A Jensen type inequality for fuzzy integrals

In this paper, we show a Jensen type inequality for the Sugeno integral. We also discuss some conditions assuring the satisfaction of opposite inequality (reverse Jensen inequality).     

Generalization of the Jensen inequality for pseudo-integral

Generalizations of the Jensen integral inequality for the pseudo-integral are proven. There are considered two cases of the real semiring with pseudo-operations: one, when pseudo-operations are defined by monotone and continuous function g (then the pseudo-integral reduces on g-integral), and the second semiring ([a,b],max,⊙), where the pseudo-multiplication ⊙ is generated.     

Wirtinger-based integral inequality: Application to time-delay systems

In the last decade, the Jensen inequality has been intensively used in the context of time-delay or sampled-data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities (LMIs). However, it is also well-known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature, reducing this gap is a relevant issue and always an open problem. In this paper, we propose an alternative inequality based on the Fourier Theory, more precisely on the Wirtinger inequalities. It is shown that this resulting inequality encompasses the Jensen one and also leads to tractable LMI conditions. In order to illustrate the potential gain of employing this new inequality with respect to the Jensen one, two applications on time-delay and sampled-data stability analysis are provided.     

Bootstrap方法在Jensen Alpha估计中的运用

詹森差别表现指数(Jensen Alpha)是使用较多的一种评价投资组合业绩的指标。进一步分析Bootstrap方法的基本原理,探讨将Bootstrap运用于Jensen Alpha估计的具体过程,从而解决了运用普通最小二乘法(OLS)估计Jensen Alpha存在的缺陷,提高了Jensen Alpha估计的稳健性,对于投资组合业绩的评价具有一定的参考意义。     

基于Jensen 不等式的2-D 区间时变时滞系统的稳定与控制

考虑具有区间时变时滞二维(2-D) 离散系统的时滞相关稳定性和控制问题. 选取含有时滞上下界的Lyapunov 函数, 对其差分时考虑到所有项, 结合2-DJensen 不等式, 由线性矩阵不等式给出系统新的时滞相关稳定性准则. 准则中含有更少的待定变量, 降低了数值计算负担, 并且比一些已有结果具有更小的保守性. 基于稳定性准则, 由状态反馈实现了系统的稳定控制. 最后, 通过数值算例表明了所得结果的有效性和优越性.

     

A complete algebraic solvability test for the nonstrict Lyapunov inequality

For arbitrary equally sized square complex matrices A and Q (Q Hermitian), the paper provides a complete algebraic test for verifying the existence of a Hermitian solution X of the nonstrict Lyapunov inequality A^*X + XA + Q 0. If existing, we exhibit how to construct a solution. Our approach involves the validation problem for the linear matrix inequality Σ_j=1^k (A_j^*X_jB_j + B_j^*X_j^*A_j) + Q> 0 in X_j, for which we provide an algebraic solvability test and a construct solutions if the kernels of A_j or, dually, those of B_j form an isotonic sequence.     

A new generalization of the Waring distribution

A tetraparametric univariate distribution generated by the Gaussian hypergeometric function that includes the Waring and the generalized Waring distributions as particular cases is presented. This distribution is expressed as a generalized beta type I mixture of a negative binomial distribution, in such a way that the variance of the tetraparametric model can be split into three components: randomness, proneness and liability. These results are extensions of known analogous properties of the generalized Waring distribution. Two applications in the fields of sport and economy are included in order to illustrate the utility of the new distribution compared with the generalized Waring distribution.     

Matrix Riccati inequality: Existence of solutions

We consider the various criteria (including Hamiltonian and frequency-domain ones) for solvability of the matrix Riccati inequality, arising from control problems. Some criteria are given in terms of symplectic algebra, which is also used as an important technical tool.     

A majorization inequality and its application to distributed Kalman filtering

In the analysis of a recently proposed distributed estimation algorithm based on the Kalman filtering and on gossip iterations, we needed to apply a new inequality which is valid for i.i.d. matrix valued random processes. This inequality can be useful in the analysis of the convergence rate of general jump Markov linear systems.In this paper, we present this inequality. This is based on the theory of majorization and on its use in the analysis of the singular values. Finally we will show the impact of this inequality on the performance analysis of gossip based distributed Kalman filters.     

A generalization of concavity for finite differences

The concept of concavity is generalized to discrete functions, u, satisfying the n^th-order difference inequality, (−1)^n−kΔⁿu(m) ≥ 0, M = 0, 1,..., N and the homogeneous boundary conditions, u(0) = … = u(k−1) = 0, u(N + k + 1) = … = u(N + n) = 0 for some k “1, …, n − 1”. A piecewise polynomial is constructed which bounds u below. The piecewise polynomial is employed to obtain a positive lower bound on u(m) for m = k, …, N + k, where the lower bound is proportional to the supremum of u. An analogous bound is obtained for a related Green's function.     

A new refinement of the Janous-Gmeiner inequality for a triangle

In this paper, the authors give a new refinement of the Janous-Gmeiner inequality for a triangle by making use of certain analytical techniques for systems of nonlinear algebraic equations. Some other closely-related geometric inequalities are also considered.     

A variational inequality for measurement feedback almost-dissipative control

In this paper, an optimal measurement feedback control problem that yields an almost- (or practically) dissipative closed loop system is considered. That is, the aim is to consider optimal control problems which, when solved, yield a closed loop system which almost satisfies a dissipation inequality. The main idea is that by weakening the required dissipation inequality, a broader class of open loop systems and controllers are admissible, leading to broader application. In obtaining the main results of this paper, dynamic programming is applied to the optimal control problem of interest to derive a variational inequality that generalizes the information state based partial differential equation associated with measurement feedback nonlinear dissipative control. This variational inequality can in principle be used to derive the optimal controller. In the special case of certainty equivalence, an explicit solution of the variational inequality exists and is a functional of the solution of the corresponding optimal state feedback almost-dissipative control problem.     

