Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ(x,t) for one dimensional Hamiltonians \(H=-\partial_{x}^{2}+V(x)\) where V (for example modeling barriers or wells) and ψ(x,0) are compactly supported.We show that the dispersive part of ψ(x,t) is the Borel sum of its asymptotic series in powers of t ^?1/2, t→∞. The remainder, the difference between ψ and the Borel sum, i.e., the exponential part of the transseries of ψ, is a convergent expansion of the form \(\sum_{k=0}^{\infty}g_{k}\Gamma_{k}(x)e^{-\gamma_{k} t}\), where Γ _k are the Gamow vectors of H, and iγ _k are the associated resonances; generically, all g _k are nonzero. For large k, γ _k ～const?klog?k+k ² π ² i/4. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way.The decomposition allows for calculating ψ for moderate and large t, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions.The analytic structure of ψ is perhaps surprising: in general (even in simple examples such as square wells), ψ(x,t) turns out to be C ^∞ in t but nowhere analytic on ?⁺. In fact, ψ is t-analytic in a sector in the lower half plane and has the whole of ?⁺ a natural boundary. In the dual space, we analyze the resurgent structure of ψ.     

Structural,thermal and electrical properties of the semiconductor system Ag(1−x)CuxInSe2

This paper presents a study of bulk samples synthesized of the Ag_1−xCu_xInSe₂ semiconductor system. Structural, thermal and electrical properties, as a function of the nominal composition (Cu content) x=0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 were studied. The influence of x on parameters such as melting temperature, solid phase transition temperature, lattice parameters, bond lengths, crystallite size t (coherent domain), electrical resistivity, electrical mobility and majority carrier concentration was analyzed. The electrical parameters are analyzed at room temperature. In general, it is observed that the properties of the Ag_1−xCu_xInSe₂ system for x≤0.4 are dominated by n-AgInSe₂, while for x>0.4, these are in the domain of p-CuInSe₂. The crystallite size t in the whole composition range (x) is of the order of the nanoparticles. Secondary phases (CuSe, Ag₂Se and InSe) in small proportion were identified by XRD and DTA.     

The green function of an infinite,fluid loaded membrane

In this paper the response of a fluid loaded plane structure (a membrane) to a concentrated line force excitation is considered in great detail. The normalized velocity response—here called the Green function G—depends upon a dimensionless range x₀=k_m|x|, where k_m is the free wavenumber on the membrane in a vacuum, on the Mach number $\text{M=}\text{k}{}_0\text{k}{}_{\mathrm{m}}$, the ratio of wave phase speed ω/k_m on the unloaded membrane to the sound speed ω/k₀, and on a parameter ? which can be regarded as a measure of fluid loading at the “coincidence” condition M=1. In the analogous problem involving a thin elastic plate, the corresponding parameter is independent of frequency and plate thickness and may be regarded as an intrinsic measure of fluid loading; moreover, in cases of common interest (steel in water, aluminium in air) that parameter is small. In the present paper, the asymptotic structure of G(x₀, M, ?) is therefore sought in the limit ? → 0. Naturally, no single asymptotic expansion can be expected to be valid throughout the (x₀, M) plane, and the programme therefore involves the delineation of regions of that plane in which distinct asymptotic results apply, the construction and discussion of those results, and the asymptotic matching (according to the procedures of the method of matched asymptotic expansions) of results holding in adjoining regions. The Fourier integral for G is broken into surface wave and acoustic components, and the asymptotic structure obtained for each. Previously obtained results for the behaviour at large distances are recovered, with a demonstration that very large distances indeed (x₀ ? ?^?2) may be needed for their validity for some ranges of M; and the drive point behaviour, of G(x₀=0, M, ?) as ? → 0 qua function of M, is shown to correspond to that already discussed in the literature. Elsewhere, in the covering of the whole (x₀, M) plane by different asymptotic expressions, a wide variety of analytical results is found, reflecting the achievement in different regions of different balances among the five competing physical mechanisms represented in the model: namely, structural stiffness, structural inertia, fluid pressures, fluid compressibility and fluid inertia. These different balances give rise to a wide variety of expressions for the phase and amplitude of the surface wave and acoustic components which can now be used to isolate the dominant structural and acoustic mechanisms at any point in the (x₀, M) plane.     

Magnetic structure of TmCu2Ge2

TmCu₂Ge₂ compound crystallizes in the tetragonal ThCr₂Si₂-type crystal structure. The neutron diffraction reveals the presence of an incommensurate antiferromagnetic order below T_N=2.5 K. The Tm magnetic moment of 5.0(1) μ_B at 0.47 K is parallel to the c-axis. The order is described by the propagation vector k=[k_x, k_x, 0], where k_x=0.117(3). The increase of the values of the components k_x near the Néel temperature is observed.     

X-ray determination of the linear compressibilities of (SN)x

The linear compressibilities of (SN)_x have been determined up to 22 kbar using a hydrostatic-pressure X-ray cell: at 1 bar, k_a = 5.5;k_b = 2.2; k_c = 5.0; k_[201] = 3.8 —; all in units of 10^-12cm²/dyne. The results are compared with other chain-like solids and confirm the anisotropic three-dimensional bonding character of (SN)_x. Pressure dependence of the electrical conductivity is discussed.     

