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.

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.

This paper presents a comprehensive characterization of Menger spaces and star-Menger spaces through the lens of families of closed sets, employing nuanced modifications of the classical finite intersec¬tion property. By introducing and analyzing specific intersection pat¬terns within these families, we develop conditions that encapsulate the essence of the Menger and star-Menger covering properties. Further¬more, we explore the associated selection principles and demonstrate how they can be systematically reversed to reconstruct the topological structure of Menger and star-Menger spaces. This dual perspective not only offers an alternative viewpoint on classical results but also contributes to the ongoing effort to bridge the gap between topologi¬cal covering properties and combinatorial selection theory. Our results provide a new framework that enhances the theoretical understand¬ing of these spaces and may inspire further investigations into related classes of topological spaces.

The “Bertrand’s paradox” arises from a geometric probability problem for which J. Bertrand provides three solutions with different values of the probability sought; from here the “paradox” arises, a term that several authors attribute to J. Poincare. The interest in this issue has lasted for over a hundred years and even today it is possible to find ar- ticles in which Bertrand’s paradox, in one way or another, is take into consideration. Indeed, it as well as in the philosophical debate concern- ing the opposition between the frequentist and Bayesian approaches to Inferential Statistics, it has also used to highlight (hypothetical) logical inconsistencies of the “principle of indifference” in problems in which the total number of cases is not countable. The Principle of Indifference, whose origin is, often, attributed to P. S. Laplace, is a fundamental concept in the field of probability but also in decision-making under uncertainty. It offers a rational starting point for assigning probabilities in the absence of any information. However, it is essential to be aware of the importance of updating our beliefs as added information about the issue becomes available. In this article, in order to resolve Bertrand’s paradox, we highlight the role, often overlooked, of the random experiment (and also of the random device for its implementation) that generates the support of a probability space. Afterwards, from the problem itself it is possible to trace the class of generating events and a pre-measure on this class to complete a probability space consistent with the problem. This way of proceeding is illustrated for each of the three solutions identified by Bertrand and for another recently proposed solution. We conclude that all solutions are equally valid because, once the appropriate probability space has specified, the only correct solution will emerge in a logical and formal way.

Some backgrounds of temperature, entropy production and matter tensor are sketchy discussed.

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

In this paper, in the framework of rational extended irreversible thermodynamics with internal variables, a model for doped semiconductor crystals with dislocations is worked out, where a dislocation tensor and its gradient are introduced in the set of independent variables to describe these defect lines influencing the mechanical, thermal, electric transport properties of these media.The main equations of the model are introduced and the entropy inequality is analyzed by Liu's theorem, deriving the equations of state for the constitutive variables, the affinities, the dissipation inequality and other relations.Applying Wang's and Smith's theorems the constitutive theory and the expressions for the sources of the rate equations are carried out.According to the extended thermodynamics, a generalized Maxwell-Cattaneo-Vernotte equation for the heat flux and transport equations for the defects and charges fluxes present a relaxation time and a finite velocity for the disturbance propagation.The obtained results may have applications in several technological sectors, such as applied computer science, integrated circuits VLSI and nanotechnology (where high-frequency processes and the construction of sophisticated new materials with particular thermal properties are studied).

In this paper, we introduce a space-fractional mathematical model to describe the dynamics and interactions between vegetation and water in arid and semi-arid environments, both on flat and sloped terrains. The fractional model links two processes, such as water flows over in- clined surfaces and water spreads on flat terrain, enabling the study of pattern formation with different slopes of the domain. The first process is usually described by the Klausmeier model, and the second one by the Klausmeier-Gray-Scott model, which describes water diffu- sion. In the proposed fractional model, the fractional Caputo derivative term allows for modeling an anomalous transport phenomenon and the non-locality of the water advection process. Oscillatory dynamics and vegetation patterns are demonstrated through numerical simulations, using a migration speed derived from a stability analysis and resulting as a function of the fractional parameter. The computational results demonstrate the robustness and effectiveness of the fractional model, which captures ecological behaviors and anomalous transport mecha- nisms in different terrains.

We study optimal design problems involving variational inequalities with unilateral conditions in the domain and pointwise boundary observation. We use regularizing and penalization tehniques in the setting of the Hamiltonian approach to shape/topology optimization problems. Numerical examples are also included.

During the mold-filling process, the fiber orientation in a flowing fiber suspension plays a crucial role in determining the material properties of the resulting fiber composite. A common modelling approach introduces the second-order orientation tensor governed by the Folgar-Tucker equation, which serves as the equation of motion, combined with an appropriate closure relation. In the present work, analytical Solutions in three dimensions of the quadratically closed Folgar-Tucker equation are presented for a planar flow. The analysis reveals that the solution does not reduce to a purely two-dimensional orientation tensor. However, in the long-term limit, only two components of the orientation tensor remain independent—corresponding to the same number as in the two-dimensional case. A reconstruction of the orientation distribution function (ODF) further highlights the distinction between the fully three-dimensional orientation state and the effectively two-dimensional assumption in steady flow conditions. The analytical solution also shows that the steady-state orientation is highly sensitive to the shear rate. At very low shear rates the solution is nearly isotropic orientation tensor, whereas at very high shear rates it approaches a configuration that is strongly aligned in the velocity direction.