Expositiones Mathematicae最新文献

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.

This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i\left(C\left(S^d\right)\right)$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.

We prove that for a given unmarked trace spectrum with multiplicity, there are only a finite number of convex real projective surfaces with that spectrum up to remarking.

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric realization functor. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy groups, the long exact sequence of a fibration, and Whitehead’s theorem.

This is a companion paper to our “Cubical setting for discrete homotopy theory, revisited” in which we apply these results to study the homotopy theory of simple graphs.