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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, Comment: 24 pages, 9 figures. This paper is a natural prosecution of the work arXiv:1903.03595 and presents different enrichments and enhancements of the results there appeared. In this second version several typos have been corrected

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., Comment: 29 pages, 4 figures. Accepted for publication in The Vladimir Rabinovich Anniversary Volume of "Operator Theory: Advances and Applications" (Birkh\"auser)

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 $\hbox{Tr}\phi(A)$ of any function $\phi:\mathbb{R}\to\mathbb{R}$ which increases, together with its first two derivatives, at infinity not faster than an exponential., Comment: 22 pages

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, Comment: 23 pages

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., Comment: 17 pages

Fluctuations of the order parameters of the Gardner model for any $\alpha<\alpha_c$ are studied. It is proved that they converge in distribution to a family of jointly Gaussian random variables., Comment: 17 pages, Latex

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 \alpha is proven for all values of \alpha. Besides, we prove a useful criterion of the factorisation of all the correlation functions for a class of spin glass model., Comment: 38 pages

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 $\alpha\le 0.113$ for a large number of neurons. This result is an advance on all rigorous results in the literature and the relationship between the capacity $\alpha$ and retrieval errors obtained here for small $\alpha$ coincides with replica calculation results., Comment: Latex 36 page macros: http://www.springer.de/author/tex/help-journals.html