Energy translation and Proper-Time Eigenstates

The usual quantum mechanics describes the mass eigenstates. To describe the proper-time eigenstates, a duality theory of the usual quantum mechanics was developed. The time interval is treated as an operator on an equal footing with the space interval, and the quantization of the space-time intervals between events is obtained. As a result, one can show that there exists a zero-point time interval.Comment: 15 pages, No figur

Utility-maximization Resource Allocation for Device-to-Device Communication Underlaying Cellular Networks

Device-to-device(D2D) underlaying communication brings great benefits to the cellular networks from the improvement of coverage and spectral efficiency at the expense of complicated transceiver design. With frequency spectrum sharing mode, the D2D user generates interference to the existing cellular networks either in downlink or uplink. Thus the resource allocation for D2D pairs should be designed properly in order to reduce possible interference, in particular for uplink. In this paper, we introduce a novel bandwidth allocation scheme to maximize the utilities of both D2D users and cellular users. Since the allocation problem is strongly NP-hard, we apply a relaxation to the association indicators. We propose a low-complexity distributed algorithm and prove the convergence in a static environment. The numerical result shows that the proposed scheme can significant improve the performance in terms of utilities.The performance of D2D communications depends on D2D user locations, the number of D2D users and QoS(Quality of Service) parameters

The structure of invertible substitutions on a three-letter alphabet

AbstractWe study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as Iw∘σ1∘σ2∘⋯∘σk, where Iw is the inner automorphism associated with w, and σj∈S for 1⩽j⩽k. As a consequence, M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices

4-(4-Hydroxy­phenyl­diazen­yl)­benzonitrile

The molecule of the title compound, C13H9N3O, is achiral but forms a chiral arrangement in the crystal structure. The mol­ecule adopts an E configuration with respect to the N=N bond and is almost planar, with an r.m.s. deviation of 0.0439 Å from the plane through all atoms in the mol­ecule. The dihedral angle between the two benzene rings is 2.2 (2)°. In the crystal structure, inter­molecular O—H⋯N hydrogen bonding generates a chain