Volume 26, Issue 6

The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between shape properties of topological spaces, modeled as continuous real-valued functions defined on the spaces themselves. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance. Moreover, we show that, under our assumptions, this optimal homeomorphism is actually a diffeomorphism.

We develop a technique to construct finitely injective modules which are non-trivial, in the sense that they are not direct sums of injective modules. As a consequence, we prove that a ring R is left noetherian if and only if each finitely injective left R -module is trivial, thus answering an open question posed by Salce.

The automorphic cohomology of a reductive ℚ-group G , defined in terms of the automorphic spectrum of G , captures essential analytic aspects of the arithmetic subgroups of G and their cohomology. The subspace spanned by all possible residues and principal values of derivatives of Eisenstein series, attached to cuspidal automorphic forms π on the Levi factor of proper parabolic ℚ-subgroups of G , forms the Eisenstein cohomology which is a natural complement to the cuspidal cohomology. We show that non-trivial Eisenstein cohomology classes can only arise if the point of evaluation features a `half-integral' property. Consequently, only the analytic behavior of the automorphic L-functions at half-integral arguments matters whether an Eisenstein series attached to a globally generic π gives rise to a residual class or not.

The paper is devoted to a study of isometry groups of Polish ultrametric spaces. We explicitly describe isometry groups of spaces that are non-locally rigid and satisfy the property that distances between orbits under the action of the isometry group are realized by points. The type of group construction appearing here is a variant of the generalized wreath product. We prove that it has a natural universality and uniqueness property. As an application, we characterize Polish ultrametric spaces satisfying the above properties, whose isometry groups have uncountable strong cofinality.

This work is an answer to a problem posed by N. Kurokawa and H. Ochiai concerning the natural boundary of meromorphy of a multivariate Euler product of Igusa type. More generally, we introduce and determine the maximal domain of meromorphy of a class of multivariate non-uniform Euler products.

We present an extension of Eilenberg's variety theorem, a well-known result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the original theory. We also prove a similar result for ordered monoids.

We study some arithmetics of Hermitian modular forms of degree two by applying the spectral theory on 3-dimensional hyperbolic space. This paper presents three main results: (1) a 3-dimensional analogue of Katok–Sarnak's correspondence, (2) an analytic proof of a Hermitian analogue of the Saito–Kurokawa lift by means of a converse theorem, (3) an explicit formula for the Fourier coefficients of a certain Hermitian Eisenstein series.

We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.

This paper establishes mixed multiplicity formulas concerning the relationship between mixed multiplicities of modules and mixed multiplicities of rings via rank of modules.

The Dressian Dr( k , n ) parametrizes all tropical linear spaces, and it carries a natural fan structure as a subfan of the secondary fan of the hypersimplex Δ( k , n ). We explore the combinatorics of the rays of Dr( k , n ), that is, the most degenerate tropical planes, for arbitrary k and n . This is related to a new rigidity concept for configurations of n - k points in the tropical k - 1-torus. Additional conditions are given for k = 3. On the way, we compute the entire fan Dr(3,8).

This article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.

The modular group PSL 2 ℤ contains many subgroups that are sharply transitive on the rational projective line ℚ ∪ ∞. If the field E is a purely transcendental extension of transcendency degree | E |, then PSL 2 E contains free subgroups that are sharply transitive on the projective line E ∪ ∞. This yields wild infinite nearfields and sharply two-transitive permutation groups.