Sufficient Conditions For Convergence Research Articles

µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is as follows: Let X be a complete paranormed vector space and Ω an arbitrary metric space, then a sequence {μn} of probability measures is weakly convergent to a probability measure μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. A special case is as the following: if X is a Banach space, Ω an arbitrary metric space, then {μn} is weakly convergent to μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. Our theorems and corollaries in the article modified or generalized some recent results regarding the convergence of sequences of measures.

A modified gradient‐based iterative algorithm for solving the complex conjugate and transpose matrix equations

In this paper, we first develop the modified gradient‐based iterative (MGI) method for the complex conjugate and transpose matrix equations . By adopting the updated technique, we can make full use of the latest information to compute the next result, which leads to a faster convergence rate. In theory, we apply the real representation of a complex matrix and the vec‐operator to prove the convergence properties. Furthermore, we extend the MGI algorithm to solve the generalized complex conjugate and transpose matrix equations. Then, the necessary and sufficient conditions for convergence of the MGI algorithm are presented. Lastly, three numerical examples are introduced to testify the efficiency of our methods.

Adaptive event-triggered [formula omitted] consensus secondary control for islanded microgrids under multiple time delays

In this paper, an adaptive event-triggered H∞ consensus control scheme is proposed for the secondary layer of AC MGs. The controller design problem of topological communication network with external disturbance, multiple delay, and communication burden is considered. Therefore, using the Lyapunov-Krasovskii theory, we obtain sufficient conditions for the robust consensus convergence of multi-agents under event-triggered and multiple time delays. Because traditional event-triggered often use fixed thresholds, it is challenging to balance reducing the communication burden and maintaining excellent system performance. Therefore, we design an adaptive event-triggered method that adaptively adjusts the trigger threshold according to the consistency error. Finally, a simulation system is established. The proposed control strategy has been validated through simulation results.

Решение оптимизационной задачи оценивания моделей полносвязной линейной регрессии

<p>This article is devoted to the problem of estimating fully connected linear regression models using the maximum likelihood method. Previously, a special numerical method was developed for this purpose, based on solving a nonlinear system using the method of simple iterations. At the same time, the issues of choosing initial approximations and fulfilling sufficient conditions for convergence were not studied. This article proposes a new method for solving the optimization problem of estimating fully connected regressions, similar to the method of estimating orthogonal regressions. It has been proven that, with equal error variances of interconnected variables, estimates of b-parameters of fully connected regression are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse covariance matrix. And if the ratios of the error variances of the variables are equal to the ratios of the variances of the variables, then the b-parameter estimates are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse correlation matrix, multiplied by the specific ratios of the standard deviations of the variables. A numerical experiment was carried out to confirm the correctness of the developed mathematical apparatus. The proposed method for solving the optimization problem of estimating fully connected regressions can be effectively used when solving problems of constructing multiple fully connected linear regressions.</p>

Analysis of Shielded Harmonic and Biharmonic Systems by the Iterative Extension Method

To describe stationary physical systems, well-known boundary problems for shielded Poisson and Sophie Germain equations are used. The obtained shielded harmonic and biharmonic systems are approximated using the finite element method and fictitiously continued. The resulting problems are solved using the developed method of iterative extensions. To expedite the convergence of this method, the relationships between physical quantities on the extension of systems and additional parameters of the iterative method are employed. The formulations of sufficient convergence conditions for the iterative process utilize interdisciplinary connections with functional analysis, applying discrete analogs of the principles of function extensions while preserving norm and class. In the algorithmic implementation of the iterative extensions method, automation is applied to control the selection of the optimal iterative parameter value during information processing. In accordance with the fictitious domain methodology, solvable problems from domains with a complex geometry are reduced to problems in a rectangle in the two-dimensional case and in a rectangular parallelepiped in the three-dimensional case. But now, in the problems being solved, the minimization of the error of the iterative processes is carried out with a norm stronger than the energy norm. Then, all relative errors are estimated from above in the used norms by terms of infinitely decreasing geometric progressions. A generalization of the developed methodology to boundary value problems for polyharmonic equations is possible.

