Investigates the connection between quadrics and Christoffel duals, and zero mean curvature surfaces.
problem Understanding the relationship between quadrics and Christoffel duals, and zero mean curvature surfaces.
method Introducing para-holomorphic elliptic functions to study timelike minimal surfaces and their Christoffel duals of 1-sheeted hyperboloids.
result Curves of type change for real isothermic surfaces of mixed causal type are aligned with the real curvature line net.
Study anisotropic flows without global terms and solve dual Orlicz Christoffel-Minkowski problems.
problem Anisotropic flows without global forcing terms and dual Orlicz Christoffel-Minkowski problems.
method Existence results for dual Orlicz Christoffel-Minkowski type problems via stationary solutions of anisotropic flows.
result Existence results for a class of dual Orlicz Christoffel-Minkowski type problems.
Paper studies inverse curvature flows and solves related geometric problems.
problem Inverse curvature flows and related geometric problems.
method Analyzes a class of expanding flows with specific speeds and proves existence and convergence.
result Proves the existence and convergence of flows under certain conditions, leading to new solutions to geometric problems.
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
Paper solves a geometric problem involving mixtures of area and curvature measures.
problem Investigates a geometric problem involving mixtures of area and curvature measures.
method Establishes a gradient estimate to prove the existence of a solution.
result Proves the existence of an even, smooth, strictly convex solution for 1<p<q≤k+1. We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.
CANARI detects near-anomalies to predict future anomalies proactively.
problem Uncertainty in anomaly detection near distribution boundaries.
method Christoffel-based ANomaly Anticipation for eaRly dIscovery (CANARI) method.
result CANARI outperforms baseline methods in detecting near-anomalies and predicting future anomalies.
New flow method solves Christoffel-Minkowski problem.
problem Solving Christoffel-Minkowski problem.
method Entropy preserving curvature flow with global term.
result Entropy preserving flow solves Christoffel-Minkowski problem.
Solves capillary Lp-Christoffel-Minkowski problem in half-space.
problem Capillary surfaces in half-space geometry.
method Non-collapsing estimate for height and capillary support function.
result Extends Christoffel-Minkowski existence result.
Paper solves Christoffel-Minkowski problem in hyperbolic space.
problem Prescribing k-th horospherical p-surface area measure of h-convex domains in hyperbolic space. method Considered a fully nonlinear equation and used the full rank theorem with a viscosity approach.
result Existence of uniformly h-convex solution under appropriate assumptions. This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.
problem Prescribing curvatures for convex hypersurfaces in hyperbolic space.
method Proving a full rank theorem to establish the existence of solutions.
result Existence of solutions to the Christoffel problem and its equivalent Nirenberg-Kazdan-Warner problem on spheres.
Paper solves Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
problem Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
method Proved existence of solutions using a new full rank theorem.
result Existence of smooth, origin-symmetric, strictly horospherically convex solutions.
In this note we exhibit a constructive demonstration of the uniqueness of the Christoffel symbol.
Kernelized inverse Christoffel function improves outlier detection.
problem Outlier detection in large datasets with many features.
method Kernelized variant of the inverse Christoffel function.
result Best performance on 15 data sets compared to current methods.
Fixed angles of convex polygons lead to combinatorially rich polytopes.
problem Understanding the structure of convex polygons with fixed vertex angles.
method Combining combinatorial and geometric approaches, including dual polytopes and Schwarz-Christoffel maps.
result Fixed-angles polytopes are dual to cyclic polytopes under certain conditions.
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. A geometric flow based in the Riemann-Christoffel curvature tensor that in two dimensions has some common features with the usual Ricci flow is presented. For n dimensional spaces this new flow takes into account all the components of the intrinsic curvature. For four dimensional Lorentzian manifolds it is found that…
Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k-th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
Researchers solve a specific case of the Lp Christoffel-Minkowski problem for 1<p<k+1.
problem Solving the Lp Christoffel-Minkowski problem for 1<p<k+1. method Establishing the existence of convex bodies with prescribed k-th even p-area measure under certain conditions. result Existence of convex bodies with prescribed k-th even p-area measure on Sn under appropriate assumptions. Solves Christoffel problem for disk area measures on spheres.
problem Conditions for a measure to be a disk area measure of convex bodies.
method Integral representation and differential equation reformulation.
result Reconstructs support function from disk area measure.
Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
Unified flow solves Lp Christoffel-Minkowski problem for p>1.
problem Solving the Lp Christoffel-Minkowski problem for p>1. method Anisotropic expanding flow of smooth hypersurfaces with speed ψσk(λ)α. result The flow converges to a solution of the Lp Christoffel-Minkowski problem. The article approximates solutions to the Beltrami equation using similarity surfaces.
problem Approximating solutions to the Beltrami equation.
method Constructing similarity surfaces from polygons and analyzing their conformal uniformization.
result Holomorphic dependence of Christoffel symbols on polygons and convergence to a specific affine connection.
Classifies nets with area-preserving transformations into two types.
problem Classifying nets with area-preserving transformations.
method Classification using Combescure transformations and isotropic metric duality.
result Found two classes of nets: cone nets and Koenigs nets.
Directly proves Brioschi formula for Gaussian curvature.
problem Express Gaussian curvature in terms of local coordinates.
method Elementary proof without Christoffel symbols.
result Directly derived Brioschi formula for Gaussian curvature.
