On the Number of Affine Equivalence Classes of Spherical Tube Hypersurfaces

We consider Levi non-degenerate tube hypersurfaces in \CC^{n+1} that are $(k,n-k)$-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature $(k,n-k)$, with $n\le 2k$. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) $k=n-2$, $n\ge 7$;\linebreak (ii) $k=n-3$, $n\ge 7$; (iii) $k\le n-4$. For all other values of $k$ and $n$, except for $k=3$, $n=6$, the number of affine classes is known to be finite. The exceptional case $k=3$, $n=6$ has been recently resolved by Fels and Kaup who gave an example of a family of $(3,3)$-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels-Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels-Kaup family, and use the $j$-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces

Broadband optical gain via interference in the free electron laser: principles and proposed realizations

We propose experimentally simplified schemes of an optically dispersive interface region between two coupled free electron lasers (FELs), aimed at achieving a much broader gain bandwidth than in a conventional FEL or a conventional optical klystron composed of two separated FELs. The proposed schemes can {\it universally} enhance the gain of FELs, regardless of their design when operated in the short pulsed regime

Leveraging Social Networks in Direct Services: Are Foundations Doing All They Can?

· Social networks are critical to physical and mental health, and they shape how people see themselves and their possible futures. · Social networks represent an under-leveraged resource in social services’ efforts to alleviate poverty and other social challenges. · Foundations may be unintentionally creating barriers to practice that leverages social networks by incentivizing individually-focused, highly specific services delivered in standardized, replicable ways. · “Network-oriented” practice can help craft a new way forward that threads the needle between everything-is-different-for-everyone and everything- is-the-same-for-everyone. · By focusing funding on efforts that build and support social networks, foundations can deepen and sustain the impact of their funding

Methods of electron beam bunching

A review of electron beam bunching methods and new bunching methods are presented. Linear accelerator, storage ring and buncher technologies offer high-beam quality and short bunches necessary for operation of high efficiency, high-power, high-degree monochromatic prebunched FELs in optical and X-ray regions.Comment: 4 pages, LaTe