相关论文: Shrinking targets on Bedford-McMullen carpets
We study the Assouad and lower dimensions of self-affine sponges; the higher dimensional analogue of the planar self-affine carpets of Bedford and McMullen. Our techniques involve the weak tangents of Mackay and Tyson as well as regularity…
An elliptical invisible cloak is proposed using a coordinate transformation in the elliptical-cylindrical coordinate system, which crushes the cloaked object to a line segment instead of a point. The elliptical cloak is reduced to a…
Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…
Cloaks for underwater applications designed for actual submarine acoustic stealth are still far from the technological advancement needed for being put in practice. Several challenges are to be overcame such as dealing with weight or…
We construct Jordan arcs of prescribed conformal dimension which are minimal for conformal dimension. These curves are used to design fractal rugs, similar to Rickman's rug, that are also minimal for conformal dimension. These fractal rugs…
In the past few years, carpet cloaking attracted interests because of its feasibility at optical frequencies and potential in stealth technologies. Metasurfaces have been proposed as a method to engineer ultra-thin carpet cloaking surfaces…
We introduce a class of conformal versions of the previously introduced quasi-conformal carpet cloak, and show how to construct such conformal cloaks for different cloak shapes. Our method provides exact refractive-index profiles in closed…
The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and…
We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases this…
Based on transformation optics theory, we present a new carpet device that can be used to thermally protect a region from the invasion of external heat flux. The designed device is termed 'thermal carpet', which provides considerable…
Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.
In dynamical systems, shrinking target sets and pointwise recurrent sets are two important classes of dynamically defined subsets. In this article we introduce a mild condition on the linear parts of the affine mappings that allow us to…
The method of coordinate transformation offers a way to realize perfect cloaks, but provides less ability to characterize the performance of a multilayered cloak in practice. Here, we propose an analytical model to predict the performance…
In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum…
This paper presents a new method of constructing physical models in a geophysical inverse problem, when there are only a few possible physical property values in the model and they are reasonably known but the geometry of the target is…
In this work, we present a practical design of quasi isotropic underwater acoustic carpet cloak with pentamode microstructure. The quasi conformal transformation is not only used to obtain the required parameters, but also used to deform…
On a manifold or a closed subset of a Euclidean vector space, a retraction enables to move in the direction of a tangent vector while staying on the set. Retractions are a versatile tool to perform computational tasks such as optimization,…
Based on transformation optics (TO), this paper uses geometric divisions and linear coordinate transformations to design shrinking-shifting - and reshaping, and amplifying-shifting - and reshaping devices. The proposed devices can reshape…
Monitoring small objects against cluttered moving backgrounds is a huge challenge to future robotic vision systems. As a source of inspiration, insects are quite apt at searching for mates and tracking prey -- which always appear as small…
Constrained competitive optimization involves multiple agents trying to minimize conflicting objectives, subject to constraints. This is a highly expressive modeling language that subsumes most of modern machine learning. In this work we…