Complete convergence for the maximal partial sums without maximal inequalities

This work provides the necessary and sufficient conditions for complete convergence for the maximal partial sums of dependent random variables. The results are proved without using maximal inequalities. The main theorems can be applied to sequences of (i) m-pairwise negatively dependent random variables and (ii) m-extended negatively dependent random variables. While the result for case (i) unifies and improves many existing ones, the result for case (ii) complements the main theorem of Chen et al. [J. Appl. Probab., 2010]. Affirmative answers to open questions raised by Chen et al. [J. Math. Anal. Appl., 2014], and Wu and Rosalsky [Glas. Mat. Ser. III, 2015] are also given. Two examples illustrating the sharpness of the main result are presented.

Average-rendezvous with connectivity preservation for first-order multiagent systems with constant reference signals

Average-rendezvous problem with connectivity preservation of constant reference signals is investigated for first-order multiagent systems. According to a constructive potential function, a proportional–integral average-rendezvous algorithm is proposed to make agents converge to the average value of constant reference signals as well as the network connectivity is preserved all the time. Sufficient convergence conditions are obtained for the algorithm by designing the proper Lyapunov functions. Numerical examples illustrate the correctness of theoretical results.

Convergence of Modified S-Iteration to Common Fixed Points of Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

In this paper, we provide some sufficient conditions for the strong convergence of an improved form of modified S-iteration for approximating common fixed points of two asymptotically nonexpansive mappings defined on a closed convex subset of a uniformly convex hyperbolic spaces.

Convergent and oscillatory solutions in infinite-dimensional synchronization systems

Control systems that arise in phase synchronization problems are featured by infinite sets of stable and unstable equilibria, caused by presence of periodic nonlinearities. For this reason, such systems are often called “pendulum-like”. Their dynamics are thus featured by multi-stability and cannot be examined by classical methods that have been developed to test the global stability of a unique equilibrium point. In general, only sufficient conditions for the solution convergence are known that are usually derived for pendulum-like systems of Lurie type, that is, interconnections of stable LTI blocks and periodic nonlinearities, which obey sector or slope restrictions. Most typically, these conditions are written as multi-parametric frequency-domain inequalities, which should be satisfied by the transfer function of the system’s linear part. Remarkably, if the frequency-domain inequalities hold outside some bounded range of frequencies, then the absence of periodic solutions with frequencies in this range is guaranteed, which can be considered as a weaker asymptotical property.

Well-posedness criteria for one family of boundary value problems

This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.

Convergence of deep ReLU networks

We explore convergence of deep neural networks with the popular ReLU activation function, as the depth of the networks tends to infinity. To this end, we introduce the notion of activation domains and activation matrices of a ReLU network. By replacing applications of the ReLU activation function by multiplications with activation matrices on activation domains, we obtain an explicit expression of the ReLU network. We then identify the convergence of the ReLU networks as convergence of a class of infinite products of matrices. Sufficient and necessary conditions for convergence of these infinite products of matrices are studied. As a result, we establish necessary conditions for ReLU networks to converge that the sequence of weight matrices converges to the identity matrix and the sequence of the bias vectors converges to zero as the depth of ReLU networks increases to infinity. Moreover, we obtain sufficient conditions in terms of the weight matrices and bias vectors at hidden layers for pointwise convergence of deep ReLU networks. These results provide mathematical insights to convergence of deep neural networks. Experiments are conducted to mathematically verify the results and to illustrate their potential usefulness in initialization of deep neural networks.

Sharp sufficient conditions for mean convergence of the maximal partial sums of dependent random variables with general norming sequences

Sharp sufficient conditions for mean convergence of the maximal partial sums of dependent random variables with general norming sequences

Adaptive hybrid Kalman filter for attitude motion parameters estimation of space non-cooperative tumbling target

