2007
http://publications.mfo.de/handle/mfo/1339
Wed, 11 Sep 2024 20:08:32 GMT2024-09-11T20:08:32ZThe McKay-conjecture for exceptional groups and odd primes
http://publications.mfo.de/handle/mfo/1218
The McKay-conjecture for exceptional groups and odd primes
Späth, Britta
Let $\mathbf{G}$ be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map $F:\mathbf{G}→\mathbf{G}$ and $\mathbf{G}:=\mathbf{G}^F$, such that the root system is of exceptional type or $\mathbf{G}$ is a Suzuki-group or Steinberg’s triality group. We show that all irreducible characters of $C_G(\mathbf{S})$, the centraliser of $\mathbf{S}$ in $G$, extend to their inertia group in $N_G(\mathbf{S})$, where $\mathbf{S}$ is any $F$-stable Sylow torus of $(\mathbf{G},F)$. Together with the work in [17] this implies that the McKay-conjecture is true for $G$ and odd primes $\ell$ 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].
OWLF 2007
Mon, 01 Jan 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12182007-01-01T00:00:00ZSpäth, BrittaLet $\mathbf{G}$ be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map $F:\mathbf{G}→\mathbf{G}$ and $\mathbf{G}:=\mathbf{G}^F$, such that the root system is of exceptional type or $\mathbf{G}$ is a Suzuki-group or Steinberg’s triality group. We show that all irreducible characters of $C_G(\mathbf{S})$, the centraliser of $\mathbf{S}$ in $G$, extend to their inertia group in $N_G(\mathbf{S})$, where $\mathbf{S}$ is any $F$-stable Sylow torus of $(\mathbf{G},F)$. Together with the work in [17] this implies that the McKay-conjecture is true for $G$ and odd primes $\ell$ 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
http://publications.mfo.de/handle/mfo/482
Drawing large pictures at random : Oberwolfach Lecture 2007
Wendelin, Werner
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)?
Mon, 01 Jan 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/4822007-01-01T00:00:00ZWendelin, WernerThis 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 Lecture 2005
http://publications.mfo.de/handle/mfo/481
Geometric flows and 3-manifolds : Oberwolfach Lecture 2005
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 non-linear 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.
Mon, 01 Jan 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/4812007-01-01T00:00:00ZHuisken, 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 non-linear 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.Hölder-Differentiability of Gibbs Distribution Functions
http://publications.mfo.de/handle/mfo/1048
Hölder-Differentiability of Gibbs Distribution Functions
Keßeböhmer, Marc; Stratmann, Bernd
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 $\mathbb{R}$. 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 $\alpha$-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.
Research in Pairs 2007
Thu, 29 Mar 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10482007-03-29T00:00:00ZKeßeböhmer, MarcStratmann, BerndIn 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 $\mathbb{R}$. 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 $\alpha$-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.Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay
http://publications.mfo.de/handle/mfo/1207
Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay
Trofimchuk, Elena; Tkachenko, Victor; Trofimchuk, Sergei I.
We study positive bounded wave solutions $u(t, x) = \phi(\nu \cdot x+ct)$, $\phi(-\infty)=0$, of equation $u_t(t, x)=\delta u(t,x)−u(t,x) +g(u(t−h,x))$, $x \in \mathbb{R}^m$(*). It is supposed that Eq. (∗) has exactly two non-negative equilibria: $u_1 \equiv 0$ and $u_2 \equiv \kappa>9$. The birth function $g \in C(\mathbb{R}_+, \mathbb{R}_+)$ satisfies a few mild conditions: it is unimodal and differentiable at $0,\kappa$. Some results also require the positive feedback of $g:[g(\text{max} g),\text{max} g] \to \mathbb{R}_+$ with respect to $\kappa$. If additionally $\phi(+\infty) =\kappa$, the above wave solution $u(t,x)$ is called a travelling front. We prove that every wave $\phi(\nu \cdot x+ct)$ is eventually monotone or slowly oscillating about $\kappa$. Furthermore, we indicate $c^∗ \in \mathbb{R}_+ \cup {+\infty}$ 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 $\phi$. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass $\mathcal{G}$ of ’asymmetric’ tent maps such that given $g \in \mathcal{G}$, there exists exactly one travelling front for each fixed admissible speed.
