Symmetric Markov chains in ℤd: How fast can they move?

Summary Consider a reversible Markov chain X _n which takes values in a subset of ^d. If the steps of the chain are uniformly bounded and the invariant measure satisfies a mild regularity condition, Varopoulos, Carne and Kesten have obtained estimates on \lambda n^{1/2} )$$ " align="middle" border="0"> which exhibit a Gaussian tail in but blow up as n. Following Kesten's approach we derive bounds which are uniform in n in some special cases. Our main result, however, is an example which shows that in general the estimates of Varopoulos, Carne and Kesten are essentially best possible.Research partially supported by an S.E.R.C. (U.K.) visiting fellowship and an operating grant from N.S.E.R.C. of Canada     

Insights into dolphin sonar discrimination capabilities from human listening experiments

A variety of dolphin sonar discrimination experiments have been conducted, yet little is known about the cues utilized by dolphins in making fine target discriminations. In order to gain insights on cues available to echolocating dolphins, sonar discrimination experiments were conducted with human subjects using the same targets employed in dolphin experiments. When digital recordings of echoes from targets ensonified with a dolphinlike signal were played back at a slower rate to human subjects, they could also make fine target discriminations under controlled laboratory conditions about as well as dolphins under less controlled conditions. Subjects reported that time-separation-pitch and duration cues were important. They also reported that low-amplitude echo components 32 dB below the maximum echo component were usable. The signal-to-noise ratio had to be greater than 10 dB above the detection threshold for simple discrimination and 30 dB for difficult discrimination. Except for two cases in which spectral cues in the form of "click pitch" were important, subjects indicated that time-domain rather than frequency-domain processing seemed to be more relevant in analyzing the echoes.     

On the differentiability of maps into Lie transformation groups

Let G be a Lie transformation group on a manifold M. Then a map f: NG is differentiable, iff for every point pM the map q f(q)·p is differentiable.     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)