Primitive Illumination Systems for Families of Convex Bodies in the Plane

Let F= {C₁,C₂,...,C} be a family of ndisjoint convex bodies in the plane. We say that a set Vof exterior light sources illuminates F, if for every boundary point of any member of Fthere is a point in Vsuch that is visible from ,i.e. the open line segment joining and is disjoint from F. An illumination system Vis called primitive if no proper subset of Villuminates F. Let p_max(F) denote the maximum number of points forming a primitive illumination system for F, and letp_max(n) denote the minimum of F) taken over all families Fconsisting of ndisjoint convex bodies in the plane. The aim of this paper is to investigate the quantities p_max(F) and p_max(n).     

Approximate expressions that describe the profiles of sessile and pendent drops for the cases of small Bond numbers and strong wetting

Variational formulations of the problems of sessile and pendent drops are given taking into account the force of gravity in the axially symmetric case. Approximate expressions that describe the surface profiles of these drops by the asymptotic method for small Bond numbers have been obtained by the linearization method in the case of strong wetting.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations

We characterize the occurrence of complete or incomplete blow-up (and extinction) for a general quasilinear heat equation of the form in terms of the constitutive functions and f We assume that (u)>0 for u>0 and that f(u)0. For the positive sign + before f(u) in (HE), with f(u) superlinear as u, blow-up occurs in finite time: sup _x u(x, t) as tT<. For the negative sign, we consider the case of singular absorption: f(u) as u0. Then initially positive solutions vanish at some point in finite time (extinction), and a singularity in the equation occurs there. An important aspect of blow-up or extinction problems is the possibility of having a nontrivial extension of the solution for t>T, i.e., after the singularity occurs. If such continuation exists, we say that the blow-up (extinction) is incomplete; otherwise it is called complete. Our characterization is based on the qualitative behaviour of the family of travelling-wave solutions and a proper use of the Intersection-Comparison argument. The analysis applies to other nonlinear models, like the equations with gradient-dependent diffusivity.     

An adaptive front tracking technique for three‐dimensional transient flows

An adaptive technique, based on both surface stretching and surface curvature analysis for tracking strongly deforming fluid volumes in three‐dimensional flows is presented. The efficiency and accuracy of the technique are demonstrated for two‐ and three‐dimensional flow simulations. For the two‐dimensional test example, the results are compared with results obtained using a different tracking approach based on the advection of a passive scalar. Although for both techniques roughly the same structures are found, the resolution for the front tracking technique is much higher. In the three‐dimensional test example, a spherical blob is tracked in a chaotic mixing flow. For this problem, the accuracy of the adaptive tracking is demonstrated by the volume conservation for the advected blob. Adaptive front tracking is suitable for simulation of the initial stages of fluid mixing, where the interfacial area can grow exponentially with time. The efficiency of the algorithm significantly benefits from parallelization of the code. Copyright © 2000 John Wiley & Sons, Ltd.     

Effects of interactions and disorder on the compressibility of two-dimensional electron and hole systems

The compressibility χ of dilute two-dimensional electron and hole gases in GaAs semiconductor structures has been studied in the ranges of the interaction parameter r_s=1–2.5 and r_s=10–30 for the electron and hole system, respectively. Nonmonotonic dependence of χ^-1 with an upturn at low carrier densities is observed. Despite the large difference in r_s the behavior of χ^-1 in both systems can be accurately described by the theory of nonlinear screening of disorder by the carriers.     

Electron interaction and ‘memory’ effects in the presence of mixed disorder

We have studied the effect of electron–electron interaction in the presence of mixed disorder on the conductivity and Hall effect of a high-mobility two-dimensional electron gas in a GaAs/AlGaAs heterostructure. A parabolic, negative, temperature-dependent magnetoresistance (MR) and temperature-dependent Hall effect are observed. We show that these effects can be explained in terms of the interaction theory. In addition, a temperature independent, positive MR is observed. This classical MR is also shown to be a consequence of the mixed disorder.