Orthogonality of eigenwaves in a semi-open elastic waveguide

We consider a two-layer elastic waveguide structure such that its one layer is unbounded in the lateral direction, and the standard boundary conditions are stated on the boundary of another one. We study several kinds of eigenwaves for this structure. In general, they have a complex-valued longitudinal propagation constant (the spectral parameter). On the base of the Green formula we introduce the scalar product of two waves and prove that the system of eigenwaves of the semiopen elastic waveguide is orthogonal. We construct families of waves which belong to discrete and continuous parts of the spectrum.     

Eigenwaves in waveguides with dielectric inclusions: spectrum

We consider fundamental issues of the mathematical theory of the wave propagation in waveguides with inclusions. Analysis is performed in terms of a boundary eigenvalue problem for the Maxwell equations which is reduced to an eigenvalue problem for an operator pencil. We formulate the definition of eigenwaves and associated waves using the system of eigenvectors and associated vectors of the pencil and prove that the spectrum of normal waves forms a non-empty set of isolated points localized in a strip with at most finitely many real points.     

Eigenwaves in a lossy metal-dielectric waveguide

ABSTRACT

The problem of normal waves in a closed (shielded) regular waveguide of arbitrary cross-section is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found. We also consider properties of system of eigenvectors and associated vectors of the operator-function. Double completeness of system of eigenvectors and associated vectors with a finite defect is established.     

On guided waves in 3-D photonic crystals

The propagation of guided waves in 3-D photonic crystals is studied. In this paper, we suppose that the spectrum of reference medium (a perfectly 3-D photonic crystal) has band gaps. If a line defect is introduced along one of the lattice vectors, we prove that spectrum can be created inside the gaps of the reference spectrum if the line defect is sufficiently strong. Furthermore, we also prove that the generalized eigenfunctions corresponding to the generalized eigenvalues created by line defect are exponentially decaying away from the line defect.     

The property of orthogonality and energy transfer by three-dimensional eigenwaves in transversely isotropic laminated plates with and without contact with a liquid

The properties of eigenwaves in laminated plates with anisotropy in a transforse directon properties are studied. The most general form of the solutions, dispersion relations, power flows and generalized orthogonality relations are analysed. The similarity and difference in the properties of the waves as compared with isotropic media and ideal fluids, as well as the extension to the case of layered spaces and half-spaces, is investigated. A method of determining the coefficients for the eigenwaves radiated in the plate is proposed in the case of a problem with dynamical sources of finite size. A method of summing the series over eigenwaves is suggested.     

Linear waves at the free surface of a saturated porous media

The principal aim of this paper is to study the propagation of linear waves at the free surface of a saturated porous media. This problem is formulated as an eigenvalue problem with complex eigenvalues, and the solution is given in term of an orthogonal eigenfunction expansion, whose completeness has been taken for granted in the literature regarding the problem as a classical Sturm-Liouville problem, which is not the case due to the complex nature of the eigenvalues. The main purpose of the present work is to prove the completeness of the eigenfunctions for all possible physical values of the parameters involved, even for some values of the parameters, where previous numerical works have found abnormal behavior of the eigenvalues. In those cases if we mistakenly consider the problem as a Sturm-Liouville one, as has been done before, the eigenfunction expansion will not hold, but indeed we will prove that it does.On leave from Institute de Mecánica de los Fluidos, Universidad Central de Venezuela, Caracas, Venezuela.     

On completeness of root functions of elliptic boundary problems in a domain with conical points on the boundarySur la complétude des fonctions propres et associées d'un problème au bord elliptique dans un domaine avec points coniques sur le bord

We prove the completeness of the system of eigen and associated functions (i.e., root functions) of an elliptic boundary value problem in a domain, whose boundary is a smooth surface everywhere, except at a finite number of points, such that each point possesses a neighborhood, where the boundary is a conical surface. To cite this article: Y.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 649–654.     

Problem of small and normal oscillations of a rotating elastic body filled with an ideal barotropic liquid

An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.     

Order reduction in the integral equations of the null-field method

We reduce the system of equations of the null-field method in the problem of diffraction on a permeable scatterer to a single equation. For a special representation of the field inside the scatterer, we prove the equivalence of that equation to the original boundary value problem of diffraction. We state conditions of the convergence of the approximate solution to the exact one. We prove the completeness and closedness of a system of plane waves.     

Global Analysis of a Model for Capillary Water Waves in Two Dimensions

In this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 (the capillary waves system). The model we consider here retains most of the difficulties of the full capillary water waves system, including the delicate time resonance structure and modified scattering. It is slightly simpler, however, at the technical level, and our goal here is to present our method in this simplified situation. The analysis of the full system is presented in a preprint by the authors (2014). © 2016 Wiley Periodicals, Inc.     

