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QDD225  20/03/2017
Bertacchi D.; Coletti C.F.; Zucca F.
Global survival of branching random walks and treelike branching random walks  Abstract   The local critical parameter $lambda_s$ of continuoustime branching random walks is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter $lambda_w$ is a certain function of the reproduction rates, which we denote by $ 1/K_w$. We provide here new sufficient conditions which guarantee that the global critical parameter equals $ 1/K_w$. This result extends previously known results for branching random walks on multigraphs and general branching random walks.
We show that these sufficient conditions are satisfied by periodic treelike branching random walks.
We also discuss the critical parameter and the critical behaviour of continuoustime branching processes in varying environment.
So far, only examples where $lambda_w=1/K_w$ were known; here we
provide an example where $lambda_w>1/K_w$. 

QDD224  10/11/2016
Zucca F.; Bertacchi D.; Rodriguez P.M.
GaltonWatson processes in varying environment and accessibility percolation  Abstract   This paper deals with branching processes in varying environment, namely, whose offspring distributions depend on the
generations. We provide sufficient conditions for survival or extinction which rely only on the first and second moments of the
offspring distributions. These results are then applied
to branching processes in varying environment with selection where every particle has a realvalued label and labels can only increase along
genealogical lineages;
we obtain analogous conditions for survival or extinction. These last results can be interpreted in terms of
accessibility percolation on GaltonWatson trees, which represents a relevant tool for modeling the evolution of biological populations. 

QDD223  27/08/2016
Pellacci, B.; Verzinii, G.
Optimization of the positive principal eigenvalue for indefinite fractional Neumann problems  Abstract   We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamic. Our main result concerns the optimization of such eigenvalue with respect to the fractional order s in (0,1], the case s = 1 corresponding to the standard Neumann Laplacian: when the habitat is not too hostile in average, the principal positive eigenvalue can not have local minima for 0 < s < 1. As a consequence, the best strategy for survival is either following the diffusion with the lowest possible s, or with s = 1, depending on the size of the domain. In addition, we show that analogous results hold for the standard fractional Laplacian in the whole space , in periodic environments. 

QDD221  16/07/2016
Cipriani, F.
Noncommutative PotentialTheory: a survay  Abstract   The aim of these notes is to provide an introduction to Noncommutative Potential Theory as given at I.N.D.A.M.C.N.R.S. "Noncommutative Geometry and Applications" Lectures, Villa MondragoneFrascati June 2014. 

QDD222  16/07/2016
Pierotti, D.; Verzini, G.
Normalized bound states for the nonlinear Schrödinger equation in bounded domains  Abstract   We investigate the standing wave solutions with prescribed mass (or charge) of the nonlinear Schrödinger equation with power nonlinearity, in a bounded domain of dimension N and with Dirichlet boundary condition. Assuming that the exponent p>1 of the nonlinear term is Sobolevsubcritical, it follows by the GagliardoNirenberg inequality that there are solutions of any positive mass whenever p is less than the critical value 1+4/N. If p is equal or larger than 1+4/N, we prove that there are solutions having Morse index bounded above (by some positive integer k) only for sufficiently small masses. Lower bounds on these intervals of allowed masses are then obtained, by suitable variational principles, in terms of the Dirichlet eigenvalues of the Laplacian. 

QDD220  21/03/2016
Lember, J.; Matzinger, H., Sova, J.; Zucca, F.
Lower bounds for moments of global scores of pairwise Markov chains  Abstract   Let us consider two random sequences such
that every random variable takes values in a finite set. We consider a global similarity score that measures the homology
(relatedness) of words obtained by the random sequences. A
typical example of such score is the length of the longest common
subsequence. We study the order of central absolute rmoment of the score
in the case where the twodimensional joint process represented by the two random sequences is a Markov chain. This is a very general model involving independent
Markov chains, hidden Markov models, Markov switching models and
many more. Our main result establishes a general condition which allows to
obtain an explicit asymptotic value of the central absolute rmoment of the score. We also perform simulations indicating the validity of the condition. 

QDD219  10/12/2015
Cipriani, F.; Sauvageot, J.L.
Negative definite functions on groups with polynomial growth  Abstract   The aim of this work is to show that on a locally compact, second countable, compactly generated group G with polynomial growth and homogeneous dimension $d_h$, there exist a continuous, proper, negative definite function $ell$ with polynomial growth dimension $d_ell$ arbitrary close to $d_h$. 

QDD218  30/11/2015
Cirant, M.; Verzini, G.
Bifurcation and segregation in quadratic twopopulations Mean Field Games systems  Abstract   We consider a twopopulations ergodic Mean Field Games system, which describes Nash equilibria in differential games with identical players. In these models, each population consists of a very large number of indistinguishable rational agents, aiming at minimizing some longtime average criterion. Via the HopfCole transformation, such system reduces to a semilinear elliptic one, for normalized densities. Firstly, we discuss existence of nontrivial solutions; secondly, for selected families of nontrivial solutions, we address the appearing of segregation in the vanishing viscosity limit. 
