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Vol. 63, no. 2 (2022)

Blow-up of positive-initial-energy solutions for nonlinearly damped semilinear wave equations. Mohamed Amine Kerker
We consider a class of semilinear wave equations with both strongly and nonlinear weakly damped terms, \[ u_{tt}-\Delta u-\omega\Delta u_t+\mu\vert u_t\vert^{m-2}u_t=\vert u\vert^{p-2}u, \] associated with initial and Dirichlet boundary conditions. Under certain conditions, we show that any solution with arbitrarily high positive initial energy blows up in finite time if $m < p$. Furthermore, we obtain a lower bound for the blow-up time.
Binomial edge ideals of cographs. Thomas Kahle and Jonas Krüsemann
We determine the Castelnuovo–Mumford regularity of binomial edge ideals of complement-reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph-theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.
A compact manifold with infinite-dimensional co-invariant cohomology. Mehdi Nabil
Let $M$ be a smooth manifold. When $\Gamma$ is a group acting on $M$ by diffeomorphisms, one can define the $\Gamma$-co-invariant cohomology of $M$ to be the cohomology of the complex $\Omega_c(M)_\Gamma=\operatorname{span}\{\omega-\gamma^*\omega : \omega\in\Omega_c(M),\,\gamma\in\Gamma\}$. For a Lie algebra $\mathcal{G}$ acting on the manifold $M$, one defines the cohomology of $\mathcal{G}$-divergence forms to be the cohomology of the complex $\mathcal{C}_{\mathcal{G}}(M)=\operatorname{span}\{L_X\omega : \omega\in\Omega_c(M),\,X\in\mathcal{G}\}$. In this short paper we present a situation where these two cohomologies are infinite dimensional on a compact manifold.
Conformal vector fields on statistical manifolds. Leila Samereh and Esmaeil Peyghan
Introducing the conformal vector fields on a statistical manifold, we present necessary and sufficient conditions for a vector field on a statistical manifold to be conformal. After presenting some examples, we classify the conformal vector fields on two famous statistical manifolds. Considering three statistical structures on the tangent bundle of a statistical manifold, we study the conditions under which the complete and horizontal lifts of a vector field can be conformal on these structures.
The isometry groups of Lorentzian three-dimensional unimodular simply connected Lie groups. Mohamed Boucetta and Abdelmounaim Chakkar
We determine the isometry groups of all three-dimensional, connected, simply connected and unimodular Lie groups endowed with a left-invariant Lorentzian metric.
On the differential and Volterra-type integral operators on Fock-type spaces. Tesfa Mengestie
The differential operator fails to admit some basic structures including continuity when it acts on the classical Fock spaces or weighted Fock spaces, where the weight functions grow faster than the classical Gaussian weight function. The same conclusion also holds in some weighted Fock spaces including the Fock–Sobolev spaces, where the weight functions grow more slowly than the Gaussian function. We consider modulating the classical weight function and identify Fock-type spaces where the operator admits the basic structures. We also describe some properties of Volterra-type integral operators on these spaces using the notions of order and type of entire functions. The modulation operation supplies richer structures for both the differential and integral operators in contrast to the classical setting.
A note on the density of the partial regularity result in the class of viscosity solutions. Disson dos Prazeres, Edgard A. Pimentel, and Giane C. Rampasso
We establish the density of the partial regularity result in the class of continuous viscosity solutions. Given a fully nonlinear equation, we prove the existence of a sequence entitled to the partial regularity result, approximating its solutions. Distinct conditions on the operator driving the equation lead to density in different topologies. Our findings include applications to nonhomogeneous problems, with variable-coefficient models.
Monster graphs are determined by their Laplacian spectra. Ali Zeydi Abdian, Ali Reza Ashrafi, Lowell W. Beineke, Mohammad Reza Oboudi, and Gholam Hossein Fath-Tabar
A graph $G$ is determined by its Laplacian spectrum (DLS) if every graph with the same Laplacian spectrum is isomorphic to $G$. A multi-fan graph is a graph of the form $(P_{n_1}\cup P_{n_2}\cup \cdots \cup P_{n_k})\bigtriangledown K_1$, where $K_1$ denotes the complete graph of size 1, $P_{n_1}\cup P_{n_2}\cup \cdots \cup P_{n_k}$ is the disjoint union of paths $P_{n_i}$, $n_i\geq 1$ and $1 \leq i \leq k$; and a starlike tree is a tree with exactly one vertex of degree greater than 2. If a multi-fan graph and a starlike tree are joined by identifying their vertices of degree more than 2, then the resulting graph is called a monster graph. In some earlier works, it was shown that all multi-fan and path-friendship graphs are DLS. The aim of this paper is to generalize these facts by proving that all monster graphs are DLS.
Sharp bounds for fractional type operators with $L^{\alpha,s}$-Hörmander conditions. Gonzalo H. Ibañez-Firnkorn, María Silvina Riveros, and Raúl E. Vidal
We provide the sharp bound for a fractional type operator given by a kernel satisfying the $L^{\alpha,s}$-Hörmander condition and certain fractional size condition, $0 < \alpha < n$ and $1 < s\leq \infty$. In order to prove this result we use a new appropriate sparse domination. Examples of these operators include the fractional rough operators. For the case $s=\infty$ we recover the sharp bound of the fractional integral, $I_{\alpha}$, proved by Lacey et al. [J. Functional Anal. 259 (2010), no. 5, 1073–1097].
The isolation of the first eigenvalue for a Dirichlet eigenvalue problem involving the Finsler $p$-Laplacian and a nonlocal term. Andrei Grecu
We analyse the isolation of the first eigenvalue for an eigenvalue problem involving the Finsler $p$-Laplace operator and a nonlocal term on a bounded domain subject to the homogeneous Dirichlet boundary condition.

