We introduce and study generalized -harmonic equations (1.1). Using some ideas and techniques in studying -harmonic functions from [W1] (2007), and in studying nonhomogeneous -harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity in the generalized -harmonic equatio…
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48 results for “1-harmonic functions”
Proves a function's locally least gradient property if its level sets are minimal laminations.
problem Understanding the relationship between 1-harmonic functions and minimal laminations.
method Analyzes minimal laminations and their convergence properties, then applies to 1-harmonic functions.
result Proves a function is 1-harmonic if its level sets are minimal laminations.
On 1-Harmonic Functionsmath.DG
Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1\mathbb{R}$; and every…
New complex-valued maps found on complex geometries.
problem Finding new maps on complex geometries.
method Solving non-linear PDEs based on manifold geometry.
result Constructed new proper biharmonic and (2,1)-harmonic maps.
A method to analyze maps into circles with singularities.
problem Analyzing maps with singularities into circles.
method Renormalization of Dirichlet Lagrangian for -harmonic maps.
result Applications in Willmore energy and frame energies.
The existence and nonexistence of -harmonic functions in unbounded domains of are investigated. We prove that if the Hausdorff measure of the asymptotic boundary of a domain is zero, then there is no bounded -harmonic function of for , where $λ_1(\mathb…
The paper defines Vn-slant helices in a lightlike cone and their curvature functions.
problem Understanding Vn-slant helices in a lightlike cone Qn+1.
method Defined Vn-slant helices and their harmonic curvature functions in Qn+1, expressed differential equations, and provided conditions for being Vn-slant helices.
result Differential equations of harmonic curvature functions and necessary conditions for Vn-slant helices in Qn+1.
The actions of a half Virasoro algebra have appeared in many integrable systems. In this paper we show that there is an action of a (Half) Virasoro algebra on the space of (2+0) harmonic maps into a Lie group. This action is generated by a natural action on the frames. A similar calculation on the space-time (1+1) harm…
Notes on Harmonic Mapsmath.DG
Notes on harmonic maps between manifolds, existence and regularity covered.
problem Existence and regularity of harmonic maps between Riemannian manifolds.
method Lecture-based approach covering harmonic maps, pluriharmonic maps, and related theorems.
result Coverage of existence and regularity of harmonic maps, including Siu-Sampson formula and Donaldson-Corlette theorem.
For harmonic maps of degree 2, a similar quantitative stability estimate does not hold uniformly.
problem Investigate the quantitative stability of harmonic maps of degree 2.
method Prove a local quantitative stability result for harmonic maps of degree 2, showing dependence on the given harmonic map.
result A uniformly quantitative stability estimate does not hold for degree 2 harmonic maps.
The paper studies geometric properties of -harmonic maps and proves Liouville type results.
problem Exploring geometric properties of -harmonic maps.
method Unified geometric analytic methods, first and second variation formulas, stress-energy tensor, conservation law, monotonicity formula, asymptotic assumption, extrinsic average variational method.
result Proves Liouville type results for -harmonic maps.
New mass-type invariants for cosmological space-times.
problem Characterize de Sitter solutions in space-times with a cosmological constant.
method Introduce new mass-type invariants and prove positive mass theorems.
result 1-harmonic Mass provides new characterizations and inequalities.
Study on stability of half-harmonic maps from R to S, proving non-degeneracy and quantitative stability.
problem Stability and non-degeneracy of half-harmonic maps from R to S.
method Analyzing the kernel of the linearized operator and using quantitative rigidity estimates.
result Uniform control of deviation for half-harmonic maps near Möbius transformations and Blaschke products.
New neural network models for complex functional data analysis.
problem Complex relations between functional predictors and responses.
method Function-on-Function regression models using neural networks with continuous hidden layers.
result Demonstrated power and flexibility in handling complex functional models.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
Introduces new weighted floating functions and affine surface areas.
problem Developing new mathematical concepts for convex bodies.
method Introducing weighted floating functions and weighted functional affine surface areas.
result New relations to traditional and classical affine surface areas.
Develops methods for selecting and estimating smooth functional coefficients in high-dimensional multivariate functional data.
problem Functional predictor selection and estimation of smooth functional coefficients in high-dimensional multivariate functional data.
method Functional group-sparse regression methods in a generic Hilbert space of infinite dimension.
result Consistency of estimation and selection (oracle property) under infinite-dimensional Hilbert spaces.
Neural networks can approximate functionals on RKHS with error bounds.
problem Approximating functionals on RKHS using neural networks.
method Interpolating orthogonal projections in RKHS using point evaluations.
result Explicit error bounds for various kernels (inverse multiquadric, Gaussian, Sobolev).
FFBO optimizes functions as inputs and outputs, improving on existing BO methods.
problem Optimizing functions as both inputs and outputs in complex systems.
method Function-on-function Gaussian process (FFGP) model with a separable operator-valued kernel, scalar upper confidence bound (UCB) acquisition function, and scalable functional gradient ascent algorithm (FGA).
result FFBO outperforms existing methods in synthetic and real-world data.
Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circ…
The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.
problem Bounding the Fridman function for hyperbolic manifolds.
method Analyzing the relationship between the Fridman function and the injectivity radius function.
result The Fridman function is bounded above by the injectivity radius function for certain hyperbolic manifolds.
