Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.
problem Understanding Willmore surfaces in asymptotically Schwarzschild 3-manifolds.
method Application of Lyapunov-Schmidt reduction method.
result End of the manifold is foliated by area-constrained Willmore spheres.
Paper proves existence of magnetic geodesics on sphere.
problem Existence of closed K-magnetic geodesics on S2. method Lyapunov-Schmidt reduction and local variational formulation.
result Existence and multiplicity of closed K-magnetic geodesics. Tackles variational problems with curvature and fractional Brezis-Nirenberg equations, overcoming lack of compactness.
problem Variational problems with curvature and fractional Brezis-Nirenberg equations.
method Refined techniques and Lyapunov-Schmidt method for ODE systems, blow-up analysis for elliptic equations.
result Existence and multiplicity of solutions with qualitative properties.
Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.
problem Finding entire solutions to magnetic Ginzburg-Landau equations in 4D.
method Using Lyapunov-Schmidt reduction.
result Existence of entire solutions and saddle type solutions with specific zero sets.
From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the correspondi…
The paper proves the existence of stable spheres in asymptotically flat 3-manifolds.
problem Existence of stable spheres in asymptotically flat 3-manifolds.
method Lyapunov-Schmidt reduction
result Existence of an asymptotic foliation of (M,g) by stable constant mean curvature spheres. Study local foliations of surfaces with constant mean curvature and constant expansion in space-time.
problem Characterize surfaces with constant mean curvature and constant expansion in space-time.
method Use Lyapunov Schmidt reduction in an n+1 dimensional manifold to construct and prove the uniqueness of foliations.
result Construct and prove the uniqueness of local foliations of surfaces with constant mean curvature and constant expansion.
Refines geometric center of mass analysis for Einstein field equations.
problem Analyzing the geometric center of mass of Willmore surfaces in initial data for Einstein field equations.
method Refined Lyapunov-Schmidt analysis to study geometric center of mass of area-constrained Willmore surfaces.
result The geometric center of mass agrees with the Hamiltonian center of mass under specific conditions.
The conformal Willmore functional (which is conformal invariant in general Riemannian manifold (M,g)) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds (R3,gε) -where gε is a metric close and asymptotic to the euclidean o…
New cylindrical solutions found for Grushin-type problem.
problem Critical Grushin-type problem on CR sphere.
method Local Pohozaev identities for non-degeneracy, Lyapunov-Schmidt reduction for solutions.
result New type of multi-bubbling cylindrical solutions constructed.
We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form H(X)=1+A∣X∣−γ for ∣X∣ large, when A<0 and γ∈(0,2). Such surfaces are close to sections of unduloids with small necksize, fold…
We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on th…
CR structure on S³ with non-compact solutions to CR Yamabe problem.
problem Existence of non-compact solutions to CR Yamabe problem.
method Deforming standard CR structure of S³, using Lyapunov-Schmidt method.
result Existence of a blowing-up sequence of solutions.
The paper classifies noncollapsed translators in 4D space.
problem Classifying entire convex translators in 4D space.
method Developed Fredholm theory and used Lyapunov-Schmidt reduction.
result The one-parameter family of translators is uniquely determined.
We are concerned with hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing sph…
We associate to a parametrized family f of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} β(f) which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index β(f) is derived from the index bundle of the linearization of the …
We consider the multi-bump solutions of the following fractional Nirenberg problem \begin{equation}\label{01} (-Δ)^s u=K(x)u^{\frac{n+2s}{n-2s}}, \;\;\;\;u>0\;\;\text{ in }\mathbb{R}^n, \end{equation} where s∈(0,1) and n>2+2s. If K is a periodic function in some k variables with 1≤k<2n−2s, we pr…
Normal forms for equivariant maps in infinite dimensions established.
problem Establishing normal forms for equivariant maps in infinite-dimensional manifolds.
method Inspired by Lyapunov-Schmidt reduction and Kuranishi method, uses Slice Theorem for Fréchet manifolds.
result Abstract moduli spaces of equivariant maps are locally modeled on quotient by a compact group.
We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and a…
This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Möbius degeneration of the tori. In the first paper the construction was perfo…
Constructs foliations of critical surfaces for Hawking energy in asymptotically flat initial data sets.
problem Positivity and rigidity of Hawking quasi-local energy in asymptotically flat spacetimes.
method Lyapunov-Schmidt reduction within a Willmore-foliation framework.
result Existence and uniqueness of foliations by Hawking surfaces, positivity and large-sphere limit of Hawking energy.
Normal forms and symplectic reduction for gauge field theory in infinite dimensions.
problem Understanding the structure of moduli spaces in gauge field theory.
method Establishing normal forms for equivariant maps and developing singular symplectic reduction in infinite dimensions.
result The reduced phase space decomposes into smooth manifolds each with a natural symplectic structure.
Study on hemisphere threshold for Escobar functional on Riemannian manifolds, revealing mass and boundary invariant behaviors.
problem Analyzing the hemisphere threshold for the Escobar functional on compact Riemannian manifolds.
method Near-threshold landscape organization by boundary invariants, exact evaluation of weighted profile moments, Lyapunov-Schmidt correction, and blow-up analysis.
result At threshold, blow-ups concentrate at umbilic points with vanishing mass and gradient, leading to compactness and hemispherical rigidity.
