Paper constructs multivalued harmonic functions on R^3 using twistor methods.
problem Constructing multivalued harmonic functions on R^3.
method Twistor methods to construct multivalued harmonic functions.
result Found a family of multivalued harmonic functions with branching sets as ellipses and quadratic growth at infinity.
We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Na…
We outline a cohomological treatment for multivalued (classical) action functionals. We point out that an application of Takens' theorem, after Zuckerman, Deligne and Freed, allows to conclude that multivalued functionals yield globally defined variational equations.
The paper explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.
problem Existence and properties of eigenfunctions for spherical conical metrics.
method Application of multivalued harmonic maps to spheres and algebraic constructions.
result New criteria and examples of metrics with many 2-eigenfunctions.
Bayesian method models multivalued power data from wind farms.
problem Accurate modeling of power curves with multivalued relationships.
method Overlapping mixture of probabilistic regression models.
result Model accurately represents practical power data.
Defines finite type Multivalued Shape using hyperspaces.
problem No specific problem stated; focuses on definition.
method Construction over metric compacta using hyperspaces.
result Defines finite type Multivalued Shape.
This article is a survey of the Novikov problem of the structure of leaves of the foliations induced by a collection of closed 1-forms in a compact manifold M. Equivalently, this is to the study of the level sets of multivalued functions on M. To date, this problem was thoroughly investigated only for M=Tn …
Framework for universal graph function approximators outperforms existing methods.
problem Graph classification and separation of graph classes.
method Inspired by persistent homology, dependency parsing, and multivalued functions, the framework constructs universal approximators on graph isomorphism classes.
result Achieves state-of-the-art performance on four graph datasets.
We extend the work of Simon and Wickramasekera, who constructed a large class of C1,μ multivalued solutions to the minimal surface equation, to produce C1,μ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the La…
Generalization is one of the most important issues in machine learning problems. In this study, we consider generalization in restricted Boltzmann machines (RBMs). We propose an RBM with multivalued hidden variables, which is a simple extension of conventional RBMs. We demonstrate that the proposed model is better than…
This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our…
Two accelerated extragradient methods converge at O(1/k) rate for co-hypomonotone inclusions.
problem Solving co-hypomonotone inclusions with sum of Lipschitz and multivalued operators.
method Developed two Nesterov's accelerated extragradient methods for co-hypomonotone inclusions.
result Achieve O(1/k) last-iterate convergence rates on the residual norm. Extends holomorphic functions on complex manifolds to larger spaces.
problem Extending holomorphic functions on complex manifolds.
method Proving the existence of a larger space B(S,X) for continuous maps that allows holomorphic continuation. result Bounded holomorphic functions on C(S,X) can be extended to holomorphic functions on B(S,X). In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.
Let M be a closed n-dimensional manifold, n > 2, whose first real cohomology group H 1 (M ; R) is non-zero. We present a general method for constructing a Morse 1-form α on M , closed but non-exact, and a pseudo-gradient X such that the differential ∂ X of the Novikov complex of the pair (α, X) has at leas…
The Gauss Image Measure uniquely identifies dual convex bodies up to dilation.
problem Identifying dual convex bodies based on their Gauss Image Measure.
method Analyzing the Gauss Image Measure and its properties to establish the uniqueness of dual bodies.
result Dual convex bodies are equal up to a dilation on each path-connected component of the support of the measure.
New methods for estimating causal effects with limited overlap, using Stable Probability Weighting.
problem Estimating causal effects with limited overlap in multivalued treatments.
method Stable Probability Weighting (SPW) and Finite-Sample Stable Probability Weighting (FPW) methods.
result SPW and FPW provide practical solutions for estimating and inferring causal effects with limited overlap.
We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold Σ of a Riemannian manifold M. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of …
Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.
problem Creating explicit solutions for p-harmonic functions and harmonic morphisms. method Using joint eigenfunctions of the Laplace-Beltrami and conformality operators.
result Induces solutions on dual non-compact Riemannian symmetric spaces.
We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in e…
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
Extends Fatou theorem to bounded harmonic maps.
problem Classical Fatou theorem for bounded harmonic functions.
method Extending theorem to bounded harmonic maps.
result Identifies bounded harmonic maps on unit disk with bounded measurable functions on boundary.
