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We consider the attraction–repulsion chemotaxis system in a smoothly bounded domain Ω⊆R2. When the system is parabolic–elliptic–parabolic–elliptic, we establish the finite time blow-up conditions of nonradial solutions by making a differential inequality on the moment of solutions. Apart from that, in some special cases, the solutions of the system are globally bounded without blow-up. Our results extend some known conclusions in the literature.

The well-posedness of the Neumann and Robin problems for the Korteweg–de Vries equation is studied with data in Sobolev spaces. Using the Fokas unified transform method, the corresponding linear problems with forcing are solved and solution estimates are derived. Then, using these, an iteration map is defined, and it is proved to be a contraction in appropriate solution spaces after the needed bilinear estimates are derived.

In this paper, we get the invariant domain of 1D non-isentropic gas dynamics equations. First of all, we construct the elementary waves with the characteristic analysis method. According to the characteristic of elementary waves, we divide the u–p plane into five areas. By analyzing the structure of Riemann solutions in each area, we find a new convex bounded domain where if the Riemann data belong to the domain, then the Riemann solutions also belong to the domain. Moreover, it is used to prove the uniform boundedness of approximate solutions built by the difference scheme, so it is the basis for the Riemann problem to be applied to the Cauchy problem.

In this paper, we consider the initial value problem for the mixed Schrodinger equation. For the Schwartz initial data q0(x)∈S(R), by defining a general analytical domain and two reflection coefficients, we ever found an unified long-time asymptotic formula via the Deift–Zhou nonlinear steepest descent method. In this paper, under essentially minimal regularity assumptions on initial data in a much weak weighted Sobolev space q0(x)∈H2,2(R), we apply the ∂ steepest descent method to obtain long-time asymptotics for the mixed Schrodinger equation. In the asymptotic expression, the leading order term O(t−1/2) comes from the dispersive part qt + iqxx and the error order O(t−3/4) comes from a ∂ equation.

This paper is concerned with the stationary approximations of the stochastic wave equations defined on Rn with critical exponents. For a class of nonlinear diffusion terms, we prove the existence and uniqueness of tempered pullback attractors of the random wave equations driven by a stationary process as an approximation to the white noise. For a linear multiplicative noise, we prove the upper semicontinuity of these attractors as the step size of the approximations approaches zero. The asymptotic compactness of the solutions on Rn is established by the idea of energy equations and the uniform estimates on the tails of the solutions.

The ∂-dressing method is developed to study the three-component coupled Hirota (tcCH) equations. We first start from a ∂-problem and construct a new spectral problem. Based on the recursive operator, we successfully derive the tcCH hierarchy associated with the given spectral problem. In addition, the soliton solutions of the tcCH equations are first obtained via determining the spectral transform matrix in the ∂-problem. Finally, one-, two-, and three-soliton solutions are analyzed to discuss the dynamic phenomena of the tcCH equations. It is remarked that the interaction between solitons depends on whether the characteristic lines intersect.

This paper is devoted to studying the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/modified Camassa–Holm equation, and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation as its special case. We first show that the gFQXL/gCH-mCH equation is locally well-posed in Besov spaces. Then, we present a blow-up criterion and a precise blow-up scenario, and utilizing the fine structure and the H1-norm conservation, we establish a finite time blow-up result with respect to the initial data. Finally, peakons are derived in an explicit formula.

A two-component-two-dimensional coupled with one-component-three-dimensional (2C2Dcw1C3D) flow may also be called a real Schur flow (RSF), as its velocity gradient is uniformly of real Schur form, the latter being the intrinsic local property of any general flows. The thermodynamic and `vortic' fine structures of RSF are exposed and, in particular, the complete set of equations governing a (viscous and/or driven) 2C2Dcw1C3D flow are derived. The Lie invariances of the decomposed vorticity 2-forms of RSFs in $d$-dimensional Euclidean space $\mathbb{E}^d$ for any interger $d\ge 3$ are also proven, and many Lie-invariant fine results, such as those of the combinations of the entropic and vortic quantities, including the invariances of the decomposed Ertel potential vorticity (and their multiplications by any interger powers of entropy) 3-forms, then follow.

