In the framework of the Global Regularity Problem for the incompressible Navier-Stokes (NS) equations in the whole space R 3 , Li and Sinai in [J.Eur.Math.Soc., 10:267-313, 2008] proved the existence of smooth complex solutions that become singular ("blow-up") in a finite time.We report new results obtained by computer simulations on the behavior of complex solutions with support of Li-Sinai type and of real flows related to them.For the complex solutions the simulations indicate that the class of initial data leading to a blow-up is much larger than that considered by Li and Sinai.The real flows show some remarkable properties, such as a sharp increase of the total enstrophy and a concentration of high values of velocities and vorticity in small regions.We conclude with a discussion on the perspectives of a real blow-up in the framework of the Li-Sinai approach.

About 45 years ago Mitchell Feigenbaum made a beautiful mathematical discovery.So beautiful and unexpected that for a while, many mathematicians refused to believe it.His discovery attracted a lot of public attention.It is quite rare when a mathematical discovery fascinates the general public.In the last 50 years, we can remember only two other stories: Wiles's proof of Fermat's Last Theorem and Perelman's solution to the Poincaré conjecture.With time, the public interest heated by Mitchell's enigmatic personality cooled down.But the mathematical importance of Feigenbaum's work has only increased.In these short notes we will try to reflect on the background of Mitchell's discovery,

We report results, obtained by computer simulations, guided by a theoretical analysis, on a new solution of the incompressible Navier-Stokes equations (NS) in $\R^{3}$, with no boundary conditions, which arises in connection with the contribution of Li and Sinai to the "global regularity problem", i.e., the problem whether smooth solutions in absence of forcing can become singular at a finite time. The problem is open since the pioneering work of J. Leray \cite{Leray} in 1937, and, in spite of many brilliant contributions, it is still open and in the list of the Clay millennium prize.

In this paper, some facts related Joel L. Lebowitz are mentioned. In addition, some of the known theory concerning the statistical physics of freely fluctuating two-dimensional crystals subject to a non-linear elastic hamiltonian is described. Particular topics that are discussed include the existence of two-dimensional crystals in relation to the Hohenberg–Mermin–Wagner theorem, the crumpling transition for freely suspended crystalline membranes and the renormalization of elastic moduli. Although much of what is known has been uncovered since the mid-80s, the topic has become of interest once again due to the discovery of graphene and other two-dimensional crystals. The field is vast so it is the aim of this note to describe some of its fundamental properties.

On 9 January 2018, the renowned mathematician Professor Robert Adol’fovich Minlos passed away at the age of 86. An eminent researcher and outstanding teacher, he was a world-renowned specialist in the area of functional analysis, probability theory and contemporary mathematical physics.

By applying methods of statistical physics Li and Sinai (J Eur Math Soc 10:267–313, 2008) proved that there are complex solutions of the Navier–Stokes equations in the whole space \({\mathbb R}^{3}\) which blow up at a finite time. We present a review of the results obtained so far, by theoretical work and computer simulations, for the singular complex solutions, and compare with the behavior of related real solutions. We also discuss the possible application of the techniques introduced in (J Eur Math Soc 10:267–313, 2008) to the study of the real ones.

In this survey, we outline several results on the distribution of ${\cal B}$-free integers and explore a random process naturally associated to them. We show how, notwithstanding the rigid ergodic properties of this process (zero entropy, pure point spectrum, no weak mixing), it exhibits a central limit theorem resembling a theorem by Beck on the circle rotation by a quadratic surd. We explain the connection of the random process to the distribution of ${\cal B}$-free integers in short intervals, with particular emphasis on their variance and higher moments.

We construct eigenfunctions of a modified Laplace operator in certain two-dimensional polygonal domains with Dirichlet boundary condition. A special property of domains is the existence of a lattice on which our operator can be defined. This allows us to give explicit expressions for eigenfunctions.

Let \((X_n)_{n=1}^{\infty }\) be a sequence of independent identically distributed random variables. We study the normalized partial sums and the corresponding renormalization group flow in the space of probability densities. We prove the convergence to stable limit laws under suitable assumptions on the initial density.