Expositiones Mathematicae最新文献

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series ${\sum }_{n=1}^{\infty }\frac{a\left(n\right)}{n^s}$. We provide a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener–Ikehara theorem with an error term.

We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes $\Omega \times R$ in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected $\Omega \subset R^2$. These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section $\Omega$.

This paper has an expository nature. We compare the spectral properties (such as boundedness and compactness) of three families of semi-infinite matrices and point out similarities between them. The common feature of these families is that they can be understood as matrices of some linear operations on appropriate Hardy spaces.

We answer a question of Samir Siksek, asked at the open problems session of the conference “Rational Points 2022”, which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set $S$ of perfect integer powers, using Mihăilescu’s theorem, we construct a polynomial $f_S\in Z\left[x\right]$ such that the set $f_S\left(Z\right)$ contains a perfect integer power if and only if it belongs to $S$. We first discuss the easier case where we restrict to all powers with the same exponent. In this case, the constructed polynomials are inspired by Runge’s method and Fermat’s Last Theorem. Therefore we can construct a polynomial–exponential Diophantine equation whose solutions are determined in advance.

In this paper, we give a complete answer to the question: “Under what conditions the product of three harmonic homologies of the real projective space $P^n\left(R\right)$ is a harmonic homology again?” Among other things, we prove the three harmonic homologies theorem in $P^n\left(R\right)$ by which the product of three harmonic homologies with collinear centers is again a harmonic homology if and only if the hyperplanes are polars of the centers with respect to a quadric. It is shown that the three reflections theorem, the three inversions theorem, notably Pascal’s theorem and Miquel’s theorem in Laguerre geometry are special cases of this theorem.

We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu et al. (2021). In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.

For a semi-linear Schrödinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude, obeying different scaling limits. The corresponding nets of approximate solutions represent actual generalised solutions for the singular-perturbed Schrödinger equation. The behaviour of such nets is investigated, comparing the distinct scaling regimes that yield, respectively, the Hartree equation with point interaction Hamiltonian vs the ordinary Hartree equation with the free Laplacian. In the second case, the distinguished regime admitting a generalised solution in the Colombeau algebra is studied, and for such a solution compatibility with the classical Hartree equation is established, in the sense of the Colombeau generalised solution theory.

We provide a new definition of Hardy space atoms that avoids use of coordinates to formulate the moment condition. Thus the new theory can be used in abstract settings such as spaces of homogeneous type. We give applications of this theory to the definition of and study of smooth functions on spaces of homogeneous type.

We establish that $\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{a}_m\overline{a_n}\frac{mn}{{(max(m,n))}^3}\le \frac{4}{3}\sum _{m=1}^{\infty }{\left|a_m\right|}^2$holds for every square-summable sequence of complex numbers $a=(a_1,a_2,\dots )$ and that the constant $4/3$ cannot be replaced by any smaller number. Our proof is rooted in a seminal 1911 paper concerning bilinear forms due to Schur, and we include for expositional reasons an elaboration on his approach.