Study Bergman kernels for Gevrey potentials on Kähler manifolds.
problem Analyzing the asymptotic behavior of Bergman kernels for potentials with Gevrey regularity.
method Using the method of \cite{BBS} to find upper bounds for Bergman coefficients.
result Improved asymptotic expansion for Bergman kernels in shrinking neighborhoods of the diagonal.
Upper bounds for Bergman kernels from smooth Kähler potentials.
problem Bounding Bergman kernels from smooth Kähler potentials.
method Using Taylor coefficients of the Kähler potential, we give upper bounds for Bergman kernels of tensor powers of a smooth positive line bundle.
result Improved off-diagonal rate of decay for analytic, quasi-analytic, and Gevrey potentials.
Sharp threshold found for metric uniqueness in Riemannian Calderón-type problems.
problem Determining metrics uniquely from Dirichlet-to-Neumann maps in Riemannian Schrödinger problems.
method Adaptation of Lassas-Uhlmann reconstruction theorem and novel Gevrey space techniques.
result Analytic metrics uniquely determine the metric up to boundary-preserving diffeomorphisms, but non-analytic metrics are not uniquely determined.
Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.
problem Analyzing Bergman projections with Gevrey weights.
method Extending direct approach to semiclassical asymptotics to Gevrey weights using Fourier integral operators.
result Gevrey symbol amplitude of asymptotic Bergman projection with Gevrey weights and Gevrey-type growth rate.
Study shows observability from a measurable set for Gevrey functions.
problem Determining observability from a subset for Gevrey functions.
method Used measurable sets and inequalities for Gevrey regular functions.
result Established observability estimates from measurable sets for Gevrey functions.
New bounds found for nodal sets on special manifolds.
problem Finding bounds for nodal sets on specific types of manifolds.
method Used polynomial upper bounds for eigenfunctions on Gevrey and quasianalytic Riemannian manifolds.
result Established new upper bounds for the size of nodal sets.
Our aim is to prove that two formal power series of importance to quantum topology are Gevrey. These series are the Kashaev invariant of a knot (reformulated by Huynh and the second author) and the Gromov norm of the LMO of an integral homology 3-sphere. It follows that the power series associated to a simple Lie algeb…
We study the problem of coupling Einstein's equations to a relativistic and physically well-motivated version of the Navier-Stokes equations. Under a natural evolution condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid is incompressible, where this condition is given …
Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein's equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These la…
Abstract: Studies differential systems on compact Lie groups, extending Greenfield and Wallach's methods.
problem Global properties of left-invariant differential systems on compact Lie groups.
method Abstract: Extends Greenfield and Wallach's methods to systems, obtaining characterizations for regularity, range closeness, and cohomology spaces.
result Abstract: Derives generalizations of results and global versions of Caetano and Cordaro's result.
Heat kernel resurgent structure from Picard-Lefschetz theory
problem Short-time heat kernel asymptotics
method Picard-Lefschetz theory
result 1-Gevrey small-time expansion
The free energy of a closed 3-manifold is a 2-parameter formal power series which encodes the perturbative Chern-Simons invariant (also known as the LMO invariant) of a closed 3-manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3-manifold is uniformly Gevrey-1. As a …
Let N be a smooth (n+l)-dimensional Riemannian manifold. We show that if V is an area-stationary union of three or more C1,μ n-dimensional submanifolds-with-boundary Mk⊂N with a common boundary Γ, then Γ is smooth and each Mk is smooth up to Γ (real-analytic in the case N is real-anal…
Analytic networks with bounded coefficients can't outperform polynomial approximations.
problem Approximation limits of neural networks with analytic activation functions under coefficient constraints.
method Deterministic analysis using comparison argument and Bernstein-type estimates.
result Networks with analytic activation functions and controlled coefficients cannot outperform classical polynomial approximation rates on non-analytic targets.
The main goal of the paper is to address the issue of the existence of Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non compact manfold. Motivated by the recent results for compact manifolds we construct Kempf's distortion function an…
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.