On automorphisms of Enriques surfaces and their entropy

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.Comment: 16 pages / v2: some refinements on the theorems and the proofs, and also attached plain text lists of Salem numbers related to the proble

Arithmeticity of Certain Symplectic Hypergeometric Groups

We give a sufficient condition on a pair of (primitive) integral polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.Comment: 31 pages, 4 tables (To appear in Duke Mathematical Journal

New Optical Splitter Design for Network Scalability and Flexibility

In this paper, we present the design of two optical splitter which are 2x3 optical splitter-combine and Multi Ratio Optical Splitter (MROS) based on waveguide with SU-8 polymer material. The technology ensures low cost fabrication and excellent performance as compared to conventional technique that uses fused coupler for optical splitter fabrication. Both optical splitter-combine and MROS are suggested to be used in passive customer access network FTTH-PON. The optical splitter-combine is used to variety the signals send to the premises and as for MROS, it is used in power management in minimizing the power loss. The problems arose such as the loss of output power is also discussed. The simulation result shows that factor such as wide-angle branching and refractive index will increase the output power loss. To obtain maximum power, the optimization of those parameters’ value must be used. This designed optical splitter is operated at wavelength 1500 nm – 1600 nm. In this study, we characterized the trend of total output power with the effect of the parameters studied. Keywords: Optical splitter-combine, Multi Ratio Optical Splitter, waveguide, SU-8 Polymer, parameter

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima