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In this paper, we answer the question: what are the greatest values α1 , α2 and the least values β1,β2 , such that the inequalities α1T (a,b)+(1−α1)H(a,b) 0 with a = b? Here, R(a,b) , T (a,b) and H(a,b) denote the NeumanSandor, centroidal and harmonic means of two positive numbers a and b , respectively. Mathematics subject classification (2010): 26D15, 26E60.

In this paper, we establish some inequalities for the Toader-Sandor mean by use of Jensen inequality and convexity. Mathematics Subject Classification: 26E60, 26D15

For x = (x 1, x 2, ..., x n ) ∈ ℝ + , the symmetric function ψ n (x, r) is defined by $$\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,$$ , where r = 1, 2, ..., n and i 1, i 2, ..., i n are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmoni convexity of ψ n (x, r) are discussed. As applications, some inequalities are established by use of the theory of majorization.

In this work the authors present some new lower and upper bounds for the functions sin x / x and x / sinh x , thus improving some inequalities put forward by Klen et al. (2010) in the paper [2] .

In the article, we establish several Lazarevic and Cusa type inequalities involving the hyperbolic sine and cosine functions with two parameters. As applications, we find some new bounds for certain bivariate means.

This paper deals with the problem of decentralized \(\mathcal{L}_{2}\)–\(\mathcal{L}_{\infty}\) filtering for a class of interconnected (or large-scale) Markovian jump systems with constant time delays. The purpose is to present delay-dependent conditions for the existence of mode-dependent decentralized filters, which guarantees that the filtering error system is stochastically stable with a prescribed \(\mathcal{L}_{2}\)–\(\mathcal{L}_{\infty}\) disturbance attenuation level. Such a purpose is achieved by using a mode-dependent centralized Lyapunov functional together with the so-called Jensen’s inequality. The obtained synthesis conditions are expressed in terms of linear matrix inequalities (LMIs), which leads to a convex design method for the concerned filters. An example including numerical and simulation results is provided finally to illustrate the effectiveness of the proposed design method.

This article addresses the problem of robust H∞ filter design of a class of Takagi-Sugeno fuzzy neutral systems with time-varying delays and norm-bounded parameter uncertainties. A fuzzy filter is constructed, which ensures both the robust stability and a prescribed H∞ performance of the filtering error system. A linear matrix inequality approach is developed, and a delay-dependent sufficient condition is obtained. A simulation example is provided to demonstrate the effectiveness of the proposed approach.

This paper addresses the H ∞ control problem for jumping delayed systems with two separable Markov processes. The objective is to design a controller such that the resulting closed-loop system is exponentially mean-square stable with a given decay rate and satisfies a prescribed H ∞ performance level. A decay-rate-dependent condition is obtained for the existence of admissible controllers in term of linear matrix inequalities. Two numerical examples are presented to demonstrate the effectiveness of the proposed method.

We prove that the Gini mean values S(a, b; x, y) are Schur harmonic convex with respect to ( x , y ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) if and only if ( a , b ) ∈ { ( a , b ) : a ≥ 0 , a ≥ b , a + b + 1 ≥ 0 } ∪ { ( a , b ) : b ≥ a , a + b + 1 ≥ 0 } and Schur harmonic concave with respect to ( x , y ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) if and only if ( a , b ) ∈ { ( a , b ) : a ≤ 0 , b ≤ 0 , a + b + 1 ≤ 0 } .

In this paper, we first give the definition of degenerate weakly (, -quasiregular mappings by using the technique of exterior power and exterior differential forms, and then, by using Hodge decomposition and Reverse Hlder inequality, we obtain the higher integrability result: for any satisfying 0 = (n, l, , ) > l, such that every degenerate weakly (, )-quasiregular mapping f (, ) belongs to (, ), that is, f is a degenerate (, )-quasiregular mapping in the usual sense.

SUMMARY This paper proposes a sliding mode observer–controller design method for uncertain Markovian jump systems with time delays and uncertain switching probabilities. Both the structures of a sliding mode observer and a sliding mode controller are given. By the mode-dependent Lyapunov functional approach, a sufficient condition for the stochastic stability of the closed-loop system is given, which can be converted into a convex optimization problem. The reachability of the sliding surfaces in both the estimation error space and the state estimate space can be ensured by the presented control scheme. Finally, the effectiveness of the proposed design method is demonstrated by a simulation example. Copyright © 2011 John Wiley & Sons, Ltd.

This article considers the problem of H∞ filter design for stochastic systems with time-varying delay. The time delay is assumed to be of interval type. Attention is focused on the design of delay-dependent filters that guarantee the asymptotic stability in mean square and a prescribed noise attenuation level in an H∞ sense for the filtering error dynamics. The delay-dependent H∞ filter design scheme is proposed in terms of a linear matrix inequality. A numerical example is used to illustrate the effectiveness of the proposed approach.

For p ∈ R the p -th power mean Mp(a,b) , arithmetic mean A(a,b) , geometric mean G(a,b) , and harmonic mean H(a,b) of two positive numbers a and b are defined by Mp(a,b) = ⎧⎨ ⎩ ( ap+bp 2 )1/p , p = 0, √ ab, p = 0, A(a,b) = (a+b)/2, G(a,b) = √ ab , and H(a,b) = 2ab/(a+b) , respectively. In this paper, we answer the questions: For α ∈ (0,1) , what are the greatest values p, r and m , and the least values q, s and n , such that the inequalities Mp(a,b) Aα(a,b)G1−α (a,b) Mq(a,b),Mr(a,b) Gα (a,b)H1−α (a,b) Ms(a,b) and Mm(a,b) Aα(a,b)H1−α (a,b) Mn(a,b) hold for all a,b > 0 ? Mathematics subject classification (2010): 26E60.

We find the greatest value α and the least value β such that the double inequality αT (a,b) + (1−α)G(a,b) 0 with a = b . Here T (a,b) , G(a,b) , and A(a,b) denote the Seiffert, geometric, and arithmetic means of two positive numbers a and b , respectively.

The synchronization problem is considered for a class of stochastic complex networks with interval time-varying delays. Based on the Lyapunov functional method and stochastic analysis theory, delay-dependent sufficient conditions are obtained, which ensure that the identical delayed networks are globally synchronized in the mean square sense. Furthermore, the 'piecewise delay' approach is improved to deal with the synchronization problem of systems with stochastic disturbances by introducing two new lemmas. The efficiency and applicability of the proposed methodology are demonstrated by two numerical examples.

This paper studies the problem of the gossip consensus algorithm with real-valued and quantized data. We study the effect of the mixing parameter on the convergence rate of the proposed gossip consensus algorithm, and show when the proposed bounds are optimized with respect to the mixing parameter. For a gossip consensus algorithm with quantized data, we prove that it can achieve the consensus almost surely, and the expected value of the final states is equal to the average of the initial states. Moreover, we provide a result characterizing the convergence performance of the distance from consensus and make a comparison with the non-quantized gossip consensus algorithm. Finally, simulation results are provided to evaluate the effectiveness of the proposed algorithm.

A robust adaptive output-feedback control scheme is proposed for a class of nonlinear systems with unknown time-varying actuator faults. Additional unmodelled terms in the actuator fault model are considered. A new linearly parameterized model is proposed. The boundedness of all the closed-loop signals is established. The desired control performance of the closed-loop system is guaranteed by appropriately choosing the design parameters. The properties of the proposed control algorithm are demonstrated by two simulation examples. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper, we prove that is geometrically convex on (0, ). As its applications, we obtain some new estimates for .

The problem of accelerating distributed average consensus by using the information of second-order neighbors in both the discrete- and continuous-time cases is addressed in this Letter. In both two cases, when the information of second-order neighbors is used in each iteration, the network will converge with a speed faster than the algorithm only using the information of first-order neighbors. Moreover, the problem of using partial information of second-order neighbors is considered, and the edges are not chosen randomly from second-order neighbors. In the continuous-time case, the edges are chosen by solving a convex optimization problem which is formed by using the convex relaxation method. In the discrete-time case, for small network the edges are chosen optimally via the brute force method. Finally, simulation examples are provided to demonstrate the effectiveness of the proposed algorithm.

In this paper, the problems of robust global exponential synchronization for a class of complex networks with time-varying delayed couplings are considered. Each node in the network is composed of a class of delayed neural networks described by a nonlinear delay differential equation of neutral-type. Since model errors commonly exist in practical applications, the parameter uncertainties are involved in the considered model. Sufficient conditions that ensure the complex networks to be robustly globally exponentially synchronized are obtained by using the Lyapunov functional method and some properties of Kronecker product. An illustrative example is presented to show the effectiveness of the proposed approach.

The problem of nonfragile robust H∞ filtering for a class of Takagi–Sugeno fuzzy time-delay systems is investigated in the paper. We design a fuzzy filter which can tolerate some level of the filter gain variations while ensuring both the robust stability and a prescribed H∞ performance level of the filtering error system. Delay-dependent sufficient conditions for the existence of such filters are obtained in terms of linear matrix inequalities, and an explicit expression of a desired filter is given.

