Unified method visualizes curvature on curves and surfaces.
problem Visualizing curvature on planar curves and surfaces of revolution.
method Tangential angle parameterization.
result Clear, consistent visualizations without arbitrary parameter tuning.
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
The paper explores connections between perimeter, area, and visual angle of convex sets.
problem Understanding geometric properties of convex sets through visual angle and related measurements.
method Establishing universal formulas and characterizing convex sets of constant width.
result Crofton's formula is the unique universal formula relating visual angle, length, and area.
The paper interprets and proves integral formulas for visual angles.
problem Integral formulas of visual angles.
method Integral Geometry approach to Crofton, Hurwitz, and Masotti formulas.
result New simpler proofs of integral formulas.
We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. As well we establish some upper and lower bounds …
For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpiński compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the …
New examples show right-angled Artin groups can have connected boundaries.
problem CAT(0) group boundaries not always path connected.
method Provided examples of right-angled Artin groups with connected boundaries.
result Right-angled Artin groups can have boundaries that are path connected.
The paper evaluates integrals of planes and their relation to convex set angles.
problem Integrals of invariant measures of pairs of planes in E3. method Expressing integrals in terms of visual angle functions of convex sets.
result Evaluation of a Crofton-type inequality deficit.
The study examines subgroups of RACGs and RAAGs, focusing on their RAAG properties.
problem Characterizing subgroups of right-angled Coxeter and Artin groups that are themselves RAAGs.
method Analyzes specific classes of subgroups and uses quasi-isometry and commensurability properties.
result Characterizes finite-index visual RAAG subgroups of 2-dimensional RACGs and provides new examples of RACGs commensurable to RAAGs.
Study of loxodromes on twisted surfaces in 3D space.
problem Characterizing loxodromes on surfaces with varying curvature.
method Analysis of loxodromes on twisted surfaces in Euclidean 3-space.
result Construction of examples to visualize loxodromes on twisted surfaces.
Paper uses random projection to preserve subspace structure for efficient data analysis.
problem Efficiently analyzing data with low-dimensional structure.
method Compressed Subspace Learning (CSL) framework based on Johnson-Lindenstrauss property.
result Random projection preserves the UoS structure of data, enabling efficient analysis.
DFKI Cabin Simulator tests visual monitoring functions in vehicles.
problem Validating novel human-vehicle interfaces and driver assistance systems.
method Driving simulator with in-cabin mock-up and camera system.
result Validation of in-cabin monitoring functions for advanced driver assistance and automated driving.
A framework visualizes embedding spaces of neural survival analysis models using anchor directions.
problem Visualizing complex embeddings in neural survival analysis models.
method Estimating anchor directions through clustering or user-supplied concepts, revealing relationships with raw inputs and survival times.
result Visualization strategies reveal how anchor directions relate to raw clinical features and survival time distributions.
In this work we simulate null geodesics for the Bonnor massive dipole metric by implementing a symbolic-numerical algorithm in Sage and Python. This program is also capable of visualizing in 3D, in principle, the geodesics for any given metric. Geodesics are launched from a common point, collectively forming a cone of …
Visualizes movement control optimization landscapes to understand why it's hard and how to make it easier.
problem Understanding and optimizing movement control problems in animation research.
method Novel visualizations of high-dimensional control optimization landscapes.
result Trajectory optimization becomes increasingly ill-conditioned with longer trajectories, while parameterizing control as partial target states can act as an efficient preconditioner.
We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape features in a visually intuitive manner. This method have implications in shape an…
We generalize arc coordinates for maximal representations on a pair of pants.
problem Maximal representations of reflection groups on hyperbolic surfaces.
method Introducing geometric parameters and reflections in Siegel space.
result Natural parametrization of maximal representations into PSp(4, R).
Crochet models show hyperbolic geometry properties.
problem Visualizing hyperbolic geometry concepts.
method Triangulated hyperbolic plane models.
result Demonstrates various hyperbolic geometry properties.
Deep learning reduces artifacts in limited angle X-ray microscopy.
problem Artifacts in image reconstruction from limited angle data in TXM.
method Training a U-Net deep neural network from synthetic data to reduce artifacts.
result Significant improvement in image quality and structural similarity.
The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and v…
Mobile app uses CNN to help visually impaired cross streets.
problem Helping visually impaired cross streets safely.
method LytNetV2 deep learning model for pedestrian traffic light detection.
result 96% classification accuracy and 6.15 degree angle error.
Study boundary actions of CAT(0) spaces and their C∗-algebras.
problem Investigate boundary actions of CAT(0) spaces and their associated C∗-algebras. method Topological dynamics and C∗-algebras, focusing on actions of specific groups and their properties. result Established (strongly) pure infiniteness results for reduced crossed product C∗-algebras of boundary actions. As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like propertie…
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our constructi…
Applied statisticians use sequential regression procedures to produce a ranking of explanatory variables and, in settings of low correlations between variables and strong true effect sizes, expect that variables at the very top of this ranking are truly relevant to the response. In a regime of certain sparsity levels, …
Study defines a new boundary for CAT(0) groups, invariant under quasi-isometries.
problem Defining boundaries for CAT(0) groups when visual boundaries are not well-defined.
method Introducing a sublinear function κ to define κ-Morse boundaries, showing invariance under quasi-isometries.
result κ-Morse boundaries are invariant and metrizable for CAT(0) groups.
