Constructs new explicit proper r-harmonic functions on Thurston geometries.
problem Developing proper r-harmonic functions on specific 3D geometries.
method Explicit construction of new functions for Thurston geometries.
result Explicit construction of new proper r-harmonic functions on various Thurston geometries.
New method constructs complex-valued r-harmonic functions on Riemannian manifolds.
problem Constructing complex-valued r-harmonic functions on Riemannian manifolds.
method Introducing a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds and applying it to specific semisimple Lie groups.
result The method successfully constructs complex-valued r-harmonic functions on various Riemannian manifolds, including specific Lie groups.
The paper studies polyharmonic hypersurfaces in space forms, proving their minimal properties and characterizing specific cases.
problem Characterizing and understanding polyharmonic hypersurfaces in space forms.
method Analyzing hypersurfaces of order r (briefly, r-harmonic) in space forms Nm+1(c), focusing on c≤0 and Sm+1. result Proves that r-harmonic hypersurfaces in Nm+1(c) are minimal if c≤0 and mean curvature and shape operator are constant. Polyharmonic, or r-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper r-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion i:Sn−1(R)→Sn i…
The paper proves existence and instability of weak r-harmonic maps.
problem Existence and stability of weak r-harmonic maps. method Construction of critical points and analysis of stability.
result Existence and instability of weak r-harmonic maps restricted to specific dimensions. Constructs biharmonic and r-harmonic submanifolds in cohomogeneity one manifolds.
problem Constructing biharmonic and r-harmonic submanifolds. method Using cohomogeneity one manifolds, the normal index of submanifolds is studied, and new examples are provided.
result Constructs metrics on the sphere with biharmonic non-minimal hypersurfaces.
We construct new explicit proper r-harmonic functions on the standard n-dimensional sphere S^n and hyperbolic space H^n for any r\ge 1 and n\ge 2.
The paper explores polyharmonic hypersurfaces in pseudo-Riemannian space forms.
problem Characterizing polyharmonic hypersurfaces in pseudo-Riemannian space forms.
method Analyzing hypersurfaces with specific properties under given conditions.
result Existence of new families of proper r-harmonic hypersurfaces.
The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.
problem Existence and classification of polyharmonic helices of order r.
method Analytical and geometric approaches, including Lie group theory and Euclidean sphere analysis.
result Complete classification of proper r-harmonic helices in Sol_3 and new examples in Bianchi-Cartan-Vranceanu spaces.
Study classifies triharmonic surfaces in 3D homogeneous spaces.
problem Classifying triharmonic surfaces in 3D homogeneous spaces.
method Classification through isoparametric and CMC surfaces.
result Complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces.
New method for triharmonic maps to spheres in various dimensions.
problem Creating triharmonic maps to spheres in different dimensions.
method Construction method based on eigenmaps and suitable deformations.
result Existence of triharmonic maps from Rm∖{0} into spheres. The paper explores polyharmonic curves in semi-Riemannian manifolds.
problem Investigating polyharmonic curves in semi-Riemannian manifolds.
method Analyzing Frenet curves in semi-Riemannian manifolds of various types.
result Existence, non-existence, and classification results for polyharmonic curves.
The study characterizes and constructs polynomial harmonic morphisms on spheres.
problem Characterizing and constructing polynomial harmonic morphisms on spheres.
method Characterization and construction of polynomial harmonic morphisms using eigenfamilies.
result Strong restrictions and classification of polynomial harmonic morphisms in low dimensions.
The study describes the geometry of surfaces and their representations in SL(3,R).
problem Understanding the geometry of surface group representations into SL(3,R).
method Proving asymptotic formulas and harmonic map convergence for equivariant maps.
result The geometry of the image is weakly convex and a (one-third) translation surface.
The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called ES−r-energy funct…
Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane R2 with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potent…
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.
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 extends mixability theory to function-valued forecasts, proving various loss functions are mixable.
problem Efficient aggregation of functional and probabilistic forecasts in online prediction games.
method Adapting mixable and exponentially concave loss functions to function-valued forecasts.
result Various loss functions used for probabilistic forecasting are mixable (exp-concave).
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…
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.
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.
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 C2-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 φ-convex function and deduce some basic properties of φ-convex function and geodesic φ-convex function. We also introduce the concept of geodesic φ-convex set and φ-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.