© 2016 American Mathematical Society. We investigate conditions that are sufficient to make the Extalgebra of an object in a (triangulated) category into a Frobenius algebra, and compute the corresponding Nakayama automorphism. As an application, we prove the conjecture that hdet(μA) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.

By analyzing a data set of 2.92 fb$^{-1}$ of $e^+e^-$ collision data taken at $\sqrt s= 3.773~\rm GeV$ and 106.41$\times 10^{6}$ $\psi(3686)$ decays taken at $\sqrt s= 3.686~\rm GeV$ with the BESIII detector at the BEPCII collider, we measure the branching fraction and the partial decay width for $\psi(3770)\to\gamma\chi_{c0}$ to be ${\mathcal B}(\psi(3770)\to\gamma\chi_{c0})=(6.88\pm0.28\pm0.67)\times 10^{-3}$ and $\Gamma[\psi(3770)\to\gamma\chi_{c0}]=(187\pm8\pm19)~\rm keV$, respectively. These are the most precise measurements to date.

We extract the $e^+e^-\rightarrow \pi^+\pi^-$ cross section in the energy range between 600 and 900 MeV, exploiting the method of initial state radiation. A data set with an integrated luminosity of 2.93 fb$^{-1}$ taken at a center-of-mass energy of 3.773 GeV with the BESIII detector at the BEPCII collider is used. The cross section is measured with a systematic uncertainty of 0.9%. We extract the pion form factor $|F_\pi|^2$ as well as the contribution of the measured cross section to the leading order hadronic vacuum polarization contribution to $(g-2)_\mu$. We find this value to be $a_\mu^{\pi\pi,\rm LO}(600-900\;\rm MeV) = (368.2 \pm 2.5_{\rm stat} \pm 3.3_{\rm sys})\cdot 10^{-10}$.

We study $D_{s}^{+}$ decays to final states involving the $\eta'$ with a 482$\,$pb$^{-1}$ data sample collected at $\sqrt{s}$ = 4.009$\,$GeV with the \mbox{BESIII} detector at the BEPCII collider. We measure the branching fractions $\mathcal{B}(D^+_{s}\rightarrow \eta'X)$ = (8.8$\pm$1.8$\pm$0.5)$\%$ and $\mathcal{B}(D_{s}^{+}\rightarrow \eta'\rho^{+})$ = ($5.8\pm1.4\pm0.4$)$\%$ where the first uncertainty is statistical and the second is systematic. In addition, we estimate an upper limit on the non-resonant branching ratio $\mathcal{B}(D_{s}^{+}\rightarrow \eta'\pi^+\pi^0)<5.1\%$ at the 90$\%$ confidence level. Our results are consistent with CLEO's recent measurements and help to resolve the disagreement between the theoretical prediction and CLEO's previous measurement of $\mathcal{B}(D_{s}^{+}\rightarrow \eta'\rho^{+})$.

By analyzing 2.93 fb$^{-1}$ data collected at the center-of-mass energy $\sqrt s=3.773$ GeV with the BESIII detector, we measure the absolute branching fraction of the semileptonic decay $D^+\rightarrow\bar K^0 e^{+}\nu_{e}$ to be ${\mathcal B}(D^{+}\rightarrow\bar K^0 e^{+}\nu_{e})=(8.59 \pm 0.14 \pm 0.21)\%$ using $\bar K^0\to K^0_S\to \pi^0\pi^0$, where the first uncertainty is statistical and the second systematic. Our result is consistent with previous measurements within uncertainties.