A duality principle corresponding to the parabolic equations

Our principal problem consists in finding threshold values for a family of blow-up solutions of nonlinear evolution problems depending on a real parameter λ. We seek to obtain the solution of the problem in a constructive and unified approach which would provide the answer for the following general question: Is there is a constructive method for finding the threshold values? We illustrate our approach by means of an example of finding a threshold value for a family of parabolic equations with nonlinearity indefinite in sign. In the main result, the threshold value is expressed via explicitly specified dual variational principles of a new type.     

On a pathology in indefinite metric inner product space

A pathology related to an indefinite metric, which has been pointed out by Ito in connection with construction of a two dimensional quantum field model at a finite cutoff, is mathematically analyzed in a simple model. It is found for a model Hamiltonian with a parameter in an indefinite metric inner product space that eigenvalues with a complete set of eigenvectors changes suddenly from positive integers to negative integers as a parameter crosses a critical value (the Hamiltonian being skew selfadjoint with absolutely continuous spectrum on a pure imaginary axis at the critical value of the parameter), if a fixed (positive definite Hilbert space) topology is used in the completion of the underlying indefinite metric inner product space. However it is also found that if the topology is varied with the parameter of the Hamiltonian in the manner similar to analytic continuation, then the Hamiltonian keeps positive integer eigenvalues with a complete set of eigenvectors.     

Total reflection of electromagnetic waves propagating from an isotropic medium to an indefinite metamaterial

The existence conditions for total reflection and the corresponding critical angle at the interface separating an isotropic medium and an indefinite metamaterial for TE- and TM-polarized electromagnetic waves are obtained. For different kinds of indefinite metamaterial, there appear different total reflection phenomena. Particularly, the anomalous total reflection in which the incident angle is smaller than the critical angle and the Brewster’s angle can be smaller than the critical angle can occur for anti-cutoff medium. Furthermore, the omnidirectional total reflection exists for the always cutoff medium and anti-cutoff medium. The total reflection depends on the thickness of indefinite metamaterial when the indefinite metamaterial is finite, and the photon tunneling phenomenon can occur when the thickness of indefinite metamaterial is smaller than wavelength.     

über eine natürliche Regularisierung von Zweipunktfunktionen

A field belonging to a Heisenberg type particle equation with operators derived from the elements of a commutator ring is quantized according to Fermi statistics. The resulting anticommutator is shown to be more general than the Lehmann-Källén type. It contains a density function for every mass value, that may be used to regularize, with the help of an indefinite metric, theδ-function without implications for the mass spectrum.     

Kochen-Specker Theorem in Krein Space

The well known Kochen-Specker’s theorem (Kochen and Specker J. Math. Mech. 17:59–87, 1967) is devoted to the problem of hidden variables in quantum mechanics. In the paper we present a geometric proof for an indefinite analogy of Kochen-Specker’s theorem. On the real three-dimensional Krein space there exists unique two-valued probability measure.     

Stability of operator splitting methods for systems with indefinite operators: Advection–diffusion–reaction systems

This brief paper presents an A-stability result for operator splitting type time integration methods applied to advection–diffusion–reaction equations with possibly indefinite source terms. These results extend our earlier work on diffusion–reaction systems [D.L. Ropp, J.N. Shadid, Stability of operator splitting methods for systems with indefinite operators: reaction–diffusion systems, J. Comput. Phys. 203 (2) (2005) 449–466]. The A-stability result presents sufficient conditions that control both low and high wave number instabilities. A corollary shows that if L-stable methods are used for the diffusion term the high wave number instability will be controlled more easily. Numerical results are presented that verify second-order convergence for the operator splitting methods and demonstrate control of instabilities on a chemotaxis problem by use of an L-stable diffusion integrator.     

The R-function method for the free vibration analysis of thin orthotropic plates of arbitrary shape

Free flexural vibrations of homogeneous, thin, orthotropic plates of an arbitrary shape with mixed boundary conditions are studied using the R-function method. The proposed method is based on the use of the R-function theory and variational methods. In contrast to the widely used methods of the network type (finite differences, finite element, and boundary element methods), in the R-function method all the geometric information given in the boundary value problem statement is represented in an analytical form. This allows one to seek a solution in a form of some formulas called a solution structure. These solution structures contain some indefinite functional components that can be determined by using any variational method. A method of constructing the solution structures satisfying the required mixed boundary conditions for eigenvalue plate bending problems is described. Numerical examples for the vibration analysis of orthotropic plates of complex geometry with mixed boundary conditions for illustrating the aforementioned R-function method and comparison against the other methods are made to demonstrate its merits.     

An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation

We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa–Holm equation is integrable by the inverse spectral transform in the multi-peakon case.     

