Exact sum rules for inhomogeneous drums

We derive general expressions for the sum rules of the eigenvalues of drums of arbitrary shape and arbitrary density, obeying different boundary conditions. The formulas that we present are a generalization of the analogous formulas for one dimensional inhomogeneous systems that we have obtained in a previous paper. We also discuss the extension of these formulas to higher dimensions. We show that in the special case of a density depending only on one variable the sum rules of any integer order can be expressed in terms of a single series. As an application of our result we derive exact sum rules for the homogeneous circular annulus with different boundary conditions, for a homogeneous circular sector and for a radially inhomogeneous circular annulus with Dirichlet boundary conditions.     

Reflection amplitudes of boundary Toda theories and thermodynamic Bethe ansatz

We study the ultraviolet asymptotics in A_n affine Toda theories with integrable boundary actions. The reflection amplitudes of non-affine Toda theories in the presence of conformal boundary actions have been obtained from the quantum mechanical reflections of the wave functional in the Weyl chamber and used for the quantization conditions and ground-state energies. We compare these results with the thermodynamic Bethe ansatz derived from both the bulk and (conjectured) boundary scattering amplitudes. The two independent approaches match very well and provide the non-perturbative checks of the boundary scattering amplitudes for Neumann and (+) boundary conditions.     

A Cahn-Hilliard model in a domain with non-permeable walls

We consider in this article a Cahn-Hilliard model in a bounded domain with non-permeable walls, characterized by dynamic-type boundary conditions. Dynamic boundary conditions for the Cahn-Hilliard system have recently been proposed by physicists in order to account for the interactions with the walls in confined systems and are obtained by writing that the total bulk mass is conserved and that there is a relaxation dynamics on the boundary. However, in the case of non-permeable walls, one should also expect some mass on the boundary. It thus seems more realistic to assume that the total mass, in the bulk and on the boundary, is conserved, which leads to boundary conditions of a different type. For the resulting mathematical model, we prove the existence and uniqueness of weak solutions and study their asymptotic behavior as time goes to infinity.     

The Independence on Boundary Conditions for the Thermodynamic Limit of Charged Systems

We study systems containing electrons and nuclei. Based on the fact that the Thermodynamic limit exists for systems with Dirichlet boundary conditions, we prove that the same limit is obtained if one imposes other boundary conditions such as Neumann, periodic, or elastic boundary conditions. The result is proven for all limiting sequences of domains which are obtained by scaling a bounded open set, with smooth boundary, except for isolated edges and corners. Work partially supported by EU grant HPRN-CT-2002-00277.     

Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2-dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to nontrivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.     

Nonreflecting boundary conditions for discrete waves

We introduce a new class of nonreflecting boundary conditions for lattice models, which minimizes reflections at artificial boundaries. Exact integrodifferential boundary conditions for finite chains and half-spaces are obtained by means of Green’s functions for initial value problems. Truncating the resulting integrals in time, we obtain absorbing boundary conditions. Numerical tests illustrate the ability of these conditions to suppress reflections.     

Boundary conditions for the junction of a normal metal with multiband superconductors with unusual types of superconducting pairing

The boundary conditions for the junction of a normal metal with multiband superconductors with unusual types of superconducting pairing have been obtained on the basis of the tight-binding equations. These boundary conditions have been obtained beyond the effective mass approximation. They make it possible to take into account the nonparabolic and anisotropic spectrum of the normal excitations in a superconductor and their multiband structure, as well as unusual types of symmetries of the superconducting order parameter. These boundary conditions have been used to calculate the conductivities of junctions of a normal metal with a high-temperature superconducting pnictide for different orientation angles of the interface with respect to the pnictide crystallographic axes.     

General Boundary Conditions for a Majorana Single-Particle in a Box in (1 + 1) Dimensions

We consider the problem of a Majorana single-particle in a box in (1 + 1) dimensions. We show that the most general set of boundary conditions for the equation that models this particle is composed of two families of boundary conditions, each one with a real parameter. Within this set, we only have four confining boundary conditions—but infinite not confining boundary conditions. Our results are also valid when we include a Lorentz scalar potential in this equation. No other Lorentz potential can be added. We also show that the four confining boundary conditions for the Majorana particle are precisely the four boundary conditions that mathematically can arise from the general linear boundary condition used in the MIT bag model. Certainly, the four boundary conditions for the Majorana particle are also subject to the Majorana condition.     

Coherent transmission and angular structure of light scattering by monolayer films of polymer dispersed liquid crystals with inhomogeneous boundary conditions

We consider monolayer polymer films with oriented droplets of a nematic liquid crystal (LC). Relations for the coherent transmission coefficients of a layer of oriented ellipsoidal droplets and for the intensity of light scattered by monolayers of spherical and spheroidal droplets have been obtained. The amplitude-phase screen model and the interference approximation of the theory of multiple wave scattering have been used. To describe light scattering by an individual ellipsoidal droplet with inhomogeneous surface binding, we have developed an anomalous diffraction approximation. For monolayers of spherical LC droplets, the coherent scattering coefficients and the angular scattering structure have been analyzed. The internal structure of nematic droplets have been calculated by the relaxation method based on the solution of the minimization problem of the free energy volume density. We have studied basic regular features of light scattering by a monolayer with homogeneous and inhomogeneous boundary conditions at the LC-polymer interface. We show that, for films that contain droplets with inhomogeneous boundary conditions of the tangentially normal type, the angular structure of the scattered light is asymmetric with respect to the polar scattering angle.     

