Maximin share and minimax envy in fair-division problems

For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envy-minimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.     

Characterization theorems for the projective space and vector bundle adjunction

 We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves on projective manifolds. Received: 16 May 2002 / Revised version: 18 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 14E30, 14J40, 14J45     

MGM Optimal convergence for certain (multilevel) structured linear systems

We present a multigrid algorithm to solve linear systems whose coefficient metrices belongs to circulant, Hartley or τ multilevel algebras and are generated by a nonnegative multivariate polynomial f. It is known that these matrices are banded (with respect to their multilevel structure) and their eigenvalues are obtained by sampling f on uniform meshes, so they are ill‐conditioned (or singular, and need some corrections) whenever f takes the zero value. We prove the proposed metod to be optimal even in presence of ill‐conditioning: if the multilevel coefficient matrix has dimension ni at level i, i = 1, … , d, then only ni operations are required on each iteration, but the convergence rate keeps constant with respect to N(n) as it depends only on f. The algorithm can be extended to multilevel Toeplitz matrices too.     

Separation of Variables for Bi-Hamiltonian Systems

We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called N manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel'fand–Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.     

Fe–Mn–Al–C Alloys: a Study of Their Corrosion Behaviour in SO2 Environments

The corrosion reaction of four Fe–Mn–Al alloys exposed to a cycling, dry–humid, SO₂ (0.001% by volume) polluted atmosphere was studied. ICEMS, XPS, AES-SAM and transmission Mössbauer spectroscopy at different temperatures were employed to characterize the corrosion products. The analytical results indicate that (i) ferrihydrite is the main component of the rust; (ii) there is an abundant presence of Mn²⁺ and SO₃ ^2–/SO₄ ^2– on the top of the corrosion layer, the concentration of SO₄ ^2– increasing with the number of cycles; and (iii) the magnetic hyperfine pattern exhibited by the series of low-temperature spectra of the rust is quite different from that observed in the rust formed under similar corrosive environments on iron and weathering steel. This latter finding is correlated with a slow rate of transformation of the Fe³⁺ species formed at the early stages of corrosion into -FeOOH, the usual final product of this type of corrosion processes. The sulphate anions, abundant inside the electrolyte during the wet periods, could be incorporated to the ferrihydrite structure being responsible for the Mössbauer spectral pattern recorded from the corrosion products at low temperatures.     

On the geometry of master-slave synchronization

In 1990, Pecora and Carroll reported the observation that one can synchronize the orbits of two identical dynamical systems, which may be chaotic, by feeding state variables of one of them to the other one with no feedback, a phenomenon often called master-slave synchronization. We report here some results on the theory of master-slave synchronization for maps and flows, which are all inspired by a similar geometric and coordinate independent point of view to the one introduced in master-slave synchronization by Tresser, Worfolk, and Bass. Our results are variations on the theme that projection often can compensate for expansion.(c) 2002 American Institute of Physics.     

Asymptotic Information-Theoretic Detection of Dynamical Organization in Complex Systems

The identification of emergent structures in complex dynamical systems is a formidable challenge. We propose a computationally efficient methodology to address such a challenge, based on modeling the state of the system as a set of random variables. Specifically, we present a sieving algorithm to navigate the huge space of all subsets of variables and compare them in terms of a simple index that can be computed without resorting to simulations. We obtain such a simple index by studying the asymptotic distribution of an information-theoretic measure of coordination among variables, when there is no coordination at all, which allows us to fairly compare subsets of variables having different cardinalities. We show that increasing the number of observations allows the identification of larger and larger subsets. As an example of relevant application, we make use of a paradigmatic case regarding the identification of groups in autocatalytic sets of reactions, a chemical situation related to the origin of life problem.     

EEG Fractal Analysis Reflects Brain Impairment after Stroke

Stroke is the commonest cause of disability. Novel treatments require an improved understanding of the underlying mechanisms of recovery. Fractal approaches have demonstrated that a single metric can describe the complexity of seemingly random fluctuations of physiological signals. We hypothesize that fractal algorithms applied to electroencephalographic (EEG) signals may track brain impairment after stroke. Sixteen stroke survivors were studied in the hyperacute (<48 h) and in the acute phase (∼1 week after stroke), and 35 stroke survivors during the early subacute phase (from 8 days to 32 days and after ∼2 months after stroke): We compared resting-state EEG fractal changes using fractal measures (i.e., Higuchi Index, Tortuosity) with 11 healthy controls. Both Higuchi index and Tortuosity values were significantly lower after a stroke throughout the acute and early subacute stage compared to healthy subjects, reflecting a brain activity which is significantly less complex. These indices may be promising metrics to track behavioral changes in the very early stage after stroke. Our findings might contribute to the neurorehabilitation quest in identifying reliable biomarkers for a better tailoring of rehabilitation pathways.