The Thurston spine's properties are studied in relation to Morse-Smale complexes.
problem Understanding the Thurston spine's local properties and their global implications.
method Analyzes the Thurston spine as a subset of Teichmüller space and studies its local properties in relation to the systole function.
result The Thurston spine satisfies properties analogous to Morse-Smale complexes, demonstrating its topological significance.
New method improves submodular maximization for machine learning applications.
problem Inexact monotonicity in submodular functions limits traditional algorithms' performance.
method Introduces monotonicity ratio as a continuous version of monotonicity, leading to improved approximation guarantees.
result Improved approximation ratios for movie recommendation, quadratic programming, and image summarization.
Machine learning predicts liquid water properties from cluster data.
problem Accuracy of bulk properties from machine-learned potentials is limited by training data.
method Local, atom-centred descriptors enable prediction of bulk properties from cluster data.
result Excellent agreement with experimental and theoretical counterparts of liquid water properties.
The paper studies sprays on Hamel-Funk functions and their properties.
problem Analyzing the properties of sprays and functions in geometric contexts.
method Using Funk metrics to construct sprays and proving the existence of Funk functions.
result The existence of local and global Funk functions on certain spray manifolds.
Paper defines Farey Recursive Functions and explores their properties.
problem Understanding recursive functions on rationals.
method Defined and studied Farey Recursive Functions using Farey graph.
result Farey Recursive Functions naturally connect to 2-bridge knots and links.
In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings i…
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …
In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like.
The article studies critical points of a new energy functional in higher dimensions.
problem Investigating critical points of a new energy functional in higher dimensions.
method Holomorphic deformations, closed and open properties, differential of the functional.
result Properties of critical points under holomorphic deformations are closed and open.
Study functionals on almost complex structures for Yau's Challenge.
problem Yau's Challenge on compact C-manifolds. method Variational properties of functionals on almost complex structures.
result Variational properties could be used to tackle Yau's Challenge.
Study uses property elicitation to understand how fairness regularizers affect optimal decisions.
problem Understanding how fairness regularizers change the optimal decision in predictive algorithms.
method Property elicitation to analyze the relationship between loss, regularization, and optimal decision.
result Necessary and sufficient condition for when a property changes with the addition of a regularizer.
Establishes relationships between prudence and stability properties of risk functionals.
problem Stability properties of risk functionals
method General relationships and preservation of prudence under cash-additive hulls and inf-convolutions
result General methods for constructing prudent risk measures
Paper explores RKHS properties for derivative and integral operators.
problem Establishing sufficient conditions for reproducing property in RKHS.
method Establishing reproducing property for combinations of composition operators.
result Provides framework for regularized learning algorithms involving function values, gradients, or operators.
The staircase property aids deep learning by guiding hierarchical feature learning.
problem Understanding how hierarchical structure influences deep learning performance.
method Defined and proved the staircase property for Boolean hypercube functions, and showed its learnability by layerwise stochastic coordinate descent.
result Staircase functions can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks.
The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable prop…
We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem: Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on…
We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications…
Study shows infoGAN's generalization error bound for two-layer networks.
problem Understanding generalization error in infoGAN for two-layer neural networks.
method Analyzes the difference between empirical and population objective functions, derives Rademacher complexity bounds.
result Derives error bound for infoGAN's generalization error in a two-layer network.
We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on …
In this paper, we study volume growth, Liouville theorem and the local gradient estimate for f-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.
The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…
Study expands classical harmonic function results to Riemannian manifolds.
problem Classical harmonic function properties in domains of Riemannian manifolds.
method Generalized classical results to Riemannian manifolds, including pinched negative curvature.
result Generalized results for Riemannian manifolds, including pinched negative curvature.
The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.
Introduces Hurewicz fibrations for embedding maps of orbifold charts.
problem No specific problem stated; focuses on new concept definition.
method Defines E-fibration embedding and studies its properties.
result Introduces and studies properties of E-fibration embedding.
The paper studies continuous submodular functions and their optimization.
problem Maximizing continuous submodular functions in poly. time.
method Characterization of continuous submodularity, operations preserving it, and algorithms for constrained maximization.
result Continuous submodularity is equivalent to a weak DR property, leading to continuous DR-submodular functions with the full DR property.
The authors define a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming i…
The paper proves rigidity of certain Dirac operators using theta functions.
problem Rigidity of twisted Dirac operators on specific bundles.
method Lefschetz formula, Atiyah-Bott localization, theta function properties.
result Lefschetz numbers are constant under certain conditions, proving operator rigidity.