Research in Pairs 2007
Wed, 28 Mar 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12072007-03-28T00:00:00ZTrofimchuk, ElenaTkachenko, VictorTrofimchuk, Sergei I.We study positive bounded wave solutions $u(t, x) = \phi(\nu \cdot x+ct)$, $\phi(-\infty)=0$, of equation $u_t(t, x)=\delta u(t,x)−u(t,x) +g(u(t−h,x))$, $x \in \mathbb{R}^m$(*). It is supposed that Eq. (∗) has exactly two non-negative equilibria: $u_1 \equiv 0$ and $u_2 \equiv \kappa>9$. The birth function $g \in C(\mathbb{R}_+, \mathbb{R}_+)$ satisfies a few mild conditions: it is unimodal and differentiable at $0,\kappa$. Some results also require the positive feedback of $g:[g(\text{max} g),\text{max} g] \to \mathbb{R}_+$ with respect to $\kappa$. If additionally $\phi(+\infty) =\kappa$, the above wave solution $u(t,x)$ is called a travelling front. We prove that every wave $\phi(\nu \cdot x+ct)$ is eventually monotone or slowly oscillating about $\kappa$. Furthermore, we indicate $c^∗ \in \mathbb{R}_+ \cup {+\infty}$ 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 $\phi$. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass $\mathcal{G}$ of ’asymmetric’ tent maps such that given $g \in \mathcal{G}$, there exists exactly one travelling front for each fixed admissible speed.Secondary heat flow between confocal ellipses: an application of extended thermodynamics
http://publications.mfo.de/handle/mfo/1196
Secondary heat flow between confocal ellipses: an application of extended thermodynamics
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.
Research in Pairs 2007
Tue, 27 Mar 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11962007-03-27T00:00:00ZBarbera, ElviraMüller, IngoMuch 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.Zeta functions of 3-dimensional p-adic Lie algebras
http://publications.mfo.de/handle/mfo/1185
Zeta functions of 3-dimensional p-adic Lie algebras
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 $\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\mathbb{Z}_p$. The formula for the zeta function is given in terms of Igusa's local zeta function associated to this form.
Research in Pairs 2007
Mon, 26 Mar 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11852007-03-26T00:00:00ZKlopsch, BenjaminVoll, ChristopherWe give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the $p$-adic integers $\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\mathbb{Z}_p$. 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
http://publications.mfo.de/handle/mfo/1174
Self-dual polygons and self-dual curves
Fuks, Dmitrij B.; Tabachnikov, Serge
Research in Pairs 2007
Sun, 25 Mar 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11742007-03-25T00:00:00ZFuks, Dmitrij B.Tabachnikov, SergeNonlinear matroid optimization and experimental design
http://publications.mfo.de/handle/mfo/1163
Nonlinear matroid optimization and experimental design
Lee, Jon; Onn, Shmuel; Weismantel, Robert; Berstein, Yael; Maruri-Aguilar, Hugo; Riccomagno, Eva; Wynn, Henry P.
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.
Research in Pairs 2007
Sat, 24 Mar 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11632007-03-24T00:00:00ZLee, JonOnn, ShmuelWeismantel, RobertBerstein, YaelMaruri-Aguilar, HugoRiccomagno, EvaWynn, Henry P.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.On test sets for nonlinear integer maximization
http://publications.mfo.de/handle/mfo/1152
On test sets for nonlinear integer maximization
Lee, Jon; Onn, Shmuel; Weismantel, Robert
A finite test set for an integer maximation problem enables us to verify whether a feasible point atains the global maximum. We establish in this paper several general results that apply to integer maximization problems with nonlinear objective functions.
Research in Pairs 2007
Fri, 23 Mar 2007 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11522007-03-23T00:00:00ZLee, JonOnn, ShmuelWeismantel, RobertA finite test set for an integer maximation problem enables us to verify whether a feasible point atains the global maximum. We establish in this paper several general results that apply to integer maximization problems with nonlinear objective functions.