Bezout's theorem and Cohen-Macaulay modules

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y: If they intersect properly in an arithmetically Cohen-Macaulay subscheme of positive dimension then X and Y are arithmetically Cohen-Macaulay. The module version of this result implies splitting criteria for reflexive sheaves. Received August 26, 1999 / Published online March 12, 2001     

Quantum phase expectation values of a mesoscopic Josephson junction from quantum current measurements

A simple way to acquire information on the mean values of the phase operators sinϕ and cosϕ of an ultrasmall Josephson junction prepared in an arbitrary pure or not state is reported. Our proposal exploits the recently predicted occurrence of current spikes in the I-V characteristic of a mesojunction irradiated by a quantum single-mode low-intensity coherent electromagnetic field. A necessary condition for the validity of our treatment is presented and discussed. Received 11 April 2001     

First-principles density-functional theory calculations of electron-transfer rates in azurin dimers

We have conceived and implemented a new method to calculate transfer integrals between molecular sites, which exploits few quantities derived from density-functional theory electronic structure computations and does not require the knowledge of the exact transition state coordinate. The method uses a complete multielectron scheme, thus including electronic relaxation effects. Moreover, it makes no use of empirical parameters. The computed electronic couplings can then be combined with estimates of the reorganization energy to evaluate electron-transfer rates that are measured in kinetic experiments: the latter are the basis to interpret electron-transfer mechanisms. We have applied our approach to the study of the electron self-exchange reaction of azurin, an electron-transfer protein belonging to the family of cupredoxins. The transfer integral estimates provided by the proposed method have been compared with those resulting from other computational techniques, from empirical models, and with available experimental data.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Rabi oscillations in the dynamics of a fully quantized SQUID ring-electromagnetic field system

In this paper, a study based on a fully quantum-mechanical model for the coupling between a super-conducting quantum interference device (an rf SQUID) and a single-mode quantized electromagnetic field is presented. The aim of this work is the prediction of the existence of some interesting phenomena, characterized by more than one Rabi frequency, occurring in the dynamics of the total coupled system when our analysis is developed inside a reduced low-lying energy 4D Hilbert subspace.     

On the minimal free resolution of general forms

Let and let be the ideal of generically chosen forms of degrees . We give the precise graded Betti numbers of in the following cases:

• ;
• and is even;
• , is odd and ;
• is even and all generators have the same degree, , which is even;
• is even and ;
• is odd, is even, and .

We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e., one for which the Hilbert function is maximal, given and the socle degree) when is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree, then there are no redundant summands, and we present some evidence for this conjecture.

     

Water-mediated electron transfer between protein redox centers

Recent experimental and theoretical investigations show that water molecules between or near redox partners can significantly affect their electron-transfer (ET) properties. Here we study the effects of intervening water molecules on the electron self-exchange reaction of azurin (Az), by performing a conformational sampling on the water medium and by using a newly developed ab initio method to calculate transfer integrals between molecular redox sites. We show that the insertion of water molecules at the interface between the copper active sites of Az dimers slightly increases the overall ET rate, while some favorable water conformations can considerably enhance the ET kinetics. These features are traced back to the interplay of two competing factors: the electrostatic interaction between the water and protein subsystems (mainly opposing the ET process for the water arrangements drawn from MD simulations) and the effectiveness of water in mediating ET coupling pathways. Such an interplay provides a physical basis for the found absence of correlation between the electronic couplings derived through ab initio electronic structure calculations and the related quantities obtained through the Empirical Pathways (EP) method. In fact, the latter does not account for electrostatic effects on the transfer integrals. Thus, we conclude that the water-mediated electron tunneling is not controlled by the geometry of a single physical pathway. We discuss the results in terms of the interplay between different ET pathways controlled by the conformational changes of one of the water molecules via its electrostatic influence. Finally, we examine the dynamical effects of the interfacial water and check the validity of the Condon approximation.     

The minimal resolution conjecture on a general quartic surface in P3

Musta?? has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in ${\mathbb{P}}^3$ this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.