A note on spin rescalings in post-Newtonian theory

Usually, the reduced mass $\mu$ is viewed as a dropped factor in $\mu l$ and $\mu h$, where $l$ and $h$ are dimensionless Lagrangian and Hamiltonian functions. However, it must be retained in post-Newtonian systems of spinning compact binaries under a set of scaling spin transformations $\mathbf{S}_i=\mathbf{s}_iGM^{2}$ because $l$ and $h$ do not keep the consistency of the orbital equations and the spin precession equations but $(\mu/M) l$ and $(\mu/M) h$ do. When another set of scaling spin transformations $\mathbf{S}_i=\mathbf{s}_iG\mu M$ are adopted, the consistency of the orbital and spin equations is kept in $l$ or $h$, and the factor $\mu$ can be eliminated. In addition, there are some other interesting results as follows. The next-to-leading-order spin-orbit interaction is induced in the accelerations of the simple Lagrangian of spinning compact binaries with the Newtonian and leading-order spin-orbit contributions, and the next-to-leading-order spin-spin coupling is present in a post-Newtonian Hamiltonian that is exactly equivalent to the Lagrangian formalism. If any truncations occur in the Euler-Lagrangian equations or the Hamiltonian, then the Lagrangian and Hamiltonian formulations lose their equivalence. In fact, the Lagrangian including the accelerations with or without truncations can be chaotic for the two bodies spinning, whereas the Hamiltonian without the spin-spin term is integrable.Comment: 10 pages, 1 figur

Functional inequalities on path space over a non-compact Riemannian manifold

We prove the existence of the O-U Dirichlet form and the damped O-U Dirichlet form on path space over a general non-compact Riemannian manifold which is complete and stochastically complete. We show a weighted log-Sobolev inequality for the O-U Dirichlet form and the (standard) log-Sobolev inequality for the damped O-U Dirichlet form. In particular, the Poincar\'e inequality (and the super Poincar\'e inequality) can be established for the O-U Dirichlet form on path space over a class of Riemannian manifolds with unbounded Ricci curvatures. Moreover, we construct a large class of quasi-regular local Dirichlet forms with unbounded random diffusion coefficients on the path space over a general non-compact manifold

Revisit on "Ruling out chaos in compact binary systems"

Full general relativity requires that chaos indicators should be invariant in various spacetime coordinate systems for a given relativistic dynamical problem. On the basis of this point, we calculate the invariant Lyapunov exponents (LEs) for one of the spinning compact binaries in the conservative second post-Newtonian (2PN) Lagrangian formulation without the dissipative effects of gravitational radiation, using the two-nearby-orbits method with projection operations and with coordinate time as an independent variable. It is found that the actual source leading to zero LEs in one paper [J. D. Schnittman and F. A. Rasio, Phys. Rev. Lett. 87, 121101 (2001)] but to positive LEs in the other [N. J. Cornish and J. Levin, Phys. Rev. Lett. 89, 179001 (2002)] does not mainly depend on rescaling, but is due to two slightly different treatments of the LEs. It takes much more CPU time to obtain the stabilizing limit values as reliable values of LEs for the former than to get the slopes (equal to LEs) of the fit lines for the latter. Due to coalescence of some of the black holes, the LEs from the former are not an adaptive indicator of chaos for comparable mass compact binaries. In this case, the invariant fast Lyapunov indicator (FLI) of two-nearby orbits, as a very sensitive tool to distinguish chaos from order, is worth recommending. As a result, we do again find chaos in the 2PN approximation through different ratios of FLIs varying with time. Chaos cannot indeed be ruled out in real binaries.Comment: 5 pages, 3 figure

Symplectic structure of post-Newtonian Hamiltonian for spinning compact binaries

The phase space of a Hamiltonian system is symplectic. However, the post-Newtonian Hamiltonian formulation of spinning compact binaries in existing publications does not have this property, when position, momentum and spin variables $[X, P, S_1, S_2]$ compose its phase space. This may give a convenient application of perturbation theory to the derivation of the post-Newtonian formulation, but also makes classic theories of a symplectic Hamiltonian system be a serious obstacle in application, especially in diagnosing integrability and nonintegrability from a dynamical system theory perspective. To completely understand the dynamical characteristic of the integrability or nonintegrability for the binary system, we construct a set of conjugate spin variables and reexpress the spin Hamiltonian part so as to make the complete Hamiltonian formulation symplectic. As a result, it is directly shown with the least number of independent isolating integrals that a conservative Hamiltonian compact binary system with both one spin and the pure orbital part to any post-Newtonian order is typically integrable and not chaotic. And conservative binary system consisting of two spins restricted to the leading order spin-orbit interaction and the pure orbital part at all post-Newtonian orders is also integrable, independently on the mass ratio. For all other various spinning cases, the onset of chaos is possible.Comment: 7 pages, no fig

A Trust-based Pollution Attack Prevention Scheme in Peer-to-Peer Streaming Networks

