arXiv research

Classifies Lie algebras and related spacetimes for a specific type of symmetry.

The Landau-Lifshitz equation is derived as the reduction of a geodesic flow on the group of maps into the rotation group. Passing the symmetries of spatial isotropy to the reduced space is an example of semidirect product reduction by stages.

We argue that Horava-Lifshitz (HL) gravity provides the minimal holographic dual for Lifshitz-type field theories with anisotropic scaling and dynamical exponent z. First we show that Lifshitz spacetimes are vacuum solutions of HL gravity, without need for additional matter. Then we perform holographic renormalization …

In this paper, we bring in General Landau-Lifshitz-Bloch equation and prove that it admits a local strong solution.

By using the geometric concept of PDEs with prescribed curvature representations, we show that the 1+2 dimensional Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up $H^3({\bf R}^2)$-solutions to the 1+…

Tropical geometry aids in computing topological quantum field theories.

We show that holographic renormalization of relativistic gravity in asymptotically Lifshitz spacetimes naturally reproduces the structure of gravity with anisotropic scaling: The holographic counterterms induced near anisotropic infinity take the form of the action for gravity at a Lifshitz point, with the appropriate …

We devise an algorithm which allows one to count the number of Killing vectors for a Lorentzian manifold of dimension 3. Our algorithm relies on the principal traces of powers of the Ricci tensor and branches intricately according to the values of differential invariants arising from the compatibility conditions of the…

In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into $S^n$ is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…

We generalize Penrose's notion of conformal infinity of spacetime, to situations with anisotropic scaling. This is relevant not only for Lifshitz-type anisotropic gravity models, but also in standard general relativity and string theory, for spacetimes exhibiting a natural asymptotic anisotropy. Examples include the Li…

New topological quantum gravity theories linked to Ricci flow.

We consider instanton solutions of Euclidean Horava-Lifshitz gravity in four dimensions satisfying the detailed balance condition. They are described by geometric flows in three dimensions driven by certain combinations of the Cotton and Ricci tensors as well as the cosmological-constant term. The deformation curvature…

We study long wave limits for general Schrodinger maps systems into Kahler manifolds with a constraining potential vanishing on a Lagrangian submanifold. We obtain KdV type systems set on the tangent space of the submanifold. Our general theory is applied to study the long wave limit of the Gross-Pitaevskii equation, a…

Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.

We study the existence problem of harmonic maps with potential from $\mathbb{R}^2$ into $S^2$. For a specific class of potential functions on $S^2$, we give the sufficient and necessary conditions for the existence of equivariant solutions of this problem. As an application, we generalize and improve the results on the…

We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on u, u_x, u_{xx}, u_{xxx} for the Krichever-Novikov equation u_t=u_{xxx}-3u_{xx}^2/(2u_x)+p(u)/u_x+au_x in the case when the polynomial p(u)=4u^3-g_2u-g_3 has distinct roots. We prove that there is a universal prolongation algebra…

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the wor…

The paper analyzes symmetries of Vaidya-Bonner geodesics.

New method detects symmetries beyond affine transformations.

Humans take advantage of real world symmetries for various tasks, yet capturing their superb symmetry perception mechanism with a computational model remains elusive. Motivated by a new study demonstrating the extremely high inter-person accuracy of human perceived symmetries in the wild, we have constructed the first …

Classifies symmetries of non-flat 3-webs around a point.

Geometric mechanism mimics physics' symmetry breaking.

Method improves deep learning models for datasets with mixed approximate symmetries.

Symmetry in loss functions constrains model parameters, leading to specific learning outcomes.

Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…

New symmetry dimensions for higher order ODEs are identified.

Generalizes symmetries of curved manifolds.

Study of continuous symmetries in Nahm data and BPS monopoles.

New framework discovers non-affine continuous symmetries in neural networks.

Researchers discover symmetries in Ricci flows and use them to find invariant solutions.

This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of…

New symmetry found in colored Alexander polynomial.

Symmetry in finance is a neglected but potentially valuable concept.

Symmetry of neural network densities can be determined from correlation functions.

Symmetries in shrinking Ricci solitons spread outward.

Clarifies relation between Pfaffian fibrations and relative algebroids.

Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.

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

Noether's theorem clarifies how symmetries in neural networks influence learning.

Researchers create BPS monopoles with any desired symmetry breaking.

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

SymPE breaks symmetries in equivariant networks, improving performance across various tasks.

Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.

We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in …

The study examines symmetries in spaces with positive or non-negative curvature.

Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.

We propose to impose symmetry in neural network parameters to improve parameter usage and make use of dedicated convolution and matrix multiplication routines. Due to significant reduction in the number of parameters as a result of the symmetry constraints, one would expect a dramatic drop in accuracy. Surprisingly, we…

Study on symmetries of differential equations using gauge transformations and coverings.