Study Kähler metrics with constant scalar curvature using coupled equations.
problem Finding Kähler metrics with constant scalar curvature.
method Solving a system of elliptic equations for a Kähler metric and a closed (1,1)-form, proving higher order estimates and smooth convergence.
result Smooth convergence to a cscK metric coupled to a harmonic (1,1)-form under uniform estimates.
Study on G2 structures and their moduli space in 7D.
problem Understanding moduli space of G2 structures with torsion. method Solving a system of partial differential equations related to the Bianchi identity.
result Moduli space of solutions is finite-dimensional.
Derives monotonicity formulas for elliptic gradient systems and classifies finite Morse index solutions.
problem Qualitative behavior of solutions to coupled elliptic gradient systems.
method Derives monotonicity formulas for local and nonlocal systems of equations.
result Derives monotonicity formulas for entire solutions of coupled elliptic gradient systems.
New solutions found for elliptic systems with mixed couplings.
problem Existence of fully nontrivial solutions to elliptic systems with mixed couplings.
method Study of fully nontrivial solutions to the system with mixed couplings in a bounded or unbounded domain.
result New existence and multiplicity results of fully nontrivial solutions.
The paper proves existence of solutions for Einstein-type elliptic systems on AE manifolds.
problem Analyzing semi-linear systems of partial differential equations motivated by the conformal formulation of Einstein constraint equations.
method Proving existence theorems under suitable conditions, including smallness assumptions on free parameters.
result Existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations.
The paper provides estimates for eigenvalues of elliptic differential problems.
problem Computing eigenvalue estimates for elliptic differential problems.
method Analytical computation of eigenvalues for specific types of elliptic differential equations.
result Universal estimates of eigenvalues and gaps between consecutive eigenvalues are derived.
Proves inequality linking function deviation to gradient norm on compact manifolds.
problem Analyzing coupled elliptic systems on compact manifolds.
method Develops a new Poincaré-Sobolev inequality with a density-free reference average.
result Poincaré constant depends on the density's gradient norm.
We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of sym…
The paper describes Seiberg-Witten equations in all dimensions.
problem Formulating Seiberg-Witten equations in all dimensions.
method Develops elliptic systems for Seiberg-Witten equations in various dimensions, using connections, spinors, and odd degree forms.
result Proves a system of equations for Seiberg-Witten in all dimensions, modulo gauge.
Establishes interior regularity results for a broad class of two-dimensional nonlinear elliptic systems using a unified abstract framework.
problem Interior regularity results for two-dimensional nonlinear elliptic systems
method A unified abstract framework built around a Campanato-type discrete iteration scheme coupled with a Caccioppoli-type estimate
result Local Hölder continuity of the map u is established, with an explicit Hölder exponent that optimally attains the classical Morrey--Campanato threshold dictated by the Lebesgue integrability of the source term f Geometric flow method finds static extensions for axisymmetric data.
problem Bartnik's static metric extension conjecture under axisymmetry.
method Geometric flow coupled with Weyl-Papapetrou formalism.
result Axisymmetric static extensions found for various data.
Paper classifies minimal graph transformations into new families of surfaces.
problem Classifying minimal graph transformations into new families of surfaces.
method Formulated and solved a coupled system of partial differential equations, reduced to solving an ordinary differential equation.
result Established rigorous equivalence to a modified problem for a harmonic function, yielding new families of minimal surfaces.
Study on optimal partitions and nodal solutions for the Yamabe equation.
problem Existence and structure of optimal partitions for the Yamabe equation.
method Analysis of a weakly coupled elliptic system related to the Yamabe equation.
result Existence of least energy sign-changing solutions with precisely two nodal domains.
Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.
problem Proving well-posedness for the Bartnik static extension problem near Schwarzschild spheres.
method Introduced a geodesic gauge to formulate governing equations as coupled elliptic and transport equations; used Bochner-measurable functions for transport equations.
result Established local well-posedness for arbitrary Bartnik data near Schwarzschild spheres, including those with small mean curvature.
