Klein瓶上的双奇异地图

一个地图的每条边如果不是环就是割边(即该边的两边是同一个面的边界)，则称之为双奇异地图，本文研究Klein瓶上带根双奇异地图的计数问题，得到了此类地图以边数、平面环数、手柄上本质环数和又帽上本质环数为参数的计数公式，并得到了部分计数显式。     

Bisingular maps on the torus

A map is bisingular if each edge is either a loop or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps on the sphere and the torus, and we also present formula for such maps with four parameters: the root-valency,the number of isthmus, the number of planar loops and the number of essential loops.     

On the multiplication groups of semifields

We investigate the multiplicative loops of finite semifields. We show that the group generated by the left and right multiplication maps contains the special linear group. This result solves a BCC18 problem of A. Drápal. Moreover, we study the question of whether the big Mathieu groups can occur as multiplication groups of loops.     

Chaos in a bienzymatic cyclic model with two autocatalytic loops

A general bienzymatic cyclic system including two autocatalytic loops is studied and used as a basic design principle for modelling extracellular matrix turnover. Using classical enzyme kinetic rates, the model is described by a set of four ordinary differential equations and numerically studied by bifurcation diagrams and Poincaré sections. We observe limit-cycle oscillations and chaotic behaviors arising from period-doubling cascades or intermittency. Chaotic oscillations originate from distinct strange attractors that undergo boundary and internal crisis. For some parameter values, the system presents several bistable areas, where a limit cycle coexists with another one or with a strange attractor. The dynamics are qualitatively modified when the weight of the autocatalytic loops on the system varies, resulting in the change in the number of attractors.     

Characterization of Trimmed NURBS Surface Boundaries Using Topological Criteria

This paper presents an approach based on topology for the determination of characteristics and properties of curves used in the trimming of NURBS surfaces. Through discrete subdivision and topological criteria, a method is presented to determine characteristics of the boundary; such as whether the set of trimming curves forms a set of closed loops, whether trimming curves contain singularities or self intersections, and whether the boundary is simply connected. A surface mesh partitionning the parameter space is used, formed of isoparametric lines in both parametric directions. Topological properties of the cells of this mesh and their intersections with the trimming curves allow to localize the boundary. Topological treatment of this localization allows to define the interior and exterior of the face, and to refine the boundary localization. Singularities and self intersections of the boundary as well as voids in the face are investigated through the study of topological properties of neighbors. As an application, an algorithm for point localization is presented that very rapidly allows to determine whether a given point in parameter space lies inside, on the boundary or outside of the trimmed surface.     

BISINGULAR MAPS ON SOME SURFACES

A map is bisingular if each edge is either a loop (This paper only considersplanar loop) or an isthmus (i.e., on the boundary of the same face). This paper studies thenumber of rooted bisingular maps on the sphere and the torus, and also presents formulaefor such maps with three parameters: the root-valency, the number of isthmus, and thenumber of planar loops.     

Loops in Reeb Graphs of 2-Manifolds

Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.     

Local spectral gaps on loop spaces

We prove Poincaré inequalities w.r.t. the distributions of Brownian bridges on sets of loops with jumps of limited size over compact Riemannian manifolds. Moreover, we study the asymptotic behavior of the second Dirichlet eigenvalues as the time parameter T of the underlying Brownian bridge tends to 0. This behavior depends crucially on the geodesics contained in the set of loops considered. In particular, for different choices of a Riemannian metric on the base manifold, qualitatively different asymptotic behaviors can occur. The proof of the basic Poincaré inequality is based on the construction of the Brownian bridge by consecutive bisection of the parametrization interval.     

二维不可压缩磁流体方程边界层分离的条件

该文研究了二维不可压缩磁流体方程的解,其中要求磁流体的速度满足Dirichlet边界条件、磁场在边界上的值与时间无关. 利用Taylor展开式和不可压缩流的结构分歧理论, 得到了磁流体方程发生边界层分离的条件, 它取决于外力、初值和磁场在边界上的取值, 并且该条件可以预测磁流体边界层分离发生的时间与地点.     

