Paper proves a Liouville theorem for solitons with constant curvature.
problem Understanding harmonic functions on specific geometric structures.
method Proved a Liouville theorem without gradient estimates.
result Finite dimensionality of harmonic functions with polynomial growth.
The paper classifies hypersurfaces in H2imesH2 with constant curvature.
problem Classifying hypersurfaces in H2imesH2 with constant sectional curvature. method Analyzing the geometry of H2imesH2 and constructing specific examples. result Examples of hypersurfaces in H2imesH2 with non-constant product angle function. Unique time function found for certain spacetimes with constant curvature.
problem Finding unique time functions with constant curvature in specific spacetimes.
method Proving existence of a unique foliation by hypersurfaces with constant scalar curvature.
result Existence of a unique time function with isochrones of constant scalar curvature.
The paper studies curves of constant-ratio in pseudo-Galilean space.
problem Characterizing curves of constant-ratio in pseudo-Galilean space.
method Analyzing spacelike curves with constant-ratio in terms of curvature functions.
result Characterization of special curves of constant-ratio in pseudo-Galilean space.
Reconstructs piecewise constant functions from geodesic integrals.
problem Recovering piecewise constant functions from X-ray data.
method Injectivity proof using variations through geodesics, improved for simple manifolds.
result Explicit formulas for function values near the boundary and stability analysis.
Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.
problem Null Penrose inequality on a null hypersurface.
method Global existence of constant mass aspect function foliation on a nearly spherically symmetric incoming null hypersurface in a vacuum perturbed Schwarzschild spacetime.
result Geometry of the constant mass aspect function foliation compared to the spherically symmetric foliation in the Schwarzschild spacetime.
Universal functions and metrics with constant curvature on domains.
problem Finding universal functions and metrics with constant curvature.
method Proving Runge-type theorems and universality results for locally univalent functions, refining Heins' result.
result Existence of universal conformal metrics with constant curvature on hyperbolic domains.
Study of Lagrangian submanifolds with constant angle functions in nearly Kähler S³×S³.
problem Characterizing Lagrangian submanifolds with constant angle functions in nearly Kähler S³×S³.
method Analysis of angle functions, classification theorems, and constructions.
result Classification of Lagrangian submanifolds with constant angle functions.
Functions with constant geodesic X-ray transform are restricted to manifolds with specific geometrical properties.
problem Existence of functions with constant geodesic X-ray transform on manifolds.
method Analyzing the geometrical properties of manifolds based on the existence of such functions.
result Functions with constant geodesic X-ray transform impose specific geometrical restrictions on the manifold.
Geodesic tomography identifies piecewise constants on convex manifolds.
problem Determining piecewise constant functions on nontrapping manifolds.
method Iterating local uniqueness results based on geodesic integrals.
result Piecewise constant functions are uniquely determined by their geodesic integrals.
A quaternionic version of Picard's theorem limits how many values a slice regular function can avoid.
problem How many values can a non-constant slice regular function of a quaternionic variable avoid?
method Investigates slice regular functions of quaternionic variables, extending the classical Picard theorem.
result A non-constant slice regular function of a quaternionic variable can avoid at most one value, similar to the classical Picard theorem.
This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension n≥3 with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they a…
Sharp HLS inequality on bounded domains with applications to curvature and isoperimetric constants.
problem Sharp Hardy-Littlewood-Sobolev inequality on bounded domains.
method Extension operator and suitable test functions.
result Existence of extremal functions and abstract domains with zero scalar curvature and larger isoperimetric constant.
Proves existence of curves with constant curvature in a sphere.
problem Existence of curves with constant geodesic curvature in a Riemannian 2-sphere.
method Develops a min-max scheme for a weighted length functional.
result Proves existence for almost every prescribed curvature.
In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary dif…
The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
problem Rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
method Characterizations and rigidity investigations for hypersurfaces with constant weighted shifted mean curvatures or ratios.
result Rigidity results for hypersurfaces with constant linear combinations of weighted shifted mean curvatures and radially symmetric shifted mean curvatures.