Vol. 64, no. 2 (2022)
Special volume: Mathematical Congress of the Americas 2021

Editors: Guillermo Cortiñas (managing editor, Universidad de Buenos Aires); Renato Iturriaga (CIMAT, Guanajuato, Mexico); Andrea Nahmod (University of Massachusetts, Amherst, USA); Jimmy Petean (CIMAT, Guanajuato, Mexico); Arturo Pianzola (University of Alberta, Canada and Universidad CAECE, Argentina); Cecilia Salgado (University of Groningen, Netherlands and Universidade Federal do Rio de Janeiro, Brazil); Soledad Torres (CIMFAV-INGEMAT, Universidad de Valparaíso, Chile); Bernardo Uribe (Universidad del Norte, Colombia); Steven Weintraub (Lehigh University, Bethlehem, USA).

Preface. Guillermo Cortiñas
Univariate rational sums of squares. Teresa Krick, Bernard Mourrain, and Agnes Szanto
Given rational univariate polynomials $f$ and $g$ such that $\gcd (f,g)$ and $f/\gcd(f,g)$ are relatively prime, we show that $g$ is non-negative at all the real roots of $f$ if and only if $g$ is a sum of squares of rational polynomials modulo $f$. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial $g$ is non-negative at the real roots of a non-zero polynomial $f$ when the above assumption is satisfied.
A note on Bernstein–Sato ideals. Josep Àlvarez Montaner
We define the Bernstein–Sato ideal associated to a tuple of ideals and we relate it to the jumping points of the corresponding mixed multiplier ideals.
On essential self-adjointness of singular Sturm–Liouville operators. S. Blake Allan, Fritz Gesztesy, and Alexander Sakhnovich
Considering singular Sturm–Liouville differential expressions of the type \[ \tau_{\alpha} = -(d/dx)x^{\alpha}(d/dx) + q(x), \quad x \in (0,b), \, \alpha \in \mathbb{R}, \] we employ some Sturm comparison-type results in the spirit of Kurss to derive criteria for $\tau_{\alpha}$ to be in the limit-point and limit-circle case at $x=0$. More precisely, if $\alpha \in \mathbb{R}$ and, for $0 < x$ sufficiently small, \[ q(x) \geq [(3/4)-(\alpha/2)]x^{\alpha-2}, \] or, if $\alpha\in (-\infty,2)$ and there exist $N\in\mathbb{N}$ and $\varepsilon>0$ such that, for $0 < x$ sufficiently small, \begin{align*} & q(x)\geq[(3/4)-(\alpha/2)]x^{\alpha-2} - (1/2) (2 - \alpha) x^{\alpha-2} \sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{-1} \\ & \quad\quad\quad +[(3/4)+\varepsilon] x^{\alpha-2}[\ln_{1}(x)]^{-2}, \end{align*} then $\tau_{\alpha}$ is nonoscillatory and in the limit-point case at $x=0$. Here iterated logarithms for $0 < x$ sufficiently small are of the form \[ \ln_1(x) = |\ln(x)| = \ln(1/x), \qquad \ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in \mathbb{N}. \] Analogous results are derived for $\tau_{\alpha}$ to be in the limit-circle case at $x=0$. We also discuss a multi-dimensional application to partial differential expressions of the type \[ - \operatorname{div} |x|^{\alpha} \nabla + q(|x|), \quad \alpha \in \mathbb{R}, \, x \in B_n(0;R) \backslash \{0\}, \] with $B_n(0;R)$ the open ball in $\mathbb{R}^n$, $n\in \mathbb{N}$, $n \geq 2$, centered at $x=0$ of radius $R \in (0, \infty)$.
Inequivalent representations of the dual space. Tepper L. Gill, Douglas Mupasiri, and Erdal Gül
We show that there exist inequivalent representations of the dual space of $\mathbb{C}[0,1]$ and of $L_p[\mathbb{R}^n]$ for $p \in [1,\infty)$. We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if $\mathcal{A}$ is a proper closed subspace of $\mathbb{C}[0,1]$, there always exists a closed subspace $\mathcal{A}^\bot$ (with the same definition as for $L_2[0,1]$) such that $\mathcal{A} \cap \mathcal{A}^\bot = \{0\}$ and $\mathcal{A} \oplus \mathcal{A}^\bot = \mathbb{C}[0,1]$. Thus, the geometry of $\mathbb{C}[0,1]$ can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator $A$ on $\mathbb{C}[0,1]$ has a uniquely defined adjoint $A^*$ defined on $\mathbb{C}[0,1]$, and both can be extended to bounded linear operators on $L_2[0,1]$. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to $L_p[\mathbb{R}^n]$. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on $\mathbb{C}[0,1]$ is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.