arXiv research

The paper examines translating solitons and their relation to Lagrangian mean curvature flows with zero Maslov class.

We show the existence of constant mean curvature surfaces in the homology classes of closed 3-manifolds.

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

This note finds closed-form solutions for mean-risk portfolios using a specific type of mixture distribution.

Proves optimal regularity for sphere minimizers in 3-sphere.

This paper extends neural collapse to class-imbalanced datasets using an unconstrained ReLU feature model.

In recent years, more machine learning algorithms have been applied to odor classification. These odor classification algorithms usually assume that the training datasets are static. However, for some odor recognition tasks, new odor classes continually emerge. That is, the odor datasets are dynamically growing while b…

Local classification of surfaces and hypersurfaces with radial mean curvature.

The Matérn covariance function is a popular choice for prediction in spatial statistics and uncertainty quantification literature. A key benefit of the Matérn class is that it is possible to get precise control over the degree of mean-square differentiability of the random process. However, the Matérn class possesses e…

Study finds polynomial convergence rate for Farey sequences linked to Riemann hypothesis.

Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.

A mean function in reproducing kernel Hilbert space, or a kernel mean, is an important part of many applications ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given finite samples, an empirical average is the standard estimate for the true kernel mean. We show that this e…

Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.

E$^2$CM uses class means for efficient early exits in neural networks.

New findings extend rigidity results to broader classes of manifolds.

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature…

Generalizing the construction of the Maslov class for a Lagrangian embedding in a symplectic vector space, we prove that it is possible to give a consistent definition of this class for any Lagrangian submanifold of a Calabi-Yau manifold. Moreover, we prove that this class can be represented by the contraction of the K…

Deep nets exhibit 'Neural Collapse' during training's final phase, simplifying decision-making.

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier conditi…

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

The paper bounds the $L^2$-norm of Euler class for foliations on 3-manifolds.

We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for the ambient space, there exists a large class of closed initial hypersurfaces, …

We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a cert…

In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a Kähler-Einstein manifold. An immediate consequence is that in Kähler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.

New quantity helps map homotopy classes in complex spaces.

Defines CAMC discrete nets and their properties.

Paper establishes sufficient condition for comparing linear combinations of infinite-mean risks.

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

Unified framework for shrinkage, thresholding, and regularization in normal mean estimation and linear regression.

This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.

A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is proved.

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…

Study minimizes Willmore energy with constraints on surface properties.

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant co…

Residual flows are shown to approximate MMD well.

Study timelike meridian surfaces in Minkowski 4-space with specific properties.

We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.

We prove the existence of a new class of constant mean curvature cylinders with an arbitrary number of umbilics by unitarizing the monodromy of Hill's equation.

Paper extends stochastic dominance for compound binomial distributions.

Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.

We investigate a Gaussian mixture model (GMM) with component means constrained in a pre-selected subspace. Applications to classification and clustering are explored. An EM-type estimation algorithm is derived. We prove that the subspace containing the component means of a GMM with a common covariance matrix also conta…

In this paper, we presented a novel semi-supervised one-class classification algorithm which assumes that class is linearly separable from other elements. We proved theoretically that class is linearly separable if and only if it is maximal by probability within the sets with the same mean. Furthermore, we presented an…

Learning from class-imbalanced data continues to be a common and challenging problem in supervised learning as standard classification algorithms are designed to handle balanced class distributions. While different strategies exist to tackle this problem, methods which generate artificial data to achieve a balanced cla…

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and…

New, algebraic surfaces found in curved spaces.

In this paper we solve the Björling problem for the class of immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed function…

In this paper, we find a condition on $(α, β)$-metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.