arXiv research

Randomly initialized wide neural networks with zero-mean activations are nearly independent, potentially solving AI interpretability limits.

New technique trains deep neural networks without normalization or minibatch statistics.

Study duality of zero mean curvature surfaces in Heisenberg group.

Classifies surfaces with zero mean curvature in a light cone.

Solves surface problem in 3D light cone.

Study classifies zero mean curvature surfaces with planar curvature lines.

Paper investigates reflection principles for zero mean curvature surfaces in isotropic 3-space.

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this …

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found …

New framework for zero mean curvature surfaces in isotropic 3-space.

Classifies zero mean curvature surfaces in Lorentz-Minkowski space.

New examples of mixed-type zero-curvature graphs found.

Riemann zero mean curvature examples in the Lorentz-Minkowski space are surfaces with zero mean curvature foliated by circles contained in parallel planes. In contrast to the Euclidean case, this family of surfaces presents new and rich features because of the variety of types of circles. In this paper, we give a geome…

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

Study learns dynamics of linear systems from multiple short trajectories.

New, algebraic surfaces found in curved spaces.

The paper shows how Scherk-type surfaces can be decomposed into helicoids.

In this work, we study a class of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}_2^4$ whose profile curves lie in two-dimensional planes. We solve the differential equation that characterizes the rotational surfaces with zero mean curvature to determine the profile curves of such rotational surfaces. The…

We construct triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space, with the same topology as the triply periodic minimal surfaces in the Euclidean 3-space, called Schwarz rPD surfaces.

We construct embedded triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space with the same topology as the Schwarz D surface in the Euclidean 3-space.

Formula for spacelike submanifolds in warped products.

New method for initializing RBM weights without datasets.

Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space $\boldsymbol L^3$ which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space $\boldsymbol E^3$ and maximal surfaces in Lorentz-Minko…

Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like poi…

Study maxfaces and minfaces converging to surfaces with folded singularities.

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been…

Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.

In this paper we investigate relations between solutions to the minimal surface equation in Euclidean $3$-space $\mathbb{E}^3$, the zero mean curvature equation in Lorentz-Minkowski $3$-space $\mathbb{L}^3$ and the Born-Infeld equation under Wick rotations. We prove that the existence conditions of real solutions and i…

Study on surfaces in neutral space forms with zero mean curvature.

The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied w…

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter $N$-dimensional s…

Ordinary stochastic neural networks mostly rely on the expected values of their weights to make predictions, whereas the induced noise is mostly used to capture the uncertainty, prevent overfitting and slightly boost the performance through test-time averaging. In this paper, we introduce variance layers, a different k…

This article addresses the modeling of reverberant recording environments in the context of under-determined convolutive blind source separation. We model the contribution of each source to all mixture channels in the time-frequency domain as a zero-mean Gaussian random variable whose covariance encodes the spatial cha…

Paper relaxes independence assumption for non-centered data.

We consider the optimization of a quadratic objective function whose gradients are only accessible through a stochastic oracle that returns the gradient at any given point plus a zero-mean finite variance random error. We present the first algorithm that achieves jointly the optimal prediction error rates for least-squ…

Study Born-Infeld solitons and solve Björling problem for them.

With several concrete examples of zero mean curvature surfaces in $\boldsymbol{R}^3_1$ containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first …

The paper proves a Moser-Trudinger inequality for zero-mean functions in 2D.

Linear Transformer Block combines MLP and linear attention for near-optimal ICL in linear regression.

New connections found on zero-mean multivariate normal distributions.

Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.

We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.

Improved bound for Gaussian mechanism in differential privacy.

Unbounded convex domains have zero mean curvature on disconnected boundaries.

Deep Learning has revolutionized vision via convolutional neural networks (CNNs) and natural language processing via recurrent neural networks (RNNs). However, success stories of Deep Learning with standard feed-forward neural networks (FNNs) are rare. FNNs that perform well are typically shallow and, therefore cannot …

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-spac…

Batch normalization (BN) is a technique to normalize activations in intermediate layers of deep neural networks. Its tendency to improve accuracy and speed up training have established BN as a favorite technique in deep learning. Yet, despite its enormous success, there remains little consensus on the exact reason and …