Negative and positive Goos–Hänchen shifts of a light beam transmitted from an indefinite medium slab

We study the lateral shift of a light beam transmitted from an indefinite medium slab. The analytic expressions for the transmission coefficient and the lateral shift are obtained by using the stationary-phase method, and different conditions for the occurrence of a negative lateral shift are deduced. It is shown that the thickness of the indefinite medium strongly affects the properties of the lateral shift. For a larger thickness, the sign of the lateral shift depends only on the signs of the permittivity and permeability, while, for a smaller thickness, the sign of the lateral shift is determined by the combined effect of the incident angle of the light, the signs of the permittivity and permeability, and the thickness of the indefinite medium slab. In addition, the lateral shift at the transmitted resonance points is found to be enhanced greatly, and the sign of the lateral shift around the transmitted resonance points is strongly dependent on the permittivity and permeability of the indefinite medium. Some examples are given to validate the obtained theoretical results. PACS 41.20.Jb; 42.25.Gy; 42.25.Bs; 78.20.Ci; 78.68.+m     

Coupling Biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction

The interaction between a fluid and a poroelastic structure is a complex problem that couples the Navier–Stokes equations with the Biot system. The finite element approximation of this problem is involved due to the fact that both subproblems are indefinite. In this work, we first design residual-based stabilization techniques for the Biot system, motivated by the variational multiscale approach. Then, we state the monolithic Navier–Stokes/Biot system with the appropriate transmission conditions at the interface. For the solution of the coupled system, we adopt both monolithic solvers and heterogeneous domain decomposition strategies. Different domain decomposition methods are considered and their convergence is analyzed for a simplified problem. We compare the efficiency of all the methods on a test problem that exhibits a large added-mass effect, as it happens in hemodynamics applications.     

Scattering Theory for Quantum Fields¶with Indefinite Metric

In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag–Ruelle theory do not carry over to the case of indefinite metric [4], we propose an axiomatic framework for the construction of in- and out-states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out-fields, called the “form factor functional”, which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework. Received: 13 September 1999/ Accepted: 1 August 2000     

滑线式电桥电阻测量条件和灵敏度数据处理的探讨

运用不确定度,推出了滑线式电桥测电阻的最佳测量条件,指出了该条件应用时出现的问题,提出了解决办法：另外对电桥灵敏度测量的等效方法给出了证明,对灵敏度数据处理方法进行了补充。     

Light transmission through a stack of right- and left-hand cholesteric liquid crystal layers with indefinite dielectric and magnetic permittivities. I. Reflection peculiarities

Optical properties of a stack of layers of right- and left-hand cholesteric liquid crystal (CLC) with locally indefinite dielectric and magnetic permittivities are investigated. The problem is solved by the modified Ambartsumian??s layer addition method. The peculiarities of reflection spectra of this system are studied and it is shown that, in contrast to a single layer of CLC, this system has multiple photonic band gaps (PBG). It is shown that the indefinite character of the dielectric and magnetic permittivities essentially affects the PBG map of the system. These multiple PBG can find wide application, in particular, in technologies of production of displays.     

Airy函数乘积的积分

本文提出对∫xⁿA^(p)B^(q)dx型不定积分普遍方法,其中A,B都是含有参变量λ的Airy方程y″=(λ+x)y的解,p,q是微分阶数,n是非负整数。当p,q≤1,A与B所含的参数不相同时,本文导出了上述积分普遍公式。当A=B时,又提出了与Albright完全不相同的求积方法。作为它的应用,本文讨论了处于三角势阱中电子的波函数正交归一问题,并计算了势阱中能级间的跃迁几率。 关键词：     

Wick-Ordered Entire Functions of the Indefinite Metric Free Field

We present a simple and general method for constructing Wick-ordered entire functions of free fields with an indefinite metric, based on using an appropriate generalization of the Paley–Wiener–Schwartz theorem.     

Some singular errors near the contact line singularity, and ways to resolve both

The history of resolving the contact line singularity is briefly reviewed, with the emphasis on the traps posed by unstable solutions and the ways to resolve the problem by evoking microscopic physics to compute indefinite parameters of self-similar mesoscopic solutions.     

Walker 4-manifolds with proper almost complex structures

It is shown that a Walker 4-manifold, endowed with a canonical neutral metric depending on three arbitrary functions, admits a specific almost complex structure (called proper) and an associated opposite almost complex structure. We study when these two almost complex structures are integrable and when the corresponding Kähler forms are symplectic. The conditions for the canonical neutral metric to be Kähler imply that the three arbitrary functions in the metric are all harmonic with respect to two coordinate variables, and we obtain a useful method of constructing indefinite Kähler 4-manifolds. Petean’s example of a nonflat indefinite Kähler–Einstein 4-manifold is a special case of this construction.     

Light propagation in anisotropic metamaterials. I. Dispersion surfaces

The peculiarities of solutions of the wave dispersion equation in an unbounded indefinite medium for arbitrarily oriented optical axes are considered. All possible dispersion surfaces arising in such media are described, and the conditions of their existence are obtained. It is shown that only some specific pairs of surfaces are possible. The dispersion equation for the boundary problem is obtained, as well as the dependence of dispersion curves on the orientation of optical axis is investigated. The nonreciprocity of the wave refraction is shown.