6d Dirac fermion on a rectangle; scrutinizing boundary conditions, mode functions and spectrum

We investigate localization and mass spectrum of spin-1/2 fermionic field on a scalar thick brane configuration with Poincaré symmetry, which arises in a 5-dimensional theory of gravity coupling to a self-interacting scalar field in a regular Riemannian manifold. It is shown that, in the case of no Yukawa coupling of scalar-fermion, there is no existence of localized massless zero mode for both left and right chiral fermions. By introducing appropriate coupling, we can obtain the localized modes as well as a continuous spectrum of massive Kaluza-Klein (KK) modes for left and right chiral fermions. It is found that there exists a mass gap between the massless zero mode and the excited modes for left chiral fermions or right chiral fermions. The total number of bound states is determined by the Yukawa coupling constant and the parameter of the thick brane solution. Depending on the sign of the coupling constant, only one category of zero modes, namely, either the left or right chiral zero mode can be trapped on the brane.

Conformal theory of the two-dimensional O( N) model with ordinary, extraordinary, and special boundary conditions

We compute the vacuum polarization for a massless, conformally coupled scalar field on the covering space of global, four-dimensional, anti-de Sitter space-time. Since anti-de Sitter space is not globally hyperbolic, boundary conditions must be applied to the scalar field. We consider general Robin (mixed) boundary conditions for which the classical evolution of the field is well-defined and stable. The vacuum expectation value of the square of the field is not constant unless either Dirichlet or Neumann boundary conditions are applied. We also compute the thermal expectation value of the square of the field. For Dirichlet boundary conditions, both thermal and vacuum expectation values approach the same well-known limit on the space-time boundary. For all other Robin boundary conditions (including Neumann boundary conditions), the vacuum and thermal expectation values have the same limit on the space-time boundary, but this limit does not equal that in the Dirichlet case.

We report our study on the properties of the topological structures present in the QCD medium around the critical temperature $T_c$. We use dynamical domain wall fermion configurations on lattices of size $32^3 \times 8$ to detect the topological structures through the zero and near-zero modes of the overlap operator. We show that the properties of the zero modes of the QCD Dirac operator agree well with that of calorons with non-trivial holonomy. We also show how the zero modes move around when changing the fermionic boundary conditions, and how the locality of the solutions also depends on the boundary conditions. This supports the presence of instanton-dyons in the hot QCD medium around $T_c$, where the distance between dyons control the shape and extent of the zero modes.

We study localization of various matter fields on a non-Z2-symmetric scalar thick brane in a pure geometric Weyl integrable manifold in which variations in the length of vectors during parallel transport are allowed and a geometric scalar field is involved in its formulation. It is shown that, for spin 0 scalar field, the massless zero mode can be normalized on the brane. Spin 1 vector field cannot be normalized on the brane. And there is no spinor field which can be trapped on the brane for the case of no Yukawa-type coupling. By introducing the appropriate Yukawa coupling, the left or right chiral fermionic zero mode can be localized on the brane.

The sign-regularity properties of Green's functions for a special class of mixed boundary-value problems

In this research, a new analytical solution for finding the buckling loads of thin orthotropic rectangular plates with different boundary conditions is derived. The methods currently known in the literature for finding the buckling loads of plates are mainly numerical. Although some plates with specific boundary conditions have analytical solutions, a comprehensive analytical method providing analytical solutions that fit all possible combinations of boundary conditions is lacking. The solution method in this study is based on the development of a static solution for a plate. The physical meaning of buckling is the loss of stiffness, and it is found as the value of the in-plane loading intensity at which a zero force on the plate surface will generate infinite displacement. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. The solution presented for this plate will fit different boundary conditions of the deflection, slope, shear force, and bending moment along the common edges of neighboring plates. Using this new method, exact buckling loads and the buckling modes of many new cases of classical boundary conditions are found (cases involving clamped, simply supported, guided, and free boundaries). Results are given for several stiffness ratios in both directions of the plate, as well as for unidirectional and several cases of bidirectional loading.

Model development and analysis of the evolution of coal permeability under different boundary conditions

We define the chiral zero modes' phase space of the G = SU(n) Wess–Zumino–Novikov–Witten (WZNW) model as an (n − 1)(n + 2)-dimensional manifold q equipped with a symplectic form Ωq involving a Wess–Zumino term ρ which depends on the monodromy M and is implicitly defined (on an open dense neighbourhood of the group unit) by This classical system exhibits a Poisson–Lie symmetry that evolves upon quantization into a Uq(sln) symmetry for q a primitive even root of 1. For each (non-degenerate, constant) solution of the classical Yang–Baxter equation we write down explicitly a ρ(M) satisfying equation () and invert the form Ωq, thus computing the Poisson bivector of the system. The resulting Poisson brackets (PB) appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously in Furlan et al (2000 Preprint hep-th/0003210). We argue that it is advantageous to equate the determinant D of the zero modes' matrix (ajα) to a pseudoinvariant under permutations q-polynomial in the SU(n) weights, rather than to adopt the familiar convention D = 1. A finite-dimensional 'Fock space' operator realization of the factor algebra q/h where h is an appropriate ideal in q for qh = −1, is briefly discussed.

