Design and Fabrication of Composite Smart Structures with High Electric and Mechanical Performances for Future Mobile Communication

In this paper, we have developed a load-bearing outer skin for antennas, which is termed a composite smart structure (CSS). The CSS is a multilayer composite sandwich structure in which antenna layers are inserted. A direct-feed stacked patch antenna is considered. A design procedure including the structure design, material selection, and design of antenna elements in order to obtain high electric and mechanical performances is presented. An optimized honeycomb thickness is selected for efficient radiation and impedance characteristics. High gain conditions can be obtained by placing the outer facesheet in the resonance position, which is at about a half wavelength distance from the ground plane. The measured electrical performances show that the CSS has a great bandwidth (over 10%) and a higher gain than an antenna without a facesheet and has excellent mechanical performances, owing to the composite laminates and honeycomb cores. The CSS concept can be extended to give a useful guide for manufacturers of structural body panels and for antenna designers.     

Domain Structure of Bulk Ferromagnetic Crystals in Applied Fields Near Saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is in the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a multidomain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp interface energy, giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field.     

Analysis of the radiation properties of a planar antenna on a photonic crystal substrate

This paper is concerned with the rigorous investigation of the radiation properties of a planar patch antenna on a photonic crystal substrate. Under the assumptions that the driving frequency of the antenna lies within the band gap of the photonic crystal substrate and that the crystal satisfies a symmetry condition, we prove that the power radiated into the substrate decays exponentially. To do this, we reduce the radiation problem to the study of the well‐posedness of a weakly singular integral equation on the patch antenna, and to the study of the asymptotic behaviour of the corresponding Green's function. We also provide a mathematical justification of the use of a photonic crystal substrate as a perfect mirror at any incidence angle. Copyright © 2001 John Wiley & Sons, Ltd.     

Approach to Ground State and Time-Independent Photon Bound for Massless Spin-Boson Models

It is widely believed that an atom interacting with the electromagnetic field (with total initial energy well-below the ionization threshold) relaxes to its ground state while its excess energy is emitted as radiation. Hence, for large times, the state of the atom + field system should consist of the atom in its ground state, and a few free photons that travel off to spatial infinity. Mathematically, this picture is captured by the notion of asymptotic completeness. Despite some recent progress on the spectral theory of such systems, a proof of relaxation to the ground state and asymptotic completeness was/is still missing, except in some special cases (massive photons, small perturbations of harmonic potentials). In this paper, we partially fill this gap by proving relaxation to an invariant state in the case where the atom is modelled by a finite-level system. If the coupling to the field is sufficiently infrared-regular so that the coupled system admits a ground state, then this invariant state necessarily corresponds to the ground state. Assuming slightly more infrared regularity, we show that the number of emitted photons remains bounded in time. We hope that these results bring a proof of asymptotic completeness within reach.     

Localization of the identities of the number field

The image of the group of identities of the number field under a mapping into the reduced multiplicative group of a local field is investigated. The asymptotic behavior of this image under a cyclotomic p-extension of the ground field is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 5–11, 1976.     

A Piezoelectric Finite Shell Element for the Geometrically Nonlinear Analysis of Sensors

A geometrically nonlinear finite element formulation to analyze piezoelectric shell structures is presented. The formulation is based on the mixed field variational functional of Hu–Washizu. Within this variational principle the independent fields are displacements, electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when using a three dimensional material law. It is remarked that no simplification regarding the constitutive relation is assumed. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent resultant stress and resultant dielectric displacement fields. The shell structure is modeled by a reference surface with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electrical degree of freedom, which is the difference of the electric potential through the shell thickness. The developed mixed hybrid shell element fulfills the in–plane, bending and shear patch tests, which have been adopted for coupled field problems. A numerical investigation of a smart antenna demonstrates the applicability of the piezoelectric shell element under the consideration of geometrical nonlinearity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Properties of the solitonic potentials of the heat operator

We study properties of the purely solitonic τ-function and potential of the heat equation in detail. We describe the asymptotic behavior of the potential and establish the ray structure of this asymptotic behavior on the plane (x₁, x₂) in dependence on the parameters of the potential.     

The Diffraction of an Acoustic Wave by an Embedded Quarter-Space

The subject of this paper is the diffraction of a plane harmonicwave when it falls upon a quarter-space with different transmissionproperties from the rest of unbounded space. A matching procedureallows asymptotic expressions for the field on the two planeinterfaces to be calculated in a fairly simple way. In orderto obtain these expressions, certain assumptions are made aboutthe asymptotic form of the field on the interface. These assumptionsare plausible and lead to consistent results. We begin with the problem of wave propagation in two weldedquarter-spaces due to excitation on the plane boundary. Thisproblem has an exact solution and provides an illustration ofthe method of matching asymptotic fields (not the method ofmatched asymptotic expansions). We then move on to the problemof a plane wave normally incident on our embedded quarter-spaceand derive exact expressions for the asymptotic field on theinterfaces. Finally, we include an analysis of oblique incidence.     

Effect of a protection zone in the diffusive Leslie predator-prey model

In this paper, we consider the diffusive Leslie predator-prey model with large intrinsic predator growth rate, and investigate the change of behavior of the model when a simple protection zone Ω₀ for the prey is introduced. As in earlier work [Y. Du, J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations 229 (2006) 63-91; Y. Du, X. Liang, A diffusive competition model with a protection zone, J. Differential Equations 244 (2008) 61-86] we show the existence of a critical patch size of the protection zone, determined by the first Dirichlet eigenvalue of the Laplacian over Ω₀ and the intrinsic growth rate of the prey, so that there is fundamental change of the dynamical behavior of the model only when Ω₀ is above the critical patch size. However, our research here reveals significant difference of the model's behavior from the predator-prey model studied in [Y. Du, J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations 229 (2006) 63-91] with the same kind of protection zone. We show that the asymptotic profile of the population distribution of the Leslie model is governed by a standard boundary blow-up problem, and classical or degenerate logistic equations.     

