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Register a new userThe Log-Behavior of $sqrt[n]{p(n)}$ and $sqrt[n]{p(n)/n}$
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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 difference operator respect to $n$. Desalvo and Pak pointed out that their approach to proving the log-concavity of $p(n)$ may be employed to prove a conjecture of Sun on the log-convexity of ${r(n)}_{ngeq 61}$, as long as one finds an appropriate estimate of $Delta^2 log r(n-1)$. In this paper, we obtain a lower bound for $Delta^2log r(n-1)$, leading to a proof of this conjecture. From the log-convexity of ${r(n)}_{ngeq61}$ and ${sqrt[n]{n}}_{ngeq4}$, we are led to a proof of another conjecture of Sun on the log-convexity of ${sqrt[n]{p(n)}}_{ngeq27}$. Furthermore, we show that $limlimits_{n rightarrow +infty}n^{frac{5}{2}}Delta^2logsqrt[n]{p(n)}=3pi/sqrt{24}$. Finally, by finding an upper bound of $Delta^2 logsqrt[n-1]{p(n-1)}$, we prove an inequality on the ratio $frac{sqrt[n-1]{p(n-1)}}{sqrt[n]{p(n)}}$ analogous to the above inequality on the ratio $frac{p(n-1)}{p(n)}$.
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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 $n = 50$.
A direct experimental evidence of the occurrence of the weak reaction $Lambda nprightarrow nnp$ in nuclei has been obtained by the FINUDA experiment. Three events have been found that can be attributed to $^{7}_{Lambda}$Li and $^{9}_{Lambda}$Be two nucleon-induced non mesonic weak decays. The kinematic analysis of such events is presented here.
We studied the $^{12}$C(p,2p+n) reaction at beam momenta of 5.9, 8.0 and 9.0 GeV/c. For quasielastic (p,2p) events we reconstructed {bf p_f} the momentum of the knocked-out proton before the reaction; {bf p_f} was then compared (event-by-event) with {bf p_n}, the measured, coincident neutron momentum. For $|p_n|$ > k$_F$ = 0.220 GeV/c (the Fermi momentum) a strong back-to-back directional correlation between {bf p_f} and {bf p_n} was observed, indicative of short-range n-p correlations. From {bf p_n} and {bf p_f} we constructed the distributions of c.m. and relative motion in the longitudinal direction for correlated pairs. After correcting for detection efficiency, flux attenuation and solid angle, we determined that 49 $pm$ 13 % of events with $|p_f|$ > k_F had directionally correlated neutrons with $|p_n|$ > k$_F$. Thus short-range 2N correlations are a major source of high-momentum nucleons in nuclei.
Recently, a new technique for measuring short-range NN correlations in nuclei (NN SRCs) was reported by the E850 collaboration, using data from the EVA spectrometer at the AGS at Brookhaven Nat. Lab. In this talk, we will report on a larger set of data from new measurement by the collaboration, utilizing the same technique. This technique is based on a very simple kinematic approach. For quasi-elastic knockout of protons from a nucleus ($^{12}$C(p,2p) was used for the current work), we can reconstruct the momentum {bf p$_f$} of the struck proton in the nucleus before the reaction, from the three momenta of the two detected protons, {bf p$_1$} and {bf p$_2$} and the three momentum of the incident proton, {bf p$_0$} : {bf p$_f$} = {bf p$_1$} + {bf p$_2$} - {bf p$_0$} If there are significant n-p SRCs, then we would expect to find a neutron with momentum -{bf p$_f$} in coincidence with the two protons, provided {bf p$_f$} is larger than the Fermi momentum $k_F$ for the nucleus (${sim}$220 MeV/c for $^{12}$C). Our results reported here confirm the earlier results from the E850 collaboration.
We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, and the highest degree term condition. This unifies and extends the two earlier notions of spherical functors and split P^n-functors. We construct the P-twist of such F and prove it to be an autoequivalence. We then give a criterion for F to be a P^n-functor which is stronger than the definition but much easier to check in practice. It involves only two conditions: the strong monad condition and the weak adjoints condition. For split P^n-functors, we prove Segals conjecture on their relation to spherical functors. Finally, we give four examples of non-split P^n-functors: spherical functors, extensions by zero, cyclic covers, and family P-twists. For the latter, we show the P-twist to be the derived monodromy of associated Mukai flop, the so-called `flop-flop = twist formula.
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