Equilibrium problems on Riemannian manifolds with applications

We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds.National Natural Science Foundation of ChinaNatural Science Foundation of Guizhou Province (China)Dirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa

Proximal point algorithms on Hadamard manifolds: linear convergence and finite termination

In the present paper, we consider inexact proximal point algorithms for finding singular points of multivalued vector fields on Hadamard manifolds. The rate of convergence is shown to be linear under the mild assumption of metric subregularity. Furthermore, if the sequence of parameters associated with the iterative scheme converges to 0, then the convergence rate is superlinear. At the same time, the finite termination of the inexact proximal point algorithm is also provided under a weak sharp minima-like condition. Applications to optimization problems are provided. Some of our results are new even in Euclidean spaces, while others improve and/or extend some known results in Euclidean spaces. As a matter of fact, in the case of exact proximal point algorithm, our results improve the corresponding results in [G. C. Bento and J. X. Cruz Neto, Optim., 63 (2014), pp. 1281–1288]. Finally, several examples are provided to illustrate that our results are applicable while the corresponding results in the Hilbert space setting are not.National Natural Science Foundation of ChinaZhejiang Provincial Natural Science Foundation of ChinaDirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa

Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper

New strong convergence method for the sum of two maximal monotone operators

This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis

JUNO Conceptual Design Report

The Jiangmen Underground Neutrino Observatory (JUNO) is proposed to determine the neutrino mass hierarchy using an underground liquid scintillator detector. It is located 53 km away from both Yangjiang and Taishan Nuclear Power Plants in Guangdong, China. The experimental hall, spanning more than 50 meters, is under a granite mountain of over 700 m overburden. Within six years of running, the detection of reactor antineutrinos can resolve the neutrino mass hierarchy at a confidence level of 3-4$\sigma$, and determine neutrino oscillation parameters $\sin^2\theta_{12}$, $\Delta m^2_{21}$, and $|\Delta m^2_{ee}|$ to an accuracy of better than 1%. The JUNO detector can be also used to study terrestrial and extra-terrestrial neutrinos and new physics beyond the Standard Model. The central detector contains 20,000 tons liquid scintillator with an acrylic sphere of 35 m in diameter. $\sim$17,000 508-mm diameter PMTs with high quantum efficiency provide $\sim$75% optical coverage. The current choice of the liquid scintillator is: linear alkyl benzene (LAB) as the solvent, plus PPO as the scintillation fluor and a wavelength-shifter (Bis-MSB). The number of detected photoelectrons per MeV is larger than 1,100 and the energy resolution is expected to be 3% at 1 MeV. The calibration system is designed to deploy multiple sources to cover the entire energy range of reactor antineutrinos, and to achieve a full-volume position coverage inside the detector. The veto system is used for muon detection, muon induced background study and reduction. It consists of a Water Cherenkov detector and a Top Tracker system. The readout system, the detector control system and the offline system insure efficient and stable data acquisition and processing.Comment: 328 pages, 211 figure

Measurement of the inclusive and dijet cross-sections of b-jets in pp collisions at sqrt(s) = 7 TeV with the ATLAS detector

The inclusive and dijet production cross-sections have been measured for jets containing b-hadrons (b-jets) in proton-proton collisions at a centre-of-mass energy of sqrt(s) = 7 TeV, using the ATLAS detector at the LHC. The measurements use data corresponding to an integrated luminosity of 34 pb^-1. The b-jets are identified using either a lifetime-based method, where secondary decay vertices of b-hadrons in jets are reconstructed using information from the tracking detectors, or a muon-based method where the presence of a muon is used to identify semileptonic decays of b-hadrons inside jets. The inclusive b-jet cross-section is measured as a function of transverse momentum in the range 20 < pT < 400 GeV and rapidity in the range |y| < 2.1. The bbbar-dijet cross-section is measured as a function of the dijet invariant mass in the range 110 < m_jj < 760 GeV, the azimuthal angle difference between the two jets and the angular variable chi in two dijet mass regions. The results are compared with next-to-leading-order QCD predictions. Good agreement is observed between the measured cross-sections and the predictions obtained using POWHEG + Pythia. MC@NLO + Herwig shows good agreement with the measured bbbar-dijet cross-section. However, it does not reproduce the measured inclusive cross-section well, particularly for central b-jets with large transverse momenta.Comment: 10 pages plus author list (21 pages total), 8 figures, 1 table, final version published in European Physical Journal