Optimized Tapered Fiber Decorated by Ag Nanoparticles for Raman Measurement with High Sensitivity

A tapered fiber decorated by Ag nanoparticles is prepared as a surface-enhanced Raman scattering (SERS) substrate. There are two key parameters during the preparation process, the fiber cone angle and the density of decorated AgNPs on the fiber tip surface. Their theoretical analysis on the forming mechanism and the optimization process is studied in detail. The tapered fibers with angles from 0.5 to 30° are successfully prepared, with a chemical method in a small tube using a bending interface. AgNPs with different densities are decorated on the surface of the tapered fibers with an electrostatic adsorption method. The optimized tapered fiber SERS probe with an angle of 12° and AgNPs density of 26.67% provides the detection of Rhodamine 6G (R6G) with 10−10 mol/L.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

Restricted classes of veronese type ideals and algebras

We study ideals which are generated by monomials of degree [Formula: see text] in the polynomial ring in [Formula: see text] variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese type, squarefree Veronese ideals or [Formula: see text]-spread Veronese ideals. In this paper we focus on [Formula: see text]-bounded [Formula: see text]-spread Veronese ideals and on Veronese ideals of bounded support. Their powers as well as their fiber cone are also considered.

Homology, mixed multiplies and Hilbert coefficients of the fiber cone

In this paper, we compare the depth of the fiber cone and the associated graded ring. To achieve this, we construct a bi-graded complex corresponding to a bi-graded, Noetherian, Hilbert filtration. The vanishing of the homology modules of this complex helps us to compare the depth of the fiber cone of the filtration and the depth of the corresponding associated graded ring. We also give a formula for the fiber coefficients in terms of the lengths of certain homology modules. We give an upper bound for the first fiber coefficient and show that when this bound is attained, the fiber cone has good depth.