We prove a version of the ergodic theorem for an action of an amenable group, where a Følner sequence need not be tempered. Instead, it is assumed that a function satisfies certain mixing conditions.

]]>In this manuscript, we generalize Lewis’s result about a central series associated with the vanishing off subgroup. We write for the vanishing off subgroup of , and for the terms in this central series. Lewis proved that there exists a positive integer such that if , then . Let . He also showed that if , then either or . We show that if for , where is the -th term in the lower central series of , then .

]]>In this short note we first extend the validity of the spectral radius formula, obtained by M. Anoussis and G. Gatzouras, for Fourier–Stieltjes algebras. The second part is devoted to showing that, for the measure algebra on any locally compact non-discrete Abelian group, there are no non-trivial constraints among three quantities: the norm, the spectral radius, and the supremum of the Fourier–Stieltjes transform, even if we restrict our attention to measures with all convolution powers singular with respect to the Haar measure.

]]>We establish new oscillation criteria for nonlinear differential equations of second order. The results here make some improvements of oscillation criteria of Butler, Erbe, and Mingarelli [2], Wong [8, 9], and Philos and Purnaras [6].

]]>Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials of order with over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

]]>Let be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators on variable Hardy spaces , where . As an application, we show that non-convolution type Calderón–Zygmund operators are bounded on if and only if , where and is the regular exponent of kernel of . Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

]]>Suppose that and is an integer. Let be the self-similar measure defined by . Assume that for some with and . We prove that if , then there are at most mutually orthogonal exponential functions in and is the best possible. If , then there are any number of orthogonal exponential functions in .

]]>Let be a -group and let be an irreducible character of . The codegree of is given by . If is a maximal class -group that is normally monomial or has at most three character degrees, then the codegrees of are consecutive powers of . If and has consecutive -power codegrees up to , then the nilpotence class of is at most 2 or has maximal class.

]]>We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another.

As an application, we provide a short proof (modulo Bass–Serre theory) of Vinogradov’s result that the free product of left-orderable groups is left-orderable.

]]>We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves.

]]>We prove that a generic homogeneous polynomial of degree is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order for .

]]>Let be a sequence of expanding matrices with , and let be a sequence of digit sets with , where , , and are positive integers for all . If , then the infinite convolution is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set such that is an orthonormal basis for .

]]>Let be a quasi-finite endomorphism of an algebraic variety defined over a number field and fix an initial point . We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set satisfies a strong gap principle.

]]>Every left-invariant ordering of a group is either discrete, meaning there is a least element greater than the identity, or dense. Corresponding to this dichotomy, the spaces of left, Conradian, and bi-orderings of a group are naturally partitioned into two subsets. This note investigates the structure of this partition, specifically the set of dense orderings of a group and its closure within the space of orderings. We show that for bi-orderable groups, this closure will always contain the space of Conradian orderings—and often much more. In particular, the closure of the set of dense orderings of the free group is the entire space of left-orderings.

]]>Taking into account the effects of patch structure and nonlinear density-dependent mortality terms, we explore a class of almost periodic Nicholson’s blowflies model in this paper. Employing the Lyapunov function method and differential inequality technique, some novel assertions are developed to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recently published literatures. Particularly, an example and its numerical simulations are arranged to support the proposed approach.

]]>For a split semi-simple group scheme and a principal -bundle on a relative curve , we study a natural obstruction for the triviality of on the complement of a relatively ample Cartier divisor . We show, by constructing explicit examples, that the obstruction is nontrivial if is not simply connected, but it can be made to vanish by a faithfully flat base change, if is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if is a smooth curve, and the singular locus of is finite over .

]]>The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and the box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad and the Assouad dimension. For common models of random fractal sets, we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition, and we compute the threshold for the Gromov boundary of Galton–Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton–Watson tree result.

]]>We study Fourier transforms of regular holonomic -modules. In particular, we show that their solution complexes are monodromic. An application to direct images of some irregular holonomic -modules will be given. Moreover, we give a new proof of the classical theorem of Brylinski and improve it by showing its converse.

]]>Let be a closed convex cone in which is spanning, i.e., and pointed, i.e., . Let be an -semigroup over and let be the product system associated to . We show that there exists a bijective correspondence between the units of and the units of .

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