arXiv research

The paper investigates how class mean vectors enhance neural network classification.

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with parallel mean c…

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

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…

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curva…

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature an…

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

The study classifies timelike meridian surfaces in Minkowski 4-space.

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.

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.

An odd vector field $Q$ on a supermanifold $M$ is called homological, if $Q^2=0$. The operator of Lie derivative $L_Q$ makes the algebra of smooth tensor fields on $M$ into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called character…

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

Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.

The study finds larger gaps in mean curvature for biharmonic submanifolds in spheres.

Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.

We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal bundles that preserves the mean curvature vector fields. We prove that if both Gaus…

This paper extends neural collapse to imbalanced data under cross-entropy loss.

The paper studies special surfaces in 4D space forms with specific geometric properties.

The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting…

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the …

Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.

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

New results on non-existence and rigidity of spacelike submanifolds in spacetimes.

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we su…

Following widely used in visual recognition concept of relative attributes, the article establishes definition of the relative PCA attributes for a class of objects defined by vectors of their parameters. A new rating model (RELARM) is built using relative PCA attribute ranking functions for rating object description a…

We study the class of spacelike surfaces in the four-dimensional Minkowski space whose mean curvature vector at any point is a non-zero spacelike vector or timelike vector. These surfaces are determined up to a motion by eight invariant functions satisfying some natural conditions. The subclass of Chen surfaces is char…

This paper presents a general coding method where data in a Hilbert space are represented by finite dimensional coding vectors. The method is based on empirical risk minimization within a certain class of linear operators, which map the set of coding vectors to the Hilbert space. Two results bounding the expected recon…

Paper establishes inequality for submanifolds in real space forms with semi-symmetric non-metric connection.

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres w…

Estimates mean of random vector with near-optimal error in all directions.

We study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces whose constant mean curvature is non-zero and also that every compact maximal hyper…

The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.

Estimates mean of distributed vectors with sparsification and spatial/temporal correlations.

We develop a unified approach for classification and regression support vector machines for data subject to right censoring. We provide finite sample bounds on the generalization error of the algorithm, prove risk consistency for a wide class of probability measures, and study the associated learning rates. We apply th…

We develop time-uniform confidence spheres for estimating means of random vectors.

Deep linear networks exhibit collapsing features and classifiers across datasets.

In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition …

We investigate the local geometry of a class of Kähler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $α$, which we call the pluri-mean curvature.…

We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an example of a vector field of constant mean curvature on the Lobachevsky $(n+1)$ s…

Generalizes Reilly inequality to varifolds and analyzes equality cases.

We give a complete description of all hypersurfaces of the product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\R^{n+2}\supset \Sf^n\times \R$ and $\Le^{n+2}\supset \Hy^n\times \R$. We prove that any such hyp…

Proves existence of graph on torus with prescribed curvature.

The paper bounds the mean absolute error in DNN vector-to-vector regression.

We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almos…

This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow e…

Decomposes submanifolds with special tensors into simpler parts.

We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their position and tangent vectors. Among this family, we characterize minimal, constant m…

We give a conformal representation in terms of meromorphic data for a certain class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean curvature vector is either lightlike or zero at each point. This representation extends simultaneously the Weierstrass representation for minimal surfaces in Euclidea…