Polynomials constant on a hyperplane and CR maps of spheres

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D'Angelo, proved in two dimensions by D'Angelo, Kos and Riehl and in three dimensions by the authors. The current work builds upon these results to settle the conjecture in all dimensions. We also give a complete description of all polynomials in dimensions 4 and higher for which the sharp bound is obtained. The results prove the sharp degree bounds for monomial CR mappings of spheres in all dimensions.Comment: 17 pages, 10 figures; accepted to Illinois J. Math., added 3 figures and improved expositio

On Take It or Leave It Offers in Common Agency

If the agent's preference relation satisfies a strict monotonicity condition in common agency under the asymmetric information, the set of all equilibrium allocations in the menu game where menus of contracts are allowed coincides with the set of all equilibrium allocations in the single contract game where only single contracts are allowed.take it or leave it offers, menus, common agency, robust equilibrium allocations, mixed-strategy equilibrium

Exact sequences in the cohomology of a group extension

In [J. of Alg. 369: 70-95, 2012], the authors constructed a seven term exact sequence in the cohomology of a group extension G of a normal subgroup N by a quotient group Q with coefficients in a G-module M. However, they were unable to establish the precise link between the maps in that sequence and the corresponding maps arising from the spectral sequence associated to the group extension and the G-module M. In this paper, we show that there is a close connection between [J. of Alg. 369: 70-95, 2012] and our two earlier papers [J. of Alg. 72: 296-334, 1981] and [J. Reine Angew. Math. 321: 150-172, 1981]. In particular, we show that the results in the two papers just quoted entail that the maps of [J. of Alg. 369: 70-95, 2012] other than the obvious inflation and restriction maps do correspond to the corresponding ones arising from the spectral sequence.Comment: 13 page

Angular momentum content of a virtual graviton

We show that a virtual graviton has a J=0 component, which serves to cancel the J=2, J_z=0 component when the graviton is on shell. In contrast, a massive graviton has no J=0 component either on or off shell. This difference is responsible for the van Dam-Veltman-Zakharov discontinuity.Comment: 5 pages. Reference adde

An Additional Application of the Space Interferometry Mission to Gravitational Microlensing Experiments

Despite the detection of a large number of gravitational microlensing events, the nature of Galactic dark matter remains very uncertain. This uncertainty is due to two major reasons: the lens parameter degeneracy in the measured Einstein timescale and the blending problem in dense field photometry. Recently, consideration has been given to routine astrometric followup observations of lensing events using the {\it Space Interferometry Mission} (SIM) as a means of breaking the lens parameter degeneracy in microlensing events. In this paper, we show that in addition to breaking the lens parameter degeneracy, SIM observations can also be used to correct for nearly all types of blending. Therefore, by resolving both the problems of the lens parameter degeneracy and blending, SIM observations of gravitational lensing events will significantly better constrain the nature of Galactic dark matter.Comment: 17 pages, including 4 figures and no table, substantially modified from the original version, resubmitted to MNRA

Evaluations of some Toeplitz-type determinants

In this paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+\delta_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\ \det{[|j-k|+\delta_{jk}]_{1\leq j,k\leq n}}&= \begin{cases} \frac{1+(-1)^{(n-1)/2}n}{2}&\text{if}\ 2\nmid n,\\ \frac{1+(-1)^{n/2}}{2}&\text{if}\ 2\mid n, \end{cases} \end{align*} where $\delta_{jk}$ is the Kronecker delta. For complex numbers $a,b,c$ with $b\not=0$ and $a^2\not=4b$, and the sequence $(w_m)_{m\in\mathbb Z}$ with $w_{k+1}=aw_k-bw_{k-1}$ for all $k\in\mathbb Z$, we establish the identity $$\det[w_{j-k}+c\delta_{jk}]_{1\le j,k\le n} =c^n+c^{n-1}nw_0+c^{n-2}(w_1^2-aw_0w_1+bw_0^2)\frac{u_n^2b^{1-n}-n^2}{a^2-4b},$$ where $u_0=0$, $u_1=1$ and $u_{k+1}=au_k-bu_{k-1}$ for all $k=1,2,\ldots$.Comment: 22 pages.Add parts (ii) and (iii) of Theorem 1.

American VocalArts Quintet, Saturday 13, 1988

This is the concert program of the American VocalArts Quintet performance on Saturday, February 13, 1988 at 8:00 p.m., at the Concert Hall, 855 Commonwealth Avenue. Works performed were Counting Song by Zoltán Kodály, Thus starts the Kalevala by Z. Kodály, Maiden Beauty by Z. Kodály, Vor der Tür by Johannes Brahms, Es rauschet das Wasser by J. Brahms, Der Jäger und sein Liebchen by J. Brahms, Romancero Gitano by Mario Castelnuovo-Tedesco, and Zigeunerlieder, Op. 103 by J. Brahms. Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund