1,206 research outputs found
Perturbation analysis of Markov modulated fluid models
We consider perturbations of positive recurrent Markov modulated fluid
models. In addition to the infinitesimal generator of the phases, we also
perturb the rate matrix, and analyze the effect of those perturbations on the
matrix of first return probabilities to the initial level. Our main
contribution is the construction of a substitute for the matrix of first return
probabilities, which enables us to analyze the effect of the perturbation under
consideration
Existence and Uniqueness of Perturbation Solutions to DSGE Models
We prove that standard regularity and saddle stability assumptions for linear approximations are sufficient to guarantee the existence of a unique solution for all undetermined coefficients of nonlinear perturbations of arbitrary order to discrete time DSGE models. We derive the perturbation using a matrix calculus that preserves linear algebraic structures to arbitrary orders of derivatives, enabling the direct application of theorems from matrix analysis to prove our main result. As a consequence, we provide insight into several invertibility assumptions from linear solution methods, prove that the local solution is independent of terms first order in the perturbation parameter, and relax the assumptions needed for the local existence theorem of perturbation solutions.Perturbation, matrix calculus, DSGE, solution methods, Bézout theorem; Sylvester equations
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
Unusual square roots in the ghost-free theory of massive gravity
A crucial building block of the ghost free massive gravity is the square root
function of a matrix. This is a problematic entity from the viewpoint of
existence and uniqueness properties. We accurately describe the freedom of
choosing a square root of a (non-degenerate) matrix. It has discrete and (in
special cases) continuous parts. When continuous freedom is present, the usual
perturbation theory in terms of matrices can be critically ill defined for some
choices of the square root. We consider the new formulation of massive and
bimetric gravity which deals directly with eigenvalues (in disguise of
elementary symmetric polynomials) instead of matrices. It allows for a
meaningful discussion of perturbation theory in such cases, even though certain
non-analytic features arise.Comment: 24 pages; minor changes, final versio
Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schr\"odinger Operators
We announce three results in the theory of Jacobi matrices and Schr\"odinger
operators. First, we give necessary and sufficient conditions for a measure to
be the spectral measure of a Schr\"odinger operator -\f{d^2}{dx^2} +V(x) on
with and boundary condition.
Second, we give necessary and sufficient conditions on the Jacobi parameters
for the associated orthogonal polynomials to have Szeg\H{o} asymptotics.
Finally, we provide necessary and sufficient conditions on a measure to be the
spectral measure of a Jacobi matrix with exponential decay at a given rate.Comment: 10 page
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