arXiv research

New satellite knots counter a conjecture about Lorenz knots.

We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots code…

This is a review article on Lorenz knots.

This article is a survey on Lorenz knots. We describe the original construction, prove several classical properties, in particular the fact that the closure of a positive braid is a fibered knot, and describe Ghys'correspondance between modular knots and Lorenz knots. We also prove two new properties, namely that follo…

New satellite knots found that can't be represented by positive braids with full twists.

We exhibit low-dilatation families of surface homeomorphisms among monodromies of Lorenz knots.

The unknotting number of a positive braid with n strands and k intersections is known to be equal to (k-n+1)/2. We consider Lorenz knots (which are positive braids) and, using a different method, find their unknotting numbers in terms of their positions on the Lorenz attractor.

Computed linking number of modular knots and Lorenz links.

We show that the zeroes of the Alexander polynomial of a Lorenz knot all lie in some annulus whose width depends explicitly on the genus and the braid index of the considered knot.

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit form…

New findings on T-links derived from torus links.

Improved algorithm for modular links provides upper volume bounds.

Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their …

Paper simplifies link classification in 3-sphere using braids and templates.

We describe the Williams zeta functions and the twist zeta functions of sub-Lorenz templates generated by renormalizable Lorenz maps, in terms of the corresponding zeta-functions of the sub-Lorenz templates generated by the renormalized map and by the map that determines the renormalization type.

Study Alexander polynomials of modular knots, revealing finite and infinite coefficient properties.

Study shows volumes of knot complements are bounded by linear functions of geodesic periods.

Study of Lorenz links and T-links, showing equivalence and unique presentations.

The study connects twist positivity to L-space knots and concordance.

In paper "A new twist on Lorenz links" (Journal of Topology 2(2009), 227-248) Joan Birman and Ilya Kofman prove the coincidence of the class of Lorenz links and the class of twisted links. The proof in that work is algebraic. We will identify this class in terms of grid diagrams and provide a transparent geometric argu…

In this article the Lorenz dynamical system is revived and revisited and the current state of the art results for one step ahead forecasting for the Lorenz trajectories are published. Multitask learning is shown to help learning the hard to learn z trajectory. The article is a reflection upon the evolution of neural ne…

During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic p…

The fact that the modular template coincides with the Lorenz template, discovered by Ghys, implies modular knots have very peculiar properties. We obtain a generalization of these results to other Hecke triangle groups. In this context, the geodesic flow can never be seen as a flow on a subset of $S^3$, and one is led …

Proves stability of Minkowski space-time in Einstein-Yang-Mills system.

The integrability of the geodesic flow on the three-folds $\mathcal M^3$ admitting $SL(2,\mathbb R)$-geometry in Thurston's sense is investigated. The main examples are the quotients $\mathcal M^3_Γ=Γ\backslash PSL(2,\mathbb R)$, where $Γ\subset PSL(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the correspon…

The aim of this paper is to construct natural geometrical objects on the 1-jet space J^1(T,R^5), where $T/subset R$, like a non-linear connection, a generalized Cartan connection, together with its d-torsions and d-curvatures, a jet electromagnetic d-field and a jet Yang-Mills energy, starting from the given Lorenz atm…

Using tax and census data, we demonstrate that the distribution of individual income in the USA is exponential. Our calculated Lorenz curve without fitting parameters and Gini coefficient 1/2 agree well with the data. From the individual income distribution, we derive the distribution function of income for families wi…

Unified four trade-off curves for assessing generative model proximity.

VSE estimates complex processes from noisy measurements without a model.

Stability of Minkowski space-time in higher dimensions proven for arbitrary small perturbations.

Stochastic parameterizations account for uncertainty in the representation of unresolved sub-grid processes by sampling from the distribution of possible sub-grid forcings. Some existing stochastic parameterizations utilize data-driven approaches to characterize uncertainty, but these approaches require significant str…

We consider concepts and models for measuring inequality in the distribution of resources with a focus on how inequality varies as a function of covariates. Lorenz introduced a device for measuring inequality in the distribution of income that indicates how much the incomes below the u$^{th}$ quantile fall short of the…

Physics-constrained neural nets solve EM fields of charged particle beams.

Climate projections suffer from uncertain equilibrium climate sensitivity. The reason behind this uncertainty is the resolution of global climate models, which is too coarse to resolve key processes such as clouds and convection. These processes are approximated using heuristics in a process called parameterization. Th…

DD-SP uses ML to improve SP for Lorenz 96 systems, outperforming LR and DD-P.

In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We also analyze some natural Morse decompositions of the global attractor of the syst…

Combining LETKF and RC improves chaotic system prediction from noisy, sparse data.

EnEMF uses Epanechnikov kernel for high-dimensional filtering, improving accuracy and robustness.

In this paper, the performance of three deep learning methods for predicting short-term evolution and for reproducing the long-term statistics of a multi-scale spatio-temporal Lorenz 96 system is examined. The methods are: echo state network (a type of reservoir computing, RC-ESN), deep feed-forward artificial neural n…

We study the performance of sparse regression methods and propose new techniques to distill the governing equations of dynamical systems from data. We first look at the generic methodology of learning interpretable equation forms from data, proposed by Brunton et al., followed by performance of LASSO for this purpose. …

New method calibrates predictions in chaotic systems using variational inference.

Proposes a new method for data assimilation using closed-form conditional diffusion models.

A new ML-based filter improves data assimilation for nonlinear systems.

The paper classifies hyperbolic and satellite T-links formed by twisting.

Analog methods improve forecast accuracy in complex models.

CW-EDMD improves prediction accuracy by learning local Koopman models for different state-space regions.

Stability of Minkowski space-time in Einstein-Yang-Mills system proven.