Revista de la Unión Matemática Argentina

Decidable objects and molecular toposes

Matías Menni — Wed, 17 Jul 2024 00:00:00 -0300

We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particular, we show that if $\mathcal{S}$ is a Boolean topos, then, for every hyperconnected essential geometric morphism $p : \mathcal{E} \rightarrow \mathcal{S}$ such that the leftmost adjoint $p_{!}$ preserves finite products, $p$ is molecular and $p^* : \mathcal{S} \rightarrow \mathcal{E}$ coincides with the full subcategory of decidable objects in $\mathcal{E}$. We also characterize the reflections between categories with finite limits that induce molecular maps between the respective presheaf toposes. As a corollary we establish the molecularity of certain geometric morphisms between Gaeta toposes.

Extinction time of an epidemic with infection-age-dependent infectivity

Anicet Mougabe-Peurkor, Ibrahima Dramé, Modeste N'zi, Étienne Pardoux — Thu, 12 Sep 2024 00:00:00 -0300

This paper studies the distribution function of the time of extinction of a subcritical epidemic, when a large enough proportion of the population has been immunized and/or the infectivity of the infectious individuals has been reduced, so that the effective reproduction number is less than one. We do that for a SIR/SEIR model, where infectious individuals have an infection-age-dependent infectivity, as in the model introduced in Kermack and McKendrick's seminal 1927 paper. Our main conclusion is that simplifying the model as an ODE SIR model, as it is largely done in the epidemics literature, introduces a bias toward shorter extinction time.

On hyponormality and a commuting property of Toeplitz operators

Houcine Sadraoui, Borhen Halouani — Mon, 16 Sep 2024 00:00:00 -0300

In this work we give sufficient conditions for hyponormality of Toeplitz operators on a weighted Bergman space when the analytic part of the symbol is a monomial and the conjugate part is a polynomial. We also extend a known commuting property of Toeplitz operators with a harmonic symbol on the Bergman space to weighted Bergman spaces.

Genus and book thickness of reduced cozero-divisor graphs of commutative rings

Edward Jesili, Krishnan Selvakumar, Thirugnanam Tamizh Chelvam — Mon, 23 Sep 2024 00:00:00 -0300

For a commutative ring $R$ with identity, let $\langle a\rangle$ be the principal ideal generated by $a\in R$. Let $\Omega(R)^*$ be the set of all nonzero proper principal ideals of $R$. The reduced cozero-divisor graph $\Gamma_r(R)$ of $R$ is the simple undirected graph whose vertex set is $\Omega(R)^*$ and such that two distinct vertices $\langle a\rangle$ and $\langle b\rangle$ in $\Omega(R)^\ast$ are adjacent if and only if $\langle a \rangle\nsubseteq\langle b\rangle$ and $\langle b\rangle\nsubseteq\langle a\rangle$. In this article, we study certain properties of embeddings of the reduced cozero-divisor graph of commutative rings. More specifically, we characterize all Artinian nonlocal rings whose reduced cozero-divisor graph has genus two. Also we find the book thickness of the reduced cozero-divisor graphs which have genus at most one.

Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

Luciana F. Martins, Kentaro Saji, Samuel P. dos Santos, Keisuke Teramoto — Mon, 23 Sep 2024 00:00:00 -0300

Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of this divergence, in particular the boundedness about these invariants, represent the geometry of the surface and the curve. In this paper, we study the boundedness and orders of several geometric invariants near a singular point of a surface which is a suspension of a singular curve in the plane, and those of the curves passing through the singular point. We evaluate the orders of the Gaussian and mean curvatures, as well as those of the geodesic and normal curvatures, and the geodesic torsion for the curve.

Complete presentation and Hilbert series of the mixed braid monoid $MB_{1,3}$

Zaffar Iqbal, Muhammad Mobeen Munir, Abdul Rauf Nizami — Wed, 25 Sep 2024 00:00:00 -0300

The Hilbert series is the simplest way of finding dimension and degree of an algebraic variety defined explicitly by polynomial equations. The mixed braid groups were introduced by Sofia Lambropoulou in 2000. In this paper we compute the complete presentation and the Hilbert series of the canonical words of the mixed braid monoid $MB_{1,3}$.

One-sided EP elements in rings with involution

Cang Wu, Jianlong Chen, Yu Chen — Mon, 07 Oct 2024 00:00:00 -0300

This paper investigates the one-sided EP property of elements in rings with involution. Let $R$ be a ring with involution $\ast$. Then $a \in R$ is said to be left (resp. right) EP if $a$ is Moore–Penrose invertible and $aR \subseteq a^{\ast}R$ (resp. $a^{\ast}R \subseteq aR$). Many properties of EP elements are extended to one-sided versions. Some new characterizations of EP elements are presented in relation to the absorption law for Moore–Penrose inverses.

Evolution of first eigenvalues of some geometric operators under the rescaled List's extended Ricci flow

Shahroud Azami, Abimbola Abolarinwa — Wed, 09 Oct 2024 00:00:00 -0300

Let $(M,g(t), e^{-\phi}d\nu)$ be a compact weighted Riemannian manifold and let $(g(t),\phi(t))$ evolve by the rescaled List's extended Ricci flow. In this paper, we study the evolution equations for first eigenvalues of the geometric operators, $-\Delta_{\phi}+cS^{a}$, along the rescaled List's extended Ricci flow. Here $\Delta_{\phi}=\Delta-\nabla\phi.\nabla$ is a symmetric diffusion operator, $\phi\in C^{\infty}(M)$, $S=R-\alpha|\nabla \phi|^{2}$, $R$ is the scalar curvature with respect to the metric $g(t)$ and $a, c$ are some constants. As an application, we obtain some monotonicity results under the rescaled List's extended Ricci flow.

The $w$-core–EP inverse in rings with involution

Dijana Mosić, Huihui Zhu, Liyun Wu — Wed, 09 Oct 2024 00:00:00 -0300

The main goal of this paper is to present two new classes of generalized inverses in order to extend the concepts of the (dual) core–EP inverse and the (dual) $w$-core inverse. Precisely, we introduce the $w$-core–EP inverse and its dual for elements of a ring with involution. We characterize the (dual) $w$-core–EP invertible elements and develop several representations of the $w$-core–EP inverse and its dual in terms of different well-known generalized inverses. Using these results, we get new characterizations and expressions for the core–EP inverse and its dual. We apply the dual $w$-core–EP inverse to solve certain operator equations and give their general solution forms.

The limit case in a nonlocal $p$-Laplacian equation with dynamical boundary conditions

Eylem Öztürk — Thu, 14 Nov 2024 00:00:00 -0300

In this paper we deal with the limit as $p\to \infty$ for the nonlocal analogous to the $p$-Laplacian evolution with dynamic boundary conditions. Our main result demonstrates this limit in both the elliptic and parabolic cases. We are interested in smooth and singular kernels and show the existence and uniqueness of a limit solution. We obtain that the limit solution of the elliptic problem turns out to be also a viscosity solution of a corresponding problem. We prove that the natural energy functionals associated with this problem converge, in the sense of Mosco, to a limit functional and therefore we obtain convergence of solutions to the evolution problems in the parabolic case. For the limit problem, we provide examples of explicit solutions for some particular data.