Operators which have a closed quasi-nilpotent part

We find several conditions for the quasi-nilpotent part of a bounded operator acting on a Banach space to be closed. Most of these conditions are established for semi-Fredholm operators or, more generally, for operators which admit a generalized Kato decomposition. For these operators the property of having a closed quasi-nilpotent part is related to the so-called single valued extension property.

     

Ordinary differential operators in Hilbert spaces and Fredholm pairs

Let be a path of bounded operators on a real Hilbert space, hyperbolic at . We study the Fredholm theory of the operator . We relate the Fredholm property of to the stable and unstable linear spaces of the associated system . Several examples are included to point out the differences with respect to the finite dimensional case, in particular concerning the role of the spectral flow. We define a general class of paths A for which many properties typical of the finite dimensional framework still hold. Our motivation is to develop the linear theory which is necessary for the set-up of Morse homology on Hilbert manifolds. Received: 9 March 2001; in final form: 1 March 2002 / Published online: 2 December 2002     

A note on Taylor boundary conditions for accurate image restoration

In recent years, several efforts were made in order to introduce boundary conditions for deblurring problems that allow to get accurate reconstructions. This resulted in the birth of Reflective, Anti-Reflective and Mean boundary conditions, which are all based on the idea of guaranteeing the continuity of the signal/image outside the boundary. Here we propose new boundary conditions that are obtained by suitably combining Taylor series and finite difference approximations. Moreover, we show that also Anti-Reflective and Mean boundary conditions can be attributed to the same framework. Numerical results show that, in case of low levels of noise and blurs able to perform a suitable smoothing effect on the original image (e.g. Gaussian blur), the proposed boundary conditions lead to a significant improvement of the restoration accuracy with respect to those available in the literature.     

Electrochemical genosensors for the detection of Bonamia parasite. Selection of single strand-DNA (ssDNA) probes by simulation of the secondary structure folding

A post-PCR nucleic acid work by comparing experimental data, from electrochemical genosensors, and bioinformatics data, derived from the simulation of the secondary structure folding and prediction of hybridisation reaction, was carried out in order to rationalize the selection of ssDNA probes for the detection of two Bonamia species, B. exitiosa and B. ostreae, parasites of Ostrea edulis.Six ssDNA probes (from 11 to 25 bases in length, 2 thiolated and 4 biotinylated) were selected within different regions of B. ostreae and B. exitiosa PCR amplicons (300 and 304 bases, respectively) with the aim to discriminate between these parasite species. ssDNA amplicons and probes were analyzed separately using the “Mfold Web Server” simulating the secondary structure folding behaviour. The hybridisation of amplicon-probe was predicted by means of “Dinamelt Web Server”. The results were evaluated considering the number of hydrogen bonds broken and formed in the simulated folding and hybridisation process, variance in gaps for each sequence and number of available bases. In the experimental part, thermally denatured PCR products were captured at the sensor interface via sandwich hybridisation with surface-tethered probes (thiolated probes) and biotinylated signalling probes. A convergence between analytical signals and simulated results was observed, indicating the possibility to use bioinformatic data for ssDNA probes selection to be incorporated in genosensors.     

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.     

Self-assembling of multi-jets by pyro-electrohydrodynamic effect for high throughput liquid nanodrops transfer

Destabilization of liquid film by electro-hydro-dynamics (EHD) pressure is achieved through the pyroelectric effect on a polar dielectric crystal. We show that by destabilizing the liquid film, periodical self-assembled multi-jets are obtained. The multi-jets operate simultaneously and could be exploited to dispense liquids with nanolitre drops. Such multiple self-assembled liquid jets have significant potential applicability for high-throughput liquid transfer by this novel pyro-EHD ink-jet approach. Since the method avoids the use of nozzles and electrodes, it is especially suitable for highly viscous liquids. Here we present and discuss the new multi-jet process and the results obtained with a liquid polymer (PDMS).     

“Zero” temperature stochastic 3D ising model and dimer covering fluctuations: A first step towards interface mean curvature motion

We consider the Glauber dynamics for the Ising model with “+” boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “−” spins disappears within a time τ₊, which is at most L²(log L)^c and at least L²/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between “+” and “−” domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ₊ ≃ const × L², conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ₊ can be shown to be of order L² without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like for L large, as conjectured in [2]. © 2011 Wiley Periodicals, Inc.     

A Supramolecular Approach to Antimicrobial Surfaces

In this paper, we report on the preparation of Imidazole-functionalized glass surfaces, demonstrating the ability of a dinuclear Cu(II) complex of a macrocyclic ligand to give a “cascade” interaction with the deprotonated forms of grafted imidazole moieties. In this way, we realized a prototypal example of an antimicrobial surface based on a supramolecular approach, obtaining a neat microbicidal effect using low amounts of the described copper complex.