Attitude motion parameters estimation of the space non-cooperative tumbling target is the premise of the on-orbit target capture task. However, existing methods cannot balance the estimation accuracy and efficiency. In this paper, an adaptive hybrid Kalman filter is presented to estimate the attitude, angular velocity and inertia motion parameters of the space non-cooperative tumbling target. First, the dynamic model of the non-cooperative tumbling target is derived, and the nonlinear discrete state-space equation is obtained. Then, a hybrid Kalman filter is designed by combining the extended Kalman filter and the unscented Kalman filter. The adaptive technique is introduced to track time-varying uncertain measurement noise caused by complex space disturbances. After that, based on the first-order linearization technique, the sufficient condition for the local convergence of the proposed method is proved. Finally, the numerical simulation experiment was conducted to compare the proposed method with the adaptive extended Kalman filter and the adaptive unscented Kalman filter. The experimental results show that the estimation accuracy of the proposed method is much higher than that of the adaptive extended Kalman filter and slightly lower than that of the adaptive unscented Kalman filter, and the iteration-average Central-Processing-Unit time is 64.2% smaller than that of the adaptive unscented Kalman filter and slightly longer than that of the adaptive extended Kalman filter. The proposed method achieves a much higher accuracy with a slight loss of iteration-average Central-Processing-Unit time.

The Weighted, Relaxed Gradient-Based Iterative Algorithm for the Generalized Coupled Conjugate and Transpose Sylvester Matrix Equations

By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the real representation of a complex matrix as a tool, the necessary and sufficient conditions for the convergence of the WRGI algorithm are determined. Also, some sufficient convergence conditions of the WRGI algorithm are presented. Moreover, the optimal step size and the corresponding optimal convergence factor of the WRGI algorithm are given. Lastly, some numerical examples are provided to demonstrate the effectiveness, feasibility and superiority of the proposed algorithm.

Iterative Process for a Common Fixed Point of a Finite Family of Asymptotically k-Pseudocontractive Maps

Let $K$ be a closed convex nonempty subset of a Hilbert space $H$ and let $\{T_i\}_{i=1}^N$ be a finite family of Asymptotically k-pseudocontractive maps from $K$ into itself with $F = ∩_{i=1}^NF(T_i) not empty. Sufficient conditions for the strong convergence of the sequence of successive approximations generated by a Picard-like process to a common fixed point of the family are proved.

On the asymptotic stabilization of the nonautonomous evolution equations in Hilbert spaces

This paper deals with the practical feedback stabilization of a class of evolution equations in Hilbert spaces. By using the Lyapunov function method, we give sufficient conditions for the global uniform practical convergence of the closed-loop system. Based on the global null-controllability, we construct an output feedback controller that guarantees global uniform practical stability of uncertain systems. A numerical application is given to show the applicability of our theoretical results.

Convergence analysis of deep residual networks

Various powerful deep neural network architectures have made great contributions to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets are the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at studying the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix–vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notion of activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. We also conduct experiments on benchmark machine learning data to illustrate the potential usefulness of the results.

On transforming hybrid nonlinear control problems with model uncertainty and input disturbance to mixed integer-linear programs

Control of multi-input multi-output (MIMO) hybrid nonlinear dynamic systems which are affine in control inputs is studied in this paper. It is assumed that each control input of the system can be continuous or discrete with a bound constraint. It is also assumed that the controller is discrete-time and updates the control(s) with a constant frequency. Based on these assumptions, a theorem which represents the necessary and sufficient condition for uniform convergence of the sequence of samples of the state vector of system towards the desired point is presented and proved. As a result of this theorem, a (mixed integer-) linear programming for calculation of control(s) is proposed. Two well-known nonlinear hybrid control problems with multiple inputs are solved by using the proposed method and the results are presented.

Two Point Iterative Schemes for Nondifferentiable Equations in Banach Space

The local as well as the semi-local convergence analysis is established for a certain single step-two point iterative scheme defined on a Banach space setting. These schemes converge to a locally unique solution of a nonlinear equation. Both types of convergence are based on w-type continuity and majorizing functions and sequences. An auxiliary fixed linear operator is utilized to assure the existence of inverses of the linear operators involved as well as the initial points of the iterative scheme. The local analysis provides the radius of convergence, error estimates and information on the uniqueness of the solution. Moreover, the semi local analysis provides sufficient convergence conditions, error estimates and uniqueness of the solution results. Numerical examples further validate the theoretical results.

A Sufficient Condition for Uniform Convergence of Trigonometric Series with p-Bounded Variation Coefficients

In this paper we consider trigonometric series with p-bounded variation coefficients. We presented a sufficient condition for uniform convergance of such series in case p>1. This condition is significantly weaker than these obtained in the results on this subject known in the literature.