arXiv research

The paper shows translating solitons in $\mathbb{R}^4$ have $SO(2)$ symmetry.

New proof shows symmetry for certain curved surfaces in higher dimensions.

We prove, in all dimensions $n\geq 2$, that there exists a convex translator lying in a slab of width $π\secθ$ in $\mathbb{R}^{n+1}$ (and in no smaller slab) if and only if $θ\in[0,\fracπ{2}]$. We also obtain convexity and regularity results for translators which admit appropriate symmetries and study the asymptotics a…

The symmetries of paths in a manifold $M$ are classified with respect to a given pointwise proper action of a Lie group $G$ on $M$. Here, paths are embeddings of a compact interval into $M$. There are at least two types of symmetries: Firstly, paths that are parts of an integral curve of a fundamental vector field on $…

Constructing solutions to geometric flows with rotational symmetry.

Classification of groups as symmetries of infinite translation surfaces.

The paper classifies and describes translators in $SL(2,\mathbb{R})$ under specific symmetry conditions.

Book explores infinite translation surfaces, challenging traditional geometry.

Wavelet Networks learn from raw time-series data, outperforming conventional CNNs.

New methods prove existence of rotating shapes moving in space.

New Lagrangian and special Lagrangian examples found in complex space.

In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in $\mathbb{R}^{m+1}$. We also obtain an upper bound to the maximum height that a compact embedded translator in $\mathbb{R}^{3}$ can achieve. On the other hand, we study graphical perturbation…

The height functions of K^(1/4)-flow translators in Euclidean space R^3 solve the unimodular Hessian equation. We explicitly and geometrically determine the moduli space of all helicoidal K^(1/4)-flow translators, which are generated from planar curves by the action of helicoidal groups.

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one full turn of the …

New translation equivariant neural processes improve spatio-temporal data modeling.

For the class of systems of PDEs, for which infinitesimal translations (with respect to some (in)dependent variables) possess specific finite-dimensional invariant subspaces of the space of generalized symmetries of the system considered. We establish when there exist generalized symmetries from these subspaces, which …

Classifies and constructs translators for curvature flows.

This work relaxes GNN symmetries to approximate automorphisms, improving model performance.

We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé's system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dep…

The decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger type theorem for continuous translation invariant and SO(n)-equivariant tensor valuations is also given. As an application, symme…

Study shows instability of certain MOTSs with continuous symmetry.

Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant variables) on specific finite-dimensional subspaces of the space of generalized symmetries of some system of partial differential equations, we explicitly determine the dependance of coefficients of generalized symmetries fro…

Invariances to translation, rotation and other spatial transformations are a hallmark of the laws of motion, and have widespread use in the natural sciences to reduce the dimensionality of systems of equations. In supervised learning, such as in image classification tasks, rotation, translation and scale invariances ar…

Paper defines new topological invariants for DP tangles.

Study invariant $λ$-translators in Lorentz-Minkowski space.

The paper proves nonexistence results for translating solitons in r-mean curvature flow.

We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…

The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the intera…

Probabilistic models often have parameters that can be translated, scaled, permuted, or otherwise transformed without changing the model. These symmetries can lead to strong correlation and multimodality in the posterior distribution over the model's parameters, which can pose challenges both for performing inference a…

We prove that any holomorphic locally homogeneous geometric structure on a complex torus, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true is any dimension. In higher dimension we prove it here for nilpotent models. We also prove that in any dimension the trans…

Paper analyzes CycleGAN solutions and symmetries.

A $λ$-translating soliton with density vector $\vec{v}$ is a surface $Σ$ in Euclidean space ${\mathbb R}^3$ whose mean curvature $H$ satisfies $2H=2λ+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map of $Σ$. In this article we study the shape of a compact $λ$-translating soliton in terms of its boundary. If $Γ$ is …

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that result from focusing on properties of the square torus one by one. Af…

L-CNNs maintain gauge symmetry on non-Abelian lattice theories.

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, w…

This paper aims to incorporate passive symmetries in machine learning for better generalization.

Leveraging the intrinsic symmetries in data for clear and efficient analysis is an important theme in signal processing and other data-driven sciences. A basic example of this is the ubiquity of the discrete Fourier transform which arises from translational symmetry (i.e. time-delay/phase-shift). Particularly important…

Proposes learning invariances in neural networks using a weight-space approach.

Stability of catenoid in hyperbolic space proven without symmetry assumptions.

The study constructs minimal surfaces in a product space with specific properties.

A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwi…

New mesh network preserves symmetries in deep learning.

Recent machine learning methods make it possible to model potential energy of atomic configurations with chemical-level accuracy (as calculated from ab-initio calculations) and at speeds suitable for molecular dynam- ics simulation. Best performance is achieved when the known physical constraints are encoded in the mac…

Many loss functions in representation learning are invariant under a continuous symmetry transformation. For example, the loss function of word embeddings (Mikolov et al., 2013) remains unchanged if we simultaneously rotate all word and context embedding vectors. We show that representation learning models for time ser…

This work extends PAC-Bayesian learning guarantees to non-compact symmetries and non-invariant data.

The effectiveness of Convolutional Neural Networks stems in large part from their ability to exploit the translation invariance that is inherent in many learning problems. Recently, it was shown that CNNs can exploit other invariances, such as rotation invariance, by using group convolutions instead of planar convoluti…

Study connects symmetries in dynamical systems to phase plane representations.

The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.