Proposes landmark selection for kernel methods.
problem Selecting important landmarks from large training sets.
method Deterministic and randomized adaptive algorithm for landmark selection.
result Landmarks are related to the minima of kernelized Christoffel functions.
Study on geodesic completeness for Type A surfaces.
problem Determining geodesic completeness for Type A surfaces.
method Analyzing constant Christoffel symbols to determine geodesic completeness.
result A method to determine if a set of constant Christoffel symbols can model a complete surface.
Log-concave coefficient sequences for two-bridge knots proved.
problem Proving log-concavity of Alexander polynomial coefficient sequences for alternating knots.
method Introducing a polynomial Δ(t) associated to Christoffel words and proving its log-concavity. result Strong Fox conjecture for two-bridge knots proved.
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
New capillary Christoffel-Minkowski problem solved for half-space.
problem Finding strictly convex capillary hypersurfaces from capillary functions.
method Established analogous result in capillary setting for half-space.
result Capillary functions lead to strictly convex capillary hypersurfaces.
The space of polygons up to similarity is studied using the Schwarz-Christoffel formula.
problem Understanding the space of polygons up to similarity.
method Using the Schwarz-Christoffel formula to show homeomorphism and proving embeddings for specific polygons.
result The space of labelled simple polygons up to similarity is homeomorphic to R^(2n-4) for n=3,4,5.
This paper optimizes sampling for least-squares approximation.
problem Optimizing sampling for least-squares approximation in arbitrary linear spaces.
method Introducing the Christoffel function to construct near-optimal random sampling strategies.
result The number of samples scales log-linearly in the dimension of the approximation space.
Author finds the solutions of the Christoffel problem for open and closed surfaces in Riemannian space. The Christoffel problem is reduced to the problem of construction the continuous G-deformations preserving the sum of principal radii of curvature for every point of surface in Riemannian space. G-deformation transfe…
New method connects leverage scores and kernel density, revealing a decreasing relationship.
problem Understanding the relationship between leverage scores and kernel density.
method Introducing regularized Christoffel functions to study leverage scores for kernel methods.
result Quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels.
New invariants found for a specific type of mapping in generalized Riemannian space.
problem Understanding transformations of Christoffel symbols in generalized Riemannian space.
method Analysis of third type almost geodesic mappings.
result Obtained new invariants that are projective parameters and tensors.
The paper studies curvature equations and their solvability.
problem Solving curvature type equations and their Dirichlet problems.
method General class of fully nonlinear curvature equations, Christoffel-Minkowski problem, degenerate equations.
result Solvability of curvature type equations and Dirichlet problems.
Study geodesics on Siegel-Jacobi ball with balanced metric.
problem Understanding geodesics on the Siegel-Jacobi ball.
method Determined Christoffel symbols, studied geodesic equations, calculated covariant derivatives.
result Explicit formulas for geodesics on Siegel-Jacobi ball.
Derives Levi-Civita connection formulas for specific geometries.
problem Determining geometric invariants of Lorentzian manifolds.
method Explicitly derives Christoffel symbols in terms of adapted frame fields.
result Formulas for geometric invariants of Lorentzian manifolds.
Improves data-driven reachability estimation for complex systems.
problem Estimating reachable states in complex dynamical systems with unknown parameters.
method Uses Christoffel functions and conformal prediction to improve sample efficiency and robustness.
result Guaranteed convergence to the true reach set with improved sample efficiency and robustness.
In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structure given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various ge…
Derives a formula for the k-th covariant derivative of tensor fields.
problem Finding a formula for the k-th covariant derivative of tensor fields.
method Introducing symbols P and Q depending on Christoffel symbols, deriving a formula (3.1).
result Derives a formula for the k-th covariant derivative of tensor fields.
We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all iso…
Christoffel function characterizes the corruption a bounded-degree certificate cannot remove in robust halfspace learning.
problem Robust halfspace learning under malicious noise
method Sum-of-Squares degree of outlier-removal certificate
result Christoffel function bounds the corruption a bounded-degree certificate cannot remove
Anisotropic expanding flow of convex hypersurfaces converges to a soliton under certain conditions.
problem Anisotropic expanding curvature flows of convex hypersurfaces in Euclidean space.
method Proving the existence and convergence of a unique smooth and uniformly convex solution to the flow under specific conditions.
result The flow converges to a soliton which solves an elliptic equation when parameters are within a suitable range.
The paper applies a capillary John ellipsoid theorem to solve capillary curvature problems.
problem Solving capillary curvature problems in Euclidean half-spaces.
method Applying a capillary John ellipsoid theorem to derive non-collapsing estimates and gradient estimates.
result Established existence of solutions to capillary curvature problems in certain ranges of p and q. Study curvature properties in special manifolds using specific tensors.
problem Investigate curvature conditions in 2-quasi-Einstein manifolds.
method Analyze Riemann-Christoffel curvature tensor and its linear combinations with Ricci tensor.
result Satisfy pseudosymmetry type curvature conditions in certain manifolds.
Solves curvature equations using parabolic flows in various spaces.
problem Finding strictly convex, spacelike solutions to curvature equations.
method Employing curvature flows without global terms in Riemannian and Lorentzian spaces.
result Solves a broad class of elliptic prescribed curvature equations.
New proof shows origin-centred balls are unique solutions to curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Local Brunn-Minkowski inequality and Alexandrov-Fenchel inequality.
result Origin-centred balls are the only solutions to curvature and related problems.