Study examines harmonic functions in sub-Riemannian and RCD settings.
problem Characterizing harmonic functions in sub-Riemannian and RCD settings.
method Analyzes weak and strong asymptotically mean value harmonic functions.
result Weakly amv-harmonic functions are equivalent to harmonicity in Carnot groups.
New p-harmonic and harmonic morphisms found on Lie groups.
problem Constructing explicit p-harmonic and harmonic morphisms on Lie groups.
method Using the method of eigenfamilies to construct explicit complex-valued p-harmonic functions and harmonic morphisms.
result Explicit complex-valued p-harmonic functions and harmonic morphisms constructed on non-compact classical Lie groups.
Constructs explicit p-harmonic functions on Grassmannians and flag manifolds.
problem Finding proper p-harmonic functions on Grassmannians and flag manifolds. method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper p-harmonic functions on compact real Grassmannians and non-descending functions on real flag manifolds. Study Z2 harmonic functions with singularities on flat space.
problem Understanding Z2 harmonic functions with point singularities.
method Analyzes Z2 harmonic functions on R2 with point singularities. result Characterizes Z2 harmonic functions on R2 with point singularities. Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.
problem Classifying polynomial growth solutions to drift-harmonic equations on specific types of manifolds.
method Inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.
result All drift-harmonic functions with polynomial growth asymptotically separate variables and dimensions of spaces are computed.
Examines harmonic functions on Riemannian cones, focusing on Lioville's theorem.
problem Behavior of harmonic functions on Riemannian cones.
method Analyzes harmonic functions and applies Lioville's theorem.
result Discusses the behavior of harmonic functions on Riemannian cones.
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
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.
The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in …
Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…
New proof of Yamabe invariant for RP^3 using harmonic functions.
problem Yamabe invariant of RP3 method Using harmonic functions
result New proof of Yamabe invariant for RP3 Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. The study derives formulas for functionals on surface with boundary under harmonic Ricci flow.
problem Understanding functionals along harmonic Ricci flow on surfaces with boundaries.
method Derivation of formulas for functionals under harmonic Ricci flow.
result Established formulas for functionals on surface with boundary.
Gradient estimates for special harmonic functions on manifolds.
problem Estimating gradients of (p,V)-harmonic functions on Riemannian manifolds. method Using Moser iteration method, volume comparison theorem, and Sobolev embedding theorem.
result Explicit global gradient estimates for positive entire (p,V)-harmonic functions. Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal ℏ-power series solution and Borel summability. result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
Dirac-harmonic maps are uncoupled under certain conditions.
problem Understanding the uncoupling of Dirac-harmonic maps.
method Critical points of a super-symmetric energy functional, with focus on harmonic maps.
result Dirac-harmonic maps are uncoupled under minimality assumption.
The paper constructs gluing maps for harmonic maps between Riemannian manifolds.
problem Constructing harmonic maps between Riemannian manifolds.
method Gluing construction of extended harmonic maps.
result Construction of gluing maps for harmonic maps under specific conditions.
Study harmonic functions on submanifolds and their cones.
problem Understanding harmonic functions on minimal submanifolds and their cylindrical cones.
method Using the method from [6, 7], analyzing polynomial growth.
result Dimensions of harmonic functions match if cone growth degree differs.
f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map betwe…
Study polynomial growth harmonic functions on infinite penny graphs.
problem Finite-dimensional property of polynomial growth harmonic functions on infinite penny graphs.
method Asymptotically sharp dimensional estimate for ancient solutions of the heat equation.
result Proved the asymptotically sharp dimensional estimate.
Study harmonic function growth on curved spaces, proving inequalities.
problem Understanding growth rates of harmonic functions on curved manifolds.
method Applied a double-sided Price inequality to estimate growth rates.
result Effective estimates for harmonic function growth rates on curved manifolds.
The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.
problem Investigating Liouville type properties of harmonic forms on Riemannian manifolds.
method Normalized integral Ricci curvature and BiRic curvature.
result Established Liouville theorems for p-harmonic function, p-harmonic 1 form, and harmonic q form (with q≥2). Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps