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We offer a reader-friendly introduction to the attracting edge problem (also known as the "triangle conjecture") and its most general current solution of Limic and Tarr\`es (2007). Little original research is reported; rather this article…
In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on M\"obius…
The horizontal dynamics of a bouncing ball interacting with an irregular surface is investigated and is found to demonstrate behavior analogous to a random walk. Its stochastic character is substantiated by the calculation of a permutation…
The edge-reinforced random walk (ERRW) is a random process on the vertices of a graph that is more likely to cross the edges it has visited in the past. Depending on the strength of the reinforcement, the ERRW of a single particle can…
The existence of unique scaling in a crossover regime between viscous and inertial hydrodynamic regimes is revealed for homogeneous, isotropic, incompressible, spinodal turbulence which is characterized, to begin with, by three different…
The adhesive contact between elastic solids with randomly rough, self affine fractal surfaces is studied by molecular dynamics (MD) simulations. The interfacial binding energy obtained from the simulations of nominally flat and curved…
We consider the coupled motion of a free rigid body immersed in an inviscid compressible isentropic fluid. By means of a vanishing viscosity limit, we obtain the local-in-time existence of a dissipative measure-valued solution to the model.…
Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two…
In this study we give definitions and characterizations of transversal surfaces of timelike ruled surfaces. We study some special cases such as the striction curve is a geodesic, an asymptotic line or a line of curvature. Moreover, we…
On R^n endowed with a riemannian metric of bounded nonpositive curvature, the weakly convex closed subsets are topologically trivial. The stability of such subsets under intersection characterizes the euclidean spaces.
We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with…
The effect of self-affine roughness on solid contact is examined with molecular dynamics and continuum calculations. The contact area and normal and lateral stiffnesses rise linearly with the applied load, and the load rises exponentially…
We report results of investigations of a high-speed drainage of thin aqueous films squeezed between randomly nanorough surfaces. A significant decrease in hydrodynamic resistance force as compared with predicted by Taylor's equation is…
We investigate the properties of spontaneous currents generated at surfaces and interfaces of d-wave superconductors using the self-consistent quasiclassical Eilenberger equations. The influence of the roughness and reflectivity of the…
Matter density is formally infinite at the location of caustic surfaces, where dark matter sheet folds in phase-space. The caustics separate regions with different number of streams and the volume elements change the parity by turning…
In this paper, we prove the following "Weak Bounded Negativity Conjecture", which says that given a complex smooth projective surface $X$, for any reduced curve $C$ in $X$ and integer $g$, assume that the geometric genus of each component…
Many turbulent flows encountered in nature -- seas, oceans and rivers -- are bounded by a deformable free surface. A question that remained to be fully explored is to what extent the underlying turbulent flow field can be revealed solely by…
In this talk we use nonlinear realizations to study the spontaneous partial breaking of rigid and local supersymmetry.
We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present…
In this paper, we introduce a definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, that is to deform the metric discrete conformally…