Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

Betti numbers for numerical semigroup rings

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.Comment: 22 pages; v2: updated references. To appear in Multigraded Algebra and Applications (V. Ene, E. Miller Eds.

Fiber cones of ideals with almost minimal multiplicity

Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed

Robust algorithm for head-dependent analysis of water distribution systems

In a water distribution system (WDS), low pressure situations can arise for a variety of reasons. For example, over time, the demands may exceed the capacity of the WDS or some major components e.g. pumps and trunk mains may not be operational while repairs are carried out. The extent to which demands are satisfied in a WDS is governed by the actual amount of pressure in the system and pressure-deficient operating conditions are not uncommon. Despite this, the conventional modelling approach works well only under fully satisfactory pressure regimes, with results for subnormal pressure conditions often being inaccurate, misleading and even unreasonable. Sample results for a prototype FORTRAN computer program which can perform head-driven analysis using a globally convergent Newton-Raphson scheme are presented. Extensive testing has shown that the program is efficient and robust. It is demonstrated here that head-driven analysis is not computationally more time consuming than demand-driven analysis and that different functions for head-dependent nodal outflows can produce significantly different results

On fiber cones of m-primary ideals

Two formulas for the multiplicity of the fiber cone F(I) = circle plus(infinity)(n=0) I-n/mI(n) of an in-primary ideal of a d-dimensional Cohen-Macaulay local ring (R, in) are derived in terms of the mixed multiplicity e(d-1) (m vertical bar I), the multiplicity e(I), and superficial elements. As a consequence, the Cohen-Macaulay property of F(I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of I and lengths of certain ideals. We also characterize the Cohen-Macaulay and Gorenstein properties of fiber cones of in-primary ideals with a d-generated minimal reduction J satisfying l(I-2/Ji) = I or l(Im/Jm) = 1