We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles ≤π), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotien…
Transformer models are compared to curved spacetime in General Relativity.
problem Understanding the geometric structure of Transformer models.
method Geometric analogy to General Relativity, focusing on attention and curvature effects.
result Transformer token embeddings exhibit curvature, affecting their evolution over layers.
Study boundary actions on CAT(0) spaces, proving topological freeness.
problem Understanding boundary actions on CAT(0) spaces.
method Verification of freeness of Myrberg points on boundaries.
result Large class of boundary actions are topologically free.
Study constant angle surfaces in 4D Minkowski space, proving their properties.
problem Characterize surfaces in 4D Minkowski space with constant angle between tangent planes.
method Define complex angle, prove curvature properties, use PDE methods, analyze special cases.
result Constant angle surfaces have vanishing Gauss and normal curvatures; not complete for ψeq0 [π/2]. Defines Kahler angle for a broader context.
problem Generalizing results about Kahler angle.
method Provides a general definition of Kahler angle.
result Generalized results about Kahler angle.
Study angle structures on pseudo 3-manifolds, proving existence for some cases.
problem Determining if hyperbolic 3-manifolds can have angle structures.
method Examined triangulated pseudo 3-manifolds with area-curvature angle structures, establishing sufficient and necessary conditions.
result Compact hyperbolic 3-manifolds with totally geodesic boundary can have angle structures.
Deep learning predicts preterm birth risk with improved accuracy.
problem Improving accuracy in predicting spontaneous preterm deliveries.
method U-Net segmentation network for automatic extraction of cervical length and anterior cervical angle.
result Combined markers reduce false-negative ratio from 30% to 18%
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
The paper examines rigidity in geometric actions of Coxeter groups on Croke-Kleiner spaces.
problem The rigidity of geometric actions of Coxeter groups compared to their quasi-isometric counterparts.
method Study of right-angled Coxeter groups acting geometrically on Croke-Kleiner spaces.
result Right-angled Coxeter groups have more rigid geometric actions than their quasi-isometric counterparts.
Jets from boosted heavy particles have a typical angular scale which can be used to distinguish them from QCD jets. We introduce a machine learning strategy for jet substructure analysis using a spectral function on the angular scale. The angular spectrum allows us to scan energy deposits over the angle between a pair …
Stiefel-Whitney classes of moment-angle manifolds are trivial.
problem Analyzing the topological properties of moment-angle manifolds.
method Proving triviality of Stiefel-Whitney classes for moment-angle manifolds, including partial quotients.
result Stiefel-Whitney classes of moment-angle manifolds are trivial.
In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…
Uniqueness of quasi-roots explored in right-angled Artin groups.
problem Uniqueness of quasi-roots in right-angled Artin groups.
method Introducing quasi-roots and studying their uniqueness.
result Uniqueness of quasi-roots established in right-angled Artin groups.
This paper presents KeypointNet, an end-to-end geometric reasoning framework to learn an optimal set of category-specific 3D keypoints, along with their detectors. Given a single image, KeypointNet extracts 3D keypoints that are optimized for a downstream task. We demonstrate this framework on 3D pose estimation by pro…
Study proves existence of weak mean curvature flow with contact angle.
problem Existence of weak mean curvature flow with prescribed contact angle.
method Compactness theorem for varifolds and Ilmanen's regularization extended to capillarity.
result Existence of weak mean curvature flow with contact angle for general θ. Improved volume estimates for right-angled polyhedra in hyperbolic space.
problem Estimating volumes of right-angled polyhedra in hyperbolic space.
method Combining Andreev theorem and Atkinson's results, improved upper volume estimates.
result Upper volume estimates for both compact and ideal right-angled polyhedra improved.
This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.
problem Understanding the historical context and philosophical implications of angles and solid angles.
method Historical review and analysis of mathematical and philosophical works.
result Questions raised by Euler about angles and solid angles are timeless and relevant to modern mathematics.
Inner product-based convolution has been a central component of convolutional neural networks (CNNs) and the key to learning visual representations. Inspired by the observation that CNN-learned features are naturally decoupled with the norm of features corresponding to the intra-class variation and the angle correspond…
Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
problem Uniqueness of smooth structures on real moment-angle manifolds.
method Arguments from calculus applied to results from complex moment-angle manifolds.
result Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
Defines Lorentzian angles for null vectors in geometry.
problem Handling angles involving null vectors in geometry.
method Introduces Lorentzian angles for null directions.
result Provides a proof for the Lorentzian Gauss-Bonnet theorem.
Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3-manifolds. We prove, by a constructive argument, that every veering triangulation admits positive angle structures, recovering a result of Hodgson, Rubins…
We provide a congruence theorem for minimal surfaces in S5 with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in S5 with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we wi…