arXiv research

We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dy…

Study contact geometry of energy hypersurface in symmetric 3-body problem on S^2.

We will show that the period $T$ of a closed orbit of the planar circular restricted three-body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, $2 T=kπ+\int_Ωg$ where $k$ is an integer, $Ω$ is the region enclosed by the periodic orbit and $g:\mathbb{R}^2\to \m…

The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The appro…

We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a three-body problem in the Euclidean plane. The potential is $-I/Δ^2$ where $I$ is the triangle's moment of inertia and $Δ$ its area. The reduction method uses the Jacobi-Maupertuis metric, following the author's earlier p…

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce inv…

We show that any bounded zero-angular momentum solution for the Newtonian three-body problem must suffer infinitely many eclipses, or collinearities, provided that it does not suffer a triple collision. Motivation for the result comes from the dream of building a symbolic dynamics for the three-body problem, one whose …

We investigate the Cartan and Finsler geometry of the rotating Kepler problem, a limit case of the restricted three body problem that arises if the mass of the one of the primaries goes to zero. We show that the Hamiltonian for the rotating Kepler problem can be regarded as the Legendre transform of a certain family of…

Novel methods for splitting Gaussian mixtures improve uncertainty propagation in nonlinear systems.

Geometric reduction of the Newtonian planar three-body problem is investigated in the framework of equivariant Riemannian geometry, which reduces the study of trajectories of three-body motions to the study of their moduli curves, that is, curves which record the change of size and shape, in the moduli space of oriente…

This monograph describes a Riemannian geometric reduction approach to the three-body problem. The fundamental theorems are presented in the introductory part, whereas their proofs are provided in later chapters where specific topics are analyzed in more detail. The basic idea is to reduce the kinematic and dynamics of …

Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The formula is based on the fact that the space of similarity classes of triangles …

In this paper we describe a 1-dimensional family of initial conditions Σthat provides reduced periodic solution of the three body problem. This family Σcontains a bifurcation point and extend the periodic solution described in (Perdomo, http://arxiv.org/pdf/1507.01100.pdf). This 1-dimensional family is the union of two…

Investigates the rotating Kepler problem for energy values ≤ -3/2.

In this paper we define a small variation of the Taylor method and a formula for the global error of this new numerical method that allows us to keep track of the round-off error and does not require previous knowledge of the exact solution. As an application we provide a rigorous proof of the construction/existence of…

The study of Reeb dynamics on contact manifolds without periodic orbits.

New method for studying $t$-dependent Hamilton equations on cosymplectic manifolds.

We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules. In particular, we define two classes of SympNets: the LA-SympNets composed of linear and activation modules, and the G-SympNets composed of gradient modules. Cor…

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. Our goal here is to explain how analysis of the Laplacian on the globally symmetric space $\SL(3,\RR)/\…

New weighted surface area measures for convex bodies with applications.

New periodic solution found in 4-body problem, not part of expected geometrical family.

ACA method improves gradient estimation for neural ODEs, reducing error and training time.

New method improves estimation of complex models from conditional moment restrictions.

In this study, importance of user inputs is studied in the context of personalizing human activity recognition models using incremental learning. Inertial sensor data from three body positions are used, and the classification is based on Learn++ ensemble method. Three different approaches to update models are compared:…

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…

The study restricts groups in graph of groups structures.

Study finds periodic orbits in a complex gravitational system.

Study expands classical harmonic function results to Riemannian manifolds.

Two strategies for embedding new data points from proximity data are explored.

We study the model of interacting agents proposed by Chatterjee et al that allows agents to both save and exchange wealth. Closed equations for the wealth distribution are developed using a mean field approximation. We show that when all agents have the same fixed savings propensity, subject to certain well defined app…

Sparse Polyak improves high-dimensional statistical estimation.

We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression, and constrained single index model. We review some of the theoretical properties of the least …

Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.

We consider a novel formulation of the multi-armed bandit model, which we call the contextual bandit with restricted context, where only a limited number of features can be accessed by the learner at every iteration. This novel formulation is motivated by different online problems arising in clinical trials, recommende…

In this work an iterative algorithm based on unsupervised learning is presented, specifically on a Restricted Boltzmann Machine (RBM) to solve a perfect matching problem on a bipartite weighted graph. Iteratively is calculated the weights $w_{ij}$ and the bias parameters $θ= ( a_i, b_j)$ that maximize the energy funct…

Generalization is one of the most important issues in machine learning problems. In this study, we consider generalization in restricted Boltzmann machines (RBMs). We propose an RBM with multivalued hidden variables, which is a simple extension of conventional RBMs. We demonstrate that the proposed model is better than…

We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optim…

Active-set algorithm improves Cox regression for shape-restricted covariates.

Study restricts causal graphs with expert knowledge.

Can one reduce the size of a graph without significantly altering its basic properties? The graph reduction problem is hereby approached from the perspective of restricted spectral approximation, a modification of the spectral similarity measure used for graph sparsification. This choice is motivated by the observation…

We introduce techniques for exploring the functionality of a neural network and extracting simple, human-readable approximations to its performance. By performing gradient ascent on the input space of the network, we are able to produce large populations of artificial events which strongly excite a given classifier. By…

Given a finite set of unknown distributions or arms that can be sampled, we consider the problem of identifying the one with the maximum mean using a $δ$-correct algorithm (an adaptive, sequential algorithm that restricts the probability of error to a specified $δ$) that has minimum sample complexity. Lower bounds for …

Hardware-accelerated RBM solves large combinatorial problems and integer factorization.

Method predicts RMST from censored data using pseudo-observations and super learner.

SAGA is a fast incremental gradient method on the finite sum problem and its effectiveness has been tested on a vast of applications. In this paper, we analyze SAGA on a class of non-strongly convex and non-convex statistical problem such as Lasso, group Lasso, Logistic regression with $\ell_1$ regularization, linear r…

We consider the problem of classification when inputs correspond to sets of vectors. This setting occurs in many problems such as the classification of pieces of mail containing several pages, of web sites with several sections or of images that have been pre-segmented into smaller regions. We propose generalizations o…

In his book Mickelsson notices that the infinite-dimensional Grassmannian manifold of Segal and Wilson admits a Spin^c structure and after this he naturally considers the problem of defining a Dirac operator on it. Mickelsson gives a possible candidate for such an operator but unfortunately it proves out to be badly di…

Despite their exceptional flexibility and popularity, the Monte Carlo methods often suffer from slow mixing times for challenging statistical physics problems. We present a general strategy to overcome this difficulty by adopting ideas and techniques from the machine learning community. We fit the unnormalized probabil…