A lower bound in Nehari's theorem on the polydisc

By theorems of Ferguson and Lacey (d=2) and Lacey and Terwilleger (d>2), Nehari's theorem is known to hold on the polydisc D^d for d>1, i.e., if H_\psi is a bounded Hankel form on H^2(D^d) with analytic symbol \psi, then there is a function \phi in L^\infty(\T^d) such that \psi is the Riesz projection of \phi. A method proposed in Helson's last paper is used to show that the constant C_d in the estimate \|\phi\|_\infty\le C_d \|H_\psi\| grows at least exponentially with d; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc

Composition Operators and Endomorphisms

If $b$ is an inner function, then composition with $b$ induces an endomorphism, $\beta$, of $L^\infty(\mathbb{T})$ that leaves $H^\infty(\mathbb{T})$ invariant. We investigate the structure of the endomorphisms of $B(L^2(\mathbb{T}))$ and $B(H^2(\mathbb{T}))$ that implement $\beta$ through the representations of $L^\infty(\mathbb{T})$ and $H^\infty(\mathbb{T})$ in terms of multiplication operators on $L^2(\mathbb{T})$ and $H^2(\mathbb{T})$. Our analysis, which is based on work of R. Rochberg and J. McDonald, will wind its way through the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert $C^*$-modules

The Dirichlet-to-Robin Transform

A simple transformation converts a solution of a partial differential equation with a Dirichlet boundary condition to a function satisfying a Robin (generalized Neumann) condition. In the simplest cases this observation enables the exact construction of the Green functions for the wave, heat, and Schrodinger problems with a Robin boundary condition. The resulting physical picture is that the field can exchange energy with the boundary, and a delayed reflection from the boundary results. In more general situations the method allows at least approximate and local construction of the appropriate reflected solutions, and hence a "classical path" analysis of the Green functions and the associated spectral information. By this method we solve the wave equation on an interval with one Robin and one Dirichlet endpoint, and thence derive several variants of a Gutzwiller-type expansion for the density of eigenvalues. The variants are consistent except for an interesting subtlety of distributional convergence that affects only the neighborhood of zero in the frequency variable.Comment: 31 pages, 5 figures; RevTe

Does wage rank affect employees' well-being?

How do workers make wage comparisons? Both an experimental study and an analysis of 16,000 British employees are reported. Satisfaction and well-being levels are shown to depend on more than simple relative pay. They depend upon the ordinal rank of an individual's wage within a comparison group. “Rank” itself thus seems to matter to human beings. Moreover, consistent with psychological theory, quits in a workplace are correlated with pay distribution skewness

Modeling and characterization of the SPIDER half-wave plate

Spider is a balloon-borne array of six telescopes that will observe the Cosmic Microwave Background. The 2624 antenna-coupled bolometers in the instrument will make a polarization map of the CMB with approximately one-half degree resolution at 145 GHz. Polarization modulation is achieved via a cryogenic sapphire half-wave plate (HWP) skyward of the primary optic. We have measured millimeter-wave transmission spectra of the sapphire at room and cryogenic temperatures. The spectra are consistent with our physical optics model, and the data gives excellent measurements of the indices of A-cut sapphire. We have also taken preliminary spectra of the integrated HWP, optical system, and detectors in the prototype Spider receiver. We calculate the variation in response of the HWP between observing the CMB and foreground spectra, and estimate that it should not limit the Spider constraints on inflation