Nonlinear interaction between a moving vehicle and a plate elastically mounted on a tunnel

The paper presents a theoretical study of the interaction between a nonlinear model of a moving vehicle (velocity v) and a plate elastically mounted in a tunnel. An efficient approach to the solution of the problem of vehicle-slab-track-tunnel-soil interaction is developed on the basis of a coupling of the Finite Element and Integral Transform methods (FEM and ITM). According to this approach the tunnel which may have an arbitrary shape and a portion of the surrounding soil is modelled by Finite Elements while the soil (half-space) is described by the ITM. The corresponding solution is found using the solutions for the uniform half-space and for the continuum with a cylindrical cavity. To exploit the invariance of the structure in longitudinal direction x, for this direction additionally to the time-frequency transform (tω) a space-wavenumber transform (xk_x) is used. The case of a half-space is analyzed using a Fourier transform also in the second horizontal direction (yk_y) and an analytical solution for the z-direction on the basis of exponential functions. In the case of the infinite continuum with the cavity a Fourier series (for the circumferential direction) and a series of cylindrical functions (for the radial direction) are used for the solution regarding the cross-sectional coordinates (y,z). The tunnel structure, that may have an arbitrary shape, and a portion of the surrounding soil will be modelled by Finite Elements in a (xk_x,tω) transformed domain.In order to observe the boundary conditions at the surface of the half-space as well as at the surface of the cavity, the superposition of the two solutions has to be performed after an inverse Fourier transform (IFT) in the (k_x,y,z,ω)-domain.The solution for the complete system floating slab-track-tunnel-half-space is obtained in the (k_x,y,z,ω)-domain. Once the system transfer function H(ω^′) (with ω^′=ω+vk_x) for this complete coupled system is found, the displacements of the plate can be calculated in time domain by the IFT of the product p(ω^′)H(ω^′), i.e. the convolution of the loading p(t) with the impulse response function h(t), which completely represents the behavior of the coupled system. In the context of the coupling of systems in a relative movement the problems of differential algebraic equations (DAE) have to be observed.     

Relativistic deuteron wave function from the ed → enp reaction cross section

The reaction d(e, e'p)n has been studied theoretically using three variants of the relativistic deuteron wave function (WF): (i) a light-front WF depending on two variables, k_⊥ and x; (ii) a light-front WF depending only on a specific combination of k_⊥ and x, having the meaning of a relative momentum; (iii) a WF depending on the nucleon-spectator momentum (the Gross WF). It is shown that, if the coincidence experiment is carried out under specific kinematical conditions, these three variants lead to qualitatively different predictions for the cross section. At a fixed value of the electron scattering angle θ_e, the conditions concerned are reduced to some relation between the final electron energy E′ and the proton emission angle θ_p.     

On-shell T-matrices in multiple scattering

The transition operator T for the scattering of a particle from N potentials V_j(x) can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in 〈k|T(k₀²±i0)|k′〉, |k| = |k′| = k₀, then each t_j is also on-shell.     

Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity

We consider the nonlinear Schrödinger equation: (1) $${{i\partial u} \mathord{\left/ {\vphantom {{i\partial u} {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} = - \Delta u - \left| u \right|^{{4 \mathord{\left/ {\vphantom {4 N}} \right. \kern-\nulldelimiterspace} N}} uandu\left( {0,.} \right) = \varphi \left( . \right),$$ whereu:[0,T)×? ^N →?. For any given pointsx ₁,x ₂,...,x _k in ? ^N , we construct a solution of Eq. (1),u(t), which blows up in a finite timeT at exactlyx ₁,x ₂,...,x _k . In addition, we describe the precise behavior of the solutionu(t) whent→T, at the blow-up points {x ₁,x ₂,...,x _k } and in ? ^N ?{x ₁,x ₂,...,x _k }.     

Low-temperature thermal conductivity of GexAs40−xS60 glasses

Thermal conductivity, k, of Ge_xAs_40−xS₆₀ glasses has been measured between 4.5 and 300 K. A variation of k with the composition is found and dependences of its parameters on the mean coordination numbers are established. Analyses have been made within but also independently of the soft-potential model. Some correlations and scaling laws are proposed.     

Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.     

An extension of the fourier convolution theorem in terms of poisson brackets

The Fourier convolution theorem is extended to cover nonstationary and inhomogeneous phenomena. The Fourier transforms of input and transfer functions, F and K, are assumed to be slowly varying functions of x and t. The first-order corrections to the usual convolution theorem are given by Poisson brackets of F and K. These are calculated over k, ω, x and t. The method is applied to study induced currents in a plasma.     

Limit of Fluctuations of Solutions of Wigner Equation

We consider fluctuations of the solution W _ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W _ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, W _ε (0, x, k) = δ(x)f(k) and \({f\in {\mathcal{S}}(\mathbb{R}^d)}\), then the laws of the rescaled fluctuation \({Z_\varepsilon(t):=\varepsilon^{-1/2}[W_\varepsilon(t,x,k)-\bar{W}(t,x,k)]}\) converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered.     

