Quaderni di Dipartimento
Collezione dei preprint del Dipartimento di Matematica. La presenza del fulltext è lacunosa per i prodotti antecedenti maggio 2006.
Trovati 867 prodotti

QDD194  25/02/2015
Catino, G.
Complete gradient shrinking Ricci solitons with pinched curvature  Abstract   We prove that any ndimensional complete gradient Ricci soliton with pinched Weyl curvature is a finite quotient of R^n, R×S^{n?1} or S^n. In particular, we do not need to assume the metric to be locally conformally flat. 

QDD193  15/02/2015
Catino, G.; Mantegazza C.; Mazzieri, L.; Rimoldi M.
Locally Conformally Flat QuasiEinstein Manifolds  Abstract   In this paper we prove that any complete locally conformally flat quasiEinstein manifold of dimension n?3 is locally a warped product with (n?1)dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons. 

QDD192  15/02/2015
Catino, G.
Generalized quasiEinstein manifolds with harmonic Weyl tensor  Abstract   In this paper we introduce the notion of generalized quasiEinstein manifold, which generalizes the concepts of Ricci soliton, Ricci almost soliton and quasiEinstein manifolds. We prove that a complete generalized quasiEinstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a warped product with (n?1)dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to the one proved for gradient Ricci solitons. 

QDD190  13/01/2015
Finotelli, P.;Dulio, P.
A Mathematical Model for Evaluating the Functional Connectivity Strongness in Healthy People  Abstract   The human brain is a really complex organization of connectivity whose principal elements are neurons,
synapses and brain regions. Up to now this connectivity is not fully understood, and recent impulse
in investigating its structure has been given by Graph Theory. However, some points remain unclear,
mainly due to possible mismatching between the Mathematical and the Neuroscientic approach. It
is known that neural connectivity is classied into three categories: structural (or anatomical) connectivity,
functional connectivity and eective connectivity. The point is that these categories demand
dierent kinds of graphs, except in the case of the resting state, and sometimes topological and metrical
parameters are involved simultaneously, without a specic distinction of their roles.
In this paper we propose a mathematical model for treating the functional connectivity, based
on directed graphs with weighted edges. The function W(i; j; t), representing the weight of the edge
connecting nodes i; j at time t, is obtained by splitting the model in two parts, where dierent parameters
have been introduced step by step and rigorously motivated. In particular, there is a double role
played by the notion of distance, which, according to the dierent parts of the model, assumes a
discrete or an Euclidean meaning. Analogously, the time t appears both from a local and from a global
perspective. The local aspect relates to a specic task submitted to an health volunteer (in view of
possible future applications also to subjects aected by neurological diseases), while the global one
concerns the dierent periods in the human life that characterize the main changes in the neural brain
network. In the particular case of the resting state, we have shown that W reduces to the usually
employed probabilistic growth laws for the edge formation. We tested the correctness of our model by
means of synthetic data, where the selection of all involved parameters has been motivated according
to what is known from the available literature. It turns out that simulated outputs t well with the
expected results, which encourages further analysis on real data, and possible future applications to
neurological pathologies. 

QDD191  13/01/2015
Beretta, E.; Cerutti, M.C.; Manzoni, A.; Pierotti, D.
On a semilinear elliptic boundary value problem arising in cardiac electrophysiology  Abstract   In this paper we provide a representation formula for boundary voltage perturbations caused
by internal conductivity inhomogeneities of low volume fraction in a simplied monodomain model
describing the electric activity of the heart. We derive such a result in the case of a nonlinear
problem. Our longterm goal is the solution of the inverse problem related to the detection of regions
aected by heart ischemic disease, whose position and size are unknown. We model the presence
of ischemic regions in the form of small inhomogeneities. This leads to the study of a boundary
value problem for a semilinear elliptic equation. We rst analyze the wellposedness of the problem
establishing some key energy estimates. These allow us to derive rigorously an asymptotic formula
of the boundary potential perturbation due to the presence of the inhomogeneities, following an
approach similar to the one introduced by Capdeboscq and Vogelius in [7] in the case of the linear
conductivity equation. Finally, we propose some ideas of the reconstruction procedure that might
be used to detect the inhomogeneities. 