Regularizers change the geometric properties of loss functions in neural networks.
problem Understanding how different regularizers affect the geometric properties of loss functions in neural networks.
method Examined several regularizers, including weight decay, to determine if the regularized loss function becomes Morse.
result For certain regularizers, the regularized loss function becomes Morse, indicating a change in geometric properties.
The paper extends sequences while preserving statistical properties using a mixture model.
problem Extending sequences while retaining their statistical properties.
method Auto-regressive Sequence Extension Mixture Model (SEMM) using deep learning.
result The mixture model outperforms traditional neural networks in sequence extension with statistical property retention.
This research extends quasiplurisubharmonic functions on compact Kähler manifolds.
problem Extending quasiplurisubharmonic functions on compact Kähler manifolds.
method Using a cover of Zariski-open Stein sets with strictly plurisubharmonic potentials, the authors prove extension properties for plurisubharmonic functions.
result Any ω|_X-plurisubharmonic function on an analytic subvariety X of a compact Kähler manifold V extends to a ω-plurisubharmonic function on V.
The paper examines functional properties on manifolds with very negative curvature.
problem Functional properties on manifolds with very negative curvature.
method New Hardy-type inequalities and first and second order inequalities.
result Functional properties typically hold in manifolds with polynomially growing negative curvature.
This note establishes several integral identities relating certain metric properties of level hypersurfaces of Morse functions.
We show that a subspace S of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are that S is closed in L2(M) and that if a sequence of functions fn in S …
UDENet and ResNet can approximate any function, with ODENet showing UAP for continuous functions.
problem Approximating any function using ODENet and ResNet.
method Proved UAP for ODENet and ResNet, derived gradient, and applied to various problems.
result UDENet and ResNet can approximate any function, with ODENet showing UAP for continuous functions.
SONATA algorithm converges to solutions of nonconvex smooth functions with KL property.
problem Decentralized optimization over networks with nonconvex smooth functions and convex constraints.
method Decentralized gradient-tracking algorithm SONATA under the KL property.
result SONATA converges to stationary solutions at R-linear rate for θ∈(0,1/2], sublinear rate for θ∈(1/2,1), and R-linear rate for θ=0. New probabilistic method constructs Kähler-Einstein metrics and suggests zero-free properties of zeta functions.
problem Existence and explicit formulas for Kähler-Einstein metrics on Fano varieties.
method Probabilistic construction involving canonical random point processes.
result Zero-free properties of Archimedean zeta functions and their relation to Langlands program.
Researchers examine global properties of a scalar curvature functional to solve the prescribed Ricci curvature problem.
problem Solving the prescribed Ricci curvature problem for homogeneous metrics.
method Examining global properties of the scalar curvature functional, focusing on its critical points and maximum.
result Conditions for a global maximum of the scalar curvature functional on a general homogeneous space.
A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work asks which properties are elicitable, …
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
Introduces Floer functions and Floerfolds for intrinsic properties.
problem Complex transformation of Hessian under chart transition.
method Introduces Floer functions and Floerfolds to address intrinsic properties.
result Floer functions and Floerfolds provide intrinsic conditions for Hessian.
The paper examines how isoparametric foliations affect the Pompeiu property in compact Riemannian manifolds.
problem The Pompeiu property in compact Riemannian manifolds with isoparametric foliations.
method Analysis of the spectrum of the Laplacian and specific calculations for the round sphere.
result Conditions on the spectrum of the Laplacian under which level domains of isoparametric functions fail the Pompeiu property.
We explore the efficacy of using a novel activation function in Artificial Neural Networks (ANN) in characterizing exoplanets into different classes. We call this Saha-Bora Activation Function (SBAF) as the motivation is derived from long standing understanding of using advanced calculus in modeling habitability score …
We prove a Liouville property for any f-harmonic function with polynomial growth on a complete noncompact smooth metric measure space (M,g,e−fdv) when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.
The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the self-concordant-like property, this concept has heretofore remained unexploited in convex optimization. To this end, we deve…
The paper investigates how activation functions impact the training of Neural ODEs, leading to global convergence.
problem Challenges in training Neural ODEs, particularly gradient computation accuracy and convergence analysis.
method Investigates the impact of activation functions on the training dynamics of Neural ODEs.
result Establishes global convergence of Neural ODEs under gradient descent in overparameterized regimes.
Cubic-regularized Newton's method (CR) is a popular algorithm that guarantees to produce a second-order stationary solution for solving nonconvex optimization problems. However, existing understandings of the convergence rate of CR are conditioned on special types of geometrical properties of the objective function. In…
Proves regularity of extremal function on compact Kähler manifolds.
problem Regularity of extremal function on compact Kähler manifolds.
method Local property analysis and equivalence of continuity and Hölder continuity.
result Equivalence of classical notions of local L-regularity and locally Hölder continuous property.