多元统计分析中一类矩阵迹函数最小化问题的有效算法

研究来源于多元统计分析中的一类矩阵迹函数最小化问题$\min c+ tr(AX)+\sum\limits_{j=1}^{m}tr(B_j X C_jX^{T}),\ \ {\rm s. t.} \ X^TX=I_p,$其中$c$为常数, $A\in R^{p\times n}\ (n\geq p)$, $B_j\in R^{n\times n}, C_j\in R^{p\times p}$为给定系数矩阵. 数值实验表明已有的Majorization算法虽可行, 但收敛速度缓慢且精度不高. 本文从黎曼流形的角度重新研究该问题, 基于Stiefel流形的几何性质, 构造一类黎曼非单调共轭梯度迭代求解算法, 并给出算法收敛性分析.数值实验和数值比较验证所提出的算法对于问题模型是高效可行的.     

Probabilities of incidence between lines and a plane curve over finite fields

We study the probability for a random line to intersect a given plane curve, over a finite field, in a given number of points over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of the Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of the Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree.     

Converting red mud wastes into mesoporous ZSM-5 decorated with TiO2 as an eco-friendly and efficient adsorbent-photocatalyst for dyes removal

Red mud wastes have been converted into mesoporous zeolite socony mobile-5 (ZSM-5) followed by deposited titanium dioxide (TiO₂) nanoparticles to generate synergy adsorption-photodegradation for removal of dye removal in waste water. The amount of TiO₂ loading was varied to achieve optimum photocatalytic activity while maintaining the mesoporosity and high surface area of ZSM-5. Sol-gel method facilitated the formation of anatase TiO₂ on the ZSM-5. The fourier transform infrared spectra clarified the formation of Si–O–Ti at 957 cm^?1 by the exchanging the hydrogen ion with titanium ion, which proved by decreasing the absorption band of Si–OH and Si–O interaction at 964 and 944 cm^?1, respectively. Sol-gel method also preserved the mesopore diameter of ZSM-5 at 3.5 nm which allow the diffusion of methylene blue (MB) molecules into the pores. However, the surface area and the pore volume were slightly reduced with increasing the TiO₂ loading. The adsorption performance of samples showed that the increasing in the TiO₂ loading led to the decreasing in the adsorption capacity. All samples showed the suitability towards the pseudo second order kinetic. The Langmuir isotherm was suitable to describe the adsorption mechanism by monolayer adsorption. Mesoporosity of ZSM-5 accelerated the adsorption of dye via the increase of mass transfer in the pore channel which confirmed by the low intercept of intraparticle diffusion model at the first stage. The photocatalytic test showed that 10% TiO₂ loading on the ZSM-5 exhibited the highest methylene blue removal followed by 5% and 20% TiO₂ loading. Optimization on the amount of photocatalyst and the pH of solution indicated the reaction favoured 1 g L^?1 of catalysts and at alkaline pH. 10% TiO₂/ZSM-5 also exhibited high stability and reusability up to four reaction cycles. Photocatalytic performance of 10% TiO₂/ZSM-5 was further investigated on photodegradation of malachite green and rhodamine B organic dyes, which showed the photocatalytic efficiency of 73 and 88%, respectively. Superoxide radical, hydroxyl radical, and photogenerated electron were identified as the main active species for MB photodegradation based on the reduction of degradation rate following the addition scavenger molecules.     

Groupstrategyproofness of the egalitarian mechanism for constrained rationing problems

Motivated by applications in many economic environments, Bochet et al. (2010) generalize the classic rationing model (Sprumont 1991) as follows: there is a moneyless market, in which a non-storable, homogeneous commodity is reallocated between agents with single-peaked preferences. Agents are either suppliers or demanders. Transfers between a supplier and a demander are feasible only if they are linked, and the links form an arbitrary bipartite graph. Information about individual preferences is private, and so is information about feasible links: an agent may unilaterally close one of her links if it is in her interest to do so. For this problem they propose the egalitarian transfer solution, which equalizes the net transfers of rationed agents as much as permitted by the bilateral constraints. Furthermore, they show that the egalitarian mechanism elicits a truthful report of both preferences and links. In the variant where demanders are not strategic but demands need to be exactly met Bochet et al. (2013), they propose a similar mechanism for which truthfully reporting the peaks is a dominant strategy, but truthful reporting of links is not.The key contribution of the paper is a comprehensive study of the egalitarian mechanism with respect to manipulation by a coalition of agents. Our main result is that the egalitarian mechanism is group strategyproof : no coalition of agents can (weakly) benefit from jointly misreporting their peaks. Furthermore, we show that the egalitarian mechanism cannot be manipulated – by misreporting links or by misreporting peaks – by any coalition of suppliers (or any coalition of demanders) in the model where both the suppliers and demanders are agents. Our proofs shed light on the structure of the two models and simplify some of the earlier proofs of strategyproofness. An implication of our results is that the well known algorithm of Megiddo (1977) to compute a lexicographically optimal flow in a network is group strategyproof with respect to the source capacities and sink capacities.     

On Cones of Nonnegative Quartic Forms

Historically, much of the theory and practice in nonlinear optimization has revolved around the quadratic models. Though quadratic functions are nonlinear polynomials, they are well structured and many of them are found easy to deal with. Limitations of the quadratics, however, become increasingly binding as higher-degree nonlinearity is imperative in modern applications of optimization. In recent years, one observes a surge of research activities in polynomial optimization, and modeling with quartic or higher-degree polynomial functions has been more commonly accepted. On the theoretical side, there are also major recent progresses on polynomial functions and optimization. For instance, Ahmadi et al. (Math Program Ser A 137:453–476, 2013) proved that checking the convexity of a quartic polynomial is strongly NP-hard in general, which settles a long-standing open question. In this paper, we proceed to study six fundamentally important convex cones of quartic forms in the space of super-symmetric tensors, including the cone of nonnegative quartic forms, the sums of squared forms, the convex quartic forms, and the sums of fourth-power forms. It turns out that these convex cones coagulate into a chain in a decreasing order with varying complexity status. Potential applications of these results to solve highly nonlinear and/or combinatorial optimization problems are discussed.     

Simulation of the interaction of light with 3-D metallic nanostructures using a proper orthogonal decomposition-Galerkin reduced-order discontinuous Galerkin time-domain method

In this artice, we report on a reduced-order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3-D time-domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time-domain (DGTD) solver. With a Galerkin projection and a second order leap-frog (LF₂) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD-based ROM and assess its efficiency for 3-D nanophotonic problems.     

Preconditioned tensor format conjugate gradient squared and biconjugate gradient stabilized methods for solving stein tensor equations

This article is concerned with solving the high order Stein tensor equation arising in control theory. The conjugate gradient squared (CGS) method and the biconjugate gradient stabilized (BiCGSTAB) method are attractive methods for solving linear systems. Compared with the large-scale matrix equation, the equivalent tensor equation needs less storage space and computational costs. Therefore, we present the tensor formats of CGS and BiCGSTAB methods for solving high order Stein tensor equations. Moreover, a nearest Kronecker product preconditioner is given and the preconditioned tensor format methods are studied. Finally, the feasibility and effectiveness of the new methods are verified by some numerical examples.