Study n-superharmonic functions and their geometric applications.
problem Asymptotic behavior of n-superharmonic functions at isolated singularities. method Using Wolff potential, n-capacity estimates, and Adams-Moser-Trudinger inequality. result Strong n-capacity lower bound estimate for geometric applications. Paper introduces a new Poisson kernel for strongly pseudoconvex domains.
problem Developing a new mathematical tool for strongly pseudoconvex domains.
method Introducing a maximal plurisubharmonic function called the pluricomplex Poisson kernel.
result The pluricomplex Poisson kernel shares properties with the classical Poisson kernel and reproduces pluriharmonic functions.
Study of curved Kakeya sets using spray geometry and projective flatness.
problem Characterizing and understanding curved Kakeya sets in projective geometry.
method Introducing spray geometry to study curved Kakeya problems and characterizing behaviors using projective flatness.
result Characterization of the existence of totally geodesic surfaces and the equivalence to projective flatness.
Characterizes convergence of orbits of holomorphic semigroups in the unit disc.
problem Understanding the convergence behavior of orbits of holomorphic semigroups.
method Analyzes the shape of the starlike domain image of the Koenigs function and uses properties of quasi-geodesics.
result Characterizes convergence types (non-tangential and tangential) in terms of the domain's shape and quasi-geodesic properties.
This paper shows how to approximate CAT(-1) representations by Fuchsian ones.
problem Approximating representations of surface groups in CAT(-1) spaces.
method Constructing equivariant maps from the hyperbolic plane to CAT(-1) spaces.
result Every CAT(-1) representation can be approximated by a Fuchsian one.
New method finds Fuchsian representations dominating others in surface group representations.
problem Finding Fuchsian representations that dominate others in surface group representations.
method Straightening the pleated plane and applying strip deformations.
result There exists a Fuchsian representation that strictly dominates a given non-Fuchsian representation.
In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed comple…
In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary…
Generic Hitchin representations avoid hyperplanes in Lie algebras.
problem Properties of Hitchin representations in Lie algebras.
method Defined J(ρ) and used hyperplanes in Lie algebras to show J(ρ)∩H=∅. result Generic G-Hitchin representations avoid hyperplanes in the Lie algebra of G. Characterizes braid group actions on R and mapping class group actions on S1.
problem Understanding the rigidity of braid group actions on R and mapping class group actions on S1.
method Using the space of left orderings of B_n, isolated points, and conjugacy action of B_n.
result Characterizes actions of B_n on R that produce translation numbers agreeing with the standard action.
Improved simulation of phase transitions using hierarchical autoregressive networks.
problem Simulating phase transitions in complex systems.
method Hierarchical Autoregressive Neural (HAN) network sampling algorithm.
result Significant improvement in statistical uncertainty compared to the Wolff cluster algorithm.
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.
Extracts interpretable potential energy from Hamiltonian systems.
problem Learning an interpretable potential energy function from Hamiltonian systems.
method Constructs a neural network model of the potential and applies equation discovery to extract a closed-form algebraic expression.
result Close agreement between learned neural potentials and ground truth potentials, including correct effective potential for a central force problem.
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…
Investigates how adding a scalar potential affects Dirac-harmonic maps.
problem Analyzing the impact of scalar potential on Dirac-harmonic maps.
method Examines various geometric and analytic properties with different potentials.
result Cannot achieve certain properties with the potential term in general.
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.
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.
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. Study on Yamabe problem with potential in Euclidean space.
problem Constant scalar curvature problem with potential.
method Existence and nonexistence results for conformal equation.
result Existence and nonexistence results for radial case.
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.
Proposes a potential flow generator for generative models.
problem Improving the correctness and robustness of generative models.
method Integrates L2 optimal transport regularity into generative models. result Demonstrates effectiveness in image translation tasks.
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.
Paper assesses financial potential for enterprise development.
problem Determining financial potential for enterprise development.
method Stages of financial potential assessment based on literature analysis.
result Proposes a mechanism for managing enterprise financial potential.
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). 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…
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 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.
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.
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…