We relate the complexity of both differential and $q$-difference equations of order one and degree one and their solutions. Our point of view is to show that if the solutions are complicated, the initial equation is complicated too. In this spirit, we bound from below an invariant of the differential or $q$-difference equation, the height of its Newton polygon, in terms of the characteristic factors of a solution. The differential and the $q$-difference cases are treated in a unified way.

This paper deals with cyclic covers of a large family of rational normal surfaces that can also be described as quotients of a product, where the factors are cyclic covers of algebraic curves. We use a generalization of the Esnault–Viehweg method to show that the action of the monodromy on the first Betti group of the covering (and its Hodge structure) splits as a direct sum of the same data for some specific cyclic covers over ${\mathrm{\mathbb{P}}}^1$.

This has applications to the study of Lê–Yomdin surface singularities, in particular to the action of the monodromy on the mixed Hodge structure, as well as to isotrivial fibered surfaces.

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^{*}$. At the same time we determine conditions under which these methods converge in weak and weak* topologies on $X$ and $X^{*}$ respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in $X$. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, and de Branges–Rovnyak spaces. Our approach shows that all these applications flow from just two abstract theorems.

We first show that a generic hypersurface $V$ of degree $d\ge 3$ in the projective complex space ${\mathrm{\mathbb{P}}}^n$ of dimension $n\ge 3$ has at least one hyperplane section $V\cap H$ containing exactly $n$ ordinary double points, alias $A_1$ singularities, in general position, and no other singularities. Equivalently, the dual hypersurface $V^{\vee }$ has at least one normal crossing singularity of multiplicity $n$. Using this result, we show that the dual of any smooth hypersurface with $n,d\ge 3$ has at least a very singular point $q$, in particular a point $q$ of multiplicity $\ge n$.

Let $G$ be a group of finite rank and $\pi$ any finite set of primes. We prove that $G$ contains a characteristic subgroup $H$ of finite index such that every finite $\pi$-image of $H$ is nilpotent. Our conclusions are stronger if $G$ is also soluble.

We develop new local $T1$ theorems to characterize Calderón–Zygmund operators that extend boundedly or compactly on $L^p({\mathrm{ℝ}}^n,\mu )$, with $\mu$ a measure of power growth.

The results, whose proofs do not require random grids, have weaker hypotheses than previously known local $T1$ theorems since they only require a countable collection of testing functions. Moreover, a further extension of this work allows the use of testing functions supported on cubes of different dimensions.

As a corollary, we describe the measures $\mu$ of the complex plane for which the Cauchy integral defines a compact operator on $L^p(\mathrm{ℂ},\mu )$.

We study a class of nonlocal double phase problems with discontinuous coefficients. A local self-improving property and a higher Hölder continuity result for weak solutions to such problems are obtained under the assumptions that the associated coefficient functions are of VMO (vanishing mean oscillation) type and that the principal coefficient depends not only on the variables but also on the solution itself.

We study properties of continuous finite group actions on topological manifolds that hold true, for any finite group action, after possibly passing to a subgroup of index bounded above by a constant depending only on the manifold. These include the Jordan property, the almost fixed point property, as well as bounds on the discrete degree of symmetry. Most of our results apply to manifolds satisfying some restriction such as having nonzero Euler characteristic or having the integral homology of a sphere. For an arbitrary topological manifold $X$ such that $H_{*}(X;\mathrm{\mathbb{Z}})$ is finitely generated, we prove the existence of a constant $C$ with the property that for any continuous action of a finite group $G$ on $X$ such that every $g\in G$ fixes at least one point of $X$, there is a subgroup $H\le G$ satisfying $[G:H]\le C$ and a point $x\in X$ which is fixed by all elements of $H$.

Condition $\mathbf{\text{(Fg)}}$ was introduced in [6] to ensure that the theory of support varieties of a finite-dimensional algebra, established by Snashall and Solberg, has some similar properties to that of a group algebra. In this paper we give some easy-to-check combinatorial conditions that are equivalent to $\mathbf{\text{(Fg)}}$ for monomial $d$-Koszul algebras. We then extend this to monomial $(D,A)$-stacked algebras. We also extend the description of the Yoneda algebra of a $d$-Koszul algebra in [10] to $(D,A)$-stacked monomial algebras.

Let $G$ be a locally compact group and $\mu$ be a probability measure on $G$. We consider the convolution operator ${\lambda }_1\left(\mu \right):L_1\left(G\right)\to L_1\left(G\right)$ given by ${\lambda }_1\left(\mu \right)f=\mu *f$ and its restriction ${\lambda }_1^0\left(\mu \right)$ to the augmentation ideal $L_1^0\left(G\right)$. Say that $\mu$ is uniformly ergodic if the Cesàro means of the operator ${\lambda }_1^0\left(\mu \right)$ converge uniformly to $0$, that is, if ${\lambda }_1^0\left(\mu \right)$ is a uniformly mean ergodic operator with limit $0$, and that $\mu$ is uniformly completely mixing if the powers of the operator ${\lambda }_1^0\left(\mu \right)$ converge uniformly to $0$.

We completely characterize the uniform mean ergodicity of the operator ${\lambda }_1\left(\mu \right)$ and the uniform convergence of its powers, and see that there is no difference between ${\lambda }_1\left(\mu \right)$ and ${\lambda }_1^0\left(\mu \right)$ in these regards. We prove in particular that $\mu$ is uniformly ergodic if and only if $G$ is compact, $\mu$ is adapted (its support is not contained in a proper closed subgroup of $G$), and $1$ is an isolated point of the spectrum of $\mu$. The last of these three conditions can actually be replaced by $\mu$ being spread out (some convolution power of $\mu$ is not singular). The measure $\mu$ is uniformly completely mixing if and only if $G$ is compact, $\mu$ is spread out, and the only unimodular value in the spectrum of $\mu$ is $1$.