We consider in two dimensions, the inverse boundary problem of reconstructing the absorption and scattering coefficient of an inhomogeneous medium by probing it with diffuse light. The problem is modeled as an inverse boundary problem for the stationary linear Boltzmann equation. The information is encoded in the albedo operator. We show that we can recover the absorption and the scattering kernel from this information provided that the latter is small in an appropriate topology. We also give stability estimates and propose an approximate reconstruction procedure.

We consider several dynamic equations and present methods on how to solve these equations. Among them are linear equations of higher order, Euler-Cauchy equations of higher order, logistic equations (or Verhulst equations), Bernoulli equations, Riccati equations, and Clairaut equations. In order to solve Bernoulli dynamic equations, we define an important product on the set of positively regressive functions and give a power rule in terms of this product.

The purpose of this note is to show that microlocal techniques can also be applied to the study of the pseudospectra of matrices (1.2) such as that discussed in a recent paper by Trefethen and Chapman [18] (and generalizations). In this light we interpret the twist condition as Hormander's solvability condition on the Poisson bracket of the real and imaginary parts of the symbol of a pseudodifferential operator. The connection between Hormander's condition and pseudospectra was first made by M. Zworski in [21]. In this paper we construct pseudomodes for Berezin-Toeplitz operators under condition (1.4) on the (smooth) symbol. Although we will discuss our results in detail in the next section, we should mention some limitations of our work. The methods of Trefethen and Chapman apply to rough symbols, f, and they obtain exponentially small error terms. For analytic symbols, it is very likely that exponentially small estimates (in the Toeplitz setting) can be achieved by microlocal methods, as has been done in [10] for pseudodifferential operators. The problem of dealing with general non-smooth symbols, however, is much more challenging. Trefethen and Chapman's main theorem includes a global condition on the symbol (in addition to 1.2), and they present compelling numerical evidence that global conditions on non-smooth symbols are necessary for the existence of "good" pseudomodes [18]. This is a very interesting issue that we do not address here. On the other hand, our results for smooth symbols are fairly general and include a number of cases not covered by the results in [18] (e. g. the "Scottish flag" matrix). Furthermore, the pseudomodes we construct are localized in phase space, sharpening the localization results of [18].

The author proves the W^i,p convergence of the radial minimizers u_E = (u_E1, u_E2, u_E3) of an energy function as EPSILON goes to 0, and the zeros of the u_E1 ² + u_E2² are located roughly. In addition, the estimates of the convergent rate of u_E3 ² are presented.

We study the minimizer of the d-wave Ginzburg-Landau energy in a specific class of functions. We show that the minimizer having distinct degree-one vortices is Holder continuous. Away from vortex cores, the minimizer converges uniformly to a canonical harmonic map. For a single vortex in the vortex core, we obtain the C^1/2-norm estimate of the fourfold symmetric vortex solution. Furthermore, we prove the convergence of the fourfold symmetric vortex solution under different scales of DELTA.

The traveling wave problem for a viscous conservation law with a nonlinear source term leads to a singularly perturbed problem which necessarily involves a non-hyperbolic point. The correponding slow-fast system indicates the existence of canard solutions which follow both stable and unstable parts of the slow manifold. In the present paper we show that for the viscous equation there exist such heteroclinic waves of canard type. Moreover, we determine their wave speed up to first order in thesmall viscosity parameter by a Melnikov-like calculation after a blow-up near the non-hyperbolic point. It is also shown that there are discontinuous waves of the inviscid equation which do not have a counterpart in the viscous case.

In this paper we study self-similar solutions for nonlinear Schrodinger equations using a scaling technique and the partly contractive mapping method. We establish the small global well-posedness of the Cauchy problem for nonlinear Schrodinger equations in some non-reflexive Banach spaces which contain many homogeneous functions. This we do by establishing some a priori nonlinear estimates in Besov spaces, employing the mean difference characterization and multiplication in Besov spaces. These new global solutions to nonlinear Schrodinger equations with small data admit a class of self-similar solutions. Our results improve and extend the well-known results of Planchon [18], Cazenave and Weissler [4, 5] and Ribaud and Youssfi [20].

We define Periodic Beurling Boehmians and analytic Boehmians. We also define boundary value of an analytic Boehmian as a periodic Beurling Boehmian and prove that each periodic Beurling Boehmian x can be written as x₊ + x_- where x₊ and x_- are boundary values of certain analytic Boehmians.