Local trace formulae and scaling asymptotics in Toeplitz quantization, II

In the spectral theory of positive elliptic operators, an important role is played by certain smoothing kernels, related to the Fourier transform of the trace of a wave operator, which may be heuristically interpreted as smoothed spectral projectors asymptotically drifting to the right of the spectrum. In the setting of Toeplitz quantization, we consider analogues of these, where the wave operator is replaced by the Hardy space compression of a linearized Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz operators. We study the local asymptotics of these smoothing kernels, and specifically how they concentrate on the fixed loci of the linearized dynamics.Comment: Typos corrected. Slight expository change

Scaling asymptotics for quantized Hamiltonian flows

In recent years, the near diagonal asymptotics of the equivariant components of the Szeg\"{o} kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of this theme, here we consider the local scaling asymptotics of the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically how they concentrate on the graph of the underlying classical map

Propagating and evanescent internal waves in a deep ocean model

We present experimental and computational studies of the propagation of internal waves in a stratified fluid with an exponential density profile that models the deep ocean. The buoyancy frequency profile $N(z)$ (proportional to the square root of the density gradient) varies smoothly by more than an order of magnitude over the fluid depth, as is common in the deep ocean. The nonuniform stratification is characterized by a turning depth $z_c$, where $N(z_c)$ is equal to the wave frequency $\omega$ and $N(z < z_c) < \omega$. Internal waves reflect from the turning depth and become evanescent below the turning depth. The energy flux below the turning depth is shown to decay exponentially with a decay constant given by $k_c$, which is the horizontal wavenumber at the turning depth. The viscous decay of the vertical velocity amplitude of the incoming and reflected waves above the turning depth agree within a few percent with a previously untested theory for a fluid of arbitrary stratification [Kistovich and Chashechkin, J. App. Mech. Tech. Phys. 39, 729-737 (1998)].Comment: 13 pages, 4 figures, 4 table

Local trace formulae and scaling asymptotics in Toeplitz quantization

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szeg\"o kernel along the diagonal

Preferential utilization of endogenous arachidonate by cyclo-oxygenase in incubations of human platelets

AbstractThromboxane B2 (TXB2) and 12-hydroxy-5,8,10,14-eicosatetraenoic acid (12-HETE) formed from the endogenous and exogenous arachidonate during human platelet incubation, was evaluated by selected ion monitoring (SIM). TXB2 formed from endogenous substrate accounted for about one third of the total, whereas the great part of 12-HETE derived from exogenous arachidonate. These data indicate that under the tested conditions the pool of arachidonate that acts as substrate for cyclo-oxygenase is different from the pool that acts as substrate for lipoxygenase and that the arachidonate released from phospholipids is preferentially utilized by cyclo-oxygenase