### A 1-norm bound for inverses of triangular matrices with monotone entries.

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In this paper we construct analytic-numerical solutions for initial-boundary value systems related to the equation ${u}_{t}-A{u}_{xx}-Bu=0$, where $B$ is an arbitrary square complex matrix and $A$ ia s matrix such that the real part of the eigenvalues of the matrix $\frac{1}{2}(A+{A}^{H})$ is positive. Given an admissible error $\epsilon $ and a finite domain $G$, and analytic-numerical solution whose error is uniformly upper bounded by $\epsilon $ in $G$, is constructed.

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

It is shown that if A is a bounded linear operator on a complex Hilbert space, then $w\left(A\right)\le 1/2\left(\right|\left|A\right||+|\left|A\xb2\right|{|}^{1/2})$, where w(A) and ||A|| are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.

We survey results related to the Kreiss Matrix Theorem, especially examining extensions of this theorem to Banach space and Hilbert space. The survey includes recent and established results together with proofs of many of the interesting facts concerning the Kreiss Matrix Theorem.