Combinatorial proofs of some properties of tangent and Genocchi numbers

The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to $k$-ary trees, leading to a new generalization of the Genocchi numbers

Nuclear β+\beta^+/EC decays in covariant density functional theory and the impact of isoscalar proton-neutron pairing

Self-consistent proton-neutron quasiparticle random phase approximation based on the spherical nonlinear point-coupling relativistic Hartree-Bogoliubov theory is established and used to investigate the $\beta^+$/EC-decay half-lives of neutron-deficient Ar, Ca, Ti, Fe, Ni, Zn, Cd, and Sn isotopes. The isoscalar proton-neutron pairing is found to play an important role in reducing the decay half-lives, which is consistent with the same mechanism in the $\beta$ decays of neutron-rich nuclei. The experimental $\beta^+$/EC-decay half-lives can be well reproduced by a universal isoscalar proton-neutron pairing strength.Comment: 12 pages, 4 figure

Seeing the Unobservable: Channel Learning for Wireless Communication Networks

Wireless communication networks rely heavily on channel state information (CSI) to make informed decision for signal processing and network operations. However, the traditional CSI acquisition methods is facing many difficulties: pilot-aided channel training consumes a great deal of channel resources and reduces the opportunities for energy saving, while location-aided channel estimation suffers from inaccurate and insufficient location information. In this paper, we propose a novel channel learning framework, which can tackle these difficulties by inferring unobservable CSI from the observable one. We formulate this framework theoretically and illustrate a special case in which the learnability of the unobservable CSI can be guaranteed. Possible applications of channel learning are then described, including cell selection in multi-tier networks, device discovery for device-to-device (D2D) communications, as well as end-to-end user association for load balancing. We also propose a neuron-network-based algorithm for the cell selection problem in multi-tier networks. The performance of this algorithm is evaluated using geometry-based stochastic channel model (GSCM). In settings with 5 small cells, the average cell-selection accuracy is 73% - only a 3.9% loss compared with a location-aided algorithm which requires genuine location information.Comment: 6 pages, 4 figures, accepted by GlobeCom'1