Improved iterative methods for risk parity portfolio weights.
problem Solving for portfolio weights in risk parity allocation.
method Enhanced CCD and Newton methods, including a rescaling step and improved initial guess.
result Improved CCD method is the best, three times faster with 40% fewer iterations.
We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…
A new method combines Laplace and Variational Bayes for scalable inference.
problem Complex models and large datasets make exact inference infeasible.
method Low-Rank Variational Bayes Correction (VBC) using Laplace method and Variational Bayes correction in a lower dimension.
result The method ensures scalability in both model complexity and data size.
Unified framework for model explanation methods based on feature removal.
problem Unclear relationships and preferences among various model explanation methods.
method Characterizes removal-based explanations along three dimensions.
result Unified 26 existing methods, including widely used approaches.
This work reviews and evaluates methods for predicting prediction intervals in regression problems.
problem Calibration of prediction intervals in regression problems.
method Four classes of methods: Bayesian, ensemble, direct interval estimation, and conformal prediction.
result Conformal prediction can be used as a general calibration procedure.
Derives kernel PCA with Nyström method for scalability.
problem Scalability of kernel PCA.
method Nyström method for kernel PCA.
result Provides scalable alternative to full kernel PCA.
In this paper, the author considers the numerical computation of CVA for large systems by Mote Carlo methods. He introduces two types of stochastic mesh methods for the computations of CVA. In the first method, stochastic mesh method is used to obtain the future value of the derivative contracts. In the second method, …
Develops a fast method for pricing American options under variance gamma model.
problem Inefficient methods for pricing American options under variance gamma model.
method Inspired by quadratic approximation method, uses machine learning on pre-calculated quantities to reduce error.
result Proposed method is efficient and accurate for practical use.
New method combines spectral and sparse methods for Gaussian processes.
problem Efficiently fitting Gaussian processes to large datasets.
method Orthogonally decoupled variational Fourier features.
result Competitive performance on synthetic and real-world data.
Simple stochastic Newton and cubic Newton methods with fast convergence.
problem Minimizing large numbers of smooth and strongly convex functions.
method Stochastic Newton and cubic Newton methods with simple local linear-quadratic rates.
result Local linear-quadratic convergence results with fast adaptation to problem's curvature.
Improved spectral methods of moments for robust latent variable model learning.
problem Limited robustness of spectral methods of moments to model misspecification.
method Hierarchical approach using approximate joint diagonalization instead of tensor decomposition.
result Our method outperforms previous tensor decomposition methods in speed and model quality.
A comprehensive benchmark of 15 scRNA-seq imputation methods across various datasets and analyses.
problem Imputation of single-cell RNA sequencing data to recover latent transcriptional signals.
method Evaluation of 15 imputation methods across 30 datasets and 6 downstream analyses.
result Traditional methods generally outperform DL-based methods in scRNA-seq data analysis.
New methods using natural gradient for structured optimization.
problem Structured optimization problems.
method Structured second-order methods via natural gradient descent.
result Efficiency demonstrated on non-convex and deep learning problems.
Improved A2C method with lower variance.
problem Reducing variance in deep policy gradient methods.
method Using control variate theory, derived a new A2C formulation with lower variance.
result New A2C method has lower variance and improved performance.
Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of t…
A new method speeds up deep neural network training.
problem Nonconvex optimization in deep neural networks.
method Scaled conjugate gradient method for nonconvex optimization.
result The method converges faster and achieves lower scores in practical applications.
We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on Alternating Direction Multiplier Method (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch…
NCG methods improve shape optimization efficiency.
problem Shape optimization problems
method Nonlinear conjugate gradient methods
result NCG methods are efficient for shape optimization
Geometric methods study 3-manifold splittings.
problem Studying Heegaard splittings of 3-manifolds.
method Geometric approaches.
result Recent advances in geometric methods.
We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing problems arising from market models with multiple stochastic factors. We demons…
Proposes UTC method for stock price prediction with uncertainty quantification.
problem Lack of uncertainty estimates in stock prediction methods.
method Combines TC method with probabilistic modeling for point and uncertainty predictions.
result UTC method achieves higher returns and lower risks than baselines.
Various approaches to gene selection for cancer classification based on microarray data can be found in the literature and they may be grouped into two categories: univariate methods and multivariate methods. Univariate methods look at each gene in the data in isolation from others. They measure the contribution of a p…
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
A novel weighted feature selection method using fuzzy sets improves classification accuracy and stability.
problem Improving feature selection accuracy and stability in machine learning models.
method Combination of four feature selection methods using fuzzy sets and bootstrap.
result Our method achieved significantly higher stability than individual methods.
New method improves accuracy in computing implied volatility.
problem Computing implied volatility from the Black-Scholes model.
method Adaptive gradient descent optimizers for numerical computation.
result More accurate results compared to close form approximation and Newton-Raphson method.