We show that resonant states in scattering on asymptotically hyperbolic man-ifolds that are analytic near conformal infinity, have analytic radiation patterns at infinity. On even asymptotically hyperbolic manifolds we also show that smooth solutions of Vasy operators with analytic coefficients are also analytic. That answer a question of M.Zworski ([14] Conjecture 2). The proof is based on previous results of Baouendi-Goulaouic and Bolley-Camus-Hanouzet and for convenience of the reader we present an outline of the proof of the latter.
We investigate well-accepted formulations describing charge transport in composite cathodes of batteries. Our upscaling of carefully selected microscopic equations shows three main features: (i) a novel set of six equations equipped with nine effective parameters which systematically couple the microscale to the macroscale; (ii) the coupling of transport and flow equations allows to account for three scales: pore scale, Darcy scale, and macroscale; (iii) the upscaled equations take phase separation during Li-intercalation into account as well as specific particle configurations. The wide range of applications and interest in energy storage devices make these results a promising tool to study the influence of the microstructure on current-voltage characteristics and to optimize cathode designs.
The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey auctions -- throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the optimal bidding strategy in various realistic contexts: the maximization of the inventory bought with a given amount of cash in the framework of audience strategies, the maximization of the number of conversions/acquisitions with a given amount of cash, etc. In our model, the sequence of auctions is modeled by a Poisson process and the \textit{price to beat} for each auction is modeled by a random variable following almost any probability distribution. We show that the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and that almost-closed form solutions can be found by using a fluid limit. Numerical examples are also carried out.
We study the macroscopic limit of a chain of atoms governed by the Newton equation. It is known from the work of Blanc, Le Bris, Lions, that this limit is the solution of a nonlinear wave equation, as long as this solution remains smooth. We show, numerically and mathematically that, if the distances between particles remain bounded, it is not the case any more when there are shocks -at least for a convex nearest-neighbour interaction potential with convex derivative.
We are concerned with the non-normal Schrodinger operator H = -Delta + V on L-2 (R-n), where V is an element of W-loc(1,infinity) (R-n) and Rev vc(x)>= c vertical bar x vertical bar(2-) d for some c, d > 0. The spectrum of this operator is discrete and its real part is bounded below by -d. In general, the s-pseudospectrum of H will have an unbounded component for any epsilon > 0 and thus will not approximate the spectrum in a global sense. By exploiting the fact that the semigroup e(-tH) is immediately compact, we show a complementary result, namely that for every delta > 0, R > 0 there exists an epsilon > 0 such that the epsilon-pseudospectrum sigma(epsilon)(H) subset of {z: Re z > R} boolean OR boolean OR {z: vertical bar z-lambda vertical bar<delta} In particular, the unbounded part of the pseudospectrum escapes towards +infinity as epsilon decreases. In addition, we give two examples of non-selfadjoint Schrodinger operators outside of our class and study their pseudospectra in more detail.
The aim of this article is to prove a Beals type characterization theorem for pseudodifferential operators in Wiener spaces. The definition of pseudodifferential operators in Wiener spaces and a Calder\'on-Vaillancourt type result appear in [1]. The set of symbols considered here is the one of [1]. The Weyl calculus in infinite dimension considered here emphasizes the role of the Wick bi-symbols.
We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schr{\"o}dinger equations. First, we give global variational characterizations of each of these periodic waves, which in particular provide alternate proofs of their orbital stability with respect to same-period perturbations, restricted to certain subspaces. Second, we prove the spectral stability of the cnoidal waves against same-period perturbations (in a certain parameter range), and provide an alternate proof of this (known) fact for the snoidal waves, which does not rely on complete integrability. Third, we give a rigorous version of a formal asymptotic calculation of Rowlands to establish the instability of a class of real-valued periodic waves in 1D, which includes the cnoidal waves of the 1D cubic focusing nonlinear Schr{\"o}dinger equation, against perturbations with period a large multiple of their fundamental period. Finally, we develop a numerical method to compute the minimizers of the energy with fixed mass and momentum constraints. Numerical experiments support and complete our analytical results.