Injectivity of geodesic ray transform for piecewise constants on compact manifolds.
problem Injectivity of geodesic ray transform for piecewise constant functions.
method Injectivity of geodesic ray transform on piecewise constant functions weighted by a continuous matrix weight.
result Injectivity of the geodesic X-ray transform on piecewise constant functions.
Classifies Kähler metrics with constant holomorphic curvature.
problem Classifying Kähler metrics with constant holomorphic sectional curvature.
method Exploiting the geometry of the bundle of 1-jets of holomorphic functions.
result Local classification of Kähler metrics with constant holomorphic sectional curvature.
The paper classifies various types of hypersurfaces in a product space.
problem Classifying hypersurfaces in a specific product space.
method Analyzing hypersurfaces with constant curvatures, product angle functions, and additional conditions.
result Different types of hypersurfaces are classified based on their properties.
The paper extends a classical result to positively curved homogeneous spaces.
problem Classical results on constant functions on spheres do not extend to all positively curved homogeneous spaces.
method Proving that Lipschitz functions on positively curved homogeneous spaces are almost constant on high-dimensional submanifolds.
result Lipschitz functions on positively curved homogeneous spaces are almost constant on high-dimensional submanifolds.
Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.
problem Creating a Lipschitz map with specific local Lipschitz constants on a subset of a manifold.
method Constructs a Lipschitz map that matches a given map on a subset and has a local Lipschitz constant defined by a continuous function.
result A Lipschitz map can be constructed with a local Lipschitz constant prescribed by a continuous function.
New model OPSS allows constant approximation for maximum coverage problem.
problem Optimizing coverage functions from samples is hard.
method Proposed OPSS model with structured samples.
result Achieved constant approximation for maximum coverage problem.
Extends Lipschitz functions while preserving local constants.
problem Extending Lipschitz functions on metric spaces while maintaining local constants.
method Extends Lipschitz functions on metric spaces while locally preserving the asymptotic Lipschitz constant.
result Sobolev spaces on metric measure spaces are invariant under isomorphism of mm-structures.
Study Finsler metrics with Killing fields on constant flag curvature surfaces.
problem Characterize Finsler metrics with Killing fields on surfaces of constant flag curvature.
method Developed a normal form and method to calculate functions for spherically symmetric Finsler surfaces.
result Obtained the normal form of the Funk metric on the unit disk D^2.
The paper calculates bounds on the local Lipschitz constants of neural network layers.
problem Understanding the Lipschitz constants of neural network layers for robustness analysis.
method Analytical approach to determine upper bounds on local Lipschitz constants of affine-ReLU functions.
result The method produces tighter bounds than the standard conservative bound, especially for small perturbations.
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
Analyzes compound interest with constant payments and interest rate.
problem Examines the properties of compound interest balance and payment functions.
method Analyzes the outstanding balance and payment functions for constant payments and interest rate.
result The outstanding balance function is not generally concave in the interest rate.
Study finds nodal solutions of Yamabe equation constant along isoparametric levels.
problem Existence of nodal solutions for the Yamabe equation.
method Constant along isoparametric levels of a function.
result Existence of nodal solutions proven.
Characterizes warping functions in Einstein Poisson warped spaces.
problem Existence and nonexistence of warping functions with constant scalar curvature.
method Analyzes various dimensions of base space and constant scalar curvature conditions.
result Characterizes warping functions for different dimensions of base space.
The study classifies isoparametric hypersurfaces in specific product spaces.
problem Classifying isoparametric hypersurfaces in product spaces.
method Analyzing the angle function and constant principal curvatures.
result A complete classification of isoparametric and homogeneous hypersurfaces in SnimesSm and SnimesHm. The paper defines and analyzes a new mass function for compact manifolds.
problem Understanding the properties of metrics on compact manifolds.
method Introducing and studying the Mass Function $a \geq 0 \mapsto \xp{M}{a}$ and $\xm{M}{a}$.
result The Mass Functions are well-defined and have properties leading to applications to the Yamabe invariant.
Sharp mapping properties and regularization for X-ray transform on disks of constant curvature.
problem Sharp mapping properties and regularization of X-ray transform.
method Derive functional relations and mapping properties using elliptic differential operators.
result Theoretical possibility of regularized inversions for X-ray transform.
