A graded K-algebra R has property Np if it is generated in degree 1, has relations in degree 2 and the syzygies of order ≤ p on the relations are linear. The GreenLazarsfeld index of R is the largest p such that it satisfies the property Np. Our main results assert that (under a mild assumption on the base field) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index ≥ c+1. T...

In this study, free vibration of functionally graded rectangular plates for various types of boundary conditions has been presented . The properties of the plate are assumed as power- law form along the thickness direction , while poisson's ratio is kept constant. the linear vibration equations of functionally graded rectangular plates are derived based on first order shear deformation theory b...

We experimentally demonstrate, for the first time, propagation loss reduction via graded-index (GRIN) cladding layers in high-index-contrast (HIC) glass waveguides. We show that scattering loss arising from sidewall roughness can be significantly reduced without compromising the high-index-contrast condition, by inserting thin GRIN cladding layers with refractive indices intermediate between th...

A sample of 78 Pinus radiata D. Don logs was measured and sawn to maximise yields of 100 x 50-mm and 200 x 50-mm timber. The average bending strength and stiffness of the timber were found to depend on the basic density of the log but the lower fifth-percentile strength was more dependent on branch index. For the 28-year-old stand sampled, logs with a low branch index yielded 100 x 50-mm timber...

We use the structure of finite-dimensional graded algebras to develop theory antilinear representations finite $C_2$-graded groups. A group is a with subgroup index 2. In this acts linearly, while other coset antilinearly. introduce blocks, whose crucial component theory. Among things, we study characters and Frobenius-Schur indicators. As an example, describe $A_n \leq S_n$.

We present a method for ultra-low-loss coupling between two single-mode optical fibers with different mode-field diameters using multimode interference in a graded-index multimode optical fiber. We perform a detailed analysis of the interference effects and show that the graded-index fiber can also be used as a beam expander or condenser. The results are important for devices in which optical f...