Minimal H\"older regularity implying finiteness of integral Menger curvature

We study two families of integral functionals indexed by a real number $p > 0$. One family is defined for 1-dimensional curves in $\R^3$ and the other one is defined for $m$-dimensional manifolds in $\R^n$. These functionals are described as integrals of appropriate integrands (strongly related to the Menger curvature) raised to power $p$. Given $p > m(m+1)$ we prove that $C^{1,\alpha}$ regularity of the set (a curve or a manifold), with $\alpha > \alpha_0 = 1 - \frac{m(m+1)}p$ implies finiteness of both curvature functionals ($m=1$ in the case of curves). We also show that $\alpha_0$ is optimal by constructing examples of $C^{1,\alpha_0}$ functions with graphs of infinite integral curvature

Counter Simulations via Higher Order Quantifier Elimination: a preliminary report

Quite often, verification tasks for distributed systems are accomplished via counter abstractions. Such abstractions can sometimes be justified via simulations and bisimulations. In this work, we supply logical foundations to this practice, by a specifically designed technique for second order quantifier elimination. Our method, once applied to specifications of verification problems for parameterized distributed systems, produces integer variables systems that are ready to be model-checked by current SMT-based tools. We demonstrate the feasibility of the approach with a prototype implementation and first experiments.Comment: In Proceedings PxTP 2017, arXiv:1712.0089

The Most Luminous z~9-10 Galaxy Candidates yet Found: The Luminosity Function, Cosmic Star-Formation Rate, and the First Mass Density Estimate at 500 Myr

[abridged] We present the discovery of four surprisingly bright (H_160 ~ 26 - 27 mag AB) galaxy candidates at z~9-10 in the complete HST CANDELS WFC3/IR GOODS-N imaging data, doubling the number of z~10 galaxy candidates that are known, just ~500 Myr after the Big Bang. Two similarly bright sources are also detected in a systematic re-analysis of the GOODS-S data set. Three of the four galaxies in GOODS-N are significantly detected at 4.5-6.2sigma in the very deep Spitzer/IRAC 4.5 micron data, as is one of the GOODS-S candidates. Furthermore, the brightest of our candidates (at z=10.2+-0.4) is robustly detected also at 3.6 micron (6.9sigma), revealing a flat UV spectral energy distribution with a slope beta=-2.0+-0.2, consistent with demonstrated trends with luminosity at high redshift. The abundance of such luminous candidates suggests that the luminosity function evolves more significantly in phi_* than in L_* at z>~8 with a higher number density of bright sources than previously expected. Despite the discovery of these luminous candidates, the cosmic star formation rate density for galaxies with SFR >0.7 M_sun/yr shows an order-of-magnitude increase in only 170 Myr from z ~ 10 to z ~ 8, consistent with previous results. Based on the IRAC detections, we derive galaxy stellar masses at z~10, finding that these luminous objects are typically 10^9 M_sun. The cosmic stellar mass density at z~10 is log10 rho_* = 4.7^+0.5_-0.8 M_sun Mpc^-3 for galaxies brighter than M_UV~-18. The remarkable brightness, and hence luminosity, of these z~9-10 candidates highlights the opportunity for deep spectroscopy to determine their redshift and nature, demonstrates the value of additional search fields covering a wider area to understand star-formation in the very early universe, and highlights the opportunities for JWST to map the buildup of galaxies at redshifts much earlier than z~10.Comment: 20 pages, 12 figures, changed to match resubmitted version to Ap