Related papers: Higher dimensional conundra

We introduce the notion of almost finite dimensionality of algebras and study its connection with the classical finiteness conditions.

In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures.…

We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.

We show for a free action of a countable group $\Gamma$ on a finite-dimensional, compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of $\Gamma$.

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.

We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper.…

In this article, we study the asymptotic behavior of large solutions for a quasi-linear equation involving the p-Laplacian, defined on a sequence of finite cylindrical domains converging to an infinite cylinder. We demonstrate that the…

Let $X$ be a proper metric space and let $F$ be a finite group acting on $X$ by isometries. We show that the asymptotic dimension of $F\backslash X$ is the same as the asymptotic dimension of $X$.

A meander can be seen as a pair of transversally intersecting simple closed curves on a 2-sphere. We consider pairs of transversally intersecting simple closed curves on a closed oriented surface of arbitrary genus g. The number of such…

We consider the Einstein constraints on asymptotically euclidean manifolds $M$ of dimension $n \geq 3$ with sources of both scaled and unscaled types. We extend to asymptotically euclidean manifolds the constructive method of proof of…

We survey the distributional properties of progressively dilating sets under projection by covering maps, focusing on manifolds of constant sectional curvature. In the Euclidean case, we review previously known results and formulate some…

In the past few years we have derived asymptotic expansions for lambda_d of the dimer problem and lambda_d(p) of the monomer-dimer problem. The many expansions so far computed are collected herein. We shine a light on results in two…

Asymptotic methods for hypothesis testing in high-dimensional data usually require the dimension of the observations to increase to infinity, often with an additional condition on its rate of increase compared to the sample size. On the…

Our knowledge and ignorance concerning the geometry of quantum states are discussed.

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric…

We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.

A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting.

This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the…

Bearing the final fate of gravitational collapse in mind, we study the asymptotic structures at timelike infinity in four dimensions. Assuming that spacetimes are asymptotically stationary, we will examine the asymptotic structure of…