Complete Convergence for Randomly Weighted Sums of Random Variables Satisfying Some Moment Inequalities

For random variables and random weights satisfying Marcinkiewicz-Zygmund and Rosenthal type moment inequalities, we establish complete convergence results for randomly weighted sums of the random variables. Our results generalize those of(Thanh et al. SIAM J. Control Optim., 49,106–124(2011), Han and Xiang J. Ineq. Appl., 2016, 313(2016), Li et al. J. Ineq. Appl., 2017, 182(2017), and Wang et al. Statistics, 52, 503–518(2018).)     

On the Surrogate Gradient Algorithm for Lagrangian Relaxation

When applied to large-scale separable optimization problems, the recently developed surrogate subgradient method for Lagrangian relaxation (Zhao et al.: J. Optim. Theory Appl. 100, 699–712, 1999) does not need to solve optimally all the subproblems to update the multipliers, as the traditional subgradient method requires. Based on it, the penalty surrogate subgradient algorithm was further developed to address the homogenous solution issue (Guan et al.: J. Optim. Theory Appl. 113, 65–82, 2002; Zhai et al.: IEEE Trans. Power Syst. 17, 1250–1257, 2002). There were flaws in the proofs of Zhao et al., Guan et al., and Zhai et al.: for problems with inequality constraints, projection is necessary to keep the multipliers nonnegative; however, the effects of projection were not properly considered. This note corrects the flaw, completes the proofs, and asserts the correctness of the methods. This work is supported by the NSFC Grant Nos. 60274011, 60574067, the NCET program (No. NCET-04-0094) of China. The third author was supported in part by US National Science Foundation under Grants ECS-0323685 and DMI-0423607.     

用守恒型间断跟踪法数值模拟Richtmyer-Meshkov不稳定现象

使用茅德康所建立的守恒型间断跟踪法,数值模拟了两个关于Richtmyer-Meshkov不稳定性现象的物理实验,并且将数值模拟结果与Holmes等人在文中所获得的结果进行了比较．该文的结果与Holmes等人所得到的结果在总体上有较好的一致性．该文的数值模拟也捕捉到了非线性的压缩现象,即穿越波和反射波相互作用所产生的现象,Holmes等人指出其是导致介质界面减速的原因．但是所得到的扰动振幅和扰动增长率比Holmes等人所得到的结果略大一些．     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

分块幂等矩阵中广义Schur补的幂等性

为了讨论分块幂等矩阵中使用A(1)与A(2)的广义Schur补的幂等性问题,定义了(M/D)I=D-CA(1)B和(M/A)_O=D-CA(2)B,讨论得到了(M/D)_I=D-CA(1)B与(M/A)O=D-CA(2)B具有幂等性的充要条件,并研究了一些特殊情况,推广了J K Baksalary和Zhou J H的结论.     

Further variations on the theme of FC-groups

Groups that are FC, or more generally satisfy any of the weakenings of the FC-condition considered in de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et?al. (J. Algebra 326:218?C226, 2011), have local systems consisting of normal finite-by-nilpotent subgroups. Apart from generalizing results from de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et al. (J. Algebra 326:218?C226, 2011) to the more general context of locally (normal and finite-by-nilpotent) groups, we partially settle an open problem raised in Robinson et?al. (J. Algebra 326:218?C226, 2011) concerning the isomorphism of maximal p-subgroups, but in this more general setting of locally (normal and finite-by-nilpotent) groups.     

An augmented Lagrangian approach for sparse principal component analysis

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in the literature (Cadima and Jolliffe in J Appl Stat 22:203–214, 1995; d’Aspremont et?al. in J Mach Learn Res 9:1269–1294, 2008; d’Aspremont et?al. SIAM Rev 49:434–448, 2007; Jolliffe in J Appl Stat 22:29–35, 1995; Journée et?al. in J Mach Learn Res 11:517–553, 2010; Jolliffe et?al. in J Comput Graph Stat 12:531–547, 2003; Moghaddam et?al. in Advances in neural information processing systems 18:915–922, MIT Press, Cambridge, 2006; Shen and Huang in J Multivar Anal 99(6):1015–1034, 2008; Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006). Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic (Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006), Pitprops (Jeffers in Appl Stat 16:225–236, 1967), and gene expression data (Chin et?al in Cancer Cell 10:529C–541C, 2006), respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors. Moreover, the experiments on random data show that our method is capable of solving large-scale problems within a reasonable amount of time.     

Convergence analysis of a modified inexact implicit method for general mixed monotone variational inequalities

We consider a useful modification of the inexact implicit method with a variable parameter in Wang et al. J Optim Theory 111: 431–443 (2001) for generalized mixed monotone variational inequalities. One of the contributions of the proposed method in this paper is that the restrictions imposed on the variable parameter are weaker than the ones in Wang et al. J Optim Theory 111: 431–443 (2001). Another contribution is that we establish a sufficient and necessary condition for the convergence of the proposed method to a solution of the general mixed monotone variational inequality.     

Existence of solutions for models of shallow water in a basin with a degenerate varying bottom

We prove the existence of solutions for the great lake equations. These equations are obtained from the three-dimensional Euler equations in a basin with a free upper surface and a spatially varying bottom topography by taking a low aspect ratio, i.e., low wave speed and small wave amplitude expansion. These equations are rewritten in an abstract form by considering generalized Euler equations as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996). This paper is an extension of Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996), where the varying bottom was assumed to be nondegenerate. Here, we discuss the degenerate case and obtain similar results as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996).     