Initial Boundary Value Problem for Conservation Laws

This paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions. The main assumption is that the system under study admits a convex entropy extension. It is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either Godunov schemes or by suitable viscous approximations. Furthermore, in the case that the weak interior solutions are generated by Godunov schemes, any Lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values. This in particular gives a trace formula for the flux functions in terms of the numerical boundary data. We also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws. It is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary Riemann problems, and the converse is not true in general. The general theory is then applied to some specific examples. First, several new facts are obtained for convex scalar conservation laws. For example, we give example which show that Godunov schemes produce numerical boundary layers. It is shown that any continuous functions of density have traces on the boundary (instead of only entropy fluxes). We also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data. A generalized Oleinik entropy condition is also obtained. Next, we prove the existence of a weak solution to the initial-boundary value problem for a family of × quadratic system with a uniformly characteristic boundary condition. Received: 23 July 1996 / Accepted: 28 October 1996     

Supersymmetric Boundary Conditions in \mathcal{N}=4 Super Yang-Mills Theory

We study boundary conditions in ${\mathcal{N}}=4$ super Yang-Mills theory that preserve one-half the supersymmetry. The obvious Dirichlet boundary conditions can be modified to allow some of the scalar fields to have a “pole” at the boundary. The obvious Neumann boundary conditions can be modified by coupling to additional fields supported at the boundary. The obvious boundary conditions associated with orientifolds can also be generalized. In preparation for a separate study of how electric-magnetic duality acts on these boundary conditions, we explore moduli spaces of solutions of Nahm’s equations that appear in the presence of a boundary. Though our main interest is in boundary conditions that are Lorentz-invariant (to the extent possible in the presence of a boundary), we also explore non-Lorentz-invariant but half-BPS deformations of Neumann boundary conditions. We make preliminary comments on the action of electric-magnetic duality, deferring a more serious study to a later paper.     

Phenomenological models of microwave heating of a flat coal mass with release of absorbed heat by the convection law

Phenomenological models of electrodynamics and heat transfer in application to microwave heating are constructed. Analytically rigorous solutions to problems of heating a flat coal mass under microwave radiation are obtained. The boundary conditions correspond to convection mechanism of absorbed heat release to the ambient medium. Mathematical models of dielectric heating for homogeneous boundary conditions are solved by a method of dual integral Laplace and Fourier transform. In the presence of inhomogeneities in the boundary conditions, a quite universal method of Green functions is used. The obtained formulas have a constraint associated with constancy of electro- and thermophysical characteristics of coal fuel, or when their piecewise constant approximation is admissible. The obtained dependences form the basis for scientific support of the microwave heating technology.     

A diffusion problem in a region with a moving boundary

We consider the first linear diffusion problem in a semiinfinite region with a boundary moving in accordance with a quadratic law. For the case of uniformly decelerating motion of the boundary, the solution of the problem is obtained by the Green's function, method under the most general boundary conditions; the solution of the problem for the case of uniformly accelerated motion of the boundary is obtained by an operator method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 102–110, December, 1970.     

Tracking gauge symmetry factorizability on intervals

We track the gauge symmetry breaking pattern by boundary conditions on fifth and higher-dimensional intervals. It is found that, with Dirichlet–Neumann boundary conditions, the Kaluza–Klein decomposition in five-dimension for arbitrary gauge group can always be factorized into that for separate subsets of at most two gauge symmetries, and so is completely solvable. Accordingly, we present a simple and systematic geometric method to unambiguously identify the gauge breaking/mixing content by general set of Dirichlet–Neumann boundary conditions. We then formulate a limit theorem on gauge symmetry factorizability to recapitulate this interesting feature. Albeit the breaking/mixing, a particularly simple check of orthogonality and normalization of fields' modes in effective 4-dim picture is explicitly obtained. An interesting chained-mixing of gauge symmetries in higher dimensions by Dirichlet–Neumann boundary conditions is also explicitly constructed. This study has direct applications to higgsless/GUT model building.     

Threshold gain analysis in GaN-based photonic crystal surface emitting lasers

We have analyzed threshold gains and lasing modes in GaN-based photonic crystal (PC) surface emitting lasers (PCSELs) by using the multiple scattering method (MSM) for triangular-lattice PC patterns. Moreover, GaN-based PCSELs with different boundary shapes have been fabricated and measured. The lasing mode at the Γ band edge of GaN-based PCSELs can be identified by using the angled resolved spectroscopy and matched well to the results calculated by MSM. Threshold conditions in the GaN-based PCSELs with different boundary shapes are obtained by optical pumping and agree well with simulation results.     

Binary scattering and breakup in the three-nucleon system

We present the further development of the three-particle formalism for differential Faddeev equations. The asymptotic boundary conditions in the hyperspherical adiabatic representation have been constructed. We prove that these conditions are asymptotically equivalent to the standard Merkuriev boundary conditions. With these boundary conditions we have formulated the boundary-value problem for Faddeev equations which has the property that the binary channel and the breakup channel are explicitly orthogonal. The effective numerical scheme for solving the formulated boundary-value problem is given.     

Two-phase flow in porous media: dynamical phase transition

We study numerically the behavior of two-phase flow in porous media via the parameters capillary number and viscosity ratio, under steady-state conditions and various levels of saturation. We construct a phase diagram, where the phases are defined according to whether one or both fluids move. We establish a semi-empirical theory for the location of the phase boundaries. The steady-state conditions are obtained by implementing biperiodic boundary conditions.     

Boundary Correlation Functions of the gl(1|1) Supersymmetric Vertex Model

The gl(1|1) supersymmetric vertex model with domain wall boundary conditions (DWBC) on an N×N square lattice is considered. We derive the reduction formulae for the one-point boundary correlation functions of the model. The determinant representation for the boundary correlation functions is also obtained.