On The Hilbert Coefficients of Fiber Cones

Let （R, m） be a Cohen-Macaulay local ring of dimension d, I an mprimary ideal and K an ideal containing I. Assuming that I has minimal （almost minimal） mixed multiplicity with respect to K, we get some results on the Hilbert coefficients of fiber cones.

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

Let(A,m)be a Noetherian local ring with maximal ideal m,I an ideal of A,J an m-primary ideal of A,N a finitely generated A-module,S=■n≥0 Sn a finitely generated standard graded algebra over A and M=■n≥0 Mn a finitely generated graded S-module.We characterize the multiplicity and the Cohen-Macaulayness of the fiber cone F(M)=■n≥0 Mn/JM_(n).As an application,we obtain some results on the multiplicity and the Cohen--Macaulayness of the fiber cone■n≥0 I^(n)N/JI^(n)N.

The flame retardancy of alginate/flame retardant viscose fibers investigated by vertical burning test and cone calorimeter

In this research, the flame retardancy of neat alginate fiber, flame retardant viscose fiber （FRV） and alginate/FRV （50/50） blending fibers were investigated by vertical burning and cone calorimeter tests. The vertical burning test showed that the afterflame time of alginate fiber was 0 s, but alginate presented serious smoldering behavior with the afterglow time of 605 s and damaged length of 85 mm, while the afterglow time of FRV was 0 s. When the FRV was incorporated into alginate with the weight ratio of 50/ 50, the afterglow time and damaged length were significantly reduced to 85 s and 35 mm, indicating the smoldering of alginate can be effectively decreased. The morphology and chemical structure of the alginate residual demonstrated that it was seriously destroyed during smoldering process, which was ascribed to its relative low initial thermal degradation temperature. Based on the thermal properties analysis, alginate and FRV fibers shared the concurrence of rapid degradation in the same temperature region of 250-300 ℃, through which, the compact and stable char formed by FRV can prevent the heat transmission and suppress the smoldering of alginate. Further, the cone calorimeter results demonstrated that the time to ignition （TTI） significantly increased and peak heat release rate （PHRR） decreased for alginate/FRV （50/50） compared with FRV. With this research, a new method to overcome the smoldering of alginate was proposed by blending with FRV

On the Depth and Hilbert Series of the Fiber Cone

Let （R,m） be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ？ 1 and λ（ JKKIIkk-1 ） = 1. We show that if depth G（I） ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK（I） assuming that depth G（I） ≥ d - 1.

On Hilbert Coefficients of Filtrations

Let F be a Hilbert filtration with respect to a Cohen-Macaulay module M. When G（F, M） and FK（F,M） have almost maximal depths, the Hilbert coefficients gi（F, M） is calculated. In the general case, an upper bound for g2（F, M） is also given.

Hilbert Coefficients of Filtrations with Almost Maximal Depth

Let F be a Hilbert filtration with respect to a Cohen-Macaulay R-module M. When G(F,M) and FK(F,M) have almost maximal depths, we show that the length λ(KInM/KJIn-1M) and the reduction number rJK (F,M) are independent of J. Lower bounds for the first and second Hilbert coefficients are obtained.