The period of orbits in the restricted three-body problem depends on the enclosed region.
problem Understanding the period of orbits in the restricted three-body problem.
method Analyzing the relationship between the period and the enclosed region using the Jacobian integral.
result The period of a closed orbit is determined by the enclosed region and a function of the Jacobian integral.
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.
problem Contact geometry of energy hypersurface in symmetric 3-body problem on S^2.
method Contact geometry analysis, Moser regularization, Weinstein conjecture.
result Components of energy hypersurface are contactomorphic to R^3 or its connected sum for specific energies.
The hyperbolic plane is derived from a three-body problem in Euclidean space.
problem Constructing the hyperbolic plane from a three-body problem.
method Scale plus symmetry reduction of a three-body problem in Euclidean plane using Jacobi-Maupertuis metric.
result The hyperbolic plane and its geodesic flow are derived from a three-body problem.
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 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…
Study periodic orbits in Stark-Zeeman systems using Arnold's J+-invariant.
problem Understanding periodic orbits in Stark-Zeeman systems.
method Apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems and introduce invariants of periodic orbits based on Arnold's J+-invariant. result Study the behavior of invariants of periodic orbits in planar Stark-Zeeman systems.
Novel methods for splitting Gaussian mixtures improve uncertainty propagation in nonlinear systems.
problem Improving accuracy and efficiency in nonlinear uncertainty propagation.
method Preserving mean and covariance, novel heuristics for selecting splitting direction informed by initial uncertainty and nonlinear function properties.
result Improved accuracy and efficiency in uncertainty propagation compared to existing techniques.
Reduces 3-body problem to shape and spherical geometry.
problem Investigates the motion of three bodies in a plane.
method Geometric reduction using equivariant Riemannian geometry.
result Time parametrization of moduli curve determined by shape curve and potential function.
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.
problem Understanding periodic orbits and symplectic structures in rotating celestial mechanics.
method Ligon-Schaaf and Levi-Civita symplectic regularizations, special concave toric domain construction.
result Identification of a special concave toric domain (SCTD) for the RKP phase space.
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.
problem Existence of periodic Reeb orbits on bm-contact manifolds. method Generalization of the Weinstein conjecture, proof of periodic orbits, existence of traps.
result In dimension 3, there are infinitely many periodic orbits on the critical set.
New method for studying t-dependent Hamilton equations on cosymplectic manifolds.
problem Existence and stability of solutions of t-dependent Hamilton equations. method Develops a cosymplectic energy-momentum method for Hamilton equations with more types of symmetries.
result Provides a more general framework for studying t-dependent Hamilton equations. 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.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
SympNets identify Hamiltonian systems from data using linear, activation, and gradient modules.
problem Identifying Hamiltonian systems from data.
method Composition of linear, activation, and gradient modules; universal approximation theorems.
result SympNets can approximate arbitrary symplectic maps and generalize well to various Hamiltonian systems.
New periodic solution found in 4-body problem, not part of expected geometrical family.
problem Finding new periodic solutions in the 4-body problem not fitting the expected geometrical family.
method Analytic continuation of a numerical solution to discover a new family of solutions.
result Existence of a non-planar periodic solution for any pair of masses and integer n.
ACA method improves gradient estimation for neural ODEs, reducing error and training time.
problem Inaccurate gradient estimation methods hinder the performance of neural ODEs on benchmark tasks.
method Adaptive Checkpoint Adjoint (ACA) method that applies trajectory checkpointing, deletes redundant components, and supports adaptive solvers.
result ACA reduces error rate by half and training time by half compared to adjoint and naive methods on image classification tasks.
New method improves estimation of complex models from conditional moment restrictions.
problem Estimation of complex models from conditional moment restrictions.
method Functional Generalized Empirical Likelihood (GEL) with a practical method.
result The method achieves state-of-the-art performance on two problems.
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.
problem Realizing groups as fundamental groups of graph of groups with restricted vertex groups.
method Analyzes restrictions on groups that can be realized and applies to manifold construction.
result Places constraints on groups that can be realized in graph of groups structures.
Study finds periodic orbits in a complex gravitational system.
problem Existence of periodic solutions in a gravitational system with multiple primaries.
method Proves existence of periodic solutions when primaries move on a Hip-Hop solution.
result Proves the existence of periodic solutions in the restricted (2N+1)-body problem. Study expands classical harmonic function results to Riemannian manifolds.
problem Classical harmonic function properties in domains of Riemannian manifolds.
method Generalized classical results to Riemannian manifolds, including pinched negative curvature.
result Generalized results for Riemannian manifolds, including pinched negative curvature.