Nowadays, peer-to-peer (P2P) streaming systems have become a popular way to deliver multimedia content over the internet due to their low bandwidth requirement, high video streaming quality, and flexibility. However, P2P streaming systems are vulnerable to various attacks, especially pollution attacks, due to their distributed and dynamically changing infrastructure. In this paper, by exploring the features of various pollution attacks, we propose a trust management system tailored for P2P streaming systems. Both direct trust and indirect trust are taken into consideration when designing the trust management system. A new direct trust model is proposed. A dynamic confidence factor that can dynamically adjust the weight of direct and indirect trust in computing the trust is also proposed and studied. A novel double-threshold trust utilization scheme is given. It is shown that the proposed trust management system is effective in identifying polluters and preventing them from further sharing of polluted data chunks.Comment: to appear in Computer Network

Search for the Electric Dipole Moment and anomalous magnetic moment of the tau lepton at tau factories

Precise measurement of the Electric Dipole Moment (EDM) and anomalous magnetic moment ($g$-2) of particles are important tests of Beyond Standard Model (BSM) physics. It is generally believed that the tau lepton couples more strongly to BSM due to its large mass, and can be searched for at collider experiments. A new method to approximately reconstruct the neutrinos from the hadronic decays of $\tau^-\tau^+$ pairs produced at $e^-e^+$ tau factories is proposed. With all final state particle momenta available, observables based on matrix elements and sensitive to BSM are calculated. It is estimated that with 50 ab$^{-1}$ of data to be delivered by the $Belle$-II experiment, a tau EDM search with a 1-$\sigma$ level precision of $|d_\tau^{NP}|<2.04\times 10^{-19}$ e$\cdot$cm, and $g$-2 search with $|a_\tau^{NP}|<1.75\times 10^{-5}$ ($1.5\%$ of the SM prediction), can be expected when systematics are not considered. The new precision can effectively constrain BSM models with heavy mirror neutrinos. It can also constrain models containing a light scalar with mass at $O$(1 GeV), which can explain the current muon $g$-2 anomaly as well. The method in this work offers a new opportunity to search for BSM at current and future tau factories with high precision.Comment: v2: minor changes, refs added; v3: improved analysis, refs added, 13 pages, 10 figures, 3 tables; v4: minor changes, match to the published versio

A class of generalized positive linear maps on matrix algebras

We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum information theory, we discuss the structural physical approximation and optimality of entanglement witness associated with these maps

Uniform Spanning Forests and the bi-Laplacian Gaussian field

We construct a natural discrete random field on $\mathbb{Z}^{d}$, $d\geq 5$ that converges weakly to the bi-Laplacian Gaussian field in the scaling limit. The construction is based on assigning i.i.d. Bernoulli random variables on each component of the uniform spanning forest, thus defines an associated random function. To our knowledge, this is the first natural discrete model (besides the discrete bi-Laplacian Gaussian field) that converges to the bi-Laplacian Gaussian field

A note on the equivalence of Lagrangian and Hamiltonian formulations at post-Newtonian approximations

It was claimed recently that a low order post-Newtonian (PN) Lagrangian formulation, which corresponds to the Euler-Lagrange equations up to an infinite PN order, can be identical to a PN Hamiltonian formulation at the infinite order from a theoretical point of view. This result is difficult to check because in most cases one does not know what both the Euler-Lagrange equations and the equivalent Hamiltonian are at the infinite order. However, no difficulty exists for a special 1PN Lagrangian formulation of relativistic circular restricted three-body problem, where both the Euler-Lagrange equations and the equivalent Hamiltonian not only are expanded to all PN orders but also have converged functions. Consequently, the analytical evidence supports this claim. As far as numerical evidences are concerned, the Hamiltonian equivalent to the Euler-Lagrange equations for the lower order Lagrangian requires that they both be only at higher enough finite orders.Comment: 10 pages, 5 figures and 1 tabl

Double Thompson Sampling for Dueling Bandits

In this paper, we propose a Double Thompson Sampling (D-TS) algorithm for dueling bandit problems. As indicated by its name, D-TS selects both the first and the second candidates according to Thompson Sampling. Specifically, D-TS maintains a posterior distribution for the preference matrix, and chooses the pair of arms for comparison by sampling twice from the posterior distribution. This simple algorithm applies to general Copeland dueling bandits, including Condorcet dueling bandits as its special case. For general Copeland dueling bandits, we show that D-TS achieves $O(K^2 \log T)$ regret. For Condorcet dueling bandits, we further simplify the D-TS algorithm and show that the simplified D-TS algorithm achieves $O(K \log T + K^2 \log \log T)$ regret. Simulation results based on both synthetic and real-world data demonstrate the efficiency of the proposed D-TS algorithm.Comment: 27 pages, 5 figures, 9 tables; accepted by 30th Conference on Neural Information Processing Systems (NIPS), 201