Some Equivalent Forms of Bernoulli’s Inequality: A Survey

The main purpose of this paper is to link some known inequalities which are equivalent to Bernoulli’s inequality.

Delay-dependent Wide-area Damping Controller Synthesis Approach Using Jensen’s Inequality and Evolution Algorithm

A time-delay-dependent wide-area damping controller synthesis approach,based on Jensen’s integral inequality and evolution algorithm,is developed to suppress the adverse effect of time delay on the supplemental control of high-voltage direct current(DC)transmission systems.Initially,the state-space model of hybrid AC/DC systems with time delay is derived and the delay-dependent criteria for the stability of the closed-loop system are provided based on Jensen’s integral inequality.Subsequently,initial solutions are randomly generated to overcome the difficulty of solving the nonlinear matrix inequality.Finally,the time-delay stability upper bound of the controller is optimized using the differential evolution algorithm.In comparison to popular time-delay stable controller design methods,such as the free-weighting-matrix approach,the proposed method based on output feedback realization requires fewer decision variables and is more suitable for large-scale hybrid AC/DC systems.Three examples are introduced to verify the effectiveness of the proposed method.

On Several New Inequalities

In this paper, we find some new inequalities by Redheffer inequality, Jordan inequality and Jensen inequality. As application, we generalize one result which was given in [1].

Double Conditional Expectation

The concept of double conditional expectation is introduced. A series of properties for the double conditional expectation are obtained several convergence theorems and Jensen inequality are proved. Finally we discuss the special cases and application for double conditional expectation. Key words double conditional expectation - covergence theorem - Jensen inequality - branching chain in random environment CLC number O 211.6 Foundation item: Supported by the National Science Foundation of China (10371092) and the Foundation of Wuhan UniversityBiography: HU Di-he (1935-), male, Professor, research direction: stochastic processes and random fractals.

Jensen's Inequality for Backward Stochastic Differential Equations

Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen＇s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g（t, 0）≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen＇s inequality for gexpectation in [4, 7-9].

Ergodic Rate Analysis on Applying Antenna Selection in D2D Communication Underlaying Cellular Networks

By reusing the spectrum of a cellular network, device-to-device(D2D) communications is known to greatly improve the spectral efficiency bypassing the base station(BS) of the cellular network. Antenna selection is the most cost efficient scheme for interference management, which is crucial to D2D systems. This paper investigates the achievable rate performance of the D2D communication underlaying the cellular network where a multiple-antenna base station with antenna selection scheme is deployed. We derive an exact closed-form expression of the ergodic achievable rate. Also, using Jensen's inequality, two pairs of upper and lower bounds of the rate are derived and we validate the tightness of the two sets of bounds. Based on the bounds obtained, we analyze the ergodic achievable rate in noise-limited scenario, interference-limited high SNR scenario and larger-scale antenna systems. Our analysis shows that the presence of D2D users could be counter-productive if the SNR at cellular UE is high. Further analysis shows that the relationship between the ergodic rate and the number of antennas it positive, but keeps decreasing as the antenna number increasing. These show the inefficiency of antenna selection in D2D interference management.

A Note on Jensen’s Inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.

Asymptotic stability for impulsive functional differential equations

In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.

Robust H_∞ control for uncertain systems with heterogeneous time-varying delays via static output feedback

This paper is concerned with the problem of robust H∞ control for a novel class of uncertain linear continuous-time systems with heterogeneous time-varying state/input delays and norm-bounded parameter uncertainties. The objective is to design a static output feedback controller such that the closed-loop system is asymptotically stable while satisfying a prescribed H∞ performance level for all admissible uncertainties. By constructing an appropriate Lyapunov-Krasvskii functional, a delay-dependent stability criterion of the closed-loop system is presented with the help of the Jensen integral inequality. From the derived criterion, the solutions to the problem are formulated in terms of linear matrix inequalities and hence are tractable numerically. A simulation example is given to illustrate the effectiveness of the proposed design method,

Blow-up of Solution for the Nonlinear Sobolev-Galpern Equation

In this paper,the initial boundary value problems of the nonlinear Sobolev-Galpern equation are studied.The existence,uniqueness of local solution for the problem are obtained by means of a special Green's function and the contraction mapping principle.Finally,the blow-up of solution in finite time under some assumed conditions is proved with the aid of Jensen's inequality.

Operator Jensen's Inequality on C*-algebras

A real continuous function which is defined on an interval is said to beA-convex if it is convex on the set of self-adjoint elements,with spectra in the interval,in all matrix algebras of the unital C-algebra A.We give a general formation of Jensen’s inequality for A-convex functions.

Some Strengthed and Reversed Pachpatte Inequalities

The main purpose of the present article is to establish some new strengthed and reversed Pachpatte's type inequalities. As applications, some new type Hilbert's inequlities are generalized and strengthened.

The minimal sublinear expectations and their related properties

In this paper, we prove that for a sublinear expectation ?[·] defined on L 2(Ω, $ \mathcal{F} $ ), the following statements are equivalent: ? is a minimal member of the set of all sublinear expectations defined on L 2(Ω, $ \mathcal{F} $ )? is linearthe two-dimensional Jensen’s inequality for ? holds.Furthermore, we prove a sandwich theorem for subadditive expectation and superadditive expectation.

Some properties of g-convex functions

In this paper,we obtain that eachg-convex function is continuous and convex,and we also extend Jia and Peng’s result on the characterization of g-convex function without the bounded assumption of the value of g at the origin.

Some New Inverse-type Hilbert-Pachpatte Integral Inequalities

In this paper, some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved. The results of this paper reduce to those of Pachpatte (1998, J. Math. Anal. Appl. 226, 166–179) and Zhao and Debnath (2001, J. Math. Anal. Appl. 262, 411–418).

Stochastic synchronization of neutral-type chaotic impulse neural networks with leakage delay and Markovian jumping parameters

Purpose-The purpose of this paper is to develop a methodology for the stochastically asymptotic synchronization problem for a class of neutral-type chaotic neural networks with both leakage delay and Markovian jumping parameters under impulsive perturbations.Design/methodology/approach-The authors perform drive-response concept and time-delay feedback control techniques to investigate a class of neutral-type chaotic neural networks with both leakage delay and Markovian jumping parameters under impulsive perturbations.New sufficient criterion is established without strict conditions imposed on the activation functions.Findings-It turns out that the approach results in new sufficient criterion easy to be verified but without the usual assumption of the differentiability and monotonicity of the activation functions.Two examples show the effectiveness of the obtained results.Originality/value-The novelty of the proposed approach lies in removing the usual assumption of the differentiability and monotonicity of the activation functions,and the use of the Lyapunov functionalmethod,Jensen integral inequality,a novel Gu’s lemma,reciprocal convex and linear convex combination technique for the stochastically asymptotic synchronization problem for a class of neutral-type chaotic neural networks with both leakage delay and Markovian jumping parameters under impulsive perturbations.