Related papers: Hearing the shape of a triangle
We address the problem of one dimensional segment detection and estimation, in a regression setup. At each point of a fixed or random design, one observes whether that point belongs to the unknown segment or not, up to some additional…
In this paper, we establish two sharp quantitative results for the direct and inverse time-harmonic acoustic wave scattering. The first one is concerned with the recovery of the support of an inhomogeneous medium, independent of its…
The circumcircle of a planar convex polygon P is a circle C that passes through all vertices of P. If such a C exists, then P is said to be cyclic. Fix C to have unit radius. While any two angles of a uniform cyclic triangle are negatively…
An old theorem of Alexander Soifer's is the following: Given five points in a triangle of unit area, there must exist some three of them which form a triangle of area 1/4 or less. It is easy to check that this is not true if "five" is…
We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,...,6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $\tau$ is parametrized by…
The ability to approach a physical phenomenon and grasp its major importance is a remarkable quality of understanding. This paper presents a rather elegant and novel way of looking at the resonance phenomenon, which among others shares a…
Pattern formation of sound is predicted in a driven resonator where subharmonic generation takes place. A model allowing for diffraction of the fields (large-aspect ratio limit) is derived by means of the multiple scale expansions…
We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1} $, whose $ k $-th anisotropic curvature measure (for $ k =0, \ldots , n-1 $) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated…
Grating spectra exhibit sharp variations of the scattered light, known as grating anomalies. The latter are due to resonances that have fascinated specialists of optics and physics for decades and are nowadays used in many applications. We…
Vogt's theorem, concerning boundary angles of a convex arc with monotonic curvature (spiral arc), is taken as a starting point to establish basic properties of spirals. The theorem is expanded by removing requirements of convexity and…
The fact that every compact oriented 4-manifold admits spin$^c$ structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is…
We characterise which simplicial surfaces can be folded onto a triangle. We define a notion of folding that incorporates the non-intersection-properties of real materials. All of the surfaces foldable onto a triangle admit a…
This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also…
Decades of work on beam deformation on reflection, and especially on lateral shifts, have spread the idea that a reflected beam is larger than the incident beam. However, when the right conditions are met, a beam reflected by a multilayered…
We show how to find the shape of an equilateral caterpillar tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm-Liouville equation on this tree. We prove that in the case of a caterpillar tree the spectra…
This paper is concerned with an inverse transmission problem for recovering the shape of a penetrable rectangular grating sitting on a perfectly conducting plate. We consider a general transmission problem with the coefficient \lambda\neq 1…
The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture in one dimension, which states that a bounded measurable subset of $\mathbb{R}$…
The chirality of an object can be studied by measuring the circular dichroism, that is, the difference in absorption of light with different helicity. The chiral optical response of an object, however, can have two different origins. On the…
We prove that the spectral radius of even order nonnegative irreducible tensors is real geometrically simple. In the case when the order of the tensor is odd, or in the complex field, some conditions are given to guarantee the geometric…
Triangles are everywhere in the virtual world. The surface of nearly every graphical object is saved as a triangular mesh on a computer. Light effects and movements of virtual objects are computed on the basis of triangulations. Besides…