Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
Study on focal locus of submanifolds in Finsler manifolds, showing regularity and smoothness.
problem Investigating the focal locus of submanifolds in Finsler manifolds.
method Using the normal exponential map and extending Warner's ideas, studying connected components and smoothness of focal time maps.
result Identified an open and dense subset where focal time maps are smooth, provided they are finite.
Study on transnormal functions and their level sets on Finsler manifolds.
problem Understanding transnormal functions and their geometric properties on Finsler manifolds.
method Proving smoothness of focal varieties and regular level sets of transnormal functions.
result Focal varieties of a C2 transnormal function are smooth submanifolds and regular level sets are tubes over these varieties.
A focal representation of a generic regular curve γ in E^{m+1} consists of the centers of the osculating hyperplanes. A k-slant helix γ in E^{m+1} is a (generic) regular curve whose unit normal vector V_{k} makes a constant angle with a fixed direction U in E^{m+1}. In the present paper we proved that if γ is a k-slant…
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the hölder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete ou…
In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Mo…
The study classifies rational isoparametric functions on Damek-Ricci spaces.
problem Classifying isoparametric functions on Damek-Ricci spaces.
method Using polynomial functions divided by t for classification. result New isoparametric functions discovered and their properties studied.
Study of focal-entropy for class-imbalanced classification.
problem Understanding the focal-loss in class-imbalanced settings.
method Distributional viewpoint and information-theoretic analysis of focal-entropy.
result Focal-entropy minimizer exists and departs from data distribution.
Study focal surfaces of wave fronts with unbounded curvatures.
problem Characterizing singularities of focal surfaces near non-degenerate singular points.
method Characterizations based on types of singularities and geometrical properties of initial fronts.
result Investigation of Gaussian curvature behavior of focal surfaces.
Self-focal points on ellipsoids of dimension 3 or higher are rare.
problem Existence of self-focal points on Riemannian manifolds of dimension 3 or higher.
method Analyzing geodesics and umbilic points on ellipsoids of various dimensions.
result Ellipsoids of dimension 3 or higher with at least 4 distinct axes have no self-focal points.
Study on unfolding maps of surfaces in 3D space, proving versality conditions.
problem Investigating the versality of rotation unfolding of folding maps for surfaces in R3. method Introducing and analyzing the rotation unfolding of folding maps, proving versality conditions in terms of geometry.
result Proved conditions for the rotation unfolding to be versal, showing diffeomorphic type of tangent plane locus.
The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being A-manifolds in the sense of A.Gray but rarely Ricci-parallel (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that …
Enhances hand gesture recognition with separate networks and shared features.
problem Improving recognition accuracy of unimodal 3D-CNNs for dynamic hand gestures.
method Separate networks for each modality, collaborative learning, spatiotemporal semantic alignment loss, focal regularization.
result Improves test time recognition accuracy and state-of-the-art performance.
This paper connects billiards in ellipses to focal billiards in ellipsoids.
problem Proving the existence of isometric counterparts between billiards in ellipses and focal billiards in ellipsoids.
method Continuous transition via isometric focal billiards in a fixed ellipsoid.
result Established the connection between planar and spatial billiards.
Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.
problem Investigate differential geometry properties of framed curves, including singular points.
method Use Legendrian dualities to analyze focal surfaces and evolutes of hyperbolic framed curves.
result Show the relationship among focal surfaces, evolutes, and dual surfaces of evolutes.
Focal loss improves classification but not class-posterior probability estimation.
problem Improving class-posterior probability estimation from focal loss.
method Proved classification-calibration and derived a transformation to recover true class-posterior probabilities.
result A transformation of the confidence score from focal loss minimization allows recovery of true class-posterior probabilities.
We prove that a submanifold with parallel focal structure, which is a generalization of isoparametric and equifocal submanifolds, induces a singular Riemannian foliation of the ambient space by its parallel and focal manifolds.
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
Study of families of lines on spheres and their focal sets.
problem Characterizing families of lines on spheres and their geometric properties.
method Analyzing submanifolds of TSn and their focal sets, using symplectic structures and sectional curvatures. result Derivation of formulas relating sectional curvatures of focal sets to differences in radii of curvature of generating hypersurfaces.
Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results for these types of surfaces to become flat and we show that there is no minimal f…
The {\em focal curve} of an immersed smooth curve γ:s↦γ(s), in Euclidean space Rm+1, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of γ (t,n1,...,nm), as $C_γ(s)=(γ+c_1{\bf n}_1+c_2{\bf n}_2+...+c_m{\bf n}…
We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.
