Castelnuovo–Mumford Regularity of Some Modules

We give bounds for the Castelnuovo–Mumford regularity of the so-called sequentially κ-Buchsbaum modules and of the canonical modules of certain rings.     

Castelnuovo–Mumford Regularity and Gorensteinness of Fiber Cone

In this article, we study the Castelnuovo–Mumford regularity and Gorenstein properties of the fiber cone. We obtain upper bounds for the Castelnuovo–Mumford regularity of the fiber cone and obtain sufficient conditions for the regularity of the fiber cone to be equal to that of the Rees algebra. We obtain a formula for the canonical module of the fiber cone and use it to study the Gorenstein property of the fiber cone.     

Castelnuovo–Mumford Regularity of Associated Graded Modules and Fiber Cones of Filtered Modules

We give bounds on the Castelnuovo–Mumford regularity of the associated graded module of an arbitrary good filtration and of its fiber cone. These bounds extend previous results of Rossi–Trung–Valla and Linh.     

Non-linear behaviour of Castelnuovo–Mumford regularity

It is shown that the Castelnuovo–Mumford regularity of lex-segment ideals of ordinary powers as well as of Frobenius powers of a homogeneous polynomial ideal is a function of polynomial type of huge degree.     

Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity

Mathematical Programming - The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer...     

Cohen–Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals

Let 𝕂 be a field, and let R = 𝕂[x ₁,…, x _n ] be the polynomial ring over 𝕂 in n indeterminates x ₁,…, x _n . Let G be a graph with vertex-set {x ₁,…, x _n }, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J ^(k) and J ^[k], respectively. In this paper, we give necessary and sufficient conditions for R/J ^k , R/J ^(k), and R/J ^[k] to be Cohen–Macaulay. We also study the limit behavior of the depths of these rings.     

Sums of products of Apostol–Bernoulli numbers

By expressing the sums of products of the Apostol?CBernoulli polynomials in terms of the special values of multiple Hurwitz?CLerch zeta functions at non-positive integers, we obtain the sums of products identity for the Apostol?CBernoulli numbers which is an analogue of the classical sums of products identity for Bernoulli numbers dating back to Euler.     

Sums of Integers

We will use a good idea on differences and polynomial patterns to develop a number of formulas involving sums of numbers that arise frequently in mathematics.Among these are the sum of the firstn integers     

Sur la régularité de Castelnuovo–Mumford des idéaux,en dimension 2

We give a bound on the Castelnuovo–Mumford regularity of a homogeneous ideal I, in a polynomial ring A, in terms of the number of variables and the degrees of generators, when the dimension of $A/I$ is at most two. This bound improves the one obtained by Caviglia and Sbarra in [G. Caviglia, E. Sbarra, Characteristic-free bounds for the Castelnuovo–Mumford regularity, Prépublication, math.AC/0310122]. In the continuation of the examples constructed in Chardin and D'Cruz [M. Chardin, C. D'Cruz, Castelnuovo–Mumford regularity: examples of curves and surface, J. Algebra 270 (2003) 347–360], we use families of monomial curves to construct homogeneous ideals showing that these bounds are quite sharp. To cite this article: M. Chardin, A.L. Fall, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Discrete stochastic approximations of the Mumford–Shah functional

We propose a new Γ-convergent discrete approximation of the Mumford–Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford–Shah functional in any dimension.     

Elementary Theory of the Alexander–Conway Polynomial

Mathematical Notes -