The catwalk task: Reflections and synthesis: Part 1

In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional.     

复Finsler流形上的Koppelman-Leray-Norguet公式

利用不变积分核(Berndtsson核),复Finsler度量和联系于Chern-Finsler联络的非线性联络,研究复Finsler流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的积分表示,得到了(p,q)型微分形式的Koppelman-Leray-Norguet公式和■-方程的解．作为应用,利用复Finsler度量和联系于Chern-Finsler联络的非线性联络,给出了Stein流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的Koppelman- Leray-Norguet公式以及■-方程的解,并且得到了Stein流形上实非退化强拟凸多面体上(p,q)型微分形式的积分表示式和■-方程的解．     

Geometry of Optimal Paths around Focal Singular Surfaces in Differential Games

We investigate a special type of singularity in non-smooth solutions of first-order partial differential equations, with emphasis on Isaacs’ equation. This type, called focal manifold, is characterized by the incoming trajectory fields on the two sides and a discontinuous gradient. We provide a complete set of constructive equations under various hypotheses on the singularity, culminating with the case where no a priori hypothesis on its geometry is known, and where the extremal trajectory fields need not be collinear. We show two examples of differential games exhibiting non-collinear fields of extremal trajectories on the focal manifold, one with a transversal approach and one with a tangential approach.     

The catwalk task: Reflections and synthesis: Part 2

In this article we recount our experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, we individually reflect on our own mathematical work on the catwalk task. Second, we reflect on our experiences working with a group of community college students on the catwalk task and our interpretations of their mathematical thinking. In so doing we also detail a number of innovative and novel student-generated representations of the catwalk photos. Finally, we each individually reflect on the entire experience with the catwalk problem, as mathematics learners, as teachers, and as professionals.     

On the mechanism of pressure increase with increasing porosity of the media compressed in conical and cylindrical targets

The effect of porosity on the solution of the problem on a shock wave entering a porous medium is studied. The dependence of the wave parameters appearing under shock compression of porous graphite, aluminum, and polytetrafluoroethylene in a steel target with a cavity in the shape of a truncated cone on the relative initial density is investigated. For a steel target with a cylindrical cavity filled with graphite, the effect of increasing the peak pressure on the symmetry axis in the case when the solid graphite is replaced with porous one is investigated.     

Survival of contact processes on the hierarchical group

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.     

The motion of a ball on a grassy lawn

The motion of a ball of constant spherical shape on a grassy lawn represented by a set of deformable rods with one end rigidly attached to a plane, while the other end is free and can contact the ball, is investigated. The contact area on the ball surface is represented by a spherical sector, whose dimensions and location on the ball surface depend on the position of the centre of the ball and its orientation. The projection of the contact area onto the plane is a circle, whose centre and radius vary as the centre of the ball moves. When the rods come into contact with the ball, they experience longitudinal and flexural strains, as well as impact actions on the boundary of the contact area. The translational–rotational motion of a ball on a lawn of rods without friction, as well as motion with viscous friction at the points of contact of the ends of the rods with the ball, is considered.     

Some congruences on trioids

We present the least idempotent congruence on the trioid with a commutative operation, the least semilattice congruence on the trioid with an idempotent operation, and the least separative congruence on the trioid with a commutative operation. Also we construct different examples of trioids.     

可定向曲面上两个地图的交叉数

In this paper, we discuss the crossing numbers of two one-vertex maps on orientable surfaces. By using a reductive method, we give the crossing number of two one-vertex maps with one face on an orientable surface and the crossing number of a one-vertex map with one face and a one-vertex map with two faces on an orientable surface. This provides a lower bound for the crossing number of two general maps on an orientable surface.     

Comparative study on control concepts of a robot manipulator with multiple-link/joint flexibilities

If one is dealing with active vibration suppression on a highly nonlinear flexible system, various techniques are needed. On the one hand a suitable dynamic model of the system is required. And on the other hand intelligent model based control concepts are necessary for active vibration damping. We deal with a basic model, where the flexibilities are approximated with linear springs and dampers, a so called lumped element model (LEM). For the control design we propose a control structure with two degrees of freedom (2DoF) for solving the tracking problem, based on the LEM. Such an approach allows designing the feedforward part independently of the feedback part. Hereby the feedforward control is based on the flatness approach, while for the feedback control several strategies are studied using acceleration- and gyrosensor-measurements. The contribution is completed with a validation by measurements from a very fast trajectory on an articulated robot with two flexible links and three elastic joints. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Noncommutative Hamiltonian dynamics on foliated manifolds

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block, Getzler and Xu, and we introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure.     

The Burgers equation with periodic boundary conditions on an interval

We study the asymptotic profile of the solutions of the Burgers equation on a finite interval with a periodic perturbation on the boundary. The equation describes a dissipative medium, and the initial constant profile therefore passes into a wave with a decreasing amplitude. In the low-viscosity case, the asymptotic profile looks like a sawtooth wave (with periodic breaks of the derivative), similar to the known Fay solution on the half-line, but it has some new properties.     

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.     

Stability on Time-Dependent Domains

We explore the key differences in the stability picture between extended systems on time-fixed and time-dependent spatial domains. As a paradigm, we take the complex Swift–Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on time-dependent spatial domains in translationally invariant systems, i.e., when dilution effects are absent. In particular, we discuss the effects of a time-dependent domain on the stability of spatially homogeneous and spatially periodic base states, and explore its effects on the Eckhaus instability of periodic states. New equations describing the nonlinear evolution of the pattern wavenumber on time-dependent domains are derived, and the results compared with those on fixed domains. Pattern coarsening on time-dependent domains is contrasted with that on fixed domains with the help of the Cahn–Hilliard equation extended here to time-dependent domains. Parallel results for the evolution of the Benjamin–Feir instability on time-dependent domains are also given.     

Lower bounding schemes for flowshops with blocking in-process

While there is a considerable amount of research that deals with a flowshop with no storage constraints, few works have addressed the flowshop with blocking in-process. Scheduling problems with blocking arise in serial manufacturing processes where no intermediate buffer storage is available. This paper proposes a lower bound, which exploits the occurrence of blocking, on the total tardiness of the jobs. Its efficiency is evaluated using a branch-and-bound algorithm on several problems. A lower bound on the makespan is also derived and computational tests are presented.     

Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gelfand, making essential use of the results of the first two parts.     

A problem with nonlocal conditions for mixed-type equations

We study a problem with the Bitsadze-Samarskii conditions on the ellipticity boundary and on a segment of degeneration line and with the displacement condition on parts of boundary characteristics of the Gellerstedt equation with singular coefficient. With the help of the maximum principle we prove the uniqueness of a solution to the problem, and with the help of the method of integral equations we prove the existence of a solution to the problem.     

On closed-form integration of some Fuchsian differential equations related to a conformal mapping of circular pentagons with a cut

To solve the problem on a conformal mapping of some circular pentagons with a cut, we suggest to use special methods designed for a class of polygons in polar grids (i.e., bounded by arcs of concentric circles and segments of lines passing through the origin) and based on finding particular solutions of Fuchs type equations in the form of linear combinations of known particular solutions of some simpler equations with three singular points with indeterminate coefficients. The obtained results are first used to solve problems on a conformal mapping of circular quadrangles with a cut which belong to the class of polygons in polar grids, and then, with regard of found solutions, to pentagons of a more complicated structure, which are not polar. In all cases, we present the complete solution of the problem on parameters.