Paper proves existence and gives construction of Symphonic map between ellipsoids.
problem Existence and construction of Symphonic map between ellipsoids.
method Geometric construction using Hopf map.
result Existence and construction of Symphonic map from ellipsoid to ellipsoid.
CTEF fits ellipsoids to noisy data in any dimension.
problem Fitting ellipsoids to noisy data in arbitrary dimensions.
method Uses the Cayley transform to fit ellipsoids.
result CTEF outperforms other methods, especially when data are not uniformly distributed.
The paper uses Seshadri constants to construct symplectic ellipsoid embeddings.
problem Constructing symplectic embeddings of ellipsoids.
method Exploring weighted blow-ups and Seshadri constants to demonstrate symplectic embeddings.
result Illustrates constructions of ellipsoid fillings and embeddings.
New minimal discs and annuli found in ellipsoids.
problem Constructing minimal surfaces in ellipsoids.
method Equivariant variational methods.
result At least three distinct embedded free boundary minimal annuli in ellipsoids.
The paper develops a method to create non-asymptotic confidence ellipsoids for linear regression without strong noise distribution assumptions.
problem Constructing reliable confidence regions for linear regression with finite sample sizes and general noise distributions.
method The paper introduces the SPS EOA algorithm to create non-asymptotically guaranteed confidence ellipsoids for linear regression problems.
result The sizes of SPS outer ellipsoids are shown to decrease at the optimal rate for linear regression problems.
We discuss Darboux-Staude type of thread configurations for the ellipsoid similar to Chasles-Graves type of thread configurations for the ellipse. These threads are formed by rectilinear segments, geodesic and line of curvature segments on the considered ellipsoid and with tangents tangent to the given ellipsoid and a …
Jacobi solved geodesics on triaxial ellipsoids.
problem Finding the shortest path on a triaxial ellipsoid.
method Numerical evaluation of integrals and solving coupled equations.
result Solution for geodesics on triaxial ellipsoids.
Study finds many nonplanar minimal spheres in elongated ellipsoids.
problem Existence of nonplanar minimal spheres in elongated ellipsoids.
method Global bifurcation techniques to establish existence and quantify number.
result Arbitrarily many nonplanar minimal spheres exist in elongated ellipsoids.
Study smoothings of singular intersections of ellipsoids.
problem Smooth singular intersections of ellipsoids.
method Generic singularities of coaxial intersections of ellipsoids studied.
result Special attention to 3D case.
Optimizes ellipsoids for uncertainty regions in parameter estimation.
problem Learning minimal volume uncertainty ellipsoids for parameter estimation.
method Differentiable optimization approach using neural networks to approximate optimal ellipsoids.
result Approximately computed ellipsoids are smaller and more accurate than existing methods.
Study on functional ellipsoids to decompose the identity.
problem Decompose the identity for functional ellipsoids.
method Construct a decomposition similar to Fritz John's theorem.
result Developed a new approach to functional ellipsoids.
Discrete analogues of ellipsoids with preserved circular cross sections.
problem Constructing discrete analogues of ellipsoids with preserved geometric properties.
method A novel discretization procedure to create discrete analogues of ellipsoids composed of planar quadrilaterals.
result Discrete analogues of ellipsoids have preserved circular cross sections and can be deformed.
Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.
problem Minimal tori in ellipsoids.
method Analyzing 3D ellipsoids invariant under a 2-torus action.
result Infinitely many distinct minimal tori bifurcate from a 2-torus orbit.
We develop an efficient algorithm to find confidence ellipsoids with volume guarantees in high dimensions.
problem Finding robust confidence ellipsoids in high-dimensional data.
method Polynomial time algorithm using primal-dual structure and geometric Brascamp-Lieb inequality.
result Algorithm finds ellipsoids within a O(β)γd volume factor of best β-conditioned ellipsoid. Study smoothings of ellipsoid intersections with singularities.
problem Smooth singular intersections of ellipsoids.
method Study 3D manifolds with singularities as small covers of Coxeter polyhedral orbifolds.
result Introduce n-pyramitoid to generalize n-pyramids. 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.
Geodesic algorithms extended to arbitrary ellipsoids.
problem Computing geodesics on ellipsoids of varying eccentricity.
method Implementation of geodesic algorithms using elliptic integrals and discrete sine transform.
result Achieved high accuracy (close to machine precision) for geodesic computations.
Ellipsoids approach Gaussian distribution in high dimensions.
problem Understanding convergence of high-dimensional ellipsoids to Gaussian spaces.
method Proof of convergence in Gromov's concentration topology.
result Solid ellipsoids converge to Gaussian space in high dimensions.
The paper studies the number of normals to ellipsoids and their intersections with caustics.
problem The number of normals to an ellipsoid passing through a given point.
method Intersection points of the ellipsoid and its caustics are used to study the problem in 3D space.
result The number of normals is dependent on the position of the given point with respect to the caustics of the ellipsoid.
The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.
Study Markov staircases in symplectic embeddings of rational homology ellipsoids.
problem Symplectic embeddings of rational homology ellipsoids into the complex projective plane.
method Analysis of almost toric fibrations and Hamiltonian isotopies.
result Existence of an infinite staircase for each Markov triple.
We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…
Derives exact formula for Minkowski sum of ellipsoids in N-space.
problem Finding volume bounds for Minkowski sum of ellipsoids.
method Closed-form parametric equation derivation and volume bounds calculation.
result Upper and lower volume bounds for Minkowski sum of ellipsoids.