A new formulation is presented that extends the material distribution topology optimization method to address boundary-effect-dominated problems, where specific boundary conditions need to be imposed at solid–fluid interfaces. As an example of such a problem, we focus on the design of acoustic structures with significant viscous and thermal boundary losses. In various acoustic applications, especially for acoustically small devices, the main portion of viscothermal dissipation occurs in the so-called acoustic boundary layer. One way of accounting for these losses is through a generalized acoustic impedance boundary condition. This boundary condition has previously been proven to provide accurate results with significantly less computational effort compared to Navier–Stokes simulations. To incorporate this boundary condition into the optimization process at the solid–fluid interface, we introduce a mapping of jumps in densities between neighboring elements to an edge-based boundary indicator function. Two axisymmetric case studies demonstrate the effectiveness of the proposed design optimization method. In the first case, we enhance the absorption performance of a Helmholtz resonator in a narrow range of frequencies. In the second case, we consider an acoustically larger problem and achieve an almost-perfect broadband absorption. Our findings underscore the potential of our approach for the design optimization of boundary-effect-dominated problems.

Capacitively coupled RF discharge at low pressure has wide range of applications from diagnostic mirror cleaning in tokamak-reactors to technological processes such as materials modification, sterilization and polishing, films etching and deposition in microelectronics. Numerical modelling is applied to the plasma-assistant technologies development / optimization as well as for fundamental researches of operational modes and physics of RF plasma discharge. The problem is significantly complicated by the presence of a number of conditions necessary for the practical application of the calculation results. In particular, for physical correctness it is necessary to guarantee the continuity of the current on the electrodes (i.e., taking into account displacement currents), the ability to simulate a discharge in arbitrary geometries requires 3D calculations. The precise description of surfaces having free-forms is possible only when using an irregular (triangular) computational mesh. Important requirements for carrying out calculations are the requirements for calculation speed, the presence of a user-friendly interface, and auxiliary functionality. The requirements listed above were implemented in the KITe code we developed. Besides the simulations of low-pressure radio-frequency capacitively-coupled discharge, the code also allows for modelling impurities transport and deposition in complex geometries such as ITER diagnostic ducts. The calculations are performed on supercomputer at Ioffe Institute. In the last code modification two special boundary conditions were added to the standard model of local electrical fields calculation (Poisson's equation and Dirichlet boundary condition) in order to include considering the displacement current. The current continuity condition is equivalent to the condition of the sum of charges at both electrodes and the volume plasma charge is equal to zero, that follows from the resulting equation system. The calculation of charge transfer through external electrical circuit allows simulation of both in DC-coupled and decoupled (where the asymmetry results in presence of the DC bias) schemes. However, additional boundary conditions produce the technical issues of the solving as the equation matrix become non-symmetrical and illconditioned, especially in the case of the complex geometries. The report presents the results of discharge modeling using code KITe at some elementary geometries: the concentric spheres, plane-parallel electrodes, isolated system of charged particles. The comparison with the standard model was performed. The model convergence and stability with variations of geometries and computational mesh are investigated. The code validation by GEC and some other mock-ups is planned.

We investigate the inverse problem of determination of two unknown numerical parameters occuring linearly and nonlinearly in the higher coefficient of a linear second order elliptic equation of the diffusion–reaction type in a domain Ω diffeomorphic to a ball layer under special boundary conditions by observation in neighborhoods of the correspondent amount of points. For an analogous inverse problem under Dirichlet boundary conditions, sufficient conditions of solution uniqueness was obtained by the author formerly, but they had an abstract character and so were inconvenient for practical usage. In the paper, these conditions are extended to the case of different boundary conditions and rendered concrete for the case of the exponential type higher coefficient. The inverse problem investigated in the paper refers to research of electric processes in the Earth atmosphere in the frame of global electric circuit in the stationary approximation and arises from needs of recovering the unknown higher coefficient of the equation on the base of observation data obtained from two local transmitters.

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

It is shown that the application of the high-frequency Floquet theory to an atom interacting with a circularly polarized field is equivalent to a treatment of the free atom with special boundary conditions. With these boundary conditions, it is possible to define other equivalent formulations. Some consequences of this flexibility are illustrated for the hydrogen atom.

The steady boundary layer problems with buoyancy effect over a moving and permeable plate are studied in the present paper. By using the boundary layer approximations with similarity variables, we obtain coupled nonlinear velocity and temperature equations. Then, we derive boundary shape functions for velocity and temperature, which automatically satisfy the specified boundary conditions, including a convective boundary condition. Generally, numerical method is difficult to keep all the specific boundary conditions exactly and accurately. Because this will lead to the decline of the accuracy of numerical solution, we are motivated to develop a novel iterative algorithm which is based on the boundary shape functions. It transforms the nonlinear boundary layer problems into the initial value problems for new variables. The unknown terminal values of these variables can be determined by iteration, while the initial values of them can be arbitrarily given. The high-performance of the iterative boundary shape functions method (BSFM), which is convergent very fast, can be confirmed by numerical examples. The parametric studies in terms of the Grashof number, Prandtl number and Biot number with different suction/injection and velocity ratio values, as well as the flow instability area in terms of the Biot number and velocity ratio are carried out by the BSFM.