Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm. In alternative, one can construct different distance func...

The problem of computing the distance in the Frobenius norm of a given real irreducible tridiagonal matrix T to the algebraic variety of real normal irreducible tridiagonal matrices is solved. Simple formulas for computing the distance and a normal tridiagonal matrix at this distance are presented. The special case of tridiagonal Toeplitz matrices also is considered.

j=1 yj φ(‖x − xj‖2), x ∈ R , where φ: [0,∞) → R is some given function, (yj) n 1 are real coefficients, and the centres (xj) n 1 are points in R. For a wide class of functions φ, it is known that the interpolation matrix A = (φ(‖xj − xk‖2)) n j,k=1 is invertible. Further, several recent papers have provided upper bounds on ‖A‖2, where the points (xj) n 1 satisfy the condition ‖xj − xk‖2 ≥ δ, j ...

To avoid the undesired effects of distance concentration in high-dimensional spaces, previous work has already advocated the use of fractional p norms instead of the ubiquitous Euclidean norm. Closely related to concentration is the emergence of hub and anti-hub objects. Hub objects have a small distance to an exceptionally large number of data points while anti-hubs lie far from all other data...

Recently, in [1], it has been proved that if B is an open unit ball in a Cartesian product l2 × l2 furnished with the lp-norm ‖ · ‖ and kB is the Kobayashi distance on B, then the metric space (B,kB) is locally uniformly convex in linear sense. Our construction of domains, which are locally uniformly convex in their Kobayashi distances, is based on the ideas from [1]. Such domains play an impor...

Here Modified Rodrigues Parameters (MRPs) are used for an implementation that avoids making a small angle approximation for the attitude ambiguity. The direct averaging of MRPs is inaccurate because the distance metric between two MRPs is nonlinear to second order. However, the distance metric between two quaternions can be shown to be linear with the addition of a unit norm constraint on the q...

We describe new lower bounds for the complexity of the directed Hausdorr distance under translation and rigid motion. We exhibit lower bound constructions of (n 3) for point sets under translation, for the L 1 , L 2 and L 1 norms, (n 4) for line segments under translation , for any L p norm, (n 5) for point sets under rigid motion and (n 6) for line segments under rigid motion, both for the L 2...

A new general algorithm for computing distance transforms of digital images is presented. The algorithm consists of two phases. Both phases consist of two scans, a forward and a backward scan. The first phase scans the image column-wise, while the second phase scans the image row-wise. Since the computation per row (column) is independent of the computation of other rows (columns), the algorith...

The Johnson-Lindenstrauss Lemma is a classic result which implies that any set of n real vectors can be compressed to O(log n) dimensions while only distorting pairwise Euclidean distances by a constant factor. Here we consider potential extensions of this result to the compression of quantum states. We show that, by contrast with the classical case, there does not exist any distribution over q...