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51.
A. V. Balakrishnan 《Applied Mathematics and Optimization》1995,31(2):177-187
In this paper we generalize the Kalman-Yakubovic lemma to infinite dimensions—or, more precisely, to semigroups of operators over a Hilbert space. The proof differs substantially from the finite-dimensional version and is based on the Paley-Wiener-Helson-Lowdenslager factorization theorem. 相似文献
52.
Zukang Zheng 《中国科学A辑(英文版)》1997,40(9):932-944
The stability of testing hypotheses is discussed. Differing from the usual tests measured by Neyman-Pearson lemma, the regret
and correction of the tests are considered. After the decision is made based on the observationsX
1,X
2, ⋅⋅⋅,X
n, one more piece of datumX
n+1 is picked and the test is done again in the same way but based onX
1,X
2, ⋅⋅⋅,X
n,X
n+l There are three situations: (i) The previous decision is right but the new decision is wrong; (ii) the previous decision
is wrong but the new decision is right; (iii) both of them are right or both of them are wrong. Of course, it is desired that
the probability of the occurrence of (i) is as small as possible and the probability of the occurrence of (ii) is as large
as possible. Since the sample size is sometimes not chosen very precisely after the type I error and the type II error are
determined in practice, it seems more urgent to consider the above problem. Some optimal plans are also given.
Project supported by the National Natural Science Foundation of China and the Doctoral Programme Foundation. 相似文献
53.
Duong Minh Duc Tran Vinh Hung Nguyen Tien Khai 《Proceedings of the American Mathematical Society》2007,135(3):921-927
Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.
54.
Yanguang Li 《Journal of Dynamics and Differential Equations》2003,15(4):699-730
For finite-dimensional maps and periodic systems, Palmer rigorously proved Smale horseshoe theorem using shadowing lemma in 1988 [20]. For infinite-dimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990 [30], and Henry in 1994 [9]. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996 [17]. For infinite-dimensional autonomous systems, the current article offers such a proof. First we prove an Inclination Lemma to set up a coordinate system around a pseudo-orbit. Then we utilize graph transform and the concept of persistence of invariant manifold, to prove the existence of a shadowing orbit. 相似文献
55.
Weishi Liu 《Journal of Dynamics and Differential Equations》2006,18(3):667-691
In this paper, we consider singularly perturbed systems with multiple sets of turning points. Two types of related results contributing to geometric singular perturbation theory are obtained. The first result establishes a new class of invariant manifolds. The second result consists of several exchange lemmas that characterize the evolution of an invariant manifold passing through the vicinity of the slow manifold.Partially supported by NSF Grants DMS-0071931 and DMS-0406998. 相似文献
56.
In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball B n respectively. 相似文献
57.
This paper is concerned with a generalized Arzela–Ascoli's lemma, which has been extensively applied in almost periodic problems by the continuation theorem of degree theory. We give a counter example to show that this lemma is incorrect, and there is a gap in the proof of some existing literature, where the addressed generalized Arzela–Ascoli's lemma was used. Moreover, we make some final comments and introduce an open problem. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
58.
Optical vortices as topological objects exist ubiquitously in nature. In this paper, we use the principle of variational method and mountain pass lemma to develop some existence theorems for the stationary vortex wave solution of a coupled nonlinear Schrödinger equations, which describe the possibility of effective waveguiding of a weak probe beam via the cross‐phase modulation‐type interaction. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Additionally, as demanded by beam confinement, we prove the exponential decay of the soliton amplitude at infinity. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
59.
Radu Precup 《Journal of Mathematical Analysis and Applications》2011,373(1):138-139
We obtain existence and localization results of positive nontrivial solutions for a class of semilinear elliptic variational systems. The proof is based on variants of Schechter's localized critical point theorems for Hilbert spaces not identified to their duals and on the technique of inverse-positive matrices. The Leray-Schauder boundary condition is also involved. 相似文献
60.
In this paper, the problem of controller design for Lur’e systems guaranteeing dichotomy is investigated. On the basis of Kalman-Yakubovich-Popov (KYP) lemma and two frequency equalities, a new methodology for the dichotomy analysis of the Lur’e systems is proposed. A linear matrix inequality (LMI) based criterion is derived, which is equivalent to the Leonov’s frequency-domain one, while for the dichotomy analysis and synthesis which is more straightforward than the frequency-domain one. In virtue of this result, a dynamic output feedback controller ensuring the dichotomy property for Lur’e systems is designed. Finally a numerical example is included to demonstrate the validity and the applicability of the proposed approach. 相似文献