Our aim in this paper is to deal with the boundedness of the central Hardy-Littlewood maximal operator on Herz spaces with radial variable exponent. As a special case, we show the boundedness of the maximal operator in weighted Lebesgue spaces with radial variable exponent. We also discuss the integrability of the maximal functions in such spaces.

In this paper, we focus on solvable prehomogeneous vector spaces associated with homogeneous cones, and consider the associated zeta functions in several variables. We discuss $\mathbb{Q}$-structures of these prehomogeneous vector spaces, and give explicit formulas of functional equations of the zeta functions by using the data of homogeneous cones. The associated $b$-functions are also described explicitly.

For small-energy initial regular planar curves with generalised Neumann boundary conditions, we consider the steepest-descent gradient flow for the $L^2$-norm of the derivative of curvature with respect to arc length. We show that such curves between parallel lines converge exponentially in the $C^\infty$ topology in infinite time to straight lines.

In this article, we give examples of smooth varieties of positive characteristic whose first integral overconvergent de Rham–Witt cohomology modulo torsion is not finitely generated over the Witt ring of the base field.

$N$-Metaplectic categories, unitary modular categories with the same fusion rules as $SO (N)_2$, are prototypical examples of weakly integral modular categories generalizing the model for the Ising anyons, i.e. metaplectic anyons. A conjecture of the second author would imply that images of the braid group representations associated with metaplectic categories are finite groups, i.e. have property $F$. While it was recently shown that $SO (N)_2$ itself has property $F$, proving property $F$ for the more general class of metaplectic modular categories is an open problem. We verify this conjecture for $N$-metaplectic modular categories when $N$ is odd, exploiting their recent enumeration together with a characterization in terms of Galois conjugation and twisting. In another direction, we prove that when $N$ is divisible by 8 the $N$-metaplectic categories have 3 non-trivial bosons, and the boson condensation procedure applied to 2 of these bosons yields $\frac{N}{4}$-metaplectic categories. Otherwise stated: any 8$k$-metaplectic category is a $\mathbb{Z}_2$-gauging of a $2k$-metaplectic category, so that the $N$ even metaplectic categories lie towers of $\mathbb{Z}_2$-gaugings commencing with $2k$- or $4k$-metaplectic categories with $k$ odd.

A $C^0$-Finsler structure on a differentiable manifold is a continuous real valued function defined on its tangent bundle such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\hat M,\hat F)$, where $\hat M$ is diffeomorphic to the Euclidean plane. The structures $\hat F$ don’t have partial derivatives and they aren’t invariant by any transformation group of $\hat M$. For every $p,q \in (\hat M,\hat F)$, we determine the unique minimizing path connecting $p$ and $q$. They are line segments parallel to the vectors $(\sqrt{3}/2,1/2)$, (0,1) or $(-\sqrt{3}/2,1/2)$, or else a concatenation of two of these line segments. Moreover $(\hat M,\hat F)$ aren’t Busemann $G$-spaces and they don’t admit any bounded open $\hat F$-strongly convex subsets. Other geodesic properties of $(\hat M,\hat F)$ are also studied.

This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial 1-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow–Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.