On a control problem for the wave equation in R<Superscript>3</Superscript>

Solutions of the wave equation (waves) initiated by infinitely distant sources (controls) are considered, and the L₂-completeness of reachable sets consisting of such waves is stidued. This problem is a natural analog of the control problem for a bounded domain where the completeness (local approximate controllability) in subdomains filled with waves generated by boundary controls occurs. It is shown that, in contrast to the latter case, the reachable sets formed by waves incoming from infinity are not complete in filled subdomains and describe the associated defect. Next, extending the class of controls to a set of special polynomials, the completeness is gained. A transform defined by jumps that arise in projecting functions to reachable sets is introduced. Its relevance to the Radon transform is clarified. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 19–37.     

量词模态逻辑的代数语义学（Ⅲ）──关于不含Barcan公式的正规模态系统的情形

本文首先讨论嵌套论域语义的相应代数语义并由Ｈｕｇｈｅｓ和Ｃｒｅｓｓｗｅｌｌ在［５］中建立的关于具有嵌套论域的正规量词模态系统的关系语义完全性定理推出其相应的代数语义完全性定理：然后对于具有任意可变论域语义的正规系统，我们用Ｈｅｎｋｉｎ方法给出其关于狭义Ｋｒｉｐｋｅ语义的关系语义完全性定理，由此通过将关系语义转化为代数语义从而亦推得其代数语义完全性定理。     

On the Existence of Global Oscillation Waves for a Class of 3x3 Semilinear Hyperbolic Equations

ln this paper, we prove the global existence of oscillation waves for a class of 3 × 3 semilinear hyperbolic equations by applying the Young measures and two-scale Young measures which are associated with the solution sequence of the system.     

Problem of checking the functional completeness in the class of piecewise parallel functions

The author showed previously that it is possible to implement an arbitrary piecewise parallel function by circuits of linear elements and one nonlinear element implementing the θ-function. Thus, the functional completeness of a particular system of functions in the class of piecewise parallel functions was established. In this paper we consider the same class and study its completeness property for a system of functions whose closures contain all linear functions. We formulate and prove a criterion of functional completeness for such systems in terms of precomplete classes.     

Banach压缩映象原理与空间完备性

本文揭示了Banach压缩映象原理与空间完备性的关系,证明了在具有一致局部连通性的距离空间中,压缩映象原理与空间完备性仍然等价.     

A Nonself-Adjoint 1D Singular Hamiltonian System with an Eigenparameter in the Boundary Condition

In this paper, we study a nonself-adjoint singular 1D Hamiltonian (or Dirac type) system in the limit-circle case, with a spectral parameter in the boundary condition. Our approach depends on the use of the maximal dissipative operator whose spectral analysis is adequate for the boundary value problem. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations so that we can determine the scattering matrix of dilation. Moreover, we construct a functional model of the dissipative operator and specify its characteristic function using the solutions of the corresponding Hamiltonian system. Based on the results obtained by the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and Hamiltonian system.     

Banach压缩映象原理与空间完备性

本文进一步揭示了Ｂａｎａｃｈ压缩映象原理与完备性的关系，指出：一般地，Ｂａｎａｃｈ压缩映象原理等价于道路完备性，在一定条件下才等价于完备性，进一步给出了这一等价性成立的充要条件     

中介时序逻辑系统MTL的完备性

本文对[1]中的系统MTL进行了进一步的讨论,给出了它的模型论性质,基于中介逻辑,构造了一种可传有序模型,证明了MTL系统的完备性.     

幂函数的一些密码学性质

二元域上n数组空间上的非线性置换在分组码，杂凑函数与流密码等密码学领域中有重要应用.域GF（2n）上的幂函数提供了二元域上n数组空间上的一类非线性置换．本文着重研究幂函数的强完全性、完全性与非线性度等密码学性质．作为结果，本文证明了幂函数具有完全性；证明了具有强完全性的函数必有较高的拓扑非线性度；木文找到一类具有强完全性的幂函数；周时也定出了幂函数的代数非线性度．     

Stability to weak dissipative Bresse system

In this paper we study the Bresse system with frictional dissipation working only on the angle displacement. Our main result is to prove that this dissipative mechanism is enough to stabilize exponentially the whole system provided the velocities of waves propagations are the same. This result is significative only from the mathematical point of view since in practice the velocities of waves propagations are always different. In that direction we show that when the velocities are not the same, the system is not exponentially stable and we prove that the solution in this case goes to zero polynomially, with rates that can be improved by taking more regular initial data. Finally, we give some numerical result to verify our analytical results.