Non-constant Case Research Articles

Anomalous singularity of the solution of the vector Dyson equation in the critical case

The vector Dyson equation (VDE) is an equation of the form −1/m = z + Sm, where S is a fixed square matrix, z lies in the complex upper half-plane, and m is a vector-function of z. We consider the solution of the VDE in the case when S has a block staircase structure with (n − 1) different critical zero blocks below the strictly positive anti-diagonal and all elements right above the anti-diagonal are strictly positive. We prove that the components of m behave as fractional powers of z in the neighborhood of zero and show that the self-consistent density of states ρ(E) behaves as |E|−n−1n+1 as E tends to zero, where n2 is the number of blocks. Both constant block and non-constant block cases are considered. In the non-constant case, uniform estimates for the components of m are obtained.

The semi-dynamical reflection equation: solutions and structure matrices

Explicit solutions of the non-constant semi-dynamical reflection equation are constructed, together with suitable parametrizations of their structure matrices. Considering the semi-dynamical reflection equation with rational non-constant Arutyunov–Chekhov–Frolov structure matrices, and a specific meromorphic ansatz, it is found that only two sets of the previously found constant solutions are extendible to the non-constant case. In order to simplify future constructions of spin-chain Hamiltonians, a parametrization procedure is applied explicitly to all elements of the semi-dynamical reflection equation available. Interesting expressions for ‘twists’ and R-matrices entering the parametrization procedure are found. In particular, some expressions for the R-matrices seem to appear here for the first time. In addition, a new set of consistent structure matrices for the semi-dynamical reflection equation is obtained.

Generalized Cheeger sets related to landslides

We study the maximization problem, among all subsets X of a given domain $\Omega$ , of the quotient of the integral in X of a given function f by the integral on the boundary of X of another function g. This is a generalization of the well-known Cheeger problem corresponding to constant functions f,g. The non-constant case is motivated by applications to landslides modeling where the the supremum given by a variational blocking problem appears as a safety coefficient. We prove that this coefficient is equal to the supremum of the shape optimization problem formerly mentioned. For constant data, this amounts to studying the first eigenvalue of the 1-laplacian operator. We prove existence of optimal sets, and give some differential characterization of their internal boundary. We study their symmetry properties using the Steiner symmetrization. In dimension two, we give explicit solutions for data depending only on one variable.

V-Cycle Multigrid for Cell-Centered Finite Differences

We introduce and analyze a V-cycle multigrid algorithm for cell-centered finite difference methods applied to second-order elliptic boundary value problems. Unlike conventional cell-centered multigrid algorithms that use the natural injection operator for prolongation, we use a new prolongation operator whose energy norm we prove is bounded by 1 in the constant coefficient case and 1+Ch in the nonconstant case. We are thus able to use general finite element multigrid theory to conclude that the V-cycle either converges well or serves as a reasonably good preconditioner, respectively. While our theory does not establish optimal performance, our numerical experiments do show that the resulting algorithm converges much faster than the conventional schemes. In fact, these results show that the energy norm convergence factor is small and remains bounded uniformly in the finest mesh size, while that of the conventional algorithm grows.

Some Notes on Marshallian Supply Functions

This paper argues, in opposition to P. A. Samuelson, that Piero Sraffa's 1920s contribution to the cost controversy pleaded for abandoning the Marshallian approach in favor of a simultaneous determination of all prices. His critique distinguished those Marshallian supply functions lacking rigorous foundations from those based on sound arguments, pointing out the validity of some nonconstant cases and focusing on the analysis of the conditions that make the ceteris paribus assumption legitimate when the effects of variable returns cannot be neglected, rather than on the negligible size of these returns or on pure rhetoric, as Samuelson suggests. Sraffa's critique was soon misinterpreted along lines similar to those followed by Samuelson. Copyright 1991 by Royal Economic Society.

A central limit theorem for generalized discounting

A central limit theorem is derived for the discounted sum of a sequence of i.i.d. random variables with finite first and second moment, where the rate of interest may depend on time. For the special case of constant rate of interest GERBER (1971) obtains a Berry-Esseen result assuming finite third moment. The same result is shown to hold in the non-const ant case.