Let Ω■R^(2) be a smooth bounded domain with 0∈■Ω.In this paper,we prove that for anyβ∈(0,1),the supremum u∈W^(1,2(Ω)),∫_(Ω)^(sup) udx=0,∫_(Ω)|▽u|^(2)dx≤1∫_(Ω)e^(2π(1-β)u^(2))/|x|^(2β)dx is finite a...Let Ω■R^(2) be a smooth bounded domain with 0∈■Ω.In this paper,we prove that for anyβ∈(0,1),the supremum u∈W^(1,2(Ω)),∫_(Ω)^(sup) udx=0,∫_(Ω)|▽u|^(2)dx≤1∫_(Ω)e^(2π(1-β)u^(2))/|x|^(2β)dx is finite and can be attained.This partially generalizes a well-known work of Chang and Yang(1988)who have obtained the inequality whenβ=0.展开更多
In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the nu...In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.展开更多
In this paper, observer design for an induction motor has been investigated. The peculiarity of this paper is the synthesis of a mono-Luenberger observer for highly coupled system. To transform the nonlinear error dyn...In this paper, observer design for an induction motor has been investigated. The peculiarity of this paper is the synthesis of a mono-Luenberger observer for highly coupled system. To transform the nonlinear error dynamics for the induction motor into the linear parametric varying （LPV） system, the differential mean value theorem combined with the sector nonlinearity transformation has been used. Stability conditions based on the Lyapunov function lead to solvability of a set of linear matrix inequalities. The proposed observer guarantees the global exponential convergence to zero of the estimation error. Finally, the simulation results are given to show the performance of the observer design.展开更多
The problem of state feedback controllers for a class of Takagi-Sugeno （T-S） Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the different...The problem of state feedback controllers for a class of Takagi-Sugeno （T-S） Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem （DMVT） and convex theory. The proposed design approach is based on the mean value theorem （MVT） to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying （LPV） systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities （LMIs）. The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control （CL-FOC） of permanent magnet synchronous machine （PMSM） drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.展开更多
A sharper asymptotic formula for the mean value sum from xmodq*L′(σ+it,X)L′(1-σ-it,X)1(where the summation is over all primitive Dirichlet characters mod q and 0<σ<1) is derived by using the analytic method...A sharper asymptotic formula for the mean value sum from xmodq*L′(σ+it,X)L′(1-σ-it,X)1(where the summation is over all primitive Dirichlet characters mod q and 0<σ<1) is derived by using the analytic method and the estimate of character sums.展开更多
Conjunction of two probability laws can give rise to a possibility law. Using two probability densities over two disjoint ranges, we can define the fuzzy mean of a fuzzy variable with the help of means two random vari...Conjunction of two probability laws can give rise to a possibility law. Using two probability densities over two disjoint ranges, we can define the fuzzy mean of a fuzzy variable with the help of means two random variables in two disjoint spaces.展开更多
To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω(u) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ(z). They are periodically...To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω(u) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ(z). They are periodically on the y-axis with period proportional to inverse step length u0. It is found that they possess additional zeros off the imaginary y-axis and additionally on this axis and vanish in the limiting case u0 → 0 in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.展开更多
In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Su...In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11721101 and 11401575)。
文摘Let Ω■R^(2) be a smooth bounded domain with 0∈■Ω.In this paper,we prove that for anyβ∈(0,1),the supremum u∈W^(1,2(Ω)),∫_(Ω)^(sup) udx=0,∫_(Ω)|▽u|^(2)dx≤1∫_(Ω)e^(2π(1-β)u^(2))/|x|^(2β)dx is finite and can be attained.This partially generalizes a well-known work of Chang and Yang(1988)who have obtained the inequality whenβ=0.
文摘In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.
文摘In this paper, observer design for an induction motor has been investigated. The peculiarity of this paper is the synthesis of a mono-Luenberger observer for highly coupled system. To transform the nonlinear error dynamics for the induction motor into the linear parametric varying （LPV） system, the differential mean value theorem combined with the sector nonlinearity transformation has been used. Stability conditions based on the Lyapunov function lead to solvability of a set of linear matrix inequalities. The proposed observer guarantees the global exponential convergence to zero of the estimation error. Finally, the simulation results are given to show the performance of the observer design.
文摘The problem of state feedback controllers for a class of Takagi-Sugeno （T-S） Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem （DMVT） and convex theory. The proposed design approach is based on the mean value theorem （MVT） to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying （LPV） systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities （LMIs）. The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control （CL-FOC） of permanent magnet synchronous machine （PMSM） drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.
基金Project supported by the National Natural Science Foundation of China.
文摘A sharper asymptotic formula for the mean value sum from xmodq*L′(σ+it,X)L′(1-σ-it,X)1(where the summation is over all primitive Dirichlet characters mod q and 0<σ<1) is derived by using the analytic method and the estimate of character sums.
文摘Conjunction of two probability laws can give rise to a possibility law. Using two probability densities over two disjoint ranges, we can define the fuzzy mean of a fuzzy variable with the help of means two random variables in two disjoint spaces.
文摘To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω(u) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ(z). They are periodically on the y-axis with period proportional to inverse step length u0. It is found that they possess additional zeros off the imaginary y-axis and additionally on this axis and vanish in the limiting case u0 → 0 in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.
文摘In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.
