Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal in a polynomial ring and for every we consider the Koszul homology with respect to a sequence of of generic linear forms. The Koszul-Betti number is, by definition, the dimension of the degree part of . In characteristic , we show that the Koszul-Betti numbers of any ideal are bounded above by those of the gin-revlex of and also by those of the Lex-segment of . We show that iff is componentwise linear and that and iff is Gotzmann. We also investigate the set of all the gin of and show that the Koszul-Betti numbers of any ideal in are bounded below by those of the gin-revlex of . On the other hand, we present examples showing that in general there is no is such that the Koszul-Betti numbers of any ideal in are bounded above by those of .

     

Gas Phase of Asymmetric Nearest Neighbor Ising Model

We consider an asymmetric d-dimensional, d>1, Ising model with the pair interaction I in one direction different from the pair interaction J in all other directions. We show that for any inverse temperature the system is in the gas phase as soon as |J|<C ^–1 d ^–2(1–tanh( |I|)) with C>0 being a small numeric constant.     

Probing pentacene polymorphs by lattice dynamics calculations

We have performed a lattice dynamics calculation to compute the "inherent structures" of minimum potential energy for pentacene, starting from available X-ray data. The calculation shows that two distinct bulk crystalline phases of pentacene exist, with very subtle structural differences but clearly different phonon spectra. The method of crystal growth (from solution or vapor) is not the determining factor for obtaining either structure.     

The Relative Dispersion and Mixing of Passive Solutes in Transport in Geologic Media

The spreading of a contaminant in a heterogeneous aquifer depends on the scales of variability effectively explored by the plume. In particular, we observe two major contributions of the fluctuating velocity field in the contaminant movement: (i) the spreading caused by velocity variations of scales lesser than that of the plume size, which we will call relative spreading, and (ii) the meander-like movement of the plume as a whole caused by velocity variations of scale larger than that of the plume size. The aim of this work is to consider the effects of the finite size of the contaminant plume on the local concentration moments <C> and _C . In particular a relative concentration, which depends on the scales of variability effectively explored by the plume, is defined. First, the mathematical formulation of the problem is developed along the Lagrangian framework. In particular, the expressions for the relative mean concentration and its variance are presented. Then, the methodology is applied to the regional transport problem, where the influence of the size of the plume and the pore-scale dispersion are quantitatively assessed.     

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s 2n points in general linear position in P ⁿ , and the Koszul property of the generic Artinian Gorenstein algebra of socle degree 3.     

Nonmonotone Algorithms for Pattern Search Methods

This work deals with the solution of ill-conditioned unconstrained minimization problems by means of pattern search methods. To this end, the usual requirement of monotonic reduction of the objective function is relaxed and nonmonotone pattern search methods are proposed, which maintain the convergence properties of their monotone counterparts. Numerical experimentation on several well-known ill-conditioned functions is reported. The results highlight a class of pattern search methods which benefit very much by the introduction of nonmonotone strategies.