Convergence rate of solutions toward stationary solutions to a two-phase model with magnetic field in a half line

In this paper, we study an asymptotic behavior of a solution to the outflow problem for a two-phase model with magnetic field. Our idea mainly comes from [1] and [2] which investigate the asymptotic stability and convergence rates of stationary solutions to the outflow problem for an isentropic Navier–Stokes equation. For the two-phase model with magnetic field, we also obtain the asymptotic stability and convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method.     

Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

We characterize the long‐time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity by using a variant of the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest‐descent method of Deift and Zhou for oscillatory Riemann‐Hilbert problems. First, we formulate the IST over a single sheet of the complex plane without introducing the uniformization variable that was used by Biondini and Kova?i? in 2014. The solution of the focusing NLS equation with nonzero boundary conditions is thereby associated with a matrix Riemann‐Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well‐known modulational instability within the context of the IST. We then eliminate this growth by performing suitable deformations of the Riemann‐Hilbert problem in the complex spectral plane. The results demonstrate that the solution of the focusing NLS equation with nonzero boundary conditions remains bounded at all times. Moreover, we show that, asymptotically in time, the xt ‐plane decomposes into two types of regions: a left far‐field region and a right far‐field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. Finally, we show how, in the latter region, the modulus of the leading‐order solution, initially obtained as a ratio of Jacobi theta functions, can be reduced to the well‐known elliptic solutions of the focusing NLS equation. These results provide the first characterization of the long‐time behavior of generic perturbations of a constant background in a modulationally unstable medium. © 2017 Wiley Periodicals, Inc.     

Asymptotic behavior of Morris–Lecar system

In this paper, the asymptotic behavior of the Morris–Lecar model is studied. Firstly, the phase plane characteristic of the model is investigated, secondly, in terms of the concepts of slow manifold and fast foliation, the asymptotic structure of the Morris–Lecar model is discussed based on Tikhonov's theorem. Finally, computation and analysis results show that the spiking of action potential of Morris–Lecar neuron actually has the jump onset–jump return structure.     

Short-wavelength grazing scattering of a plane wave by a smooth periodic boundary. II. Diffraction by an infinite periodic boundary

The short-wavelength asymptotic behavior of the field near a reflecting boundary (the Fock zone and the neighborhood of the limit ray) is constructed for the problem of the diffraction of a plane wave by a smooth periodic boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Seklova AN SSSR, Vol. 173, pp. 60–86, 1990.     

Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

We study a general class of log-gas ensembles on (shifted) quadratic lattices. We prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance. We apply our general results to analyze the asymptotic behavior of a q-boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a one-dimensional section of a pullback of the two-dimensional Gaussian free field.Our approach is based on a q-analogue of the Schwinger–Dyson (or loop) equations, which originate in the work of Nekrasov and his collaborators, and extends the methods developed by Borodin, Gorin and Guionnet to quadratic lattices.     

Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Part III

We study the asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. Thanks to an additional (fixed) parameter, we show that two different critical exponents play a crucial role in the asymptotic analysis, giving an explanation of the phenomena discovered in Gazzola et al. (Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear) and Gazzola and Serrin (Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 477).     

Asymptotic behavior of the correlation coefficients of transverse momenta in the model with string fusion

In the framework of the model with fusion of quark–gluon strings on the transverse lattice, we find the asymptotic behavior of the correlation coefficients between observables in separated rapidity intervals with a high string density in a realistic case with an inhomogeneous distribution of strings in the impact parameter plane. We calculate the asymptotic forms for three types of correlations: between the average transverse momenta of particles with rapidity in these intervals, between the average transverse momentum of particles in one rapidity interval and the multiplicity of particles in another, and also between the multiplicities of charged particles in these intervals. We show that the previously found independence of the asymptotic form of the correlation coefficient between the average transverse momenta from the variance in the number of particles produced in string fragmentation holds only in the case of a uniform distribution of strings in the transverse plane. We also show that the found general expressions for the long-range correlation coefficients in the particular case with a uniform distribution of strings in the transverse plane become the formulas previously obtained by another method applicable only in this simple case.     

Delay‐dependent criteria on the global attractivity of Nicholson's blowflies model with patch structure

In this paper, we investigate the effect of delay on the asymptotic behavior of Nicholson's blowflies model with patch structure and multiple time‐varying delays. By using the fluctuation lemma and some differential inequality technique, delay‐dependent criteria are obtained for the global attractivity of the addressed system. Meanwhile, some numerical examples are given to illustrate the feasibility of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Uniqueness and local stability for the inverse scattering problem of determining the cavity

Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization are proved in this paper.     

Source mechanism model for ground motion simulation

A simple analytical model for computing ground motion in a layered half-space due to a buried seismic source is presented in this paper. The buried earthquake source is represented as a distribution of double couples varying in time as a ramp function on the fault plane. The analysis is simplified by first decoupling the governing equations into P-SV and SH problem by a coordinate transformation in the frequency-wave number domain. These two problems are solved separately and the final solution is obtained by the sum of solutions of these individual problems. Explicit expressions for ground motion in a layered half-space due to an impulsive double couple are derived. In the sequel, Green’s function for the displacement field in an infinite medium is also presented. The developed source mechanism model is also demonstrated by simulating ground motion for the Kucth earthquake (M_w = 7.7) of 26th January 2001.