We analyze a new space–time fractional diffusion equation encompassing different diffusion processes in d-dimensions. The first-order time derivative is replaced with a time derivative of the Caputo type of arbitrary order β; the spatial-fractional operator of Riesz–Feller or Riesz–Weyl type is replaced with its extension to d-dimensions, defined by means of an extended Fourier transform. The mathematical problem with the spatial-fractional operator proposed here is formulated to tackle anomalous diffusion in heterogeneous media (fractal structures) and incorporating power-law distributions. A formal solution is proposed using the Green function method which, for appropriate initial and boundary conditions, can be expressed in terms of the generalized H-function of Fox—a typical track of anomalous diffusive processes. These mathematical tools provide a new powerful framework to model anomalous diffusion and relaxation problems in heterogeneous media.

The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations and symmetries and conservation laws in Eulerian coordinates are shown. An invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed. It possesses all the finite-difference analogs of the conservation laws. Some bottom topographies require moving meshes in Eulerian coordinates, which are stationary meshes in mass Lagrangian coordinates. The developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.

We formulate the the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure's gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial, boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution's gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters including the angular speed., Comment: 48 pages, there has been a slight change in the title and minor improvements in the presentation with this replacement

In the context of homogeneous turbulence, we prove that the Rapid Distortion Theory (RDT) model for the spectral tensor preserves the symmetry, positive semidefiniteness, and integrability properties required in Cramer’s characterization of the spectral tensor of a continuous homogeneous random process. From this, a statistically valid correlation tensor is obtained that returns a Reynolds stress tensor model that satisfies realizability conditions. The number of hypotheses used is kept to a minimum, which allows a flexible use of the model in the applications. The Kelvin–Townsend equations allow us to construct the solution and prove its properties by means of a factorization approach. Since the RDT spectral tensor model is a system of transport equations plus an algebraic restriction due to incompressibility, we deal with the existence, uniqueness, and persistence of solutions in a specific set of functions by using DiPerna–Lions renormalization techniques.

We investigate mean dynamics and invariant measures for a multi-stochastic discrete sine-Gordon equation driven by random viscosity, stochastic forces, and infinite-dimensional nonlinear noise. We first show the existence of a unique solution when the random viscosity is bounded and the nonlinear diffusion of noise is locally Lipschitz continuous, which leads to the existence of a mean random dynamical system. We then prove that such a mean random dynamical system possesses a unique weak pullback mean random attractor in the Bochner space. Finally, we show the existence of an invariant measure. Some difficulties arise from dealing with the term of random viscosity in all uniform estimates (including the tail-estimate) of solutions, which lead to the tightness of a family of distribution laws of solutions.

In this paper, we derive some new blow up criterion for the 2D full compressible Navier–Stokes equations in terms of the density and the temperature in critical spaces. This is an improvement of corresponding results obtained by Wang and Zhang [Appl. Math. Comput. 232, 719–726 (2014)].

This work concerns the Kirchhoff equation −(a + b∫Ω|∇u|2dx)Δu + u = |u|p−2u, where a, b > 0, Ω⊆R3 is an exterior domain with a smooth boundary and 4 < p < 6. By establishing a global compactness result, we prove that the equation has at least one positive solution. Our result is an extension of the work by Benci and Cerami [Arch. Ration. Mech. Anal. 99(4), 283–300 (1987)].

We consider the existence of ground state solutions for nonlinear Kirchhoff type systems in the whole space RN (2 ≤ N ≤ 4) with prescribed normalization. Two cases are studied: one is L2-supercritical and the other is mixed. In the first case, assuming that the coupling coefficient is big enough, we prove the existence of a ground state solution via minimax methods. In the second case, assuming that the coupling coefficient is sufficiently small, we show the existence of a local minimizer, which is, of course, also a ground state solution.

In this paper, we give some mathematical and numerical results on the behavior of a nonsimple viscoelastic plate corresponding to anti-plane shear deformations. First, we derive briefly the equations of the considered plate, and then, we study the well-posedness problem. Second, we prove that the solutions decay exponentially at a rate proportional to the total mass of the second order. Finally, we give some numerical experiments based on the spectral method developed for multi-dimensional problems with implementation in MATLAB for one and two-dimensional spaces.

