For $h>1,$ we consider the reaction-diffusion equation:

$$ \begin{align*} \Delta_\infty^hu(x)=f(x,u(x),Du(x)),\quad x\in\Omega, \end{align*} $$

$\Delta _\infty ^h$

$f\in C(\Omega \times \mathbb {R}\times \mathbb {R}^n)$

$ 0\leq f(x,\delta t,p)\leq \Lambda (x)\delta ^{\gamma }f(x,t,p),$

$\Lambda (x)\in C(\overline {\Omega }), \, \gamma \in [0,h), \,t>0$

$\delta>0 $

$C^{({h+1})/({h-\gamma })}$

$\partial \{u>0\}\cap \Omega .$

$\gamma =h,$

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers A and B have natural density 1, then their product set $A \cdot B \;:\!=\; \{ab \;:\; a \in A, b \in B\}$ also has natural density 1. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set A of density 1 such that $A\cdot A$ has a “large” complement.

We classify almost Ricci–Bourguignon solitons on three-dimensional almost $\alpha$-cosymplectic manifolds. We study almost Ricci-Bourguignon solitons on almost $\alpha$-cosymplectic manifolds, with an emphasis on their classification and geometric properties. Key results include soliton type characterization (shrinking, steady, expanding) via the parameter $\rho$ and conditions under which these solitons become Einstein. We also show that Ricci semi-symmetric manifolds with $\eta$-parallel tensors reduce to almost cosymplectic structures. A five-dimensional example of an almost contact manifold admitting a Ricci-Bourguignon soliton has been constructed. Also, Lie-group classifications in dimension three are obtained, which are almost RB transversal solitons on almost $\alpha$-cosymplectic manifolds.

In recent years, b-symplectic manifolds have emerged as important objects in symplectic geometry. These manifolds are Poisson manifolds that exhibit symplectic behaviour away from a distinguished hypersurface, where the symplectic form degenerates in a controlled manner. Inspired by this rich landscape, E-structures were introduced by Nest and Tsygan in [NT01] as a comprehensive framework for exploring generalizations of b-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by [MS21]. We also examine the closely related concept of almost regular Poisson manifolds, as studied in [AZ17], which reveals a natural Poisson groupoid associated with these structures.

In this article, we investigate the intricate relationship between E-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the E-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.

We establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze’s basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results.

Our key new tool is the six functor formalism with solid almost $\mathcal {O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the domain of validity of the $!$-functor formalism developed in [Man22] to allow certain ‘stacky’ maps. In the language of this extended formalism, we show that if G is a p-adic Lie group, the structure map of the classifying small v-stack $B\underline {G}$ is p-cohomologically smooth.

A closed Riemannian three-manifold $(Y,g)$ equipped with a torsion spin$^c$ structure determines a family of Dirac operators $\{D_B\}$ parametrized by a $b_1(Y)$-dimensional torus $\mathbb {T}_Y$. In this paper, we develop techniques to study how the topology of the locus $\mathsf {K}\subset \mathbb {T}_Y$ corresponding to operators with non-trivial kernel (the three-dimensional analogue of the theta divisor of a Riemann surface) depends on the geometry of the metric. As a concrete example of our methods, we show that for any metric on the three-torus $Y=T^3$ for which the spectral gap $\lambda _1^*$ on coexact $1$-forms is large, after a small perturbation of the family, the locus $\mathsf {K}$ is a two-sphere.

While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin$^c$ three-manifold $(Y,\mathfrak {s})$ with a large $\lambda _1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak {s})$ in terms of the topology of the family of Dirac operators $\{D_B\}$.

In this paper we study the relationship between ideals and congruences of the tropical polynomial and Laurent polynomial semirings. We show that the variety of a non-zero prime ideal of the tropical (Laurent) polynomial semiring consists of at most one point. We also prove a result relating the dimension of an affine tropical variety and the dimension of its “coordinate semiring”.