ResNet with Focal Loss improves speech emotion recognition.
problem Speech emotion recognition using plain text features is insufficient.
method Residual Convolutional Neural Network (ResNet) trained with Focal Loss.
result Focal Loss enhances model's focus on hard examples.
In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n+2)-space. We give condition under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singulariti…
In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theo…
Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note,…
The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.
problem Bounding bandwidth and focal radius for manifolds with positive isotropic curvature.
method Using spectral properties of a twisted de Rham-Hodge operator.
result Upper bounds on bandwidth and focal radius are derived for hypersurfaces in PIC manifolds.
We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal…
FOCAL tackles offline meta-reinforcement learning with efficient task inference and behavior regularization.
problem Efficiently adapt RL algorithms to unseen tasks without interactions, addressing bootstrapping errors and robust task inference.
method FOCAL combines behavior regularization, a deterministic context encoder, and a negative-power distance metric for efficient task inference.
result FOCAL outperforms prior algorithms on meta-RL benchmarks, demonstrating computational efficiency.
Focal loss improves deep neural networks' accuracy and calibration.
problem Miscalibration in deep neural networks.
method Using focal loss and temperature scaling to improve model calibration.
result Focal loss leads to state-of-the-art calibrated models without sacrificing accuracy.
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…
Focal loss reduces model curvature for better calibration.
problem Improving model confidence in classification problems.
method Geometric interpretation of focal loss to reduce curvature.
result Focal loss reduces the curvature of the loss surface, enhancing model calibration.
We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.
Survey of Cartan's work on isoparametric hypersurfaces in spheres.
problem Understanding isoparametric hypersurfaces and their focal submanifolds.
method Review of Cartan's original papers from 1938-1940.
result Detailed description of isoparametric hypersurfaces and their focal submanifolds.
Study calculates indices and nullities of focal manifolds in spheres.
problem Calculating indices and nullities of focal manifolds in spheres.
method Analyzes isoparametric hypersurfaces in spheres with three distinct principal curvatures.
result Index equals the ambient space dimension, nullity determined by Killing vector fields.
In this note we show that a compact asymptotically harmonic manifold without focal points is either flat or a rank one locally symmetric space.
Characterizes submanifolds with minimum ratio of diameter to focal radius.
problem Finding submanifolds with the minimum ratio of extrinsic diameter to focal radius.
method Combining K. Sakamoto's classification of submanifolds with planar geodesics and A. Schur's Bow Lemma for space curves.
result Essentially round spheres or Veronese embeddings of projective spaces achieve the minimum ratio.
Method for generating new curves from plane curves on cylinders.
problem Generating new space curves from given plane curves.
method Defining a non-planar space curve on a right generalized cylinder and examining its focal curve.
result Parametric representation of the focal curve of a cylindrical curve.
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the a…
The geodesic flow on certain surfaces is shown to be ergodic.
problem Ergodicity of geodesic flows on surfaces without focal points.
method Analyzing geodesic flows on surfaces with no focal points, proving ergodicity under specific curvature conditions.
result The geodesic flow on the unit tangent bundle of a surface with no focal points is ergodic with respect to the Liouville measure.
The focal locus ΣX of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x∈X and a circle with centre O passing through x which osculates X in x. Algebraic geometry interprets the focal locus as the branching locus of the endpoi…
An equifocal submanifold M of a symmetric space N of compact type induces a foliation with singular leaves on N. In this paper we will show how to reconstruct the equifocal foliation starting from one of the singular leaves, the so-called focal manifolds. To be more concrete: The equifocal submanifold is equal to a par…
Enhanced loss function boosts fraud detection in auto insurance claims.
problem Class imbalance in auto insurance fraud detection.
method Structured three-stage training framework integrating convex surrogate, non-convex intermediate, and standard focal loss.
result Improves minority-class F1-scores and AUC compared to baseline methods.
Focalized GP improves Bayesian optimization for large datasets.
problem Efficiently scaling Bayesian optimization to large datasets and online samples.
method Proposes focalized sparse Gaussian processes and FocalBO algorithm.
result FocalBO achieves state-of-the-art performance on complex optimization problems.
The space L of oriented lines, or rays, in R3 is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on R3. In this paper we explore the relationship between the focal se…
FOCaL meta-learner estimates functional treatment effects robustly.
problem Estimating heterogeneous treatment effects from functional outcomes.
method Doubly robust meta-learner FOCaL integrating functional regression.
result Direct and robust estimation of F-CATE.
Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.
problem Estimating the focal radius of hypersurfaces in manifolds with positive curvature.
method Proved a sharp Clifford-threshold focal-radius estimate and rigidity under specific curvature conditions.
result Any closed two-sided immersion satisfies a focal radius estimate of π/4, with equality case rigid.