The study classifies isoparametric hypersurfaces in product spaces with constant angle function.
problem Classifying isoparametric hypersurfaces in product spaces with specific curvature conditions.
method Proving constant angle function and using it to classify hypersurfaces.
result Classification of isoparametric and homogeneous hypersurfaces in product spaces.
We compute the local Lipschitz constant of ReLU networks precisely.
problem Estimating the local Lipschitz constant of ReLU networks is hard.
method We use a novel approach involving the generalized Jacobian and backpropagation.
result We provide an algorithm to compute the exact Lipschitz constant of ReLU networks.
Square inscribed in a curve made of two graph functions.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
The paper studies estimating the normalizing constant using queries to a black-box function in RKHS.
problem Estimating the normalizing constant of a function in a reproducing kernel Hilbert space.
method Combines Bayesian quadrature and Bayesian optimization approaches, considering different levels of difficulty based on the parameter λ.
result The difficulty of estimating the normalizing constant varies between Bayesian quadrature and Bayesian optimization, even with noisy function evaluations.
For a closed Riemannian manifold (Mm,g) of constant positive scalar curvature and any other closed Riemannian manifold (Nn,h), we show that the limit of the Yamabe constants of the Riemannian products (M×N,g+rh) as r goes to infinity is equal to the Yamabe constant of (Mm×Rn,[g+gE]) and is …
This paper mixes constant sum and constant product market makers to improve their features.
problem Improving the balance between stable exchange rates and liquidity in automated market makers.
method Mixing and designing new methods for AMMs with specific features.
result Demonstrates new tools for creating markets with desired characteristics.
Study on constant mu-scalar curvature Kähler metrics, generalizing cscK and Kähler-Ricci solitons.
problem Existence and uniqueness of constant mu-scalar curvature Kähler metrics.
method Investigation of volume functional and study of a new K-energy.
result Fundamental constraints and existence conditions for constant mu-scalar curvature Kähler metrics.
Paper bounds integral of distance function on compact manifolds.
problem Bounding integral of distance function on compact manifolds.
method Curvature assumptions on compact Riemannian manifolds.
result Integral is bounded below by diameter, volume, and a constant.
Study minimal Lagrangian submanifolds of complex hyperquadric.
problem Classify minimal Lagrangian submanifolds of complex hyperquadric.
method Use non-integrable almost product structures and local angle functions.
result Classify minimal Lagrangian submanifolds with constant sectional curvatures and those with coinciding local angle functions.
In this paper, we consider the generalized lambda constant and the existence of ground states of the generalized Perelman's W-functional from a variational formulation. One result is concerned with the estimation of the generalized λ constant. The other results are about the existence of ground states of generalized …
The paper classifies hypersurfaces in a product of two spheres with constant curvature.
problem Classifying hypersurfaces in S2imesS2 with constant sectional curvature. method Applying the Tsinghua principle and solving the sinh-Gordon equation.
result Hypersurfaces with constant sectional curvature are parallel to minimal hypersurfaces with C=0. In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work \cite{Mon, Brendle,BE} considering constant mean curvature functions. Secondly, …
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L-Lipschitz neural networks and their density in L-Lipschitz functions. result One layer neural networks are dense in the set of all L-Lipschitz functions. Covering spaces with exponential growth have non-constant positive harmonic functions.
problem Existence of non-constant positive harmonic functions on covering spaces with exponential growth.
method Normal Riemannian covering, exponential volume growth, Lyons and Sullivan conjecture.
result Existence of non-constant positive harmonic functions on M. The study defines and constructs hypersurfaces in a product of two space forms.
problem Characterizing hypersurfaces in a product of two space forms.
method Explicit construction using parallel families of hypersurfaces and isoparametric hypersurfaces.
result Classification of hypersurfaces with constant mean curvature and constant product angle function.
New TVD estimator adapts to piecewise constant functions, improving performance.
problem Improving TVD estimator performance for piecewise constant functions.
method Investigates adaptivity of TVD estimator to piecewise constant functions and proposes a data-driven tuning parameter.
result The ideally tuned TVD estimator performs better than in the worst case for piecewise constant functions.