Diffuse-interface approximations of osmosis free boundary problems

Free boundary problems based on mass conservation and surface tension with application in osmotic swelling are the topic of this contribution. We introduce new phase-field approximations of such models, in order to numerically investigate properties of the solutions. Formal justification of the proposed approximations is provided by matched asymptotic expansions supported by numerical tests reproducing the convergence for shrinking interface thickness

On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma

We study connections between four different types of results that are concerned with vector-valued functions u : Ω→ℝ³ of class L²(Ω) on a domain Ω ⊂ ℝ³: Coercivity results in H^1(Ω) relying on div and curl, the Helmholtz decomposition, the construction of vector potentials, and the global div-curl lemma

Sound absorption by perforated walls along boundaries

We analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modelled by a periodic geometry with periodicity ε > 0. A resonator volume of thickness ε is connected with thin channels (opening ε^3) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit ε → 0 and find that the effective system can describe sound absorption

The general treatment of non-symmetric, non-balanced star circuits: On the geometrization of problems in electrical metrology

In the present note we provide the general solution of a question concerning non-symmetric AC star circuits that came up in a practical application: Given a non-symmetric AC star circuit, we need the quantities of the line voltages. For technical reasons these quantities cannot be measured directly but the phase-to-phase voltages can be. In this text we present a way to compute the needed quantities from the measured ones. We translate this problem in electrical metrology to a geometric one and present in detail a general solution that is well adapted to the practical problem. Furthermore, we solve the generalization of the problem that discusses the non-symmetric, non-balanced star circuit. In addition, we give some further remarks on the mathematical side of the initial problem

Oscillating Ornstein-Uhlenbeck processes and modelling of electricity prices

In this paper we propose an alternative model for electricity spot prices based on oscillating Ornstein-Uhlenbeck processes. This model captures the characteristics of empirical data, especially the oscillating shape of the autocorrelation function. Furthermore, we show that our model leads to explicit formulas for forwards and options on forwards

Limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero-Moser-Sutherland type and are related with β-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities k tend to ∞, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form √k∙x for k→∞ with x in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems

Some central limit theorems for random walks associated with hypergeometric functions of type BC

The spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and dimension parameter $p>q$ are Heckman-Opdam hypergeometric functions of type BC, when the double coset spaces $G//K$ are identified with the Weyl chamber $C_q^B\subset \mathbb R^q$ of type B. The associated double coset hypergroups on $C_q^B$ can be embedded into a continuous family of commutative hypergroups $(C_q^B,*_p)$ with $p\in[2q-1,\infty[$ associated with these hypergeometric functions by Rösler. Several limit theorems for random walks on these hypergroups were recently derived by Voit (2017). We here present further limit theorems when the time as well as $p$ tend to $\infty$. For integers $p$, this admits interpretations for group-invariant random walks on the Grassmannians $G/K$

A negative index meta-material for Maxwell´s equations

We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length η > 0. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like η⁻² and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order η²; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system. In the limit η → 0, we obtain a standard Maxwell system with a frequency dependent effective permeability μ^eff (ω) and a frequency independent effective permittivity ε^eff. Our formulas for these coefficients show that both coefficients can have a negative real part, the meta-material can act like a negative index material. The magnetic activity μ^eff≠1 is obtained through dielectric resonances as in previous publications. The wires are thin enough to be magnetically invisible, but, due to their connectedness property, they contribute to the effective permittivity. This contribution can be negative due to a negative permittivity in the wires

Effective Maxwell´s equations for perfectly conducting split ring resonators

We analyze the time harmonic Maxwell's equations in a geometry containing perfectly conducting split rings. We derive the homogenization limit in which the typical size of the rings tends to zero. The split rings act as resonators and the assembly can act, effectively, as a magnetically active material. The frequency dependent effective permeability of the medium can be large and/or negative

Resonance meets homogenization - Construction of meta-materials with astonishing properties

Meta-materials are assemblies of small components. Even though the single component consists of ordinary materials, the meta-material may behave effectively in a way that is not known from ordinary materials. In this text, we discuss some meta-materials that exhibit unusual properties in the propagation of sound or light. The phenomena are based on resonance effects in the small components. The small (sub-wavelength) components can be resonant to the wave-length of an external field if they incorporate singular features such as a high contrast or a singular geometry. Homogenization theory allows to derive effective equations for the macroscopic description of the meta-material and to verify its unusual properties. We discuss three examples: Sound-absorbing materials, optical materials with a negative index of refraction, perfect transmission through grated metals