G. Cook | Searcharxiv

arXiv2018

Plane sections of Fermat surfaces over finite fields

In this paper, we characterize all curves over $\mathbb{F}_q$ arising from a plane section $$ \mathcal{P} : X_3-e_0X_0-e_1X_1-e_2X_2 = 0 $$ of the Fermat surface $$ \mathcal{S} : X_0^d + X_1^d + X_2^d +X_3^d = 0, $$ where $q = p^{h} = 2d+1$ is a prime power, $p >3$, and $e_0, e_1, e_2 \in \mathbb{F}_q$. In particular, we will prove that any nonlinear component $\mathcal{G} \subseteq \mathcal{P} \cap \mathcal{S} $ is a smooth classical curve of degree $n\leqslant d$ attaining the St\"ohr-Voloch bound $$ \# \mathcal{G}(\mathbb{F}_q) \leqslant \frac{1}{2} n(n+q-1) - \frac{1}{2} i(n-2), $$ with $i \in \{0,1,2,3,n,3n\}$.

math.AG↗

arXiv1998

Stable characteristic evolution of generic 3-dimensional single-black-hole spacetimes

We report new results which establish that the accurate 3-dimensional numerical simulation of generic single-black-hole spacetimes has been achieved by characteristic evolution with unlimited long term stability. Our results cover a selection of distorted, moving and spinning single black holes, with evolution times up to 60,000M.

gr-qc↗

arXiv1994

Degree Scale Anisotropy: SP94 Results

We present results from 2 observations of the CMB performed from the South Pole during the 93-94 austral summer. Each observation employed a 3$^{\circ}$ peak to peak sinusoidal, single difference chop and consisted of a $20^{\circ}\times 1^{\circ}$ strip on the sky near our SP91 observations. The first observation used a Q-band receiver which operates in 3 bands between 38 and 45 GHz with a $1^{\circ}$ to $1_{\cdot}^{\circ }15$ FWHM beam. The second observation overlapped the first observation and used a Ka-band receiver which operates in 4 bands between 26 and 36 GHz with a $1_{\cdot}^{\circ}25$ to $1_{\cdot}^{\circ}7$ FWHM beam. Significant correlated structure is observed in all bands for each observation. The spectrum of the structure is consistent with a CMB spectrum; however, the data do not discriminate against flat or inverted spectrum point sources. The root mean square amplitude ($\pm 1σ$) of the combined (Ka+Q) data is $ΔT_{rms}=42.0_{-6.8}^{+15.8}$ $μ$K in a band of multipoles between $\ell=68^{+38}_{-32}$. A band power estimate of the CMB power spectrum, $C_\ell$, gives $\left\langle \frac{C_\ell \ell (\ell+1)}{2π}\right\rangle_B= 1.77_{-0.54}^{+1.58}\times 10^{-10}$. The band power estimates for the seperate Ka and Q-band results are larger than but consistent with the band power estimate of the combined Ka-band SP91 results.

astro-ph↗