We give some asymptotic results for the following storage allocation model: There are m primary spaces, ranked {1, 2, ... , m}, and R secondary spaces, ranked {m + 1, m + 2, ... , m + R}. Customers arrive according to a Poisson process with rate parameter lambda, a new arrival takes the lowest ranked space, and occupies it for an exponentially distributed amount of time, with mean 1/mu. Letting N-1 and N-2 be, respectively, the numbers of occupied primary and secondary spaces, we recently [15] obtained an explicit, albeit complicated, expression for the joint steady-state distribution, pi (k, r) = Prob[N-1 = k, N-2 = r]. We also obtained various asymptotic results [14] for this distribution, in the limit where rho equivalent to lambda/mu -> infinity, but with m, R = Theta(rho). Here we consider a different limit, where rho -> infinity witm = rho + O(root rho) and R = O(root rho). Then we obtain a limiting two-dimensional density as a leading order approximation to pi (k, r). We also examine various asymptotic properties of this density, including tail behaviors, and give numerical comparisons to the discrete model. Finally we show that the density satisfies a certain two-dimensional parabolic partial differential equation with appropriate boundary/corner conditions.
We formulate a model for a point defect embedded in a homogeneous multilattice crystal with an empirical interatomic potential interaction. Under a natural phonon stability assumption, we quantify the decay of the long-range elastic fields with increasing distance from the defect. These decay estimates are an essential ingredient in quantifying approximation errors in coarse-grained models and in the construction of optimal numerical methods for approximating crystalline defects.
Given a bounded open set Omega subset of R-d with Lipschitz boundary and an increasing family Gamma(t), t is an element of[0, T], of closed subsets of Omega, we analyze the scalar wave equationu <(u)double over dot> - div(A del u) = f in the time varying cracked domains Omega\Gamma(t). Here we assume that the sets Gamma(t) are contained into a prescribed (d - 1)-manifold of class C-2. Our approach relies on a change of variables: recasting the problem on the reference configuration Omega\Gamma(0), we are led to consider a hyperbolic problem of the form <(v)double over dot> - div(B del v)+ a . del v - 2b . del<(v) over dot> = g in Omega\Gamma(0). Under suitable assumptions on the regularity of the change of variables that transforms Omega\Gamma(t) into Omega\Gamma(0), we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by- product, the continuous dependence of the solutions with respect to the cracks.
We study the Aharonov-Bohm effect (the AB effect) in quantum resonances for scattering by three solenoids in two dimensions under the situation that the centers of the solenoids are placed almost in line. The resonances are shown to be generated near the real axis by the trapping trajectories when the centers are largely separated from one another. We analyze how the AB effect is reflected in the location of these resonances. The results are described by use of the backward scattering amplitude by one solenoid and depend on the ratio of the distances between the solenoids as well as on the magnetic fluxes. The method makes a full use of the information from the scattering system by one solenoid, which is known to be exactly solvable in quantum mechanics. We also discuss the case of four solenoids.
In this article, we study the speed of convergence of the supercell reduced Hartree-Fock (rHF) model towards the whole space rHF model in the case where the crystal contains a local defect. We prove that, in a "small defect" asymptotic regime, the defect energy in a supercell model converges to the full rHF defect energy with speed L-1 when the defect is charged, where L-3 is the volume of the supercell. The main contribution is identified as the Leslie-Gillan correction term when the crystal is isotropic cubic. We also prove the result in the non-isotropic case.
We consider the focusing cubic nonlinear Schrodinger equation (NLS) in the exterior O of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if E(u(0)) M(u(0)) < E(Q) M(Q) and parallel to del u(0)parallel to(2)parallel to u(0)parallel to(2) < parallel to del Q parallel to(2)parallel to Q parallel to(2), the corresponding solution to the initial value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, Q(x) denotes the ground state for the focusing cubic NLS in R-3.
