A computational method, based on l1-norm minimization, is proposed for the problem traffic flow correction. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corruptions accurately if there are only a few bad sensors which are located at certain links by the proposed method. We mathematically quantify these links that are robust to miscounts and relate them to the geometric structure of the traffic network. In a more realistic setting, besides the unhealthy link sensors, if small measure noise are present at the other sensors, our method guarantees to give an estimated traffic flow fairly close to the ground-truth. Both toy and real-world examples are provided to demonstrate the effectiveness of the proposed method.

In this paper, we consider the three-dimensional Schrdinger operator with a -interaction of strength > 0 supported on an unbounded surface parametrized by the mapping R2x(x,f(x)), where 0, and f:R2R, f 0, is a C2-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrdinger operator coincides with 142,+. We prove that for all sufficiently small > 0, its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit 0+. In particular, this eigenvalue tends to 142 exponentially fast as 0+.

We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources.

In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary condition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem. This model well describes the behavior at the far-field for many physical and engineering systems such as fluid dynamical equation and electro-magnetic equation. The results obtained here should provide some theoretical guidance for designing effective far-field boundary conditions.

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L p-L q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n>= 3, the global existence and decay rates of solutions to the Cauchy problem are