Jiu Ding

Eigenvalues of rank-one updated matrices with some applications

We prove a spectral perturbation theorem for rank-one updated matrices of special structure. Two applications of the result are given to illustrate the usefulness of the theorem. One is for the spectrum of the Google matrix and the other is for the algebraic simplicity of the maximal eigenvalue of a positive matrix.

Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjecture to multi-dimensional transformations

Abstract We prove that Ulams piecewise constant approximation algorithm is convergent for computing an absolutely continuous invariant measure associated with a piecewise C2 expanding transformation or a Jablonski transformation S: [0, 1]N ⊂ RN → [0, 1]N. This solves an extension of Ulams conjecture to multi-dimensions and generalizes the convergence result given by T.-Y. Li for one-dimensional transformations.

Representations for Moore-Penrose Inverses in Hilbert Spaces

Abstract Let H1, H2 be two Hilbert spaces, and let T : H1 → H2 be a bounded linear operator with closed range. We present some representations of the perturbation for the Moore-Penrose inverse in Hilbert spaces for the case that the perturbation does not change the range or the null space of the operator.

On the perturbation of the least squares solutions in Hilbert spaces

Abstract Let H1 and H2 be Hilbert spaces, and let T : H1 → H2 be a bounded linear operator with closed range. We present some results of the perturbation analysis for the least squares solutions to the operator equation Tx = y.

Least Squares Approximations to Lognormal Sum Distributions

In this paper, the least squares (LS) approximation approach is applied to solve the approximation problem of a sum of lognormal random variables (RVs). The LS linear approximation is based on the widely accepted assumption that the sum of lognormal RVs can be approximated by a lognormal RV. We further derive the solution for the LS quadratic (LSQ) approximation, and our results show that the LSQ approximation exhibits an excellent match with the simulation results in a wide range of the distributions of the summands. Using the coefficients obtained from the LSQ method, we present the explicit closed-form expressions of the coefficients as a function of the decibel spread and the number of the summands by applying an LS curve fitting technique. Closed-form expressions for the cumulative distribution function and the probability density function for the sum RV, in both the linear and logarithm domains, are presented

New perturbation results on pseudo-inverses of linear operators in Banach spaces

Abstract Let X and Y be Banach spaces, let T : X → Y be a bounded linear operator with closed range, and let S : X → Y be another bounded linear operator. We study conditions on S – T that guarantee the closeness of the range of S and obtain some new bounds on the pseudo-inverse of S .

The Projection Method for Computing Multidimensional Absolutely Continuous Invariant Measures

We present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewiseC2 expanding mappingS:Ω→Ω on a bounded region Ω⊂RN. The method is based on the Galerkin projection principle for solving an operator equation in a Banach space. With the help of the modern notion of functions of bounded variation in multidimension, we prove the convergence of the algorithm.

Statistical Properties of Deterministic Systems

Foundations of Measure Theory.- Rudiments of Ergodic Theory.- Frobenius-Perron Operators.- Invariant Measures-Existence.- Invariant Measures-Computation.- Convergence Rate Analysis.- Entropy.- Applications of Invariant Measures.

Nonnegative matrices, positive operators, and applications

Elementary Properties of Nonnegative Matrices Perron-Frobenius Theory of Nonnegative Matrices Stochastic Matrices Applications of Nonnegative Matrices General Theory of Positive Operators Spectral Theory of Positive Operators Markov Operators Frobenius-Perron Operators Approximation of Positive Operators Applications of Positive Operators.

ON THE CONTINUITY OF GENERALIZED INVERSES OF LINEAR OPERATORS IN HILBERT SPACES

Abstract Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. We study the continuity problem of the generalized inverse of T and related least squares solutions to the operator equation Tx = y .