Borel-fixed ideals and reduction number

The aim of this paper is to study the relationship between reduction numbers and Borel-fixed ideals in all characteristics. By definition, Borel-fixed ideals are closed under certain specializations which is similar to the strong stability. We will estimate the number of monomials which can be specialized to a given monomial. As a consequence, we obtain a combinatorial version of the well-known Eakin-Sathaye's theorem which bounds the reduction number in terms of the Hilbert function. Furthermore, we show that the bound of Eakin-Sathaye's theorem is attained by the reduction number of a lex-segment monomial ideal. This result answers a question of Conca in the affirmative. We will also show that the reduction number of the lex-segment ideal is bounded exponentially by the reduction number of the given ideal.Comment: 11 page

DESIGN OF A POLARIZATION-INDEPENDENT MMI SOI COUPLER BASED MICRORESONATOR USING SANDWICH STRUCTURES

A microresonator based on multimode interference (MMI) couplers using sandwich structures is described in this paper. The birefringence can be controlled by introducing an additional dielectric layer within a silicon channel waveguide. It is shown that a polarization-independent MMI coupler based ring resonator can be achieved by choosing appropriate structure parameters