By Giacomo Aletti, Vincenzo Capasso

ISBN-10: 3540444459

ISBN-13: 9783540444459

Those lawsuits document at the convention "Math Everywhere", celebrating the sixtieth birthday of the mathematician Vincenzo Capasso. The convention promoted principles Capasso has pursued and shared the open surroundings he's recognized for. subject sections comprise: Deterministic and Stochastic structures. Mathematical difficulties in Biology, medication and Ecology. Mathematical difficulties in and Economics. The huge spectrum of contributions to this quantity demonstrates the reality of its name: Math is all over the place, certainly.

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**Additional resources for Math everywhere: deterministic and stochastic modelling in biomedicine, economics and industry ; dedicated to the 60th birthday of Vincenzo Capasso**

**Sample text**

D. Moreover f(X0 ,Xu ) is continuous at (x, x) for u > 0 and so we have also condition (ii). Concerning condition (iii), we can take u0 = 1 and see that +∞ x,y +∞ |gu (x, y)|du ≤ sup sup f (x)|pu (x, y) − f (y)|du . 1 x,y 1 We will prove that the last term is ﬁnite. First we estimate for u > 1 |pu (x, y) − f (y)| = [pu−1 (x, z) − f (z)]p1 (z, y)dz ≤ b1 |pu−1 (x, z) − f (z)|dz = b1 var(µxu−1 − µinv ) ≤ b1 c0 exp (c1 x − c2 u) , where in the ﬁrst inequality we have used the boundedness of p1 (·, ·) and in the second inequality we have applied Theorem 4.

In this way a nonlinear partial integral diﬀerential equation for the asymptotic mean ﬁeld was obtained. Here we wish to analyze the long time behavior of a (stochastic) generalized system for a ﬁxed N . Our interest is to analyze mechanisms that are responsible for stable aggregation. We consider a population of constant size N ∈ N − {0}. From the Lagrangian point of view, we assume that the “state” of the k-th particle is dek (t) ∈ Rd , t ≥ 0. , N }, scribed by a random vector XN k {XN (t), t ∈ R+ } is a stochastic process in the state space (Rd , BRd ), on a k may describe the spacommon probability space (Ω, F, P ).

We will see that these rates strongly depend upon the dimension of the process. e. there exists a constant A > 0 such that S(x) − S(y) ≤ A x − y , ∀x , y ∈ Rd ; (A2) there exist constants M0 ≥ 0 and r > 0 such that (S(x), x) ≤ −r x , x ≥ M0 . Notice that assumption (A1) guarantees the existence of a unique strong solution to equation (1) and assumption (A2) implies the existence of an invariant measure for the process. Moreover from (A1) we deduce that the invariant measure admits a density f (·) with respect to the Lebesgue measure.

### Math everywhere: deterministic and stochastic modelling in biomedicine, economics and industry ; dedicated to the 60th birthday of Vincenzo Capasso by Giacomo Aletti, Vincenzo Capasso

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