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.
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.
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.
We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a C1,α compactness result for submanifolds, …
The paper confirms a conjecture about manifolds with positive curvature.
problem Estimating the width of manifolds with positive sectional curvature.
method Establishing an optimal Lipschitz lower bound for functions on manifolds.
result Characterization of doubly warped product metrics with positive constant curvature.
Study on Gehring link problem and width of bands in curved manifolds.
problem Width of bands in positively curved manifolds.
method Same idea applied to focal radius and rigidity of hypersurfaces.
result Sphere theorem for hypersurfaces in Sn involving focal radius and rigidity of Clifford hypersurface. We show that the focal radius of any submanifold N of positive dimension in a manifold M with sectional curvature greater than or equal to 1 does not exceed 2π. In the case of equality, we show that N is totally geodesic in M and the universal cover of M is isometric to a sphere or a projective s…
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold M: 1) the convexity radius of p, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…
The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…
The paper proves reverse inequalities in various geometric settings using curvature radius data.
problem Proving reverse Alexandrov-Fenchel inequalities in different geometric settings.
method Using curvature radius data and associated evolute or focal maps.
result Sharp reverse Alexandrov-Fenchel estimates and inequalities in smooth convex curves and hypersurfaces.
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}…
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.
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 …
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.
Bayesian method improves EEG source imaging accuracy by estimating skull conductivity.
problem Sensitivity of EEG source imaging to skull conductivity modeling.
method Bayesian uncertainty modeling to estimate skull conductivity and focal sources.
result Clear improvements in source localization accuracy and feasible skull conductivity estimates.
The paper bounds radii and curvatures in Riemannian manifolds.
problem Bounding radii and curvatures in Riemannian manifolds.
method Analyzing scalar curvature, injectivity radius, and mean curvature.
result Proves bounds on injectivity and focal radii under specific conditions.
Algorithm selects variables and bandwidths for geographically weighted regression.
problem Estimating variable subsets and bandwidths for geographically weighted regression.
method Mathematical programming-based approach integrating variable selection and bandwidth estimation.
result Proposed algorithm provides stable spatially varying patterns with competitive explanatory power.
The study counts geodesics on special manifolds without focusing points.
problem Counting geodesics on specific types of manifolds.
method Margulis-type asymptotic estimates and analysis of geodesic flow.
result The geodesic flow on these manifolds has a unique measure of maximal entropy with the Bernoulli property.
We prove a formula for the normal injectivity radius(thickness)i(K,M)for C^{1,1} compact submanifolds K^k of complete Riemannian manifolds M^n in terms of geometric focal distance and double critical points. We also prove the C^1 compactness of the set of all compact submanifolds K contained in a compact subset D of a …
We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.
Estimates users' preference for a site over others using engagement data.
problem Lack of data on users' interactions with other sites makes it hard to estimate preferences for a focal site.
method Uses Hierarchical Bayes Method with two estimation techniques: Markov Chain Monte Carlo and Stochastic Gradient with Langevin Dynamics.
result Good support found for the approach to computing personalized share of engagement.
This is a continuation of Tang and Yan, which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g=4 distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with g=6, the present paper obtains estimates on all the eigenvalues, among others, giving…
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.
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 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 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.
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.
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.
The cone projection fR(z)=z/(1+∣z∣/R) maps lines to conic arcs with specific properties.
problem Mapping lines to conic arcs with specific properties.
method Using a reciprocal lens identity and radial homeomorphism.
result The Self-Directrix Theorem and Confocal--Codirectrix Theorem.
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler 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.
Upper estimate of Dirac eigenvalue linked to hyperspherical radius.
problem Estimating the smallest eigenvalue of the Dirac operator.
method Proved an upper estimate of the smallest eigenvalue in terms of hyperspherical radius.
result Combining with known lower estimates, geometric consequences are derived.
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…
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.
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.
We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.
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.
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,…
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…
In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and…
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.
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.
The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.
problem Understanding the properties and behavior of Ricci solitons.
method Analytical proofs and estimates for various types of Ricci solitons.
result Upper bounds and estimates for conjugate radius of Ricci solitons.
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…