(Robust) Edge-based semidefinite programming relaxation of sensor network localization

Recently Wang, Zheng, Boyd, and Ye (SIAM J Optim 19:655–673, 2008) proposed a further relaxation of the semidefinite programming (SDP) relaxation of the sensor network localization problem, named edge-based SDP (ESDP). In simulation, the ESDP is solved much faster by interior-point method than SDP relaxation, and the solutions found are comparable or better in approximation accuracy. We study some key properties of the ESDP relaxation, showing that, when distances are exact, zero individual trace is not only sufficient, but also necessary for a sensor to be correctly positioned by an interior solution. We also show via an example that, when distances are inexact, zero individual trace is insufficient for a sensor to be accurately positioned by an interior solution. We then propose a noise-aware robust version of ESDP relaxation for which small individual trace is necessary and sufficient for a sensor to be accurately positioned by a certain analytic center solution, assuming the noise level is sufficiently small. For this analytic center solution, the position error for each sensor is shown to be in the order of the square root of its trace. Lastly, we propose a log-barrier penalty coordinate gradient descent method to find such an analytic center solution. In simulation, this method is much faster than interior-point method for solving ESDP, and the solutions found are comparable in approximation accuracy. Moreover, the method can distribute its computation over the sensors via local communication, making it practical for positioning and tracking in real time.     

通识化学:有机物燃烧方程式的简易配平法

在通识化学的教学层面上,面向不同背景的学生,以简单有机物完全燃烧的化学方程式的配平为主题,梳理数学方法在化学方程式配平中的应用和特点,采用跨学科的方式,让学生能从不同的角度进行思考和学习,以掌握数学方法在化学中的应用。     

On positivity of mass for black hole space-times

For any time-symmetric asymptotically flat initial slice with an apparent horizon, the associated four-momentum measured at spatial infinity is shown to be future-directed and time-like.     

Optical limiting properties of nonlinear multimode waveguides

An experimental investigation of the transmission of multimode capillary waveguides with a nonlinear absorber in the core shows an enhanced nonlinear absorption relative to the same length of bulk material. The results are consistent with partial mode filling within the cores of the waveguides. This study confirms the promising optical limiting capabilities of multimode nonlinear waveguides and implies that the mode structure should be considered in the design and evaluation of capillary array optical limiters.     

Deciphering Photoluminescence Dynamics and Reactivity of the Luminescent Metal–Metal‐Bonded Excited State of a Binuclear Gold(I) Phosphine Complex Containing Open Coordination Sites

Luminescent metal complexes having open coordination sites hold promise in the design of sensory materials and photocatalysts. As a prototype example, [Au₂(dcpm)₂)]²⁺ (dcpm = bis(dicyclohexylphosphanyl) is known for its intriguing environmental sensitive photoluminescence. By integrating a range of complementary ultrafast time‐resolved spectroscopy to interrogate the excited state dynamics, this study uncovers that the events occurring in extremely rapid timescales and which are modulated strongly by environmental conditions play a pivotal role in the luminescence behavior and photochemical outcomes. Formed independent of the phase and solvent property within ～0.15 ps, the metal–metal bonded ³5dσ*6pσ state is highly reactive possessing strong propensity toward increasing coordination number at Au^I center, and with ～510 ps lifetime in dichloromethane is able to mediate light induced C–X bond cleavage.     

Peaceman–Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman–Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.     

Mechanical properties and morphology for polypropylene composites filled with microencapsulated red phosphorus

The effects of the microencapsulated red phosphorus (MRP) content and specimen thickness on the mechanical properties of the filled polypropylene (PP) composites were studied at room temperature. The results showed that the influence of the MRP on the tensile and impact properties of the composites was significant, and the Young's modulus and impact strength of the composites increased nonlinearly while the tensile strength decreased slightly with increasing the MRP weight fraction; the tensile elongation at break increased outstandingly when the MRP weight fraction was less than 2% and then decreased with increasing the MRP weight fraction. Furthermore, the impact fracture surface was examined by scanning electronic microscope. The results suggested that the toughening effect was attributed to the absorption of impact fracture energy by the microencapsulated layer. Copyright © 2014 John Wiley & Sons, Ltd.     

Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods

In this paper, we study the Kurdyka–?ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is \(\frac{1}{2}\). The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is \(\frac{1}{2}\). This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with \(\ell _1\) regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery.