In this paper we prove a reverse Faber–Krahn inequality for the principal eigenvalue ${\mu }_1(\Omega )$ of the fully nonlinear eigenvalue problem

Here ${\lambda }_N(D^{2}u)$ stands for the largest eigenvalue of the Hessian matrix of $u$. More precisely, we prove that, for an open, bounded, convex domain $\mathrm{\Omega }\subset {ℝ}^N$, the inequality

$${\mu }_1(\mathrm{\Omega })\le \frac{{\pi }^2}{{[\text{diam}(\mathrm{\Omega })]}^2}={\mu }_1(B_{\text{diam}(\mathrm{\Omega })/2}),$$

where $\text{diam}(\mathrm{\Omega })$ is the diameter of $\mathrm{\Omega }$, holds true. The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint.

Furthermore, we discuss the minimization of ${\mu }_1(\mathrm{\Omega })$ under different kinds of constraints.

Extending the Wedderburn–Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). Our results in this direction complement those of Iovanov–Mesyan–Reyes. An extension of Bass’ theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, many equivalences and implications between which we prove. In particular, all conditions are equivalent for topological rings with a countable base of neighborhoods of zero and for topologically right coherent topological rings. Considering the rings of endomorphisms of modules as topological rings with the finite topology, we establish a close connection between the concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. We show that any topologically agreeable split abelian category is Grothendieck and semisimple. We also prove that a module $\mathrm{\Sigma }$-coperfect over its endomorphism ring has a perfect decomposition provided that either the endomorphism ring is commutative or the module is countably generated, partially answering a question of Angeleri Hügel and Saorín.

It is known that the ideals of a Leavitt path algebra $L_K(E)$ generated by $P_l(E)$, by $P_{\text{c}}(E)$, or by $P_{ec}(E)$ are invariant under isomorphism. Though the ideal generated by $P_{b^{\infty }}(E)$ is not invariant we find its “natural” replacement (which is indeed invariant): the one generated by the vertices of $P_{b_p^{\infty }}$ (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the $DCC$ topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if $H$ is a hereditary saturated subset of vertices providing an invariant ideal, its exterior $\text{ext}(H)$ in the $DCC$ topology of $E^0$ generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance $P_l$, etc.). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain ${\text{Soc}}^{(1)}()\subseteq {\text{Soc}}^{(2)}()\subseteq \cdots$, all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative.

We study the Fitting ideals over the finite layers of the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin–Selberg convolution of two modular forms $f$ and $g$. Inspired by the theta elements for modular forms defined by Mazur and Tate in [32], we define new theta elements for Rankin–Selberg convolutions of $f$ and $g$ using Loeffler–Zerbes’ geometric $p$-adic $L$-functions attached to $f$ and $g$.

Under certain technical hypotheses, we generalize a recent work of Kim–Kurihara on elliptic curves to prove a result very close to the weak main conjecture of Mazur and Tate for Rankin–Selberg convolutions. Special emphasis is given to the case where $f$ corresponds to an elliptic curve $E$ and $g$ to a two-dimensional odd irreducible Artin representation $\rho$ with splitting field $F$. As an application, we give an upper bound of the dimension of the $\rho$-isotypic component of the Mordell–Weil group of $E$ over the finite layers of the cyclotomic ${\mathbb{Z}}_p$-extension of $F$ in terms of the order of vanishing of our theta elements.

We prove that for any $L^Q$-valued Schwartz function $f$ defined on ${ℝ}^d$, one has the multiple vector-valued, mixed-norm estimate

$$\parallel f{\parallel }_{L^P(L^Q)}\lesssim \parallel Sf{\parallel }_{L^P(L^Q)}$$

valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying componentwise. Here $S:=S_{d_1}\otimes \cdots \otimes S_{d_N}$ is a tensor product of several Littlewood–Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces ${ℝ}^{d_j}$ for $1\le j\le N$, with the property that $d_1+\cdots +d_N=d$. This answers a question that came up implicitly in our recent works [2], [3], [5] and completes in a natural way classical results of Littlewood–Paley theory. The proof is based on the helicoidal method introduced by the authors in the aforementioned papers.

An algebraic system consisting of a set together with an associative binary and a ternary heap operations is studied. Such a system is termed a pre-truss and if a binary operation distributes over the heap operation on one side we call it a near-truss. If the binary operation in a near-truss is a group operation, then it can be specified or retracted to a skew brace, the notion introduced in [8]. On the other hand if the binary operation in a near-truss has identity, then it gives rise to a skew-ring as introduced in [14]. Congruences in pre- and near-trusses are shown to arise from normal sub-heaps with an additional closure property of equivalence classes that involves both the ternary and binary operations. Such sub-heaps are called paragons. A necessary and sufficient criterion on paragons under which the quotient of a unital near-truss corresponds to a skew brace is derived. Regular elements in a pre-truss are defined as elements with left and right cancellation properties; following the ring-theoretic terminology, pre-trusses in which all non-absorbing elements are regular are termed domains. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses and near-trusses as domains that satisfy the Ore condition are introduced and pre-trusses of fractions are constructed through localisation. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces.

We give an estimate that quantifies the fact that a normalized quasiconformal map whose dilatation is non-zero only on a set of small area approximates the identity uniformly on the whole plane. The precise statement is motivated by applications of the author’s quasiconformal folding method for constructing entire functions; in particular an application to constructing transcendental wandering domains given by Fagella, Godillon, and Jarque [7].

We extend ultragraph shift spaces and the realization of ultragraph ${\text{C}}^{*}$-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allows us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which—as it happens in the free group—is computable in the finitely generated case.

This approach provides a neat geometric description of (even non-(finitely generated)) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals, respectively.

In this note we are concerned with estimates for the spectral projection operator ${𝒫}_{\mu }$ associated with the twisted Laplacian $L$. We completely characterize the optimal bounds on the operator norm of ${𝒫}_{\mu }$ from $L^p$ to $L^q$ when $1\le p\le 2\le q\le \infty$. As an application, we obtain a uniform resolvent estimate for $L$.

We define a twisted $L^2$-torsion on the character variety of a $3$-manifold $M$ and study some of its properties. In the case where $M$ is hyperbolic of finite volume, we prove that the $L^2$-torsion is a real-analytic function in a neighbourhood of any lift of the holonomy representation.

Consider the equation $\text{div}({\phi }^2\nabla \sigma )=0$ in ${ℝ}^N$, where . It is well known [4, 2] that if there exists such that ${\displaystyle {\int }_{B_R}}{(\phi \sigma )}^{2}dx\le C{R}^2$ for every $R\ge 1$, then $\sigma$ is necessarily constant. In this paper we present a simple proof that this result is not true if we replace $R^2$ with $R^k$ for in any dimension $N$. This question is related to a conjecture by De Giorgi [7].