Reduction of mm-Regular Noncrossing Partitions

In this paper, we present a reduction algorithm which transforms $m$-regular partitions of $[n]=\{1, 2, ..., n\}$ to $(m-1)$-regular partitions of $[n-1]$. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures. For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers

Collaborative rating allocation

This paper studies the collaborative rating allocation problem, in which each user has limited ratings on all items. These users are termed "energy limited". Different from existing methods which treat each rating independently, we investigate the geometric properties of a user's rating vector, and design a matrix completion method on the simplex. In this method, a user's rating vector is estimated by the combination of user profiles as basis points on the simplex. Instead of using Euclidean metric, a non-linear pull-back distance measurement from the sphere is adopted since it can depict the geometric constraints on each user's rating vector. The resulting objective function is then efficiently optimized by a Riemannian conjugate gradient method on the simplex. Experiments on real-world data sets demonstrate our model's competitiveness versus other collaborative rating prediction methods

Charmless two-body B decays: A global analysis with QCD factorization

In this paper, we perform a global analysis of $B \to PP$ and $PV$ decays with the QCD factorization approach. It is encouraging to observe that the predictions of QCD factorization are in good agreement with experiment. The best fit $\gamma$ is around $79^\circ$. The penguin-to-tree ratio $|P_{\pi \pi}/T_{\pi \pi}|$ of $\pi^+ \pi^-$ decays is preferred to be larger than 0.3. We also show the confidence levels for some interesting channels: $B^0 \to \pi^0 \pi^0$, $K^+ K^-$ and $B^+ \to \omega \pi^+$, $\omega K^+$. For $B \to \pi K^\ast$ decays, they are expected to have smaller branching ratios with more precise measurements.Comment: 20 pages, 4 figures, version to appear in Phys. Rev.