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Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $mathbb{X}^{n-1} subset mathbb{P}^n(mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the `celestial horizon lies near $d geqslant 2n$. For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain: [ d ,geqslant, big(sqrt{n},{sf log},nbig)^n, ] and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain: [ d ,geqslant, big(n,{sf log},nbig)^n. ] The latter improves $d geqslant n^{2n}$ obtained by Merker in arxiv.org/1807/11309/. Admitting a certain technical conjecture $I_0 geqslant widetilde{I}_0$, the method employed (Diverio-Merker-Rousseau, Berczi, Darondeau) conducts to constant power $n$, namely to: [ d ,geqslant, 2^{5n} qquad text{and, respectively, to:} qquad d ,geqslant, 4^{5n}. ] In Spring 2019, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_0 geqslant widetilde{I}_0$, a conjecture which will be established up to dimension $n = 50$.
Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $frac{p(n-1)}{p(n)}left(1+frac{1}{n}right)>frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=sqrt[n]{p(n)/n}$ and $Delta$ be the differe
Recent Wilkinson Microwave Anisotropy Probe (WMAP) measurements have determined the baryon density of the Universe $Omega_b$ with a precision of about 4%. With $Omega_b$ tightly constrained, comparisons of Big Bang Nucleosynthesis (BBN) abundance pre
We show that the geodesic diameter of a polygonal domain with n vertices can be computed in O(n^4 log n) time by considering O(n^3) candidate diameter endpoints; the endpoints are a subset of vertices of the overlay of shortest path maps from vertices of the domain.
In population protocols, the underlying distributed network consists of $n$ nodes (or agents), denoted by $V$, and a scheduler that continuously selects uniformly random pairs of nodes to interact. When two nodes interact, their states are updated by
We present the first results from an XMM-Newton serendipitous medium-deep survey, which covers nearly three square degrees. We show the log(N)-log(S) distributions for the 0.5-2, 2-10 and 5-10 keV bands, which are found to be in good agreement with p