In this paper we consider the Interband Light Absorption Coefficient (ILAC), in a symmetric form, in the case of random operators on the d-dimensional lattice. We show that the symmetrized version of ILAC is either continuous or has a component which has the same modulus of continuity as the density of states.

It is known that both Watson-Nadaraya and Gasser-Muller types of regression estimators have some disadvantages. A smooth version of local polynomial regression estimators are proposed to remedy the disadvantages. The mean squared error and mean integrated squared errors are computed explicitly. It turns out that by suitably selecting a kernel and a bandwidth, the proposed estimator has at least...

The theory of Eisenstein series is fundamental for the spectral theory of automorphic forms. It was first developed by Selberg, and was completed by Langlands ([Lan76]; see also [MW95]). There are several known proofs for the meromorphic continuation of Eisenstein series (apart from very special cases of Eisenstein series which can be expressed in terms of Tate integrals). In all these proofs i...

In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Hölder space of positive exponent for any positive time. In Part II, we explore inverse problems that arise in attempting to construct an example of an initial velocity producing an arbitrarily poor modulus of continuity o...

This article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize Gaussian processes, sub-Gaussian processes, and random fields that are in the nth chaos of a Wiener process. An upper bound of Dudley type on the tail of the random field’s supremum is derived using a generic chaining argument; it implies similar resu...

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form ∂t−∆+ b ·∇ resp. −∆+ b ·∇ with a divergence-free drift b. We prove the Liouville theorem and Harnack inequality when b ∈ L∞(BMO) resp. b ∈ BMO−1 and provide a counterexample demonstra...

We introduce and study very general stochastic positive linear operators induced by general positive linear operators that are acting on continuous functions. These are acting on the space of real differentiable stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related stochastic Shisha–Mond type inequalities of L-type 1 ≤ q < ∞ a...

In this paper, an L°°(Ll )-error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order ft'/4 , where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for whic...

We examine solutions u = PI f to ∆u − V u = 0 on a Lipschitz domain Ω in a compact Riemannian manifold M , satisfying u = f on ∂Ω, with particular attention to ranges of (s, p) for which one has Besov-to-L-Sobolev space results of the form PI : B s (∂Ω) −→ Lps+1/p(Ω), and variants, when the metric tensor on M has limited regularity, described by a Hölder or a Dini-type modulus of continuity. We...

A nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals. A between class modulus of continuity captures the expected length of adaptive confidence intervals. Sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of ...