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B. Tirozzi

Publications and source records attributed to B. Tirozzi.

8 recordsLinked to original sources
arXiv2024

The Volterra Integrable case. Novel analytical and numerical results

In the present paper we reconsider the integrable case of the Hamiltonian $N$-species Volterra system, as it has been introduced by Vito Volterra in 1937 and significantly enrich the results already published in the ArXiv in 2019 by two of the present authors (M. Scalia and O. Ragnisco). In fact, we present a new approach to the construction of conserved quantities and comment about the solutions of the equations of motion; we display mostly new analytical and numerical results, starting from the classical predator-prey model and arriving at the general $N$-species model

nlin.SI↗
arXiv2012

Functions of noncommuting operators in an asymptotic problem for a 2D wave equation with variable velocity and localized right-hand side

We use the theory of functions of noncommuting operators (noncommutative analysis) to solve an asymptotic problem for a partial differential equation and show how, starting from general constructions and operator formulas that seem to be rather abstract from the viewpoint of differential equations, one can end up with very specific, easy-to-evaluate expressions for the solution, useful, e.g., in the tsunami wave problem.

math.AP↗
arXiv2009

Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs

We consider the adjacency matrix $A$ of a large random graph and study fluctuations of the function $f_n(z,u)=\frac{1}{n}\sum_{k=1}^n\exp\{-uG_{kk}(z)\}$ with $G(z)=(z-iA)^{-1}$. We prove that the moments of fluctuations normalized by $n^{-1/2}$ in the limit $n\to\infty$ satisfy the Wick relations for the Gaussian random variables. This allows us to prove central limit theorem for $\hbox{Tr}G(z)$ and then extend the result on the linear eigenvalue statistics $\hbox{Tr}ϕ(A)$ of any function $ϕ:\mathbb{R}\to\mathbb{R}$ which increases, together with its first two derivatives, at infinity not faster than an exponential.

math-ph↗
arXiv2006

Dynamical behavior of a large complex system

Limit theorems for a linear dynamical system with random interactions are established. These theorems enable us to characterize the dynamics of a large complex system in details and assess whether a large complex system is stable or unstable, answering an open question raised 30 years ago in the literature

math-ph↗
arXiv2006

Stability of the dynamics of an asymmetric neural network

We study the stability of the dynamics of a network of n neurons intercting linearly through a random gaussian matrix of excitatory and inhibitory type. Using the aproach developed in a previous paper we show some interesting properties of the dynamic of this system for large values of n. We got sufficient conditions for getting diverging synchronized behavior or stability.

math-ph↗
arXiv2001

Rigorous Solution of the Gardner Problem

We prove rigorously the well-known result of Gardner about the typical fractional volume of interactions between N spins which solve the problem of storing a given set of random patterns. The Gardner formula for this volume in the limit N,p \to \infty, p/N \to αis proven for all values of α. Besides, we prove a useful criterion of the factorisation of all the correlation functions for a class of spin glass model.

math-ph↗
arXiv2000

On the Critical Capacity of the Hopfield Model

We estimate the critical capacity of the zero-temperature Hopfield model by using a novel and rigorous method. The probability of having a stable fixed point is one when $α\le 0.113$ for a large number of neurons. This result is an advance on all rigorous results in the literature and the relationship between the capacity $α$ and retrieval errors obtained here for small $α$ coincides with replica calculation results.

math-ph↗