Preservation of logarithmic concavity by the Mellin transform and applications to the Schrödinger equation for certain classes of potentials

We prove that the Mellin transform of a function log-concave (convex) is, after division by (+1), where is the argument of the transform, itself log-concave (convex) in . This theorem is first applied to the moments of the ground state wave function of the Schrödinger equation where the Laplacian of the central potential has a given sign, and generalized to other situations. This is used to derive inequalities linking thel ^th derivative of the ground state wave function at the origin for angular momentuml and the expectation value of the kinetic energy, and applied to quarkonium physics. A generalization to higher radial excitations is shown to be plausible by using the WKB approximation. Finally, new bounds on ground-state energies in power potentials are obtained.     

Generalized Lax pairs,the modified classical Yang-Baxter equation,and affine geometry of Lie groups

We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=–(M)L, where is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in gV. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV ^*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.     

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.