Spot size converters (SSCs) are essential optical components that connect waveguides with different modal spot sizes for adiabatic light coupling. However, previous SSCs usually suffer from large or limited coupling efficiencies, which not conducive to develop large-scale and high-density photonic integrated circuits (PICs). Herein, we theoretically study an ultra-compact SSC a footprint of onl...

in this paper‎, ‎we introduce the notion of $(m,n)$-‎algebr‎aically compact modules as an analogue of algebraically‎ ‎compact modules and then we show that $(m,n)$-algebraically‎ ‎compactness‎ ‎and $(m,n)$-pure injectivity for modules coincide‎. ‎moreover‎, ‎further characterizations of a‎ ‎$(m,n)$-pure injective module over a commutative ring are given‎.

Let K be a locally compact hypergroup. In this paper we initiate the concept of fundamental domain in locally compact hypergroups and then we introduce the Borel section mapping. In fact, a fundamental domain is a subset of a hypergroup K including a unique element from each cosets, and the Borel section mapping is a function which corresponds to any coset, the related unique element in the fun...

We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.