QDD189  27/11/2014
Pagani, C.D.; Pierotti,D.; Verzini, G.; Zilio, A.
A nonlinear Steklov problem arising in corrosion modeling  Abstract   We investigate the existence of pairs (?, u), with ? > 0 and u harmonic function in the unit ball B ? R3 , such that the nonlinear boundary condition ?? u = ? sinh u holds on ?B. This type of exponential boundary condition arises in corrosion modeling (ButlerVolmer condition). We prove existence of global branches of nontrivial solutions in the framework of analytic bifurcation theory and investigate their properties both analytically and numerically.


QDD188  11/11/2014
Maluta, E.; Papini, P.L.
Diametrically complete sets and normal structure  Abstract   We prove that in some classes of reflexive Banach spaces every maximal diametral set must be diametrically complete, thus showing that diametrically complete sets may have empty interior also in reflexive spaces. As a consequence, we prove that, in those spaces, normal structure is equivalent to the weaker property that, for every bounded set, the absolute Chebyshev radius is strictly smaller than the diameter. Moreover we prove that, in any normed space, the two classes of diametrically complete sets and of sets of constant radius from the boundary coincide if and only if the space has normal structure. 

QDD187  15/10/2014
Finotelli, P.; Dulio, P.
A Mathematical Proposal to Evaluate Functional Connectivity Strongness in Complex Brain Networks  Abstract   The brain is a really complex organization of connectivity whose principal elements are neurons, synapses and brain regions. To date this connectivity is not fully understood. Graph Theory represents a powerful tool in the study of brain networks. Though the complex organization of connectivity in human and animal brain has found a great impulse by the use of Graph Theory, some points result to be not very clear and needed to be clarified, the weakness lies in the mismatching between the mathematical and neuroscientific approach. In this paper we focus, in particular, on two points: the concept of distance and a mathematical approach in treating functional and structural connectivity by means of the introduction of the parameter time. One of the most relevant remark we point out concerns the concept of graph in the interfield crossing Mathematics and Neuroscience. In detail, when talking about Graph Theory in brain connectivity, it should be clear that we are considering two basic categories: one static, relative to the anatomical connectivity, and the other, dynamical, concerning the functional connectivity. We believe that in order to describe them it is fundamental to introduce the concept of time, which, at present, seems to be a lack in the theory of this area of research. For example, the static category regards the anatomical neural network in particular range of the life of human beings (and animals), i.e. the synaptic connections or directed anatomical pathways derived from neural tract tracing, can be retained static only in absence of injuries or cerebral illnesses, or far from the childhood and one’s old age. The dynamical approach is involved in the other cases, in particular it is linked to the functional connectivity, i.e. the temporal correlations between remote neurophysiological events as reaction to well specific external stimuli (e.g. social paradigms, social cognitive functions or other specific tasks), it interests cerebral areas not necessarily close each other (in the sense of Euclidean distance). Aside we emphasize that the functional connectivity is very distinctive from effective connectivity, i.e. the influence one neural system exerts over another [26]. The point is that these categories demand different kinds of graphs,except the case of resting state. The integration between these two different approaches is a topic of present interest. In this paper we formalize in a mathematical way this concept and we speculate the existence of a function which can give the weight of the edges composing the graph representing the functional connectivity. This function W(i, j, t) depends on the position of nodes i, j and on the time t at which a specific task is submitted to an health volunteer (and in prospective to a subject affected by a neurological disease). Interestingly this function, in particular cases, comes down to the probability of edge formation. Basically these particular cases are the resting state and when a particular task do not affect the cerebral region to which the nodes belong to. This second case is rare since when performing a task the region of interest, ROI, are well known. 