We define higher genus Gromov-Witten invariants and establish a mathematical theory of sigma model coupled with gravity over any semi-positive symplectic manifolds. As applications, we verify the stablizing conjecture of symplectic 4-manifolds for simply connected elliptic surfaces and construct smooth 6-manifolds admi…
A new EnKF method for elliptic PDEs reduces dimensionality for accurate state estimation.
problem Elliptic PDEs in fluid flows make traditional EnKF regularization ineffective.
method Low-rank factorization of the Kalman gain based on the Jacobian spectrum.
result Inference can be performed in a low-dimensional subspace of the state space.
Study optimal partition problem for Q-curvature equations on Einstein manifolds.
problem Optimal partition problem for prescribed Q-curvature equation.
method Cohomogeneity one actions, higher order conformal operators, weakly coupled elliptic systems.
result Existence and multiplicity of least energy symmetric and sign-changing solutions.
Elliptic systems are characterized by Darboux integrability.
problem Characterizing elliptic differential systems with holomorphic solutions.
method Using a complex manifold and associated holomorphic Pfaffian system.
result Elliptic systems are Darboux integrable under generic conditions.
Study on kinetic Langevin diffusions and their couplings, showing subtle TV bounds and new non-Markovian couplings.
problem Understanding and quantifying the TV distance between solutions of kinetic Langevin diffusions with different initial values.
method Established new non-Markovian couplings for kinetic Langevin diffusions, derived from optimal coalescence trajectories, and analyzed their TV bounds.
result No Markovian coupling can capture the asymptotic decay rate of the TV distance between solutions of kinetic Langevin diffusions with different initial values.
By carrying out a rational transformation on the base curve CP1 of the Seiberg-Witten curve for N=2 supersymmetric pure SU(2)-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure SU(2)-ga…
Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.
problem Defining invariants for elliptic Weyl groups.
method Defines a set of good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants give flat invariants and structure constants of Frobenius structures.
Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.
problem Local radial rigidity of elliptic systems on Riemannian manifolds.
method Reduction to singular ordinary differential equations of Euler type.
result Local uniqueness and existence results for solutions with prescribed initial jets.
We discuss critical elliptic systems in potential form. We prove existence, multiplicity, and compactness of solutions.
Study on evolving interfaces with complex curvature and density effects.
problem Understanding the dynamics of evolving heterogeneous elastic interfaces.
method Modeling an evolving curve with a density function, analyzing the associated gradient flow evolution.
result Analysis of the preservation and asymptotic behavior of geometric properties in the evolving system.
We give new estimates for a critical elliptic system introduced by Rivière-Struwe in \cite{riviere_struwe} (see also the work of Rupflin \cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE solved by harmonic (and almost harmonic) maps from a Euclidean ball $B_1 \In \R^n$ into Riemannian manifol…
New method infers coupling in distributed dynamical systems.
problem Learning structure of coupled dynamical systems.
method Information-theoretic approach using time delay embedding theorems.
result Globally optimal approximations of distribution found.
New flow defined to solve Hull-Strominger system, with estimates and convergence results.
problem Constructing solutions to the Hull-Strominger system of equations.
method Introducing a natural extension of pluriclosed flow and using string algebroids and higher gauge theory.
result Proves global existence and convergence of the flow on special backgrounds.
We smooth out the math of fitting surfaces into 3D space.
problem Fitting surfaces into 3D space isometrically.
method Introduce elliptic regularization to the PDE system.
result The regularization leads to a natural variational interpretation.
The application of equivalence method to classify Monge-Ampère system leads to three orbits, parabolic case, hyperbolic case and elliptic case wich correspond to three types of Monge-Ampère systems. In this paper we will study the elliptic case and give a presentation of the group as a complex group.
