William O. Bray

Kinetic model for phototransduction and G-protein enzyme cascade: understanding quantal bumps during inhibition of CaM-KII or PP2B

We utilize a computer model derived directly from the phototransductive enzyme cascade involving rhodopsin (R), G-proteins (G) and ultimately the membrane channels. In photoreceptors, single photons activate single rhodopsin molecules (R*), resulting in the production of quantal bumps. A CaM-KII inhibitor may inhibit whereas calcineurin (PP2B) inhibitors may augment arrestin activity. Arrestin is known to inactivate R*. We voltage-clamped and digitized quantal bumps for comparison with the kinetic model and simulations. Inhibitors of CaM-KII and PP2B produce results consistent with altering arrestin activity. In sum, we have succeeded in constructing a mathematical description of the phototransduction cascade that is biochemically derived and useful in interpreting results.

Growth properties of the Fourier transform

In a recent paper by the authors, growth properties of the Fourier transform on Euclidean space and the Helgason Fourier transform on rank one symmetric spaces of non-compact type were proved and expressed in terms of a modulus of continuity based on spherical means. The methodology employed first proved the result on Euclidean space and then, via a comparison estimate for spherical functions on rank one symmetric spaces to those on Euclidean space, we obtained the results on symmetric spaces. In this note, an analytically simple, yet overlooked refinement of our estimates for spherical Bessel functions is presented which provides significant improvement in the growth property estimates.

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.

Inversion of the horocycle transform on real hyperbolic spaces via a wavelet-like transform

It is proved that the horocycle transform Rf on real n-dimensional real hyperbolic space H is well-defined, for f ∈ Lp(H) if and only if 1 ≤ p < 2. Further, the function f can be recovered explicitly in Lp-norm and a.e. via a suitably defined wavelet like transform on the space of horocycles.

A spectral Paley-Wiener theorem

AbstractThe Fourier inversion formula in polar form is

A characterization theorem for the Fourier transform on Rn

f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda

Theperiodgene controls courtship song cycles inDrosophila melanogaster

for suitable functionsf on ℝn, wherePλf(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andPλf related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.

Analysis of divergence : control and management of divergent processes

We define a partial Radon transform mapping functions on