle. We provide a polynomial-time algorithm that determines an r-best solution for nonlinear functions of the total weight of an independent set, where r is a constant that depends on certain Frobenius numbers of the individual weights and is independent of the size of the ground set. In contrast, we show that finding an optimal (0-best) solution requires exponential time even in a very special case of the problem.
Non-standard behavior of density estimators for sums of squared observations
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Schick, Anton; Wefelmeyer, Wolfgang
It 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.
A characterization of semisimple plane polynomial automorphisms
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Maubach, Stephan; Furter, Jean-Philippe
It 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.

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