1,721,193 research outputs found
Heat equation and convolution inequalities
Full text linkIt is known that many classical inequalities linked to con- volutions can be obtained by looking at the monotonicity in time of convolutions of powers of solutions to the heat equation, provided that both the exponents and the coefficients of diffusions are suitably cho- sen and related. This idea can be applied to give an alternative proof of the sharp form of the classical Young’s inequality and its converse, to Brascamp–Lieb type inequalities, Babenko’s inequality and Prékopa– Leindler inequality as well as the Shannon’s entropy power inequality. This note aims in presenting new proofs of these results, in the spirit of the original arguments introduced by Stam to prove the entropy power inequality
Kinetic models of opinion formation
No full textWe introduce and discuss certain kinetic models of (continuous) opinion formation involving both exchange of opinion between individual agents and diffusion of information. We show conditions which ensure that the kinetic model reaches non trivial stationary states in case of lack of diffusion in correspondence of some opinion point. Analytical results are then obtained by considering a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of opinion among individuals
Matematica Applicata
No full textIn questa nota verranno brevemente descritte le ricerche matematiche attualmente in essere nel Dipartimento di Matematica dell’Università di Pavia, nell’ambito dei settori maggiormente coinvolti nelle applicazioni.
Verrà inoltre delineata la loro collocazione internazionale e l’interazione con le altre scienze. Particolarmente rivolti alle applicazioni risultano essere le ricerche nei campi dell’Analisi Numerica, della Fisica Matematica e del Calcolo delle Probabilità
A kinetic description of mutation processes in bacteria
Full text linkTheLuria–Delbru ̈ckmutationmodelhasbeenmathematicallyfor- mulated in a number of ways. Last, a mean field picture derived from a kinetic formulation has been derived by Kashdan and Pareschi. There, the Luria–Delbru ̈ck distribution appears as the solution of a Fokker-Planck like equation obtained as the quasi-invariant asymptotics of a linear Boltzmann equation for the number density of the number of mutated cells. This paper addresses the kinetic description for the Lea–Coulson formulation, as well as for the Kendall formulation, focusing on important modeling is- sues closely linked with the distribution of the number of mutants. The paper additionally emphasizes basic principles which not only help to unify existing results but also allow for a useful extensions
Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles
No full textWe study a Fokker-Planck equation with linear diusion and super-linear drift introduced by Kaniadakis and Quarati to describe the evolution of a gas of Bose-Einstein particles. For kinetic equation of this type it is well-known that, in the physical space R3, the structure of the equilibrium Bose-Einstein distribution depends upon a parameter m∗, the critical mass. We are able to describe the time-evolution of the solution in two dierent situations, which correspond to m ≪ m∗ and m ≫ m∗ respectively. In the former case, it is shown that the solution remains regular, while in the latter we prove that the solution starts to blow up at some nite time, for which we give an upper bound in terms of the initial mass. The results are in favor of the validation of the model, which, in the supercritical regime, could produce in nite time a transition from a normal uid to one with a condensate component. The research that led to the present paper was partially supported by a grant of the group GNFM of INdA
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