In this article, we show the existence of intertwined basins of attraction for a class of prey-predator systems, which improves the extant results by deleting some crucial and hard testing conditions.

This paper studies the problem of H ∞ filtering for a class of stochastic systems with Markovian switching and mixed mode-dependent time-varying delays. By introducing slack matrix variables and using Markovian switching Lyapunov functionals, we obtain delay-dependent sufficient conditions which guarantee the existence of H ∞ filters such that the filtering error system is exponentially mean-square stable and satisfies a prescribed H ∞ performance level; the conditions are in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H ∞ filter is given. Numerical examples are provided to demonstrate the effectiveness and the reduced conservatism of our results.

This Letter considers the problem of stability analysis of a class of delayed genetic regulatory networks with stochastic disturbances. The delays are assumed to be time-varying and bounded. By utilizing Ito's differential formula and Lyapunov–Krasovskii functionals, delay-range-dependent and rate-dependent (rate-independent) stability criteria are proposed in terms of linear matrices inequalities. An important feature of the proposed results is that all the stability conditions are dependent on the upper and lower bounds of the delays. Another important feature is that the obtained stability conditions are less conservative than certain existing ones in the literature due to introducing some appropriate free-weighting matrices. A simulation example is employed to illustrate the applicability and effectiveness of the proposed methods.

The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ ℝ × ℝ is still open. In this paper, we prove that S(a, b; x, y) is Schur convex with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): a ⩾ 0, b ⩾ 0, a + b ⩾ 1}, and Schur concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): b ⩽ 0, b ⩽ a, a + b ⩽ 1} ∩ {(a, b): a ⩽ 0, a ⩽ b, a + b ⩽ 1}.

In this note, we prove the partial regularity of stationary weak solutions for the Landau-Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey’s energy to avoid the difficulties by blowing up.

This paper investigates the problems of robust stabilization and robust H"~ control for uncertain Takagi-Sugeno (T-S) fuzzy neutral systems with both discrete and distributed time delays. The uncertainties are time-varying but norm-bounded. The purpose is to design state feedback fuzzy controllers such that the resulting closed-loop system is robustly stable (for the stabilization problem) and robustly stable with an H"~ performance level (for the H"~ control problem), respectively. Sufficient conditions for the solvability of the two problems are obtained by utilizing a Lyapunov functional together with the linear matrix inequality (LMI) approach. When the LMIs are feasible, an explicit expression of a desired state feedback fuzzy controller is given. A numerical example is provided to demonstrate the applicability of the proposed design method.

This Letter investigates the problem of stability analysis for a class of high-order neural networks with time-varying delays. The delays are bounded but not necessarily differentiable. Based on the Lyapunov stability theory together with the linear matrix inequality (LMI) approach and the use of Halanay inequality, sufficient conditions guaranteeing the global exponential stability of the equilibrium point of the considered neural networks are presented. Two numerical examples are provided to demonstrate the effectiveness of the proposed stability criteria.

In this paper the authors study the properties of the so-called exponent-quasiadditive functions and an application to the generalized Grotzsch ring function of quasiconformal theory is specified.

A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59–73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such that G-S is a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such that G-S is a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G-S is a tree.

This paper investigates the problem of stochastic stabilization for stochastic neutral systems with distributed delays. The time delay is assumed to appear in both the state and measurement equations. Attention is focused on the design of linear dynamic output feedback controllers such that the resulting closed-loop system is exponentially mean-square stable. A sufficient condition for the solvability of the problem is obtained in terms of a linear matrix inequality (LMI). When this LMI is feasible, an explicit expression of a desired dynamic output feedback controller is also given. The theory developed in this paper is demonstrated via a numerical example

The aim of this work is to prove the partial regularity of the harmonic maps with potential. The main difficulty caused by the potential is how to find the equation satisfied by the scaling function. Under the assumption on the potential we can obtain the equation, however, for a general potential, even if it is smooth, the partial regularity is still open.

This Letter provides new sufficient conditions for the existence, uniqueness and global exponential stability of the equilibrium point of recurrent neural networks with time-varying delays by employing Lyapunov functions and using the Halanay inequality. The time-varying delays are not necessarily differentiable. Both Lipschitz continuous activation functions and monotone nondecreasing activation functions are considered. The derived stability criteria are expressed in terms of linear matrix inequalities (LMIs), which can be checked easily by resorting to recently developed algorithms solving LMIs. Furthermore, the proposed stability results are less conservative than some previous ones in the literature, which is demonstrated via some numerical examples.