Study shows semi-supervised learning improves human activity recognition with minimal user input.
problem Improving human activity recognition models using incremental learning.
method Three approaches: non-supervised, semi-supervised, and supervised learning were compared.
result Semi-supervised learning achieves similar accuracy to supervised learning with minimal user input.
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…
The study explores nonparametric regression with shape constraints using least squares estimation.
problem Nonparametric regression under shape constraints.
method Least squares estimation (LSE) with focus on isotonic, unimodal, convex, and additive shape-restricted regression.
result Adaptive nature of the LSE and its risk behavior, with pointwise limiting distribution theory for isotonic regression.
Two strategies for embedding new data points from proximity data are explored.
problem Embedding new data points using proximity data.
method Two competing strategies: projection and restricted reconstruction.
result Projection and restricted reconstruction can be derived from kernel methods.
Sparse Polyak improves high-dimensional statistical estimation.
problem High-dimensional statistical estimation problems with growing problem dimension.
method Sparse Polyak modifies Polyak's adaptive step size to estimate restricted Lipschitz smoothness.
result Sparse Polyak achieves optimal statistical precision with fewer iterations.
Study proposes an RBM with multivalued hidden variables to improve generalization.
problem Improving generalization in RBMs to prevent overfitting.
method Introduced an RBM with multivalued hidden variables as a simple extension of conventional RBMs.
result The proposed model outperforms conventional RBMs in contrastive divergence learning and MNIST classification.
Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.
problem Deriving critical metrics for the Einstein-Hilbert functional on compact Riemannian manifolds.
method Restricts the variational problem to an infinite-dimensional subspace.
result Derives a novel structural characterization of critical metrics.
Adapts Thompson Sampling for contextual bandits with limited context.
problem Online problems in clinical trials, recommender systems, and attention modeling.
method Adapts Thompson Sampling to a restricted context setting.
result Empirical advantages of proposed algorithms on real-life datasets.
SAGA achieves fast convergence on non-convex problems with RSC.
problem Non-convex optimization problems with restricted strong convexity.
method SAGA, a fast incremental gradient method, analyzed under RSC.
result SAGA achieves linear convergence up to statistical estimation accuracy under RSC.
Algorithm uses RBM to solve matching problems on weighted graphs.
problem Perfect matching problem on bipartite weighted graphs.
method Iterative RBM algorithm to maximize energy function and assignment.
result Algorithm successfully solves real-world matching problems.
Active-set algorithm improves Cox regression for shape-restricted covariates.
problem Improving Cox regression for shape-restricted covariates.
method Shape-restricted inference using active-set optimization for spline basis expansion.
result Active-set algorithm produces accurate linear covariate effect estimates.
Study restricts causal graphs with expert knowledge.
problem Restricting causal graphs to include expert orientation knowledge.
method Prove properties, present new orientation rules, develop algorithms.
result Shows how to uniquely represent restricted essential ancestral graphs.
Unified framework for generalized sparsity and RIP analysis.
problem Analyzing inverse problems with sparsity models.
method Proposed generalized notions of sparsity and a unified RIP framework.
result Extends RIP analysis to broader contexts including tensor products.
Framework approximates 2-Wasserstein distance for GANs training.
problem Training GANs with improved metrics and analysis.
method Approximates 2-Wasserstein distance via restricted convex potentials.
result Improved training for GANs with moment-matching property.
Improved Monte Carlo simulations using RBMs for phase transitions.
problem Slow mixing times in Monte Carlo simulations for complex systems.
method Fit unnormalized probability to a restricted Boltzmann machine and use its feature detection ability for efficient updates.
result Improved acceptance ratio and autocorrelation time near phase transition points.
Hardware-accelerated RBM solves large combinatorial problems and integer factorization.
problem Solving large combinatorial optimization and integer factorization problems.
method Logically synthesized RBM architecture, hardware acceleration, and efficient training methods.
result Hardware-accelerated RBM factorizes 16-bit numbers with 10000x speed and 32x power improvements.
Method predicts RMST from censored data using pseudo-observations and super learner.
problem Estimating RMST from right-censored data.
method Ensemble algorithm combining pseudo-observations and super learner.
result Method performs well in simulations and real data applications.
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…
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…
The paper analyzes conditions for solving low-rank matrix recovery problems with noisy measurements.
problem Low-rank matrix recovery with corrupted measurements.
method Analysis of the restricted isometry property (RIP) and local search methods.
result Sharp bounds on the maximum distance between local minimizers and the ground truth.
Optimal δ-correct algorithm for heavy-tailed distributions with minimal samples.
problem Identifying the best arm from a set of unknown distributions with heavy tails.
method Developed a δ-correct algorithm that matches lower bounds under mild restrictions.
result The proposed algorithm achieves optimal sample complexity for heavy-tailed distributions.