Nuisance Function Tuning and Sample Splitting for Optimally Estimating a Doubly Robust Functionalmath.ST
Optimally estimates a functional using nuisance function tuning and sample splitting.
problem Estimating optimal rates for a doubly robust functional.
method Combines nuisance function tuning and sample splitting strategies.
result Shows optimal rates of convergence for various estimators.
The paper proves isoparametric functions on Finsler space forms under specific conditions.
problem Understanding isoparametric functions in Finsler space forms.
method Proving transnormal functions as isoparametric functions and constructing global and local isoparametric functions using the distance function.
result Generalization of Theorem B to Finsler space forms.
Paper introduces a nonparametric functional graphical model for random functions.
problem Estimating probabilistic conditional independence in functional graphical models.
method Functional sufficient dimension reduction to relax Gaussian or copula Gaussian assumptions.
result Enhances estimation accuracy and retains probabilistic conditional independence.
Robustifies elicitable functionals to handle small distribution misspecifications.
problem Determining uniquely optimal forecasts under distributional misspecification.
method Integrates statistical robustness into elicitable functionals using Kullback-Leibler divergence.
result Robust elicitable functionals admit unique solutions at the boundary of uncertainty regions.
The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.
problem Characterizing strong Hamel functions and their symmetries in Finsler spaces.
method Analyzing geodesic spray, strong dual symmetries, and strong dynamical symmetries.
result Strong Hamel functions can be characterized in terms of strong dual symmetries and strong dynamical symmetries.
Study biharmonic functions on vector bundles with spherical symmetry.
problem Investigate biharmonic functions on vector bundles with spherically symmetric metrics.
method Analyze vertical lifts and radial functions of functions on vector bundle manifolds.
result Construct an infinite two-parameter family of proper biharmonic functions.
Two new methods improve forecasting of functional time series data.
problem Forecasting of functional time-dependent data.
method Functional Singular Spectrum Analysis (FSFA) based forecasting methods.
result Our methods outperform existing algorithms for periodic stochastic processes.
This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a da…
Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.
problem Understanding the homotopy type of stabilizers of smooth functions on surfaces.
method Analyzing the homotopy properties of stabilizers for a specific class of smooth functions.
result The homotopy type of the connected component of the identity map of the stabilizer is completely described for Morse-Bott functions.
The paper connects convex functions to p-subharmonic functions and proves their equivalence.
problem Understanding the relationship between convex functions and p-subharmonic functions.
method Average principle, variational methods, and PDE techniques.
result Convex functions on R^n are p-subharmonic for every p > 1.
A new deep neural network tackles nonlinear functional regression with improved dimensionality reduction.
problem Nonlinear functional regression in infinite-dimensional functional data analysis.
method Functional deep neural network with adaptive kernel embedding and projection steps.
result Explicit rates of approximating nonlinear smooth functionals are derived, and the network is shown to be effective in both simulated and real datasets.
New model for network analysis using functional data.
problem Existing network models treat nodes as functions, but this paper introduces functional edges.
method Transform adjacency matrix into functional adjacency tensor, apply Tucker decomposition, regularize basis matrices, and solve tensor completion problem.
result The model effectively captures community structure and handles irregular functional edge data.
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
Function trees simplify complex ML models for better understanding.
problem Understanding and interpreting machine learning model predictions.
method Representing a multivariate function as a tree of simpler functions.
result Function trees reveal the global internal structure of functions.
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing -smooth functions with positive definite Hessians.
result Injectivity of restricted gradient in positive definite region.
NeuTSFlow models continuous functions behind time series forecasting.
problem Forecasting treats time series as discrete sequences, ignoring their continuous nature.
method NeuTSFlow uses Neural Operators to learn the transition between historical and future function families.
result NeuTSFlow outperforms traditional methods in forecasting accuracy and robustness.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
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.
A new model approximates complex functions in parameter space.
problem Complex and nonlinear functional regression problems.
method Mapping-to-Parameter function model with B-spline free knot placement.
result Robust knot placement algorithms improve model performance.
The paper generalizes inequalities on almost Kähler manifolds.
problem Generalizing inequalities on almost Kähler manifolds.
method Considered Donaldson gauge functional and twisted Aubin functionals.
result Generalized inequality between Aubin functionals.
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic -convex function and deduce some basic properties of -convex function and geodesic -convex function. We also introduce the concept of geodesic -convex set and -epigraph and in…
The diameter function is a topological Morse function.
problem The relationship between systole and diameter functions on Teichmüller space.
method Mapping class group-equivariant topological Morse function approach.
result The diameter function on Teichmüller space is a topological Morse function.
Proposes an L1-regularized functional SVM for binary classification with functional covariates.
problem Binary classification with multivariate functional covariates.
method L1-regularized functional support vector machine (SVM) with an accompanying algorithm.
result The proposed classifier performs well in prediction and feature selection.
Farey Recursive Functionsmath.GT
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 paper we propose and study a family of sparsity-inducing penalty functions. Since the penalty functions are related to the kinetic energy in special relativity, we call them \emph{kinetic energy plus} (KEP) functions. We construct the KEP function by using the concave conjugate of a -distance function and …
The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.
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.