The main aim of this paper is to discuss the following two problems: Problem I: Given X ∈ H n × m (the set of all n × m quaternion matrices), Λ = diag ( λ 1 , * , λ m ) ∈ H n × m find A ∈ B S H ≥ n × n such that A X = X Λ where B S H ≥ n × n denotes the set of all n × n quaternion matrices which are bi-self-conjugate and nonnegative definite. Problem II : Given B ∈ H n × m , find B ¯ ∈ S E such that ‖ ‖ B − B ¯ ‖ ‖ Q = min A ∈ S E ‖ ‖ B − A ‖ ‖ Q , where SE is the solution set of problem I, ‖ ‖ ċ ‖ ‖ Q is the quaternion matrix norm. The necessary and sufficient conditions for SE being nonempty are obtained. The general form of elements in SE and the expression of the unique solution B ¯ of problem II are given.

This paper deals with the problem of stochastically stability and L 2 –L ∞ performance analysis for a class of interconnected Markovian jump filtering error system with constant time delays. By using mode-dependent decentralized filters and constructing mode-dependent centralized Lyapunov functional together with Jensen's inequality, the stability conditions for the filtering error system are presented which also ensures the L 2 –L ∞ performance. The obtained stability conditions are delay-dependent which may be less conserve than that delay independent one, especially when time-delay are very small. Furthermore, these conditions are expressed in terms of matrix inequalities (LMIs), which may provide the convenience for the design of the concerned filters through linear matrix inequality technique.

This paper further investigated the problem of robust H ∞ control for uncertain fuzzy neutral systems with both distributed delays and input delays. Different from literature[4], not only input delays will be carefully considered, but also how to to reducing the conservativeness in the designing of fuzzy controller will be shown by Relaxed LMI. Based on Takagi-Sugeno (T-S) fuzzy model and a useful matrix inequality, some sufficient conditions for the solvability of the problem is obtained by utilizing a proper Lyapunov functional and LMI approach.

This paper deals with the problem of robust non-fragile H∞ filtering for neutral stochastic systems with distributed delays and norm-bounded parameter uncertainties. Attention is focused on the design of a filter which is subject to gain variations, such that the filtering error system is robustly stochastically stable with a prescribed H∞ performance level for all admissible uncertainties. A delay-dependent sufficient condition for the solvability of this problem is obtained in terms of a linear matrix inequality. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

This paper considers the problem of robust delay-dependent L2–L∞ filtering for a class of Takagi–Sugeno fuzzy systems with time-varying delays. The purpose is to design a fuzzy filter such that both the robust stability and a prescribed L2–L∞ performance level of the filtering error system are guaranteed. A delay-dependent sufficient condition for the solvability of the problem is obtained and a linear matrix inequality (LMI) approach is developed. A desired filter can be constructed by solving a set of LMIs. Copyright © 2009 John Wiley & Sons, Ltd.

Through free-weighting matrix and relaxed LMI approach, this paper deals with the problem of robust Hinfin control for uncertain discrete stochastic fuzzy systems with interval time-varying delays. The uncertainties are time-varying but norm-bounded. The purpose is to design a state feedback fuzzy controller such that the resulting closed-loop system is not only robustly stochastically stable but also with a disturbance attenuation level lambda. Based on Takagi-Sugeno (T-S) fuzzy model, a delay-dependent sufficient condition for the solvability of the problem is obtained by utilizing a proper Lyapunov functional.

Let $D$ be a proper subdomain of Euclidean $n$-space $R^n\ (n\geq2)$. The following two necessary and sufficient conditions for uniform domains are obtained in this paper: (1) $D$ is a uniform domain if and only if there exists a constant $m=m(D)$ such that $k_D(x_1,x_2)\leq mj_D(x_1,x_2)$ for any $x_1, x_2\in D$, where $k_D$ is the quasi-hyperbolic metric in $D$, $j_D(x_1,x_2)=\frac12\log\left(\frac{|x_1-x_2|}{d(x_1,\partial D)}+1\right)\left(\frac{|x_1-x_2|}{d(x_2,\partial D)}+1\right)$. (2) $D$ is a uniform domain if and only if there exists a constant $M=M(D)$ such that each pair of points $x_1,x_2\in D$ can be joined by a rectifiable arc $\gamma\subset D$ which satisfies $\frac1{\left(c_2^\alpha-c_1^\alpha\right)}\int_{\gamma_{j,[c_1,c_2]}}d(x,\partial D)^{\alpha-1}\mbox{d}s\leq\frac M\alpha|x_1-x_2|^\alpha$ for any $0

Let $D$ be a Jordan domain in $\overline R^2$ and $\Gamma=\partial D$ be the boundary of $D$. Then $D$ is a quasidisk if and only if $\Gamma$ is a bounded circular distortion curve.