Weighted discrete nonlinear Chebyshev approximation

Modifications are given to let the discrete REMEZ program for nnlinear Chebyshev approximation handle weights.     

Order of approximation by electrostatic fields due to electrons

LetD be a Jordan domain in the complex plane andA _q (D) the Bers space with norm ∥ ∥ _q . IfD is the unit disk, it is known that ∥S _n 0∥₂≥π/18, whereS _n =∑ _k=1 ⁿ l/(z?z _nk ) withz _nk ∈?D, so that approximation in ∥ ∥ _q ,q<-2, is not possible. In this paper, we give an order of estimate of ∥f?S _n ∥ _q for 2<q<∞ when ?D is a sufficiently smooth Jordan curve, and prove that this order of approximation is in general best possible.     

The spectrum of the Schrödinger operator and the distribution of primes

The distribution of the eigenvalues of the Schrödinger Operator is studied. It is found that when the underlying manifold is non-compact these eigenvalues can be generated asymptotically at the same rate as the primes.     

Chebyshev approximation by products

Characterization and uniqueness of minimax approximation by the product PQ of two finite dimensional subspaces P and Q is studied. Some approximants may have no standard characterization since PQ may not be a sun, but interior points do have the standard linear characterization.     

Linear transformations that preserve certain positivity classes of matrices

For each of several S ? R^n,n, those linear transformations $\text{L}\text{: R}{}^{\mathrm{n,n}}\text{→ R}{}^{\mathrm{n,n}}$ which map S onto S are characterized. Each class is a familiar one which generalizes the notion of positivity to matrices. The classes include: the matrices with nonnegative principal minors, the M-matrices, the totally nonnegative matrices, the D-stable matrices, the matrices with positive diagonal Lyapunov solutions, and the H-matrices, as well as other related classes. The set of transformations is somewhat different from case to case, but the strategy of proof, while differing in detail, is similar.