Decomposition of Lagrangian classes on K3 surfaces

We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subsets of the Kähler cone. Using the same technique, we show that the Kähler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.https://arxiv.org/abs/2001.00202Othe

Spatial trends of noncollinear exchange coupling mediated by itinerant carriers with different Fermi surfaces

We study the exchange coupling mediated by itinerant carriers with spin-orbit interaction by both analytic and numeric approaches. The mediated exchange coupling is noncollinear and its spatial trends depend on the Fermi-surface topology of the itinerant carriers. Taking Rashba interaction as an example, the exchange coupling is similar to the conventional Ruderman-Kittel-Kasuya-Yosida type in weak coupling. On the other hand, in the strong coupling, the spiral interaction dominates. In addition, inclusion of finite spin relaxation always makes the noncollinear spiral exchange interaction dominant. Potential applications of our findings are explained and discussed

Evaluation of Formal posterior distributions via Markov chain arguments

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function $\phi$ of a parameter when the loss is quadratic. If the posterior mean of $\phi$ is admissible for all bounded $\phi$, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient conditions for recurrence of general state space Markov chains that are also of independent interest. Our main example concerns the $p$-dimensional multivariate normal distribution with mean vector $\theta$ when the prior distribution has the form $g(\|\theta\|^2) d\theta$ on the parameter space $\mathbb{R}^p$. Conditions on $g$ for strong admissibility of the posterior are provided.Comment: Published in at http://dx.doi.org/10.1214/07-AOS542 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multi-Operator Fairness in Transparent RAN Sharing by Soft-Partition With Blocking and Dropping Mechanism

Radio access network (RAN) sharing has attracted significant attention from telecom operators as a means of accommodating data surges. However, current mechanisms for RAN sharing ignore the fairness issue among operators, and hence the RAN may be under- or over-utilized. Furthermore, the fairness among different operators cannot be guaranteed, since the RAN resources are distributed on a first come, first served basis. Accordingly, the present study proposes a “soft-partition with blocking and dropping” (SBD) mechanism that offers inter-operator fairness using a “soft-partition” approach. In particular, the operator subscribers are permitted to overuse the resources specified in the predefined service-level-agreement when the shared RAN is under-utilized, but are blocked (or even dropped) when the RAN is over-utilized. The simulation results show that SBD achieves an inter-operator fairness of 0.997, which is higher than that of both a hard-partition approach (0.98) and a no-partition approach (0.6) while maintaining a shared RAN utilization rate of 98%. Furthermore, SBD reduces the blocking rate from 35% (hard partition approach) to almost 0%, whereas controlling the dropping rate at 5%. Notably, the dropping rate can be reduced to almost 0% using a newly proposed bandwidth scale down procedure.This work was supported in part by H2020 collaborative Europe/Taiwan research project 5G-CORAL under Grant 761586, and in part by the Ministry of Science and Technology, Taiwan under Contract MOST 106-2218- E-009-018

Dimensionality's blessing: Clustering images by underlying distribution

Many high dimensional vector distances tend to a constant. This is typically considered a negative "contrast-loss" phenomenon that hinders clustering and other machine learning techniques. We reinterpret "contrast-loss" as a blessing. Re-deriving "contrast-loss" using the law of large numbers, we show it results in a distribution's instances concentrating on a thin "hyper-shell". The hollow center means apparently chaotically overlapping distributions are actually intrinsically separable. We use this to develop distribution-clustering, an elegant algorithm for grouping of data points by their (unknown) underlying distribution. Distribution-clustering, creates notably clean clusters from raw unlabeled data, estimates the number of clusters for itself and is inherently robust to "outliers" which form their own clusters. This enables trawling for patterns in unorganized data and may be the key to enabling machine intelligence.Comment: Accepted in CVPR 201