In this paper, we use a global implicit function theorem for the investigation of the existence and uniqueness of a solution as well as the sensitivity of a Cauchy problem for a general integro-differential system of ...
详细信息
In this paper, we use a global implicit function theorem for the investigation of the existence and uniqueness of a solution as well as the sensitivity of a Cauchy problem for a general integro-differential system of order alpha is an element of (0, 1) of Volterra type, involving two functional parameters nonlinearly.
We generalize a recent global implicit function theorem from [8] to the case of a mapping acting between Banach spaces. Considerations related to duality mapping and to certain auxiliary functional are used in the pro...
详细信息
We generalize a recent global implicit function theorem from [8] to the case of a mapping acting between Banach spaces. Considerations related to duality mapping and to certain auxiliary functional are used in the proof together with the local implicitfunctiontheorem and mountain pass geometry. An application to integro-differential systems is given.
The expectation-maximisation (EM) algorithm framework is an important tool for statistical computation. Due to the changing nature of data, online and mini-batch variants of EM and EM-like algorithms have become incre...
详细信息
The expectation-maximisation (EM) algorithm framework is an important tool for statistical computation. Due to the changing nature of data, online and mini-batch variants of EM and EM-like algorithms have become increasingly popular. The consistency of the estimator sequences that are produced by these EM variants often rely on an assumption regarding the continuous differentiability of a parameter update function. In many cases, the parameter update function is not in closed form and may only be defined implicitly, which makes the verification of the continuous differentiability property difficult. We demonstrate how a global implicit function theorem can be used to verify such properties in the cases of finite mixtures of distributions in the exponential family, and more generally, when the component-specific distributions admit data augmentation schemes, within the exponential family. We then illustrate the use of such a theorem in the cases of mixtures of beta distributions, gamma distributions, fully visible Boltzmann machines and Student distributions. Via numerical simulations, we provide empirical evidence towards the consistency of the online EM algorithm parameter estimates in such cases.
An adaptive fuzzy control approach is proposed for a class of multiple-input-multiple-output (MIMO) nonlinear systems with completely unknown non-affine functions. The global implicit function theorem is first used to...
详细信息
An adaptive fuzzy control approach is proposed for a class of multiple-input-multiple-output (MIMO) nonlinear systems with completely unknown non-affine functions. The global implicit function theorem is first used to prove the existence of an unknown ideal implicit controller that can achieve the control objectives. Within this scheme, fuzzy systems are employed the approximate the unknown ideal implicit controller, and robustifying control terms are used to compensate the approximation errors and external disturbances. The adjustable parameters of the used fuzzy systems are deduced from the stability analysis of the closed-loop system in the sense of Lyapunov. To show the efficiency of the proposed controllers, two simulation examples are presented.
In this paper we prove a general theorem stating a sufficient condition for the inverse image of a point under a continuously differentiable map from ? n to ? k to be connected. This result is applied to the traject...
详细信息
In this paper we prove a general theorem stating a sufficient condition for the inverse image of a point under a continuously differentiable map from ? n to ? k to be connected. This result is applied to the trajectories generated by the Newton flow. Several examples demonstrate the applicability of the results to nontrivial problems.
In the present paper, under the Lipschitz condition and a monotonicity assumption, we obtain an L-p-result (p > 1), including the existence and uniqueness of the p-th power integrable solutions and a pair of relate...
详细信息
In the present paper, under the Lipschitz condition and a monotonicity assumption, we obtain an L-p-result (p > 1), including the existence and uniqueness of the p-th power integrable solutions and a pair of related p-th power estimates, for coupled forward-backward stochastic differential equations with random coefficients on small durations. A measurable global implicit function theorem has been established and plays a key role in our analysis. It is the first measurable version of implicitfunctiontheorems to our best knowledge, therefore the theorems themselves have been improved. (C) 2019 Elsevier Inc. All rights reserved.
暂无评论