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…
Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for ∂ˉ-complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type (1,1). Cappell and Mil…
We use a method, inspired by Pohozeav's work, to study asymptotic behaviors of non-variational elliptic systems in dimension n greater than two. The results apply to changing sign solutions.
Coupled entropy corrects flaws in Tsallis entropy for complex systems.
problem Misinterpretation of generalized temperature and entropy.
method Derived from generalized Pareto and Student's t distributions.
result Provides balanced measure of uncertainty for complex systems.
The paper constructs and generalizes Poisson brackets for Jacobi elliptic functions and higher-dimensional systems.
problem Understanding and generalizing Poisson brackets for Jacobi elliptic functions and higher-dimensional systems.
method Symplectic realization and bi-hamiltonian formulation for constructing and generalizing Poisson brackets.
result The Jacobi identity is satisfied only when the Plücker relations hold for these rank 2 Poisson brackets.
Optimizes control of synchronization in networked oscillators using genetic programming.
problem Optimizing control of synchronization in complex networked systems.
method Multi-objective genetic programming-based symbolic regression.
result Learned interpretable control functions for driving systems from synchronized to non-synchronized states.
We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…
Study systemic risk in multiplex networks with asymmetric coupling and feedback.
problem Understanding systemic risk in interconnected networks with asymmetric feedback.
method Analytical branching process approximation to calculate systemic risk.
result Systemic risk in multiplex networks is lower than in aggregated networks below a critical coupling strength, but increases above it.
MUSIC learns coupled systems with sparse data and incomplete physics.
problem Learning coupled systems with incomplete physical constraints and missing data.
method Sparsity induced multitask neural network framework integrating partial physical constraints with data-driven learning.
result MUSIC accurately learns solutions to complex coupled systems under data-scarce and noisy conditions.
The coupled KdV-mKdV system arises as the classical part of one of superextensions of the KdV equation. For this system, we prove its complete integrability, i.e., existence of a recursion operator and of infinite series of symmetries.
A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors …
In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space G/G0 can be embedded into the twistor space of the corresponding symmetric space G/H. Then we prove that the second elliptic system is equivalent…
Study proves global existence of harmonic maps into Lorentzian manifolds.
problem Proving existence of harmonic maps in Lorentzian manifolds.
method Investigates a parabolic-elliptic system for maps from a compact Riemann surface into a Lorentzian manifold, proving global existence under certain conditions.
result Global existence of the harmonic map heat flow into Lorentzian manifolds.
Global solutions found for a wave-Klein-Gordon system with strong couplings in divergence form.
problem Global well-posedness of a wave-Klein-Gordon system with strong couplings in divergence form.
method Constructed an auxiliary system with shifted primitives to handle the strong couplings.
result Established global well-posedness theorem for the wave-Klein-Gordon system.
In this paper, we show that the quotient space of the domain by the reflection group for an elliptic root system has a structure of Frobenius manifold for the case of codimension 1. We also give a characterization of this Frobenius manifold structure under some suitable condition.
Paper finds conditions for finite-dimensional vector spaces of cross-sections.
problem Finite-dimensional vector spaces of cross-sections for holomorphic bundles.
method Analyzes complex vector spaces of holomorphic cross-sections over elliptic orbits.
result Provides a sufficient condition for vector spaces to be finite dimensional.
We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…
We establish a general theorem improving regularity of solutions of elliptic pseudodifferential equations. It allows to resolve in a unified way the regularity issue for a broad class of nonlinear elliptic equations and systems appearing in different areas of geometry and analysis.
Novel framework detects lead-lag relationships in Chinese A-share market.
problem Detecting lead-lag relationships in the Chinese A-share market.
method Two-stage framework: long-term coupling via correlation, dynamic time warping, and rank-based metrics; high-frequency data analysis via cross-correlation, Granger causality, and regression models.
result Strongly coupled stock pairs often exhibit lead-lag effects, especially at finer time scales.