A continuous method model for solving general variational inequality

Based on the projection operator, this paper presents a continuous method model for solving general variational inequality problems (VIPs) with bound constraints. A main feature of the proposed model is that it does not involve any form of matrix information in analysing its convergence properties. Under some reasonable assumptions, the convergence results of the proposed method model are established. Numerical results on some problems show that the proposed approach is efficient and can be applied to solve large scale VIPs with bound constraints.     

A new modified Goldstein-Levitin-Polyakprojection method for variational inequality problems

In this paper, we first show that the adjustment parameter in the step size choice strategy of the modified Goldstein-Levitin-Polyak projection method proposed by He et al. for asymmetric strongly monotone variational inequality problems can be bounded away from zero by a positive constant. Under this observation, we propose a new step size rule which seems to be more practical and robust than the original one. We show that the new modified method is globally convergent under the same conditions and report some computational results to illustrate the method.     

含离散与分布时滞的不确定中立型系统鲁棒稳定性新判据

基于离散时滞分解思想,通过构造一种新的Lyapunov-Krasovskii泛函并结合Jensen不等式技巧,建立了线性矩阵不等式(LMI)形式的时滞相关鲁棒稳定性新判据.该方法允许中立时滞项的系数矩阵存在时变不确定性,增强了系统的鲁棒性能.同时针对分布时滞项难于处理的问题,构造了其分解计算泛函.数值算例表明所得结论的有效性和更低的保守性.     

Discovering inequality conditions in the analytic solution of optimization problems

Necessary and sufficient conditions for the problem of maximizing or minimizing a function subject to inequality constraints are given by a set of equalities and inequalities known as the Kuhn-Tucker conditions. These conditions can provide an analytic solution to the optimization problem if the artificial variables known as Lagrange multipliers can be eliminated. However, this is tedious to do by hand. This paper develops a computer program to assist in the solution process which combines symbolic computation and automated reasoning techniques. The program may also be useful for other problems involving algebraic reasoning with inequalities which employ general functions or symbolic parameters.     

Gender and digital usage inequality among adolescents: A comparative study of 39 countries

The paper investigates how gender exerts its influence on contemporary adolescents with respect to their access to the Information and Communication Technologies (ICTs). The focus here is on the so-called usage access. The paper's empirical basis is that of information on the ICTs usage collected for 39 countries in the framework of the 2006 wave of the Program for International Student Assessment (PISA) study. Ordinal regression modelling is used as a method for data investigation. The analysis points to the persistence of gender inequality seemingly in favour of boys. In all countries under investigation, boys report using computers and the Internet for educational purposes more often than girls. Controlling for the 2006 value of the national GDP per capita, the level of a country's gender inequality measured by the Gender Gap Index does not have any statistically significant effect on gender gap in educational use of ICTs. A sign of the gender coefficient suggest, however, that the increase in society's gender-neutrality is associated with the increase in boys' advantage over girls as regards the frequency of ICT/Internet educational use. The possibility that this advantage of boys is in fact a sign of their educational underperformance is discussed. Another possibility is also discussed, namely, that girls' decreased (in comparison with boys) frequency of using computers and the Internet for playing computer games might, counterintuitively, be the source of girls' disadvantage in the future.     

Hierarchy of LMI conditions for the stability analysis of time-delay systems

Assessing stability of time-delay systems based on the Lyapunov–Krasovskii functionals has been the subject of many contributions. Most of the results are based, first, on an a priori design of functionals and, finally, on the use of the famous Jensen’s inequality. In contrast with this design process, the present paper aims at providing a generic set of integral inequalities which are asymptotically non conservative and then to design functionals driven by these inequalities. The resulting stability conditions form a hierarchy of LMI which is competitive with the most efficient existing methods (delay-partitioning, discretization and sum of squares), in terms of conservatism and of complexity. Finally, some examples show the efficiency of the method.     

A globally convergent BFGS method for pseudo-monotone variational inequality problems

In this paper, we propose a globally convergent BFGS method to solve Variational Inequality Problems (VIPs). In fact, a globalization technique on the basis of the hyperplane projection method is applied to the BFGS method. The technique, which is independent of any merit function, is applicable for pseudo-monotone problems. The proposed method applies the BFGS direction and tries to reduce the distance of iterates to the solution set. This property, called Fejer monotonicity of iterates with respect to the solution set, is the basis of the convergence analysis. The method applied to pseudo-monotone VIP is globally convergent in the sense that subproblems always have unique solutions, and the sequence of iterates converges to a solution to the problem without any regularity assumption. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithm.     

On delay‐dependent stabilization of retarded systems—an integral‐inequality based approach

In this paper, a new integral‐inequality method is proposed by taking both the Leibniz‐Newton formula and the discussed system into account. By using a Lyapunov‐Krasovskii functional approach combined with the new integral inequality, we present the new stabilization criteria for the time‐delay system which employs a memoryless state feedback control law. We exert ourselves to find a less conservative transformation way by bringing the free weighting matrices into the integral inequality, and also these free matrices express greatly the relationships among the system variables and among the terms in the Leibniz‐Newton formula. The techniques applied here improve the results in existing literature. The results obtained by solving LMIs in this note are illustrated to be less conservative comparing with the previous ones. Moreover, numerical examples demonstrate the validity and the evident superiority of the proposed criteria. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society