In this paper, we study the energy conservation for the nonhomogeneous incompressible ideal Hall-magnetohydrodynamic system. Three types of sufficient conditions are obtained. Precisely, the first one provides ρ, u, P, and b with sufficient regularity to ensure the local energy conservation. The second one removes the regularity condition on P while requires Lp regularity on the spatial gradient of the density ∇ρ and Lr regularity on ρt. The last one removes the regularity condition on ρt while requires certain time regularity on the velocity field u. Our main strategy relies on commutator estimates in the work of Constantin et al. [Commun. Math. Phys. 165, 207–209 (1994)].

This paper mainly investigates the asymptotic behavior of non-autonomous stochastic complex Ginzburg–Landau equations on unbounded thin domains. We first prove the existence and uniqueness of random attractors for the considered equation and its limit equation. Due to the non-compactness of Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of such a stochastic equation is proved by the tail-estimate method. Then, we show the upper semi-continuity of random attractors when thin domains collapse onto the real space R.

In this work, we present the analytical approach to the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the behavior of the harmonic as well the an-harmonic parts of the oscillator. We present the an-harmonic part of the result in the form of the operator function., Comment: 40 pages, revised argument in Appendix F, results unchanged

In this paper, we study a class of nonlinear wave equations with variable exponent sources. By introducing a family of potential wells, we first prove the global existence of solutions with initial data in the potential wells and the finite time blow-up for solutions starting in the unstable sets. The boundedness and asymptotic behavior of global solutions are also concerned. Finally, we obtain the finite time blow-up with high energy initial data.

We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step towards justification of the Bogomolny-Schmit heuristics.

In the present paper, we prove the boundedness of all solutions for some impulsive pendulum-type equations by Moser’s twist theorem. We also prove the existence of quasi-periodic solutions for the impulsive pendulum-type equations.

This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction–diffusion equations with time-varying delays on thin domains. First, we prove the existence and uniqueness of the regular random attractor. Then, we prove the upper semicontinuity of the regular random attractors for the equations on a family of (n + 1)-dimensional thin domains that collapses onto an n-dimensional domain.

In this paper, we study the nonhomogeneous stochastic initial-boundary value problem for the nonlinear Schrodinger equation on an upper-right quarter plane with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the Wiener additive noise on the boundary. Our approach allows us to show the local existence and uniqueness of solutions in the space H2. The basic properties of the solutions such as the continuity and the boundary-layer behavior are also studied using the Ito calculus. Despite several technical difficulties, we believe that the approach developed in this paper can be applied to the case of a large class of noise including fractional Wiener space time white noise, homogeneous noise, and Levy noise.

A correspondence between the orbits of a system of two complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind. This allows us to transfer results well known from the theory of polygonal billiards, such as ergodicity, the existence of periodic orbits, the absence of exponential divergence, the existence of additional conservation laws, and the presence of discontinuities in the dynamics, to the corresponding systems of ordinary differential equations. It also shows that the considerable intricacy known to exist for polygonal billiards also attends these apparently simpler systems of ordinary differential equations.

We consider a double phase problem in RN with a q−1-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Using variational methods based on the topological degree and critical point theory together with the Nehari manifold method and deformation lemma, we find three nontrivial solutions: among them, the first one is the positive ground state solution, the second one is the negative ground state solution, and the third one is the least energy sign-changing solution, which changes sign only once. We also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions.

In this work, we are concerned with the existence and nonexistence of ground state solutions for the following class of quasilinear Schrodinger coupled systems taking into account periodic or asymptotically periodic potentials. The nonlinear terms are superlinear at infinity and at the origin. By using a change of variable, we turn the quasilinear system into a nonlinear system where we can establish a variational approach with a fine analysis on the Nehari method. For the nonexistence result, we compare the potentials with periodic potentials proving the nonexistence of ground state solutions.

This paper is focused on the generalized Forchheimer flows of isentropic gas, described by a system of two nonlinear degenerating differential equations of first order. We prove the existence and uniqueness of the Dirichlet problem for stationary problem. The technique of semi-discretization in time is used to prove the existence for the time-dependent problem., arXiv admin note: substantial text overlap with arXiv:1804.01409; substantial text overlap with arXiv:1608.08829 by other authors

In this paper, we first prove the existence and uniqueness of tempered pullback random attractors for a non-autonomous stochastic fractional Ginzburg–Landau equation driven by multiplicative noise with α ∈ (0, 1) in L2R3. Then, we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. Due to the lack of the compactness of Sobolev embeddings on unbounded domains, we establish the pullback asymptotic compactness of solutions in L2(R3) by the tail-estimates of solutions.

