| Communications in Mathematics |
Editors: Jaqueline Godoy Mesquita, Mariel Sáez Trumper, Rafael Potrie & Tiago Macedo
In this paper we introduce a new two-player zero-sum game whose value function approximates the level set formulation for the geometric evolution by mean curvature of a hypersurface. In our approach the game is played with symmetric rules for the two players and probability theory is involved (the game is not deterministic).
An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as the local growth of the groups of rational points of the varieties in $\mathcal{A}$ after finite field extensions. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ is coprime with $f(1)$ divided by its radical.
We consider an elliptic system of Schrödinger-Bopp-Podolsky type in a bounded and smooth domain of R3 with a non constant coupling factor. This kind of system has been introduced in the mathematical literature in [14] and in the last years many contributions appeared. In particular here we present the results in [2] and [34] which show existence of solutions by means of the Ljusternik-Schnirelmann theory under different boundary conditions on the electrostatic potential.
In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact manifolds and finite intervals, and then introduce the notion of quasi-isospectrality. We next investigate the BMT method as a systematic approach to constructing quasi-isospectral Sturm-Liouville operators on a finite interval, and apply it to several boundary value problems. Our main result shows that any two quasi-isospectral closed manifolds of odd dimension are, in fact, isospectral. In addition, we extend classical compactness results for isospectral potentials on low-dimensional manifolds to the quasi-isospectral setting via heat trace asymptotics.
The Boltzmann equation has been a driving force behind significant mathematical research over the years. Its challenging theoretical complexity, combined with a wide variety of current scientific and technological problems that require numerical simulations based on this model, justifies such interest. This work provides a brief overview of studies and advances on the solution of the linear Boltzmann equation in one- and two-dimensional spatial dimensions. In particular, relevant aspects of the discrete ordinates approximation of the model are highlighted for neutron and photon transport applications, including nuclear safeguards, nuclear reactor shielding problems, and optical tomography. In addition, a short discussion of rarefied gas dynamics problems, relevant, for instance, to the study of micro-electro-mechanical systems, and their connection with the Linearized Boltzmann Equation, is presented. A primary goal of the work is to establish as much as possible the connections between the different phenomena described by the model and the versatility of the analytical methodology, the ADO method, in providing concise and accurate solutions, which are fundamental for numerical simulations.
In this paper, we introduce the deformed homogeneous polynomials $\mathrm{R}_{n}(x,y;u|q)$. These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials $\mathrm{h}_{n}(x|q)$, the generalized Rogers-Szegö polynomials $\mathrm{r}_{n}(x,y)$, the Stieltjes-Wigert polynomials $\mathrm{S}_{n}(x;q)$, among others. Basic properties of the polynomial $\mathrm{R}_{n}$ are given, along with recurrence relations, its $q$-difference equation, and representations. Generating functions for the polynomials $\mathrm{R}_{n}(x,y;u|q)$ are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the $q$-binomial formula and the Heine transformation formula are obtained. These results are obtained via the $u$-deformed $q$-exponential operator $\mathrm{E}(yD_{q}|u)$, defined here. From this operator, we obtain for free the operators T$(yD_{q})$ the Chen, $\mathrm{R}(yD_{q})$ of Saad, $\mathcal{E}(yD_{q})$ of Exton, and $\mathcal{R}(yD_{q})$ of Rogers-Ramanujan when $u=1,q,\sqrt{q},q^2$, respectively. We introduce the deformed basic hypergeometric series ${}_{r}Φ_{s}$, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.