Strong noise sensitivity and random graphs

The noise sensitivity of a Boolean function describes its likelihood to flip under small perturbations of its input. Introduced in the seminal work of Benjamini, Kalai and Schramm [Inst. Hautes \'{E}tudes Sci. Publ. Math. 90 (1999) 5-43], it was there shown to be governed by the first level of Fourier coefficients in the central case of monotone functions at a constant critical probability $p_c$. Here we study noise sensitivity and a natural stronger version of it, addressing the effect of noise given a specific witness in the original input. Our main context is the Erd\H{o}s-R\'{e}nyi random graph, where already the property of containing a given graph is sufficiently rich to separate these notions. In particular, our analysis implies (strong) noise sensitivity in settings where the BKS criterion involving the first Fourier level does not apply, for example, when $p_c\to0$ polynomially fast in the number of variables.Comment: Published at http://dx.doi.org/10.1214/14-AOP959 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Class invariants for quartic CM fields

One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct S-units in certain abelian extensions of K, where S is effectively determined by K. It also yields class polynomials for primitive quartic CM fields whose coefficients are S-integers.Comment: 14 page

Ultrasmall volume Plasmons - yet with complete retardation effects

Nano particle-plasmons are attributed to quasi-static oscillation with no wave propagation due to their subwavelength size. However, when located within a band-gap medium (even in air if the particle is small enough), the particle interfaces are acting as wave-mirrors, incurring small negative retardation. The latter when compensated by a respective (short) propagation within the particle substantiates a full-fledged resonator based on constructive interference. This unusual wave interference in the deep subwavelength regime (modal-volume<0.001lambda^3) significantly enhances the Q-factor, e.g. 50 compared to the quasi-static limit of 5.5.Comment: 16 pages, 6 figure

Decomposing and re-composing lightweight compression schemes - and why it matters

We argue for a richer view of the space of lightweight compression schemes for columnar DBMSes: We demonstrate how even simple simple schemes used in DBMSes decompose into constituent schemes through a columnar perspective on their decompression. With our concrete examples, we touch briefly on what follows from these and other decompositions: Composition of alternative compression schemes as well as other practical and analytical implications

A STRINGENT CONSTRAINT ON ALTERNATIVES TO A MASSIVE BLACK HOLE AT THE CENTER OF NGC 4258

There is now dynamical evidence for massive dark objects at the center of several galaxies, but suggestions that these are supermassive black holes are based only on indirect astrophysical arguments. The recent unprecedented measurement of the rotation curve of maser emission sources at the center of NGC 4258, and the remarkable discovery that it is Keplerian to high precision, provides us a unique opportunity for testing alternatives to a BH (e.g., a massive cluster of stellar remnants, brown dwarfs, low-mass stars, or halo dark matter). We use a conservative upper limit on the systematic deviation from a Keplerian rotation curve to constrain the mass distribution at the galaxy center. Based on evaporation and physical collision time-scale arguments, we show that a central cluster is ruled out, *unless* the cluster consists of *extremely* dense objects with mass less than about 0.05 solar masses (e.g., low mass BHs or elementary particles). Since both of these dynamically-allowed systems are very improbable for other astrophysical reasons, we conclude that a central dense cluster at the center of NGC 4258 is *very* improbable, thus leaving the alternative possibility of a massive BH. We also show that the mass of the BH must be at least 98% of the mass enclosed within the inner edge of the masering disk (3.6*10^7 solar masses). A substantial contribution to that mass from a density cusp in the background mass distribution is excluded.Comment: Submitted to ApJ (Letters) on March 15, 1995. 11 pages including 1 figure; uuencoded, compressed postscript

BCC vs. HCP - The Effect of Crystal Symmetry on the High Temperature Mobility of Solid 4^4He

We report results of torsional oscillator (TO) experiments on solid $^4$He at temperatures above 1K. We have previously found that single crystals, once disordered, show some mobility (decoupled mass) even at these rather high temperatures. The decoupled mass fraction with single crystals is typically 20- 30%. In the present work we performed similar measurements on polycrystalline solid samples. The decoupled mass with polycrystals is much smaller, $\sim$ 1%, similar to what is observed by other groups. In particular, we compared the properties of samples grown with the TO's rotation axis at different orientations with respect to gravity. We found that the decoupled mass fraction of bcc samples is independent of the angle between the rotation axis and gravity. In contrast, hcp samples showed a significant difference in the fraction of decoupled mass as the angle between the rotation axis and gravity was varied between zero and 85 degrees. Dislocation dynamics in the solid offers one possible explanation of this anisotropy.Comment: 10 pages, 5 figures, to appear in Journal of Low Temperature Physics - special issue on Supersolidit