Let f1, . . . , fr ∈ K[x], K a field, be homogeneous polynomials and put F = ∑r i=1 yifi ∈ K[x, y]. The quotient J = K[x, y]/I, where I is the ideal generated by the ∂F/∂xi and ∂F/∂yj , is the Jacobian ring of F . We describe J by computing the cohomology of a certain complex whose top cohomology group is J .

W. H. Tutte’s 1947 paper on a ring generated by graphs satisfying a contraction-deletion relation is extended to ribbon graphs. This ring of ribbon graphs is a polynomial ring on an infinite set of one-vertex ribbon graphs. Acknowledgements: The LSU Research Experience for Undergraduates Program is supported by a National Science Foundation grant, DMS-0648064. Page 16 RHIT Undergrad. Math. J., ...

F-thresholds are defined by Mustaţǎ, Takagi and Watanabe, which are invariants of the pair of ideals on rings of characteristic p > 0. In their paper, it is proved F-thresholds equal to jumping numbers for the test ideal on regular local rings. In this note, we give formulas of F-thresholds and F-jumping coefficients on toric rings. We prove that there exists an inequality between F-jumping coe...

Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F (I) of I as reflected in their defining ideals as homomorphic images of polynomial rings over R/I and R/m respectively. In case all the higher conormal modules of I are free over R/I , we observe that: (i) G(I) is C...

Blowup ring profiles have been investigated by finding non-vortex blowup solutions of nonlinear Schrödinger equations (NLSE) (cf. [5] and [6]). However, those solutions have infinite L norm so one may not maintain the ring profile all the way up to the singularity. To find H non-vortex blowup solutions with ring profiles, we study blowup solutions of two-component systems of NLSE with nonlinear...

The goal of this paper is to take up the approaches used to deal with Jacobians and Kummer surfaces of curves of genus 2 by Cassels and Flynn [CF] and by the author [Sto1, Sto3] and extend them to hyperelliptic curves of genus 3. Such a curve C is given by an equation of the form y = f(x), where f is a squarefree polynomial of degree 7 or 8. We denote its Jacobian by J . Identifying points with...

Medium-ring ethers of various structural types have been isolated from marine organisms.1 While medium-ring ethers such as the ladder ether toxins have important implications with regard to their biological impact, it is the exquisite structures of naturally occurring medium-ring ethers that has provoked the imagination of synthetic chemists. Because eight-membered cyclic ethers are more preval...