Sturmian comparison theory for half‐linear and nonlinear differential equations via Picone identity

In this paper, Sturmian comparison theory is developed for the pair of second‐order differential equations; first of which is the nonlinear differential equations of the form (1) and the second is the half‐linear differential equations (2) where Φ_α (s ) = |s |^{α  ? 1}s and α ₁ > ? > α _m  > β  > α _{m  + 1} > ? > α _n  > 0. Under the assumption that the solution of  2 has two consecutive zeros, we obtain Sturm–Picone type and Leighton type comparison theorems for  1 by employing the new nonlinear version of Picone formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for  1 . Examples are given to illustrate the relevance of the results. Copyright © 2016 John Wiley & Sons, Ltd.     

Sampled‐data reliable stabilization of T‐S fuzzy systems and its application

In this article, based on sampled‐data approach, a new robust state feedback reliable controller design for a class of Takagi–Sugeno fuzzy systems is presented. Different from the existing fault models for reliable controller, a novel generalized actuator fault model is proposed. In particular, the implemented fault model consists of both linear and nonlinear components. Consequently, by employing input‐delay approach, the sampled‐data system is equivalently transformed into a continuous‐time system with a variable time delay. The main objective is to design a suitable reliable sampled‐data state feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop fuzzy system. For this purpose, using Lyapunov stability theory together with Wirtinger‐based double integral inequality, some new delay‐dependent stabilization conditions in terms of linear matrix inequalities are established to determine the underlying system's stability and to achieve the desired control performance. Finally, to show the advantages and effectiveness of the developed control method, numerical simulations are carried out on two practical models. © 2016 Wiley Periodicals, Inc. Complexity 21: 518–529, 2016     

Robust guaranteed cost control for discrete‐time systems via partially delay‐dependent controller with linear fractional uncertainties

In this article, a partially delay‐dependent controller is designed to analyze the guaranteed performance analysis of a class of uncertain discrete‐time systems with time‐varying delays. By constructing suitable Lyapunov–Krasovskii Functional (LKF), sufficient conditions are derived to ensure the system to be robustly stochastically stable in mean square sense by using Wirtinger‐based inequality and convex reciprocal lemma. The proper cost function is chosen to guarantee an adequate level of performance. The derived conditions are expressed in terms of linear matrix inequalities (LMIs) which can be easily solved by LMI Toolbox in MATLAB. Further, the advantage of employing the obtained results is illustrated via numerical examples. © 2016 Wiley Periodicals, Inc. Complexity 21: 113–122, 2016     

L p -Error Bounds for Hermite Interpolation and the Associated Wirtinger Inequalities

The B-spline representation for divided differences is used, for the first time, to provide L _p -bounds for the error in Hermite interpolation, and its derivatives, thereby simplifying and improving the results to be found in the extensive literature on the problem. These bounds are equivalent to certain Wirtinger inequalities. The major result is the inequality where H_Θ f is the Hermite interpolant to f at the multiset of n points Θ, and is the diameter of . This inequality significantly improves upon Beesack's inequality, on which almost all the bounds given over the last 30 years have been based. Date received: June 24, 1994 Date revised: February 4, 1996.     

Tau-functions on Hurwitz Spaces

We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere P ¹. A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering corresponding to a given meromorphic function on a Riemann surface of genus g. In genera 0 and 1 we construct a nowhere vanishing holomorphic horizontal section of this bundle (the ‘Wirtinger tau-function’). In higher genus we compute the modulus square of the Wirtinger tau-function. In particular one gets formulas for the isomonodromic tau-functions of semisimple Frobenius manifolds connected with the Hurwitz spaces H _g,N (1,⋯,1). This revised version was published online in July 2006 with corrections to the Cover Date.     

Rudolph diagrams and an analytic realization of the Vitushkin covering

An example of a three-sheeted covering over the ball in ² is constructed. The covering is the union of a four-dimensional ball with an analytic disk. The projection of the ball is a local homeomorphism ramified along the disk, whose projection is a part of an algebraic curve.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 206–224, August, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-225.     

带Neumann边值条件的非线性微分算子逆的正性

本文研究一类带Neumann边值条件的非线性微分算子.利用Wirtinger不等式,比较定理,最大值原理以及上下解方法得到了该算子的双射性和逆算子的正性结论.     

A Closer Look at the Uncertainty Relation of Position and Momentum

We consider particles prepared by a single slit diffraction experiment. For those particles the standard deviation σ _p of the momentum is discussed. We find out that σ _p =∞ is not an exception but a rather typical case. A necessary and sufficient condition for σ _p <∞ is given. Finally, the inequality σ _p Δx≥π ℏ is derived and it is shown that this bound cannot be improved.     

Picone's formula for linear non-selfadjoint impulsive differential equations

In this paper, we derive a Picone type formula for second-order linear non-selfadjoint impulsive differential equations having fixed moments of impulse actions, and obtain a Wirtinger type inequality, a Leighton type comparison theorem, and a Sturm-Picone comparison theorem for such equations. Moreover, several oscillation criteria are also derived as applications.