Approximately Minwise Independence with Twisted Tabulation

A random hash function $h$ is $\varepsilon$-minwise if for any set $S$, $|S|=n$, and element $x\in S$, $\Pr[h(x)=\min h(S)]=(1\pm\varepsilon)/n$. Minwise hash functions with low bias $\varepsilon$ have widespread applications within similarity estimation. Hashing from a universe $[u]$, the twisted tabulation hashing of P\v{a}tra\c{s}cu and Thorup [SODA'13] makes $c=O(1)$ lookups in tables of size $u^{1/c}$. Twisted tabulation was invented to get good concentration for hashing based sampling. Here we show that twisted tabulation yields $\tilde O(1/u^{1/c})$-minwise hashing. In the classic independence paradigm of Wegman and Carter [FOCS'79] $\tilde O(1/u^{1/c})$-minwise hashing requires $\Omega(\log u)$-independence [Indyk SODA'99]. P\v{a}tra\c{s}cu and Thorup [STOC'11] had shown that simple tabulation, using same space and lookups yields $\tilde O(1/n^{1/c})$-minwise independence, which is good for large sets, but useless for small sets. Our analysis uses some of the same methods, but is much cleaner bypassing a complicated induction argument.Comment: To appear in Proceedings of SWAT 201

Direct observation of 4+ to 2+ gamma transition in 8Be

The low lying states in $^8$Be are believed to have a two-alpha cluster structure and hence a large intrinsic quadrupole deformation. An earlier calculation showed a large collective enhancement in gamma transition probability between the low lying states leading to a 4$^+$ to 2$^+$ gamma branch of $\sim10^{-7}$ and a resonant radiative cross section of 134 nb for the $\alpha+\alpha$ entrance channel. We report here the first experimental evidence for this transition through a $\gamma-\alpha-\alpha$ coincidence measurement in the reaction $^4$He($\alpha,\alpha \gamma$)$^4$He using a gas target. The measured cross sections on and off the 4$^+$ resonance are 165 $\pm$ 41 (stat) $\pm$35 (sys) nb and 39 $\pm$ 25 (stat) $\pm$7 (sys) nb, respectively.Comment: Total 4 pages, 4 figures, in RevTeX format, submitted to PR

Measurement of the damping of nuclear shell effect in the doubly magic 208^{208}Pb region

The damping of the nuclear shell effect with excitation energy has been measured through an analysis of the neutron spectra following the triton transfer in the $^7$Li induced reaction on $^{205}$Tl. The measured neutron spectra demonstrate the expected large shell correction energy for the nuclei in the vicinity of doubly magic $^{208}$Pb and a small value for $^{184}$W. A quantitative extraction of the allowed values of the damping parameter $\gamma$, along with those for the asymptotic nuclear level density parameter $\tilde{a}$, has been made for the first time.Comment: 5 pages, 5 figure

Astrophysical S_{17}(0) factor from a measurement of d(7Be,8B)n reaction at E_{c.m.} = 4.5 MeV

Angular distribution measurements of $^2$H($^7$Be,$^7$Be)$^2$H and $^2$H($^7$Be,$^8$B)$n$ reactions at $E_{c.m.}\sim$~4.5 MeV were performed to extract the astrophysical $S_{17}(0)$ factor using the asymptotic normalization coefficient (ANC) method. For this purpose a pure, low emittance $^7$Be beam was separated from the primary $^7$Li beam by a recoil mass spectrometer operated in a novel mode. A beam stopper at 0$^{\circ}$ allowed the use of a higher $^7$Be beam intensity. Measurement of the elastic scattering in the entrance channel using kinematic coincidence, facilitated the determination of the optical model parameters needed for the analysis of the transfer data. The present measurement significantly reduces errors in the extracted $^7$Be(p,$\gamma$) cross section using the ANC method. We get $S_{17}$~(0)~=~20.7~$\pm$~2.4 eV~b.Comment: 15 pages including 3 eps figures, one figure removed and discussions updated. Version to appear in Physical Review