Size of orthogonal sets of exponentials for the disk

Suppose \Lambda \subseteq \RR^2 has the property that any two exponentials with frequency from $\Lambda$ are orthogonal in the space $L^2(D)$, where D \subseteq \RR^2 is the unit disk. Such sets $\Lambda$ are known to be finite but it is not known if their size is uniformly bounded. We show that if there are two elements of $\Lambda$ which are distance $t$ apart then the size of $\Lambda$ is $O(t)$. As a consequence we improve a result of Iosevich and Jaming and show that $\Lambda$ has at most $O(R^{2/3})$ elements in any disk of radius $R$

Gaussian elimination as an iterative algorithm

Gaussian elimination (GE) for solving an $n \times n$ linear system of equations $Ax=b$ is the archetypical direct method of numerical linear algebra, as opposed to iterative. In this note we want to point out that GE has an iterative side too

Quantum Calculations On The Vibrational Predissociation Of NeBr2: Evidence For Continuum Resonances

Quantum mechanical calculations on the vibrational predissociation dynamics of NeBr2 in the B electronic state have been performed and the results compared with both experimental data and other computational studies. For vibrational levels with v less than or equal to 20 we find that the vibrational state dependence of the predissociation lifetimes is in qualitative agreement with experimental measurements, as are the calculated Br-2 fragment rotational distributions. For higher vibrational levels, the B \u3c-- X excitation profiles are well represented by a sum of two Lorentzian line shapes. We attribute this result to the presence of long-lived resonances in the dissociative continuum that are reminiscent of long-lived dissociative trajectories in previous classical studies of NeBr2. (C) 2000 American Institute of Physics. [S0021-9606(00)00205-1]

An extension of Chebfun to two dimensions

An object-oriented MATLAB system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form “chebfun2” objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented