Study on potential behavior in special geometric spaces.
problem Understanding potential behavior in specific geometric spaces.
method Analyzing asymptotic behavior of p-capacitary potentials and weak Inverse Mean Curvature Flow. result Characterized the behavior of potentials in Asymptotically Conical manifolds.
Unified view of monotonicity formulas for inverse mean curvature flow and p-capacitary potentials.
problem Understanding monotonicity formulas for various geometric flows and potentials.
method Refined analysis of p-capacitary potentials and their level sets. result Strong convergence of p-capacitary potentials to inverse mean curvature flow and curvature varifolds. Find conditions for starshapedness of level sets in Heisenberg group.
problem Ensure starshapedness of level sets of p-capacitary potentials. method Examine horizontally p-harmonic functions in the Heisenberg group. result Sharp conditions for strictly starshaped level sets.
The paper proves a Minkowski inequality on specific Riemannian manifolds.
problem Establishing a Minkowski inequality on manifolds with nonnegative Ricci curvature.
method Analyzing Riemannian manifolds with nonnegative Ricci curvature and Euclidean Volume Growth.
result Validated an optimal Minkowski inequality for certain subsets.
New proof of Penrose inequality using potential theory.
problem Proving the Riemannian Penrose inequality for black holes.
method Establishing a monotonicity formula for the p-capacitary potential.
result A new proof of the Penrose inequality for black holes.
The paper derives inequalities for p-capacitary functions in 3-manifolds with nonnegative scalar curvature.
problem Deriving inequalities for p-capacitary functions in 3-manifolds with nonnegative scalar curvature. method Deriving general monotone quantities and geometric inequalities associated with p-capacitary functions in asymptotically flat 3-manifolds with nonnegative scalar curvature. result The inequalities become equalities on the spatial Schwarzschild manifolds outside rotationally symmetric spheres.
The paper establishes inequalities for p-capacitary functions in flat half-spaces.
problem Understanding p-capacitary functions in asymptotically flat half-spaces. method Establishes monotone quantities and mass-capacity inequalities.
result Sharp inequalities attain equality on a Schwarzschild half-space.
We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely…
In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set Ω⊂Rn, n≥3. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the p-capacitary potentials associated with Ω, for every p suffici…
The paper proves existence and growth estimates for inverse mean curvature flow and related p-Laplacian Green kernel decay.
problem Existence and growth estimates for inverse mean curvature flow.
method Proving new decay estimates for the Green kernel of the p-Laplacian. result Existence and optimal growth estimates for the weak inverse mean curvature flow.
New geometric quantities help classify manifolds and relate to entropy.
problem Classifying Riemannian manifolds using geometric quantities.
method Introducing and analyzing asymptotic geometric quantities like p-capacity, eigenvalues, and Maz'ya constant.
result Geometric quantities coincide with entropy in specific conditions, characterizing manifolds.
For p∈(1,2] and a bounded, convex, nonempty, open set Ω⊂R2 let μp(Ωˉ,⋅) be the p-capacitary curvature measure (generated by the closure Ωˉ of Ω) on the unit circle S1. This paper shows that such a problem of prescribing μp on a planar convex domain: "Given a finite…
In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if t…
Develops potential theory for WZW equation in Kähler potentials space.
problem Solving the Wess--Zumino--Witten equation in Kähler potentials.
method Introduces ω-harmonicity on graphs to characterize the WZW equation and uses subharmonic distance. result Shows solvability of Dirichlet problem and approximation by finite-dimensional maps.
The paper examines stability of harmonic and symphonic maps with forms and potentials.
problem Stability of harmonic and symphonic maps with forms and potentials.
method Analyzes stability of F-harmonic and F-symphonic maps with forms and potentials. result Stability conditions for harmonic and symphonic maps are established.
The paper examines stability of subelliptic harmonic maps with potential.
problem Stability of subelliptic harmonic maps with potential.
method Derived first and second variation formulas, proved stability conditions, and gave instability results.
result Subelliptic harmonic maps with potential are stable under certain curvature and potential conditions.
The paper describes flat Hessian metrics on surfaces and their potentials.
problem Understanding Hessian metrics on surfaces.
method Theoretical description and explicit construction using integrable systems.
result Explicit construction of potentials for flat Hessian metrics on surfaces.
A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential w…
We consider the problem of learning an interpretable potential energy function from a Hamiltonian system's trajectories. We address this problem for classical, separable Hamiltonian systems. Our approach first constructs a neural network model of the potential and then applies an equation discovery technique to extract…
Article provides Bernstein gradient estimates for heat equations with potential terms.
problem Gradient estimates for heat equations with potential terms on weighted Riemannian manifolds.
method Derived Bernstein type gradient estimates for two systems of heat equations with linear, exponential, and combined potentials.
result Resolves part of the problem raised by Bhattacharyya et al. in \cite{SB-1}.
In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…
We consider the geodesic equation for the generalized Kahler potential with only mixed second derivatives bounded. We show that given such two generalized Kahler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kahler potential has uniformly bounded mixed second d…
We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potent…
The paper characterizes potential functions whose level sets are orbits in mechanical systems.
problem Characterizing smooth potential energy functions on the plane with specific level set properties.
method Analyzing inverse curvature flow and properties of level sets.
result Analytic or functions with totally path-disconnected critical sets must be radial, while every compact convex set is a critical set of a Levi potential.
The paper studies m-quasi Einstein manifolds with convex potential and finds constant scalar curvature.
problem Investigating m-quasi Einstein manifolds with a convex potential function. method Analyzing integral conditions and properties of the potential vector field.
result An m-quasi Einstein manifold with a convex potential function has constant scalar curvature. Estimates classical potential from stock price data using quantum mechanics.
problem Estimating classical potential from empirical stock price data.
method Quantum mechanical model of stock price distribution, estimating potential from wave function.
result Suggests methods to evaluate classical potential for Schrodinger equation.
We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator with a nonconvex potential in terms of a distance associated with the potential. The results here can be applied to the double well potential.
Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between A-polynomial and colored Jones polynomial. method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between A-polynomial and colored Jones polynomial potential function. It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKähler metrics with a single function that is a global potential for each of the Kähler structures (a hyperKähler potential). In an earlier paper the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite-d…
We apply the potential force estimation method to artificial time series of market price produced by a deterministic dealer model. We find that dealers' feedback of linear prediction of market price based on the latest mean price changes plays the central role in the market's potential force. When markets are dominated…
New method constructs potential functions for Kähler-Einstein metrics.
problem Constructing potential functions for Kähler-Einstein metrics on pseudoconvex domains.
method Method of potential scaling.
result Existence of 1-parameter family of automorphisms for certain pseudoconvex domains.
New proof shows compact homogeneous LCK manifolds are Vaisman.
problem Proving compact homogeneous LCK manifolds are Vaisman.
method Using homogeneous LCK manifolds with potential and a new metric construction.
result Compact homogeneous LCK manifolds are Vaisman.
Study classifies static potentials on 3-manifolds, proving one-dimensionality under specific conditions.
problem Classifying the dimension of static potentials on 3-manifolds.
method Analysis of relative zero sets of static potentials, using Miao and Tam's technique.
result Proves one-dimensionality of static potentials under specific conditions.
New proof and insights on Elliptical Potential Lemma for online learning.
problem Limitations in the original proof of the Elliptical Potential Lemma.
method Proposes a new proof and new perspectives on the lemma.
result New flexibility in the type of potentials considered.
Study magnetic geodesics on Kähler potentials using variational methods.
problem Understanding magnetic geodesics on Kähler potentials.
method Variational method for a generalized Landau-Hall functional.
result Magnetic geodesic equation and its relation to a perturbed complex Monge-Ampère equation.
Study of metrics on positive-definite matrices from power potential, linking to power means.
problem Understanding metrics on positive-definite matrices derived from power potential.
method Explicit expressions for geodesics and distance function derived from Hessian of power potential.
result Geodesics and distance function converge to weighted matrix geometric mean as β tends to zero.
Extends potential theory to Carnot groups, estimating Hausdorff dimension.
problem Estimating Hausdorff dimension of polar sets in Carnot groups.
method Geometric completeness and Riesz potential inequalities in Carnot groups.
result Developed applications in CR geometry and quaternionic CR geometry.
New theorem on Lee classes for LCK manifolds with potential.
problem Determining Lee classes on LCK manifolds with potential.
method Analyzing cohomology classes of Lee forms and proving the result for Vaisman manifolds.
result The set of Lee classes on LCK manifolds with potential forms an open half-space in H1(M,R). We study the influence of an additional scalar potential on various geometric and analytic properties of Dirac-harmonic maps. We will create a mathematical wish list of the possible benefits from inducing the potential term and point out that the latter cannot be achieved in general. Finally, we focus on several potent…
PO-Flow models potential and counterfactual outcomes for personalized treatment decisions.
problem Predicting individualized treatment effects from observational data.
method Continuous normalizing flow (CNF) framework for causal inference.
result Unified approach to potential outcome prediction, treatment effect estimation, and counterfactual prediction.
The development of accurate and transferable machine learning (ML) potentials for predicting molecular energetics is a challenging task. The process of data generation to train such ML potentials is a task neither well understood nor researched in detail. In this work, we present a fully automated approach for the gene…
The study extends GBM to include stable nonzero prices and finds a pronounced potential well.
problem The standard GBM model cannot describe stable nonzero prices in financial dynamics.
method Generalized GBM with polynomial drift of order q, model selection, and Markov chain Monte Carlo ensembles of potential functions.
result The optimal model for financial data is q=2, indicating the existence of a stable price.
New method trains normalizing flows using entropy-regularized transport.
problem Training continuous normalizing flows efficiently.
method Formulates flows as gradients of scalar potentials, training only these potentials.
result Trains normalizing flows without explicit flow computation during training.
K-nearest neighbors (KNN) method is used in many supervised learning classification problems. Potential Energy (PE) method is also developed for classification problems based on its physical metaphor. The energy potential used in the experiments are Yukawa potential and Gaussian Potential. In this paper, I use both app…
Graph potentials link to topological QFTs, with computational methods.
problem Defining a topological quantum field theory using graph potentials.
method Using colored trivalent graphs and birational type to define a topological QFT.
result Graph potentials' birational type depends on the graph's homotopy type.
In this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are closed and that there are exactly two on some surfaces with constant Gaussian cur…
Bayesian inference reconstructs external potentials in DFT for many-particle systems.
problem Reconstructing external potentials in classical density-functional theory (DFT) for many-particle systems.
method Combines Bayesian inference with classical DFT to probabilistically reconstruct external potentials.
result Accurately infers external potentials and density profiles with uncertainty quantification.
The paper applies potential theory to conformal geometry, proving theorems and dimension estimates.
problem Understanding the behavior of solutions near singularities in conformal geometry.
method Linear and nonlinear potential theory applied to conformal geometry problems.
result Established Huber's type theorems and Hausdorff dimension estimates for conformal geometry.