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.

Fermions: Spin, Hidden Variables, Violation of Bell’s Inequality and Quantum Entanglement

Using real fields instead of complex ones, it was recently claimed, that all fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, related to Planck’s constant. Quantum mechanics is described with real fields and real operators. Schrodinger and Dirac equations then are solved. The solution to Dirac equation gives four, real, 2-vectors solutions ψ1=(U1D1)ψ2=(U2D2)ψ3=(U3D3)ψ4=(U4D4)where (ψ1,ψ4) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, (ψ2,ψ3) are coupled via linear combinations to represent spin-up and spin-down anti-fermions. For an incoming entangled pair of fermions, the combined solution is Ψin=c1ψ1+c4ψ4where c1and c4are some hidden variables. By applying a magnetic field in +Z and +x the theoretical results of a triple Stern-Gerlach experiment are predicted correctly. Then, by repeating Bell’s and Mermin Gedanken experiment with three magnetic filters σθ, at three different inclination angles θ, the violation of Bell’s inequality is proven. It is shown that all fermions are in a mixed state of spins and the ratio between spin-up to spin-down depends on the hidden variables.

Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Some Refinement of Holder’s and Its Reverse Inequality

Holder’s inequality, its refinement, and reverse have received considerable attention in the theory of mathematical analysis and differential equations. In this paper, we give some refinements of Holder’s inequality and its reverse using a simple analytical technique of algebra and calculus. Our results show many results related to holder’s inequality as special cases of the inequalities presented.

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.

Jensen’s Inequality Under Nonlinear Expectation Generated by BSDE with Jumps

In this paper, we study Jensen’s inequality under f-expectation, which is a nonlinear expectation generated by backward stochastic differential equations(BSDEs) with jumps. We connect f-convex functions with the viscosity solutions of a kind of integral partial differential equations(IPDEs) with non-local terms.And find that under Lipschitz condition, the f-convex function is still convex in the usual sense, i.e., the jumps shrink the range of ‘convex’ functions.

New Majorized Results on Hilbert's Integral Inequality

In this paper, some new majorized results on Hilbert’s integral inequality are showed.

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.

On an Extension and a Refinement of Van der Corput's Inequality

By introducing a parameter α, we give an extension of Van der Corput＇s inequality. Also a new strengthened version of it is considered.

THE HADAMARD INEQUALITY FOR CONVEX FUNCTION VIA FRACTIONAL INTEGRALS

In this paper, we establish several inequalities for some differantiable mappings that are connected with the Riemann-Liouville fractional integrals. The analysis used in the proofs is fairly elementary.

A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane

In this paper, by introducing some parameters and estimating the weight coefficient, we give a new Hilbert’s inequality with the integral in whole plane and with a non-homogeneous and the equivalent form is given as well. The best constant factor is calculated by the way of Complex Analysis.

GENERALIZED ROSENTHAL'S INEQUALITY FOR BANACH-SPACE-VALUED MARTINGALES

A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.

Bohr's Inequality on the Unit Ball B^n

Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.

Necessary and Sufficient Conditions of Doubly Weighted Hardy-Littlewood-Sobolev Inequality

Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.

On Reverses of the Blaschke-Santalo Inequality for Convex Bodies

Reisner proved a reverse of the Blaschke-Santal5 inequality for zonoid bodies, Bourgain and Milman showed another reverse of the Blaschke-Santal5 inequality for centered convex bodies. In this paper, two reverses of the Blaschke-Santal5 inequality for convex bodies are given by the Petty projection inequality and above two reverses. Further, using above methods, we also obtain two analogues of the Petty＇s conjecture for projection bodies, respectively.

On an Extension of Hibert's Inequality and Applications

This paper gives a new generalization of Hilbert＇s inequality with a best constant factor involving the β function. An applications, we consider the equivalent form and some particular results.

N次幂下S-凸函数和其性质研究

以凸函数、S-凸函数、平方凸函数、平方S-凸函数的定义和性质为研究基础,根据从平方凸函数到平方S-凸函数的演变推进过程,采用了凸函数概念的代数不等式形式,给出了N次幂下S-凸函数的概念,并对其判别定理和性质进行了讨论,定义并证明了N次幂下S-凸函数的Jensen型不等式。研究表明,N次幂下S-凸函数包含了凸函数、平方凸函数、平方S-凸函数的性质,是凸函数的一种广义凸函数的推广,开辟了全新的推广途径。

Refinements to Hadamard’s Inequality for Log-Convex Functions

In this paper we show that a log-convex function satisfies Hadamard's inequality, as well as we give an extension for this result in several directions.

DOOB'S INEQUALITY, BURKHOLDER-GUNDY INEQUALITY AND MARTINGALE TRANSFORMS ON MARTINGALE MORREY SPACES

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.

Inequalities relating to L_p-version of Petty's conjectured projection inequality

Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.