Constraints fulfilled by density moments and density correlations

The first set of equations we are going to consider follow from the conservation of additive quantum numbers. These constraints assume a simpler form than the corresponding relations for inclusive cross sections and inclusive correlations and can all be fulfilled consistently with any collection of higher order density correlation functions G_ms(x,k_⊥1,ν₁,…, x_m, k_⊥m,ν_m), m ≥ 2, identically zero. We will also derive a set of constraints from the requirements that the probability of any particular class of final states must be a number greater than or equal to zero, and the number of particles of certain type which appear in any region of momentum space must also be positive of zero. These relations can be fulfilled with any finite sequence of density correlations G_ms(x₁, k_⊥1,ν₁,…, x_m, k_⊥m, ν_m), m ≥ 3, identically equal to zero, or with all correlations of order m ≥ 2 identically zero.     

Variational formulation of covariant eikonal theory for vector waves

The eikonal theory of wave propagation is developed by means of a Lorentz-covariant variational principle, involving functions defined on the natural eight-dimensional phase space of rays. The wave field is a four-vector representing the electromagnetic potential, while the medium is represented by an anisotropic, dispersive nonuniform dielectric tensor D^μν(k,x). The eikonal expansion yields, to lowest order, the hamiltonian ray equations, which define the lagrangian manifold k(x), and the wave-action conservation law, which determines the wave-amplitude transport along the rays. The first-order contribution to the variational principle yields a concise expression for the transport of the polarization phase. The symmetry between k-space and x-space allows for a simple implementation of the Maslov transform, which avoids the difficulties of caustic singularities.     

Small polaron electron transport in reduced CeO2 single crystals

An investigation of the electrical properties of reduced ceria, CeO_2?x, carried out on single crystals, shows that CeO_2?x provides one of the clearest examples of hopping conduction and the small polaron mechanism. Included are conductivity and Seebeck coefficient measurements at constant x, obtained by sealing off the specimen chamber after reduction. The Seebeck coefficient is independent of temperature, suggesting that the number of carriers is constant. On the other hand, the mobility is activated, with activation energy E_h = 0.40 eV at small x and increasing to 0.52 eV at x = 0.25. The results for the mobility preexponential are consistent with the adiabatic theory of small polaron behavior. A puzzling feature of the Seebeck data as a function of x is that, for low x, the data fit the well-known Heikes formula, without a degeneracy factor of 2 for spin. Nevertheless, these data are interpreted to show that the proportion of mobile carriers decreases as x increases, presumably because of the presence of short-range ordered configurations which immobilize some carriers.     

The evolutionary public goods game on scale-free networks with heterogeneous investment

The public goods game (PGG) is generally considered as a suitable paradigm to explain ubiquitous cooperative behavior. In this study, we investigated the evolutionary PGG on scale-free networks and studied the effect of individual heterogeneity by setting the cooperator x an investment value correlated to its degree as I_x=N⋅k_x^β/∑_jk_j^β, where k_x is the degree of x, j runs over all players and β is a tunable parameter. It is shown that the cooperation level is remarkably promoted by negative values of β whereas it is highly depressed by positive values of β. Moreover, the effect of environmental noise has also been investigated. Our result may sharpen the understanding of cooperation induced by the individual diversity.     

Solution-processed high-k thin films as a resistive switching for ReRAM applications

Resistive switching characteristics of solution-processed high-k thin films (HfO_x and TaO_x) were investigated for ReRAM applications. The thickness of solution-processed high-k thin films can be easily controlled by simple spin coating. We optimized the critical thickness of solution-processed HfO_x and TaO_x thin films, for reliable ReRAM operations. A similar bipolar resistive switching behavior was observed from both solution-processed and sputter-processed HfO_x films. Furthermore, it was found that the solution-processed HfO_x and TaO_x films have a uniform resistive switching characteristic. The dominant conduction of these solution-processed films is described by Ohmic conduction in the low-resistance state. On the other hand, Ohmic conduction at low voltage and Poole–Frenkel emission at high voltage dominate in the high-resistance state. It was verified that the solution-processed HfO_x and TaO_x films have superior endurance and retention characteristics. Therefore, ReRAM devices based on solution-processed high-k materials are expected to be a promising candidate, for usage of resistive memory in glass substrate or flexible substrate based electronic devices.     

Exact solutions for a diffusion equation with a nonlinear external force

This work is devoted to investigate the solutions of the one-dimensional diffusion equation by taking the nonlinear external force F(x,t;ρ)=−k(t)x+K/x+κx|x|^α−1η[ρ(x,t)] into account. Our investigation is first performed by considering the case α=0 and η=1, which results in a Burgers like equation with a spatial and time dependent external force. After, we consider the case α≠0 and η=α+1 and show that the solution found may be expressed in terms of the q-exponential functions present in the Tsallis formalism. In addition, we also discuss the stationary solution for α and η arbitraries.