For standard graded Artinian K-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.

Perturbation bounds in numerical linear algebra are traditionally derived and expressed using norms. Norm bounds cannot reflect the scaling or sparsity of a problem and its perturbation, and so can be unduly weak. If the problem data and its perturbation are measured componentwise, much smaller and more revealing bounds can be obtained. A survey is given of componentwise perturbation theory in ...

Efficient numerical methods for pricing American options using Heston’s stochastic volatility model is proposed. Based on this model the price of a European option can be obtained by solving a two-dimensional parabolic partial differential equation. For an American option the early exercise possibility leads to a lower bound for the price of the option. This price can be computed by solving a l...

A set of easily verifiable sufficient conditions are derived for the existence of a positive (componentwise) solution of a linear nonhomogeneous system of equations in which the coefficient matrix can be partitioned into submatrices with positive elements.

For interval linear algebraic systems Ax = b, the problem is considered of componentwise evaluation of the set Σ∃∃(A,b) = {Ab | A∈A, b∈b } formed by all solutions of Ax = b when A and b vary independently in A and b, respectively. An iterative PSS algorithm is introduced that computes optimal (exact) componentwise estimates of Σ∃∃ and its convergence is proved under fairly general conditions on...

Let (R,m) be a standard graded K-algebra whose defining ideal is componentwise linear. Using Gröbner basis techniques, bounds for the depth and the regularity of the symmetric algebra Sym(m) are given. MSC 2000: 13H10, 13P10, 13D02

We prove that it is NP-hard to compute the exact componentwise bounds on solutions of all the linear systems which can be obtained from a given linear system with a nonsingular matrix by perturbing all the data independently of each other within prescribed tolerances.

We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals. INTRODUCTION Let S = K[x1, . . . ,xn] denote the polynomial ring in n variables over a field K with each degxi = 1. Let I be a monomial ideal of S and G(I) = {u1, . . . ,us} its unique minimal system of monomial generators. The Ta...