We study the ellipticity and the ``Nekhoroshev stability'' (stability properties for finite, but very long, time scales) of the Riemann ellipsoids. We provide numerical evidence that the regions of ellipticity of the ellipsoids of types II and III are larger than those found by Chandrasekhar in the 60's and that all Ri…
Study eigenvalues of ellipsoids near a sphere, comparing to sphere's.
problem Analyzing changes in Laplacian eigenvalues for ellipsoids near a sphere.
method Comparison with standard Euclidean unit sphere, under Gaussian curvature condition.
result Eigenvalues of ellipsoids near a sphere, with comparison to sphere's.
New framework for better mapping of surfaces onto ellipsoids.
problem Mapping genus-0 closed surfaces onto spheres results in large distortion.
method Combining conformal and quasi-conformal mappings onto ellipsoids.
result Achieved a variety of ellipsoidal parameterizations with bijectivity.
Infinite families of maps found on ellipsoids in various dimensions.
problem Finding harmonic self-maps on ellipsoids in multiple dimensions.
method Analyzing ellipsoids with specific conditions and proving the existence of infinitely many harmonic maps.
result Infinite families of harmonic self-maps exist on ellipsoids under given conditions.
We analyze the horizon and geodesic structure of a class of 4D off--diagonal metrics with deformed spherical symmetries, which are exact solutions of the vacuum Einstein equations with anholonomic variables. The maximal analytic extension of the ellipsoid type metrics are constructed and the Penrose diagrams are analyz…
We will establish that the VC dimension of the class of d-dimensional ellipsoids is (d^2+3d)/2, and that maximum likelihood estimate with N-component d-dimensional Gaussian mixture models induces a geometric class having VC dimension at least N(d^2+3d)/2. Keywords: VC dimension; finite dimensional ellipsoid; Gaussian m…
We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian n-manifold M having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of …
Proves existence of non-planar minimal disks in ellipsoids.
problem Existence of non-planar minimal disks in ellipsoids.
method Optimization of Steklov eigenvalues and critical metrics.
result Existence of embedded non-planar free boundary minimal disks.
The paper describes the CR umbilical locus of a real ellipsoid in complex space.
problem Characterizing the CR umbilical locus of a real ellipsoid in complex space.
method Analyzing the set of points where the ellipsoid can be osculated by a biholomorphic image of the sphere up to 6th order.
result The CR umbilical locus is the union of stable curves and a non-trivial real variety defined by sextic equations.
The paper solves the problem of fitting an ellipsoid to random points efficiently.
problem Finding an ellipsoid that passes through random Gaussian points.
method Constructing a fitting ellipsoid using a decomposition of a random matrix and graph matrix theory.
result The ellipsoid fitting problem transitions from feasible to infeasible at a sharp threshold of n∼d2/4. Study the spectrum of Poincaré operator in triaxial ellipsoids.
problem Spectrum of the Poincaré operator in triaxial ellipsoids.
method Microlocal analysis of partial differential equations and polynomial vector fields.
result Polynomial eigenvectors and large-degree asymptotics of the operator.
Nearly all Gaussian points in high dimensions lie on a common ellipsoid.
problem Finding an ellipsoid that fits a large set of Gaussian points in high dimensions.
method Analyzing a random set of Gaussian points and proving a bound on their concentration.
result The bound nearly confirms a conjecture about fitting Gaussian points to ellipsoids.
The paper improves confidence ellipsoids for ridge regression with PAC bounds.
problem Uncertainty quantification in ridge regression for insufficiently exciting inputs.
method Extension of SPS EOA algorithm to ridge regression with PAC bounds.
result Explicitly shows how regularization parameter affects region sizes and provides tighter bounds.
Developed an ellipsoidal density-equalizing map for genus-0 closed surfaces.
problem Large geometric distortion when using spherical domain for genus-0 closed surfaces.
method Developed a novel method for ellipsoidal density-equalizing maps and combined with quasi-conformal maps.
result Significantly improved surface remeshing performance for genus-0 closed surfaces.
The topological structure of the lines of principal curvature, the umbilic and partially umbilic singularities of all tridimensional ellipsoids of R4 is described.
New symplectic embedding obstructions found for polydisks into half-integer ellipsoids.
problem Obstructing symplectic embeddings of polydisks into half-integer ellipsoids.
method Combinatorial criterion developed by Hutchings to obstruct symplectic embeddings.
result Optimal inclusion conditions for symplectic embeddings of polydisks into half-integer ellipsoids.
In this paper we construct proper biharmonic submanifolds into various types of ellipsoids. We also prove, in this context, some useful composition properties which can be used to produce large families of new proper biharmonic immersions.
New algorithms solve linear bandits in high dimensions efficiently.
problem Maximizing bilinear functions over convex sets and ellipsoids.
method Two novel algorithms for solving the problem efficiently.
result First known method to implement optimistic algorithms for linear bandits in high dimensions.
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 shows convex contact spheres resemble contact ellipsoids.
problem Characterizing the structure of convex contact spheres.
method Stratification by Reeb orbit periods and analysis of spectral invariants.
result Any stratum of a convex contact sphere is an integral homology sphere, and spectral invariants coincide with action values.
The study characterizes quadrics among affine hyperspheres based on centroid collinearity of sections.
problem Characterizing quadrics among affine hyperspheres based on section centroid collinearity.
method Extending Meyer and Reisner's theorem to unbounded convex sets and identifying additional assumptions.
result Ellipsoids, paraboloids, and one sheet of a two-sheeted hyperboloid are the only quadrics satisfying the centroid collinearity condition.
Filtered conformal ellipsoids for graph-native time series
problem Joint prediction sets for multivariate time series
method Filtered conformal ellipsoids
result Sharper at-target ellipsoids than static-covariance and non-filter baselines
We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.
Minimal spheres found in ellipsoids with large axes.
problem Finding non-planar minimal spheres in ellipsoids.
method Quantitative proof of minimal spheres existence with constraints.
result Ellipsoids with large axes contain at least three non-planar minimal spheres.
Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids …