Starting from an ancient observation of Tits concerning the interpretation of symmetric groups as Chevalley groups over a (non-existing) field having only one element, we describe combinatorial geometry over this field, as well as Linear Algebra. We...
In the 1930's, Tarski introduced his plank problem at a time when the field discrete geometry was about to born. It is quite remarkable that Tarski's question and its variants continue to generate interest in the geometric as well as analytic aspects...
In the 1930's, Tarski introduced his plank problem at a time when the field Discrete Geometry was about to born. It is quite remarkable that Tarski's question and its variants continue to generate interest in the geometric and analytic aspects of c...
Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various...
Points: 417 | Comments: 33 | Author: dabreegster
In this article we discuss the role of stability functions in geometric invariant theory and apply stability function techniques to problems in toric geometry. In particular we show how one can use these techniques to recover results of Burns-Guill...
The star product usually associated to the Snyder model of noncommutative geometry is nonassociative, and this property prevents the construction of a proper Hopf algebra. It is however possible to introduce a well-defined Hopf algebra by including t...
Active Learning (AL) promises to reduce annotation cost by prioritizing informative samples, yet its reliability is undermined when labels are noisy or when the data distribution shifts. In practice, annotators make mistakes, rare categories are ambi...
The aim of this paper is to study homological properties of tropical fans and to propose a notion of smoothness in tropical geometry, which goes beyond matroids and their Bergman fans and which leads to an enrichment of the category of smooth tropica...
We study the category of KM fans - a "stacky" generalization of the category of fans considered in toric geometry - and its various realization functors to "geometric" categories. The "purest" such realization takes the form of a functor from KM fans...
The category of (abstract) fans is to the category of monoids what the category of schemes is to the category of rings: a fan is obtained by gluing spectra of monoids along open embeddings. Here we study the basic algebraic geometry of fans: coherent...
We apply convex geometry (cones, fans) to homological input (abelian categories, hearts of bounded t-structures) to construct a new invariant of an abelian category, its heart fan. This can be viewed as a `universal phase diagram' for Bridgeland stab...
An engineering drawing is a type of technical drawing that is used to convey information about an object. A common use is to specify the geometry necessary
Multi-task learning using uncertainty to weigh losses for scene geometry and semantics, Auxiliary Tasks in Multi-task Learning (⭐ 647)
Intracortical US imaging extends B-mode imaging into bone using a dedicated image reconstruction algorithm that corrects for refraction at the bone-soft tissue interfaces. It has shown promising results in a few healthy, predominantly young adults, p...
The Kerr solution is the cornerstone of General Relativity (GR) for modelling astrophysical rotating black holes and for testing GR through gravitational-wave observations and black hole imaging. Understanding how the Kerr geometry is modified in alt...
We introduce the notion of $\varepsilonη\,$-Einstein $\varepsilon\,$-contact metric three-manifold, which includes as particular cases $η\,$-Einstein Riemannian and Lorentzian (para) contact metric three-manifolds, but which in addition allows for...
We investigate the differential geometry and topology of globally hyperbolic four-manifolds $(M,g)$ admitting a parallel real spinor $\varepsilon$. Using the theory of parabolic pairs recently introduced in arXiv:1911.08658 , we first formulate the p...
We investigate the differential geometry and topology of four-dimensional Lorentzian manifolds $(M,g)$ equipped with a real Killing spinor $\varepsilon$, where $\varepsilon$ is defined as a section of a bundle of irreducible real Clifford modules sat...
A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a powerful tool i...
The forward and reverse Funk weak metrics are fundamental distance functions on convex bodies that serve as the building blocks for the Hilbert and Thompson metrics. In this paper we study Voronoi diagrams under the forward and reverse Funk metrics i...
We survey some basic geometric properties of the Funk metric of a convex set in $\mathbb{R}^n$. In particular, we study its geodesics, its topology, its metric balls, its convexity properties, its perpendicularity theory and its isometries. The Hilbe...
Three-dimensional (3D) shape measurement devices and techniques are being rapidly adopted within a variety of industries and applications. As acquiring 3D range data becomes faster and more accurate it becomes more challenging to efficiently store, t...
We prove several results showing that every locally finite Borel graph whose large-scale geometry is "tree-like" induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tr...
Following the approach of Gromov and Witten, we define invariants under deformation of stongly semipositive real symplectic six-manifolds. These invariants provide lower bounds in real enumerative geometry, namely for the number of real rational $J...
The Butterfly Theorem is explored in Taxicab Geometry....
This paper presents a framework that fully leverages the advantages of a deferred rendering approach for the interactive visualization of large-scale datasets. Geometry buffers (G-Buffers) are generated and stored in situ, and shading is performed po...
Gold clusters Aun of size n = 2 to 12 atoms were studied by the density-functional theory with an ab-initio pseudopotential and a generalized gradient approximation. Geometry optimizations starting from a number of initial candidate geometries were...
This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $λ_1$-minimal immersion), which also play important roles in spectral geometry. I...
Most state-of-the-art localization algorithms rely on robust relative pose estimation and geometry verification to obtain moving object agnostic camera poses in complex indoor environments. However, this approach is prone to mistakes if a scene conta...
Recognizing materials in real-world images is a challenging task. Real-world materials have rich surface texture, geometry, lighting conditions, and clutter, which combine to make the problem particularly difficult. In this paper, we introduce a new,...
The broad emission spectral lines emitted from AGNs are our main probe of the geometry and physics of the broad line region (BLR) close to the SMBH. There is a group of AGNs that emits very broad and complex line profiles, showing two displaced peaks...
We generalise the geometric analysis of square fishnet integrals in two dimensions to the case of hexagonal fishnets with three-point vertices. Our results support the conjecture that fishnet Feynman integrals in two dimensions, together with their a...
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491 votes, 52 comments. 281K subscribers in the geometrydash community. The longest-running community for Geometry Dash, a rhythm-platformer game by…
1.1K votes, 101 comments. 309K subscribers in the geometrydash community. The longest-running community for Geometry Dash, a rhythm-platformer game…
Automatic Identification System (AIS) messages are useful for tracking vessel activity across oceans worldwide using radio links and satellite transceivers. Such data plays a significant role in tracking vessel activity and mapping mobility patterns...
This article illustrates pedagogy through training in the handling of abstractions. Mental arithmetic is not limited to numerical calculation; one can mentally calculate primitives and simplify analytical expressions. Even if there is software that d...
We address the problem of estimating realistic, spatially varying reflectance for complex planetary surfaces such as the lunar regolith, which is critical for high-fidelity rendering and vision-based navigation. Existing lunar rendering pipelines rel...
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local sp...