A Hierarchical Cluster System Based on HortonStrahler Rules for River Networks

We consider a cluster system in which each cluster is characterized by two parameters: an "order" i , following HortonStrahler rules, and a "mass" j following the usual additive rule. Denoting by c _i,j ( t ) the concentration of clusters of order i and mass j at time t , we derive a coagulation-like ordinary differential system for the time dynamics of these clusters. Results about the existence and the behavior of solutions as   t   are obtained; in particular, we prove that   c _i,j ( t ) 0  and   N _i ( c ( t )) 0  as   t ,  where the functional   N _i (·)  measures the total amount of clusters of a given fixed order i . Exact and approximate equations for the time evolution of these functionals are derived. We also present numerical results that suggest the existence of self-similar solutions to these approximate equations and discuss their possible relevance for an interpretation of Horton's law of river numbers.     

The stress driven instability in elastic crystals: Mathematical models and physical manifestations

Summary Equilibrium equations and stability conditions for the simple deformable elastic body are derived by means of considering a minimum of the static energy principle. The energy is supposed to be sum of the volume (elastic) and the surface terms. The ability to change relative positions of different material particles is taken into account, and appropriate natural definitions of the first and second variations of the energy are introduced and calculated explicitly. Considering the case of negligible magnitude of the surface tension, we establish that an equilibrium state of a nonhydrostatically stressed simple elastic body (of any physically reasonable elastic energy potential and of any symmetry) possessing any small smooth part of free surface is always unstable with respect to relative transfer of the material particles along the surface. Surface tension suppresses the mentioned instability with respect to sufficiently short disturbances of the boundary surface and thus can probably provide local smoothness of the equilibrium shape of the crystal. We derive explicit formulas for critical wavelength for the simplest models of the internal and surface energies and for the simplest equilibrium configurations. We also formulate the simplest problem of mathematical physics, revealing peculiarities and difficulties of the problem of equilibrium shape of elastic crystals, and discuss possible manifestations of the above-mentioned instability in the problems of crystal growth, materials science, fracture, physical chemistry, and low-temperature physics.     

Laplace eigenvalues on regular polygons: A series in 1/N

For regular polygons $P_N$ inscribed in a circle, the eigenvalues of the Laplacian converge as $N\to \infty$ to the known eigenvalues on a circle. We compute the leading terms of ${\lambda }_N/\lambda$ in a series in powers of $1/N$, by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon. The $O(1/N^2)$ term comes from Hadamard?s formula, and reflects the change in area. This term disappears if we “transcribe” the polygon, scaling it to have the same area as the circle.     

Shape Optimization and Electron Bubbles

We present an analytical treatment of the shape optimization problem that arises in the study of electron bubbles. The problem is to minimize a weighted sum of a Laplace eigenvalue, volume, and surface area with respect to the shape of the domain. The analysis employs the calculus of moving surfaces and yields surprising conclusions regarding the stability of equilibrium spherical configurations. Namely, all but the lowest eigenvalue result in unstable configurations and certain combinations of parameters, near-spherical equilibrium stable configurations exist. Two-dimensional and three-dimensional problems are considered and numerical results are presented for the two-dimensional case.     

Small Oscillations of a Soap Bubble

We analyze the spectrum of small oscillations of a soap bubble surrounded by incompressible inviscid air. The analysis is based on the linearization of the exact system for the dynamics of free fluid films. The underlying model takes into account variations in the film's shape and thickness. The resulting dispersion relationship shows the fundamental interplay among these variations. In the limit of vanishing soap bubble mass, the presented dispersion relationship agrees with Rayleigh's classical formula for an oscillating liquid droplet under the influence of surface tension.     

Uniqueness in the Freedericksz transition with weak anchoring

In this paper we consider a boundary value problem for a quasi-linear pendulum equation with non-linear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition, which is the simplest opto-electronic switch, the result of competition between reorienting effects of an applied electric field and the anchoring to the bounding surfaces. A change of variables transforms the problem into the equation xττ=−f(x) for τ∈(−T,T), with boundary conditions at τ=?T, for a convex non-linearity f. By analysing an associated inviscid Burgers' equation, we prove uniqueness of monotone solutions in the original non-linear boundary value problem.This result has been for many years conjectured in the liquid crystals literature, e.g. in [E.G. Virga, Variational Theories for Liquid Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London, 1994] and in [I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London, 2003].     

Instability of the 2S electron bubbles

The 2S electron bubble placed in liquid helium has been previously believed to be spherical. We show that the 2S bubble is morphologically unstable at pressures above -1.23 bars. The 2S state being known to be radially unstable at pressures below -1.33 bars, the result leaves only a very narrow pressure range in which it can be found in a spherical configuration. Our stability analysis indicates that the 2S bubble is unstable against perturbations proportional to any of the third spherical harmonics Y(3m). Our numerical simulations show that there exist nonspherical stable configurations, such as the ones Maris and Konstantinov predicted for the 1P, 1D, and 2P electron bubbles and confirmed experimentally for the 1P. We believe that the 2S bubbles can also be produced and that our prediction will yield itself to experimental verification.     

Hadamard’s Formula Inside and Out

Our goal is to explore boundary variations of spectral problems from the calculus of moving surfaces point of view. Hadamard’s famous formula for simple Laplace eigenvalues under Dirichlet boundary conditions is generalized in a number of significant ways, including Neumann and mixed boundary conditions, multiple eigenvalues, and second order variations. Some of these formulas appear for the first time here. Furthermore, we present an analytical framework for deriving general formulas of the Hadamard type.