Haim Brezis was born on June 1, 1944, in Riom-s-Montagnes, a small town in Auvergne (France), and passed away on July 7, 2024, in Jerusalem.He was the son of a Romanian father, who had come to France in 1930, and a Jewish mother who had fled the Netherlands.Note that Haim is a Hebrew name that means life or living, being associated with the idea of vitality and energy, and is often given as a wish for a long and healthy life.His last name, Brezis, is an abbreviation of a Hebrew sentence.He also signed some of his publications with the names Ham Brezis or Ham Brzis, but we will only use Haim Brezis here (as in his functional analysis book published by Springer in 2010, see below).Haim Brezis earned his doctoral degree (doctorat d'tat) in 1972 from Universit de Paris, with two dierent theses, as was usual at that time in France, namely Problmes unilatraux (under the guidance of Gustave Choquet) and Groupes de diomorphismes et mouvement d'un fluide incompressible (under the guidance of Jacques-Louis Lions).The subject of the first thesis belongs to pure mathematics, while the second thesis was oriented towards an applied topic.In the following years, Haim Brezis dedicated his extraordinary talent especially to Nonlinear Analysis and to its applications in the study of Partial Dierential Equations that govern processes that appear in modern science.He was and remains a prominent personality of the international mathematical community, renowned for deepening existent research directions, opening new important research directions, showing impeccable rigor, and involving

Stationary flow with heat transfer in a gas is investigated within Rational Extended Thermodynamics. A rarefied gas is considered in the gap between two confocal elliptical cylinders or non-coaxial circu­lar ones. In both symmetries, internal and external cylinders are kept at two different constant temperatures and a flow in the axial direction is generated. Both Couette and Poiseuille problems are studied in these two symmetries. The solutions of the linearized field equations are determined and compared with the solutions of Classical Thermodynamic. Then, some non-linear effects are investigated. It is shown that the non-linear terms are able to describe some additional effects that are present in the Kinetic Theory but cannot be obtained within Classical Thermodynamics. In particular, non-vanishing stress tensor components and an axial heat flux are recovered in addition to the classical solutions.

Fixed domain approaches are used for some topology optimization problems of clamped Kirchhoff-Love plates: detection of a damaged zone using pointwise observation, respectively compliance minimization and the first eigenvalue maximization.We discuss the derivability with respect to functional variations of the geometry and some descent directions are proposed.Numerical tests are presented.

The aim of this paper is to introduce a new sequence convergent to the constant e.

In a previous paper, a linear theory for magnetic relaxation phe- nomena in reacting anisotropic fluid mixtures was developed, within the framework of the irreversible processes thermodynamics with internal variables. In that model, the total magnetization was split into two irreversible contributions, introducing one of them as an internal variable. In this paper, the same approach is applied to isotropic and perfect isotropic magnetizable reacting fluid mixtures. In the latter case, it is shown that in the phenomenological equations no cross effects arise between magnetic relaxation and other irreversible phenomena, since fluxes and thermodynamic forces of different tensorial character do not couple (Curie’s principle). For perfect isotropic mixtures, we also derive the equations of state and the magnetic relaxation equa- tion, which, unlike in the anisotropic case, do not contain contributions from temperature, concentrations, and their time derivatives. Special cases are treated. The results have applications in nuclear resonance, biology, medicine, and other fields.

In this contribution, we revisit the physical mathematical frame- work introduced in a previous paper, where stress-strain relations for viscoanelastic isotropic media were derived within irreversible thermo- dynamics with internal variables. We present the fundamental equa- tions and briefly discuss their conceptual implications. In subsequent paper quantitative applications of the model for the analysis of physical phenomena in heat propagation can also be extended to the more gen­eral case of an anisotropic medium. On this occasion, the model was used to investigate heat propagation phenomena, with results bench- marked against the Fourier and Cattaneo-Vernotte frameworks.

First-passage problems are considered for a Wright-Fisher diffusion process {X(t), t > 0}, which is important in population genetics and mathematical finance. Let t(x) be the first time that X(t), starting from X(0) = x, is equal to 0 or to 1. We compute the expected area covered by the process in the interval [0, t (x)]. We also compute the probability that the process will take on a value smaller than or equal to 0 before a value greater than or equal to 1 in the case when there are uniform jumps according to a Poisson process. Finally, stochastic control problems called homing problems are solved explicitly in particular cases for a controlled version of the process {X (t), t > 0}.

We consider a variational inequality in a reflexive Banach space X, governed by a history-dependent operator.The existence of a unique solution to the inequality can be proven by using a fixed point argument.Based on this fixed point property, we provide necessary and sufficient conditions which guarantee the uniform convergence of a sequence of functions to the solution of the variational inequality.We then exploit this result in the study of both a penalty method and the well-posedness analysis of the problem.Moreover, we present an example which arises in Contact Mechanics.It concerns the study of a mathematical model which describes the contact of a viscoelastic membrane with a foundation.

A Pythagorean triangle is a right triangle where the lengths of the three sides are all integers. The most famous example of a Pythagorean triangle is the 3 - 4 - 5 triangle, where the lengths of the sides are 3, 4, and 5 units, respectively. In this short note, we will study the class of Pythagorean triangles that may be associated with a square by means of a real parameter k. It turns out that the triangle 3 - 4 - 5 holds exclusive property in terms of the rationality of k.