Energy partition for the linear radial wave equation

We consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy, as time goes to infinity. We show that the exterior energy estimate, which Duyckaerts et al. obtained in odd dimensions (Duyckaerts et al., J Eur Math Soc 13:533–599, 2011; J Eur Math Soc, 2013) fails in even dimensions. Positive results for restricted classes of data are obtained.     

The single machine batching problem with identical family setup times to minimize maximum lateness is strongly NP-hard

In this paper, we consider the single machine batching problem with family setup times to minimize maximum lateness. Recently, Cheng et al. [T.C.E. Cheng, C.T. Ng, J.J. Yuan, The single machine batching problem with family setup times to minimize maximum lateness is strongly NP-hard, Journal of Scheduling 6 (2003) 483–490] proved that this problem is strongly NP-hard. This answers a long-standing open problem posed by J. Bruno and P. Downey [Complexity of task sequencing with deadlines, setup times and changeover costs, SIAM Journal on Computing 7 (1978) 393–404]. By a modification of the proof in Cheng et al. (2003), we show that this problem is still strongly NP-hard when the family setup times are identical.     

On rack cohomology

We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given in Carter et al. (J. Pure Appl. Algebra, 157 (2001) 135) are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in Carter et al. (Twisted quandle cohomology theory and cocycle knot invariants, math. GT/0108051). We also give a group-theoretical interpretation of the second cohomology group for racks.     

Drug Release Kinetics from Biodegradable Polymers via Partial Differential Equations Models

In order to achieve prescribed drug release kinetics some authors have been investigating bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers. Recently, experimental data for the release of paclitaxel have been published by Lao et al. (Lao and Venkatraman in J. Control. Release 130:9–14, 2008; Lao et al. in Eur. J. Pharm. Biopharm. 70:796–803, 2008). In Blanchet et al. (SIAM J. Appl. Math. 71(6):2269–2286, 2011) we validated a two-parameter quadratic ordinary differential equation (ODE) model against their experimental data from three representative neat polymers. In this paper we provide a gradient flow interpretation of the ODE model. A three-dimensional partial differential equation (PDE) model for the drug release in their experimental set up is introduced and its parameters are related to the ones of the ODE model. The gradient flow interpretation is extended to the study of the asymptotic concentrations that are solutions of the PDE model to determine the range of parameters that are suitable to simulate complete or partial drug release.     

Higher-order Mond-Weir converse duality in multiobjective programming involving cones

In this work,we established a converse duality theorem for higher-order Mond-Weir type multiobjective programming involving cones.This flls some gap in recently work of Kim et al.[Kim D S,Kang H S,Lee Y J,et al.Higher order duality in multiobjective programming with cone constraints.Optimization,2010,59:29–43].     

Generalized Convexity and Nonsmooth Problems of Vector Optimization

In this paper it is shown that every generalized Kuhn-Tucker point of a vector optimization problem involving locally Lipschitz functions is a weakly efficient point if and only if this problem is KT- pseudoinvex in a suitable sense. Under a closedness assumption (in particular, under a regularity condition of the constraint functions) it is pointed out that in this result the notion of generalized Kuhn–Tucker point can be replaced by the usual notion of Kuhn–Tucker point. Some earlier results in (Martin (1985), The essence of invexity, J. Optim. Theory Appl., 47, 65–76. Osuna-Gómez et al., (1999), J. Math. Anal. Appl., 233, 205–220. Osuna-GGómez et al., (1998), J. Optim. Theory Appl., 98, 651–661. Phuong et al., (1995) J. Optim. Theory Appl., 87, 579–594) results are included as special cases of ours. The paper also contains characterizations of HC-invexity and KT- invexity properties which are sufficient conditions for KT- pseudoinvexity property of nonsmooth problems.Mathematics Subject Classifications: 90C29, 26B25     

Bounded directional distance function models

Bounded additive models in data envelopment analysis (DEA) under the assumption of constant returns to scale (CRS) were recently introduced in the literature (Cooper et al. in J Product Anal 35(2):85–94, 2011; Pastor et al. in J Product Anal 40:285–292, 2013; Pastor et al. in Omega 56:16–24, 2015). In this paper, we propose to extend the so far generated knowledge about bounded additive models to the family of directional distance function (DDF) models in DEA, giving rise to a completely new subfamily of bounded or partially-bounded CRS-DDF models. We finally check the new approach on a real agricultural panel data set estimating efficiency and productivity change over time, resorting to the Luenberger indicator in a context where at least one variable is naturally bounded.     

Bounds for the price of a European-style Asian option in a binary tree model

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.     

Bifurcation analysis in an age-structured model of a single species living in two identical patches

In this paper, we consider the age-structured model of a single species living in two identical patches derived in So et al. [J.W.-H. So, J. Wu, X. Zou, Structured population on two patches: modeling dispersal and delay, J. Math. Biol. 43 (2001) 37–51]. We chose a birth function that is frequently used but different from the one used in So et al. which leads to a different structure of the homogeneous equilibria. We investigate the stability of these equilibria and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out for supporting the analytic results.     

An embedded pair of exponentially fitted explicit Runge–Kutta methods

An embedded pair of exponentially fitted explicit Runge–Kutta (RK) methods for the numerical integration of IVPs with oscillatory solutions is derived. This pair is based on the exponentially fitted explicit RK method constructed in Vanden Berghe et al., and we confirm that the methods which constitute the pair have algebraic order 4 and 3. Some numerical experiments show the efficiency of our pair when it is compared with the variable step code proposed by Vanden Berghe et al. (J. Comput. Appl. Math. 125 (2000) 107).