Related papers: Translating Annuli for Mean Curvature Flow

In the paper we consider Riemannian surfaces admitting a global expression of the Gauss curvature as the divergence of a vector field. It is equivalent to the existence of a metric linear connection of zero curvature. Such a linear…

We study closed, embedded hypersurfaces in Euclidean space evolving by fully nonlinear curvature flows, whose speed is given by a symmetric, monotone increasing, $1$-homogeneous, positive underlying speed function $F$ composed with a…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this…

We show that the standard discrete update rule of transformer layers can be naturally interpreted as a forward Euler discretization of a continuous dynamical system. Our Transformer Flow Approximation Theorem demonstrates that, under…

Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of…

We study the capabilities of the transformer architecture with varying depth. Specifically, we designed a novel set of sequence learning tasks to systematically evaluate and comprehend how the depth of transformer affects its ability to…

Given a unit vector $\textbf{v}\in\mathbb{R}^3$ and $\lambda\in\mathbb{R}$, a translating $\lambda$-soliton is a surface in $\mathbb{R}^3$ whose mean curvature $H$ satisfies $H=\langle N,\textbf{v}\rangle+\lambda,\ |\textbf{v}|=1$, where…

All wavelets can be associated to a multiresolution like structure, i.e. an incr easing sequence of subspaces of L^2(R). We consider the interaction of a wavel et and the translation operator in terms of which of the subspaces in this multi…

In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.

Traditional neural machine translation is limited to the topmost encoder layer's context representation and cannot directly perceive the lower encoder layers. Existing solutions usually rely on the adjustment of network architecture, making…

Suppose $f\in L^1(\mathbb{R}^d)$, $\Lambda\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles,…

It has been over six years since the Transformer architecture was put forward. Surprisingly, the vanilla Transformer architecture is still widely used today. One reason is that the lack of deep understanding and comprehensive interpretation…

The transformer is a state-of-the-art neural translation model that uses attention to iteratively refine lexical representations with information drawn from the surrounding context. Lexical features are fed into the first layer and…

We demonstrate a compactness result holding broadly across supervised learning with a general class of loss functions: Any hypothesis class $H$ is learnable with transductive sample complexity $m$ precisely when all of its finite…

Large Transformer models yield impressive results on many tasks, but are expensive to train, or even fine-tune, and so slow at decoding that their use and study becomes out of reach. We address this problem by leveraging sparsity. We study…

Rotationally symmetric bodies with longitudinal cross sections of parabolic shape are frequently used to model astrophysical objects, such as magnetospheres and other blunt objects, immersed in interplanetary or interstellar gas or plasma…

Most recent work on interpretability of complex machine learning models has focused on estimating $\textit{a posteriori}$ explanations for previously trained models around specific predictions. $\textit{Self-explaining}$ models where…

We study the stable translation length of homeomorphisms of a surface acting on the fine nonseparating curve graph and compare it to the stable translation lengths of its finite approximations - mapping classes relative to a finite…

The Finsleroid-Finsler space is constructed over an underlying Riemannian space by the help of a scalar $g(x)$ and an input 1-form $b$ of unit length. Explicit form of the entailed tensors, as well as the respective spray coefficients, is…

The incompressible boundary layer in the axial flow past a cylinder has been shown Tutty et. al.(\cite{tutty}) to be stabler than a two-dimensional boundary layer, with the helical mode being the least stable. In this paper the secondary…