We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable controls can be extended to analytic functions onsome square of C, and conversely, that analytic functions defined on a certain disk can be reached by using boundary controlsthat are Gevrey functions of order 2. The method of proof combines the flatness approach with some new Borel interpolation theorem in some Gevrey class witha specified value of the loss in the uniform estimates of the successive derivatives of the interpolating function.
We consider standing waves in the focusing nonlinear Schrodinger (NLS) equation on a dumbbell graph (two rings attached to a central line segment subject to the Kirchhoff boundary conditions at the junctions). In the limit of small $L^2$ norm, the ground state (the orbitally stable standing wave of the smallest energy at a fixed $L^2$ norm) is represented by a constant solution. However, when the $L^2$ norm is increased, this constant solution undertakes two bifurcations, where the first is the pitchfork (symmetry breaking) bifurcation and the second one is the symmetry preserving bifurcation. As a result of the first symmetry breaking bifurcation, the standing wave becomes more localized in one of the two rings. As a result of the second symmetry preserving bifurcation, the standing wave becomes localized in the central line segment. In the limit of large norm solutions, both standing waves are represented by a truncated solitary wave localized in either the ring or the central line segment. Although the asymmetric wave supported in the ring is a ground state near the symmetry breaking bifurcation of the constant solution, it is the symmetric wave supported in the central line segment which becomes the ground state in the limit of large $L^2$ norm. The analytical results are confirmed by numerical approximations of the ground state on the dumbbell graph.
This paper examines the optimal lending and hedging decisions of a bank facing uncertain returns on its loans. The bank's preferences are state-dependent in that the utility function depends on a state variable, i.e., the business cycle of the economy. The purpose of this paper is to complement the results of the banking literature. To characterize the bank's optimal use of financial instruments to hedge, we show that the concept of expectation dependence [13] is useful. While the current hedging literature specifies price risk as a monotonically increasing or decreasing function of the state-variable plus noise, expectation dependence provides much more general bivariate dependence structure. The bank's optimal futures position is an under-hedge or an over-hedge, depending on whether the random return on loans is positively or negatively correlated with the business cycle of the economy in the sense of expectation dependence, respectively. The bank as such takes dependencies into consideration when devising its optimal hedging strategy.
In this work, we propose a new numerical procedure for the simulation of timedependent problems based on the coupling between the finite element method (FEM) and the lattice Boltzmann method. The procedure exploits the Parareal paradigm to efficiently couple the two numerical methods, allowing independent grid size and time-step size. The motivations behind this approach are wide-ranging. In particular, one technique may be more efficient or physically more appropriate or less memory consuming than the other depending on the target of the simulation and/or on the sub-region of the computational domain. Furthermore, the coupling with FEM may circumvent some difficulties inherent to lattice Boltzmann discretization, for some domains with complex boundaries, or for some kind of boundary conditions. The theoretical and numerical framework is presented for the time-dependent heat equation in order to describe and validate numerically the methodology in a simple situation.
We prove a new polynomial lower bound on the scattering resolvent. For that, we construct a quasimode localized on a trajectory. which is trapped in the past, but not in the future. The power in the bound is expressed in terms of the maximal Lyapunov exponent on., and gives the minimal number of derivatives lost in exponential decay of solutions to the wave equation.
The aim of this article is to justify mathematically, in the two-dimensional periodic setting, a generalization of a two-phase model with pressure dependent viscosity first proposed by A. Lefebvre-Lepot and B. Maury to describe a system in one dimension of aligned spheres interacting through lubrication forces. This model involves an adhesion potential, apparent only on the congested domain, which keeps track of history of the flow. The solutions are constructed (through a singular limit) from a compressible Navier-Stokes system with viscosity and pressure both singular close to a maximal volume fraction. Interestingly, this study can be seen as the first mathematical connection between models of granular flows and models of suspensions. As a by-product of this result, we also obtain global existence of weak solutions for a system of incompressible Navier-Stokes equations with pressure dependent viscosity, the adhesion potential playing a crucial role in this result.
This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).