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough.

In this paper, we mainly devote to investigate the periodic Dullin–Gottwald–Holm equation. By overcoming the difficulties caused by the complicated mixed nonlinear structure, a very useful priori estimate is derived in Lemma 2.7. Based on Hα1-conservation and L∞-estimate of solution, some new blow-up phenomena are derived for the periodic Dullin–Gottwald–Holm equation under different initial conditions.

In this article, we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k-dimensional integer set. The well-posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally Lipschitz continuous. The mean random dynamical system associated with the non-autonomous system is shown to possess a unique tempered weak pullback mean random attractor in L2(Ω,F,l2×l2). The existence of invariant measures for the autonomous system is also derived by using the Krylov–Bogolyubov method. The difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions.

It is a well-known and classical result that spectrally stable traveling waves of a general reaction-diffusion system in one spatial dimension are asymptotically stable with exponential relaxation rates. In a series of works in the 1990s, Goodman [Trans. Am. Math. Soc. 311, 683 (1989)], Kapitula [Trans. Am. Math. Soc. 349, 1901 (1997)], and Xin [Commun. Partial Differ. Equations 17, 1889 (1992)] considered plane traveling waves for such systems and they have succeeded in showing asymptotic stability for such objects. Interestingly, the (estimates for the) relaxation rates that they have exhibited are all algebraic and dimension dependent. It was heuristically argued that as the spectral gap closes in dimensions n ≥ 2, algebraic rates are the best possible. In this paper, we revisit this issue. We rigorously calculate the sharp relaxation rates in L∞ based spaces, both for the asymptotic phase and the radiation terms. They are, indeed, algebraic, but about twice better than the best ones obtained in these early works although this can be mostly attributed to the inefficiencies of using Sobolev embeddings to control L∞ norms by high order L2 based Sobolev spaces. Finally, we explicitly construct the leading order profiles, both for the phase and for the radiation terms. Our approach relies on the method of scaling variables, as introduced in the work of Gallay and Wayne [Arch. Ration. Mech. Anal. 163, 209 (2002) and Gallay and Wayne, Philos. Trans. R. Soc. A 360, 2155 (2002)] and, in fact, provides sharp relaxation rates in a class of weighted L2 spaces as well.

In this paper, we study the inverse Dirac spectral problem defined on [0, a] with the boundary condition at a involving a Nevanlinna function f(λ). When f(λ) is known a priori, we prove that the potentials and the other boundary condition are uniquely determined in terms of appropriate partial information on the spectral data consisting of eigenvalues and normalizing constants.

In this paper, we investigate the compressible primitive equations (CPEs) with density-dependent viscosity for large initial data. The CPE model can be derived from the 3D compressible and anisotropic Navier–Stokes equations by hydrostatic approximation. Motivated by the work of Vasseur and Yu [SIAM J. Math. Anal. 48, 1489–1511 (2016); Invent. Math. 206, 935–974 (2016)], in which the global existence of weak solutions to the compressible Navier–Stokes equations with degenerate viscosity was obtained, we construct approximate solutions and prove the global existence of weak solutions to the CPE in this paper. In our proof, we first present the vertical velocity as a function of density and horizontal velocity, which plays a role in using the Faedo–Galerkin method to obtain the global existence of the approximate solutions. Then, we obtain the key estimates of lower bound of the density, the Bresch–Desjardins entropy on the approximate solutions. Finally, we apply compactness arguments to obtain global existence of weak solutions by vanishing the parameters in our approximate system step-by-step.

This paper is focused on the generalized Forchheimer flows for slightly compressible fluids, described as a system of two nonlinear degenerating partial differential equations of first order. We prove the existence and uniqueness of the Dirichlet problem for the stationary case. The technique of semidiscretization in time is used to prove the existence for the time-dependent case.