An augmented Galerkin procedure for the boundary integral method applied to mixed boundary value problems

Here we present the asymptotic error analysis for the boundary element approximation of the direct boundary integral equations for the plane mixed boundary value problem of the Laplacian. The boundary elements are defined by B-splines for the smooth parts of the boundary charges and additional singular functions at the collision points. The asymptotic error estimates include estimates for the stress intensity factors which occur as additional unknowns to be computed within the Galerkin scheme. The numerical analysis is based on the uniqueness of the problem, a coerciveness inequality, the triangular principal part and an extended shift theorem of the boundary integral operators.     

Mathematical simulation of wave fields in media with arbitrary curvilinear boundaries

The method of calculation of wave fields in media with curvilinear interfaces is proposed. After integral transformations, the solution in the explicit form is obtained. For solving the problem, we introduce the notion of the generalized thickness of a layer. This value becomes real in the case of a plane boundary and coincides with an “ordinary” thickness of a layer. If the boundary is curvilinear, the generalized thickness of a layer becomes a complex value. The use of the generalized thickness of a layer brings about the fact that conditions on infinity (Sommerfeld’s conditions) are automatically fulfilled. The conditions of appearance of loops on the travel-time curves for different boundaries have been elucidated. Some examples of calculations are given.     

一类带记忆项的非经典热方程的爆破问题

本文考虑了一类带记忆项的非经典热方程,证明解会在有限时间爆破,而且爆破只会发生在边界.主要结论是:首先利用Green函数与Banach压缩映射定理,建立了问题的经典解;其次,利用经典解,证明了解是有限时间爆破的;最后,证明了一个关于非经典热方程解的性质,利用这个性质,证明了解是在边界上爆破的.     

Blow-up in parabolic problems under Robin boundary conditions

By means of a first-order differential inequality technique, sufficient conditions are determined which imply that blow-up of the solution does occur or does not occur for the semilinear heat equation under Robin boundary conditions. In addition, a lower bound on blow-up time is obtained when blow-up does occur.     

Complete second order differential equations in Banach spaces with dynamic boundary conditions

In this paper, we deal with the mixed initial boundary value problem for complete second order (in time) linear differential equations in Banach spaces, in which time-derivatives occur in the boundary conditions. General wellposedness theorems are obtained (for the first time), which are used to solve the corresponding inhomogeneous problems. Examples of applications to initial boundary value problems for partial differential equations are also presented.     

The Treatment of Boundary Singularities in Axially Symmetric Problems Containing Discs

Axially symmetric problems (e.g. Laplace's equation in cylindricalcoordinates) containing discs possess boundary singularitiesarising from the mixed boundary conditions that occur acrossthe disc edge. A modified finite-difference method is presentedwhich effectively eliminates the inaccuracies that occur inthe standard numerical solution near such singularities. Techniquesfor developing the analytical forms of such singularities aregiven and modified finite-difference approximations are obtained.The steady-state diffusion of oxygen around a circular electrodeis taken as the model problem and a modified quadrature methodis presented for the calculation of the oxygen flux throughthe electrode.     

Tug-of-War and Infinity Laplace Equation with Vanishing Neumann Boundary Condition

We study a version of the stochastic “tug-of-war” game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Laplace equation with vanishing Neumann boundary condition.     

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system

A diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered. Hopf and steady state bifurcation analysis are carried out in details. In particular we show the existence of multiple spatially non-homogeneous periodic orbits while the system parameters are all spatially homogeneous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation.     

Uniqueness for weak solutions of a one-dimensional boundary value problem for the Vlasov-Poisson system

The Vlasov-Poisson system models a collisionless plasma. When a boundary condition is included in the problem it is known that singularities can occur but that weak solutions exist globally in time. This article shows that the weak solution is unique for a problem in one dimension with specular reflection at the boundary.     

Symmetry-breaking at positive solutions

We consider symmetry-breaking bifurcations at positive solutions of semilinear elliptic boundary value problems in a ball, with Dirichlet boundary conditions. We prove that symmetry-breaking can only occur by bifurcation of axisymmetric solutions. We also obtain sufficient conditions for such symmetry-breaking.Dedicated to Professor H. Knobloch at the occasion of his 60th birthday