New proof of the cobordism invariance of the index

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows us to treat directly general elliptic operators which are not of Dirac type.

     

A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

This paper concerns the behaviour of the apsidal angle for orbits of central force system with homogeneous potential of degree −2?α?1$-2?\alpha ?1$ and logarithmic potential. We derive a formula for the apsidal angle as a fixed end-points integral and we study the derivative of the apsidal angle with respect to the angular momentum ?. The monotonicity of the apsidal angle as function of ? is discussed and it is proved in the logarithmic potential case.     

On the problem of dynamic contact angle

Plane motion of a viscous incompressible fluid bounded by a rectangular rigid wall and a free boundary of constant form is investigated. The free boundary is in contact with the rigid wall at a point which moves along the wall, coming into contact with it at a constant rate. The asymptotics of the velocity field near the point of contact is computed under the assumption that the motion is stationary in the coordinate system attached to the moving free boundary and, that the energy is dissipated as a finite rate.     

The contact angle in inviscid fluid mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived; however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many ‘solutions’ in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense ‘weak-type solutions’; they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.     

A phase field model in electrowetting and related phenomena

The term electrowetting is commonly used for some techniques to change the shape and wetting behaviour of liquid droplets by the application of electric fields and charges. We developand analyze a model for electrowetting that combines the Navier-Stokes system for fluid flow, a phase-field model of Cahn-Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the contact angle boundary condition for Monge-Ampère equations

We give a simple proof of a result of Xinan Ma concerning a necessary condition for the solvability of the two-dimensional Monge-Ampère equation subject to the contact angle or capillarity boundary condition. Our technique works for more general Monge-Ampère equations in any dimension, and also extends to some other boundary conditions.

     

A general estimate in the invariance principle

We obtain estimates for the accuracy with which a random broken line constructed from sums of independent nonidentically distributed random variables can be approximated by a Wiener process. All estimates depend explicitly on the moments of the random variables; meanwhile, these moments can be of a rather general form. In the case of identically distributed random variables we succeed for the first time in constructing an estimate depending explicitly on the common distribution of the summands and directly implying all results of the famous articles by Komlós, Major, and Tusnády which are devoted to estimates in the invariance principle.     

Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems

We provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that - under minimal assumptions - a closed set is invariant with respect to a stochastic control system if and only if it is invariant with respect to the (associated) deterministic control system.     

Minimal surfaces in S 3 with constant contact angle

We provide a characterization of the Clifford Torus in S ³ via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S ³ with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.      

Hydrodynamic assist and the dynamic contact angle in the coalescence of liquid drops

^** Email: decentsp{at}for.mat.bham.ac.uk The coalescence of two viscous liquid drops in an inviscid gasor in a vacuum is studied using the interface formation model.In the very early stages of coalescence during the formationof the ‘liquid-bridge’ connecting the two drops,this model predicts a moving contact line and a dynamic contactangle. This paper examines the dynamic evolution of this contactangle, and for small Reynolds number and small Capillary number,relevant particularly in micro-fluidics, a non-linear differentialequation is derived for the contact angle and solved computationally.It is found that the contact angle evolution can only be evaluatedby determining information about the flow away from the contactline. This is a manifestation of so-called hydrodynamic assist,studied experimentally in the context of curtain coating byBlake et al. (1999 Experimental evidence of non-local hydrodynamicinfluence on the dynamic contact angle. Phys. Fluids, 11, 1995–2007).For small Capillary number and small Reynolds number, the free-surfaceevolution is determined for the coalescence of two cylindersof equal radius. Finally, some comments are made on experimentsin coalescence, as well as on issues arising in a computationalsolution of the full model described here.     

Minimal annuli with constant contact angle along the planar boundaries

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.     

A useful estimate in the multidimensional invariance principle

Summary An estimate of the convergence speed in the multidimensional invariance principle is obtained. Using this estimate, we can prove strong invariance principles for partial sums of independent not necessarily identically distributed multidimensional random vectors.     

A proof of the Dvoretzky-Rogers theorem

We give a new proof of the famous Dvoretzky-Rogers theorem ([2], Theorem 1), according to which a Banach spaceE is finite-dimensional if every unconditionally convergent series inE is absolutely convergent.     

A proof of the Hoggatt-Bergum conjecture

It is proved that if and are positive integers such that the product of any two distinct elements of the set

increased by is a perfect square, then has to be . This is a generalization of the theorem of Baker and Davenport for .