Advective diffusion plays a key role in the transport of salt, heat, buoys, and markers in geophysical flows, in the dispersion of pollutants and trace gases in the atmosphere, and even in the dynamics of sea ice floes influenced by winds and ocean currents. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix D*. Three decades ago, a Stieltjes integral representation for the homogenized matrix, involving a spectral measure of a self-adjoint operator, was developed. However, analytical calculations of D* have been obtained for only a few simple flows, and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted. We overcome these limitations by providing a mathematical foundation for the computation of Stieltjes integral representations of D*. We explore two different approaches and for each approach we derive new Stieltjes integral representations and rigorous bounds for the symmetric and antisymmetric parts of D*, involving the molecular diffusivity and a spectral measure μ of a self-adjoint operator that depends on the characteristics of a randomly perturbed periodic flow. In discrete formulations of each approach, we express μ explicitly in terms of standard (or generalized) eigenvalues and eigenvectors of Hermitian matrices. We develop and implement an efficient numerical algorithm that combines beneficial numerical attributes of each approach. We use this method to compute the effective behavior for model flows and relate spectral characteristics to flow geometry and transport properties.

We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works [Li-Xin, arXiv:1504.06826] and [Antonelli-Spirito, Arch. Ration. Mech. Anal., 203(2012), 499--527], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial data. Moreover, we obtain some new a priori estimates, which can avoid using the assumption that the gradient of the velocity is a well-defined function, which is indeed used directly in [Vasseur-Yu, SIAM J. Math. Anal., 48 (2016), 1489--1511; Invent. Math., 206 (2016), 935--974]. On the other hand, in the absence of damping terms, we also prove the global existence of weak solutions to the compressible QNS equations without the lower bound assumption on the dispersive coefficient, which improves the previous result due to [Antonelli-Spirito, Arch. Ration. Mech. Anal., 203(2012), 499--527]., 33pages

We use spherical coordinates to devise a new exact solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible fluid with a general density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows-that is, flows that are observed in some averaged sense in the ocean. Owing to the employment of spherical coordinates we do not need to resort to approximations (e.g. of $f$- and $\beta$-plane type) that simplify the geometry in the governing equations. Our explicit solution represents a steady purely-azimuthal stratified flow with a free surface, that---thanks to the inclusion of forcing terms and the consideration of the Earth's geometry via spherical coordinates---makes it suitable for describing the Antarctic Circumpolar Current and enables an in-depth analysis of the structure of this flow. In line with the latter aspect, we employ functional analytical techniques to prove that the free surface distortion is defined in a unique and implicit way by means of the pressure applied at the free surface. We conclude our discussion by setting out relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations.

This paper considers a nonlinear elliptic system with Hardy-Sobolev critical exponent growth. Several existence results on the positive ground state solutions to the system are established under proper conditions.This paper considers a nonlinear elliptic system with Hardy-Sobolev critical exponent growth. Several existence results on the positive ground state solutions to the system are established under proper conditions.

Free motion on a 3-sphere, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system, which exhibits the fundamental properties of superintegrability. These include separability in a variety of coordinate systems, polynomial solutions, and a particular subset of Clebsch-Gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the Askey scheme. Deriving these results from the initial formulation in spherical geometry provides the Zernike system with interest beyond its optical applications.

The analytic computation of electric and magnetic fields near corners and edges is important in many applications related to science and engineering. However, such complicated situations are hard to deal with, since they accumulate charges and consequently they mathematically represent singularities. In order to model this singular behavior, we introduce a novel method, which is related to the geometry and the analysis of the ellipsoidal coordinate system. Indeed, adopting the benefits of the corresponding coordinate surfaces, we use a general non-circular double cone, being the asymptote of a two-sided hyperboloid of two sheets with elliptic cross section, which matches almost perfectly the particular physics and captures the corresponding essential features in a fully three-dimensional fashion. To this end, our analytical technique employs the ellipsoidal geometry and adapts the ellipsoidal functions (solutions of the well-known Lame equation) so as to construct a new set of the so-called elliptic cross-sectional hyperboloidal harmonics, supplemented by the appropriate orthogonality rules on every constant coordinate surface. By first recollecting the key results of the coordinate system and the related potential functions, including the indispensable orthogonality results, we demonstrate our method to the solution of two boundary value problems in electrostatics. Both refer to a non-penetrable two-hyperboloid of elliptic cross section and its double-cone limit, the first one being charged and the second one scattering off a plane wave. Closed form expressions are derived for the related fields, while the already known formulae from the literature are readily recovered, all cases being followed by the appropriate numerical implementation. Published by AIP Publishing.