One difficulty for quaternion neural networks (QNNs) is that quaternion nonlinear activation functions are usually non-analytic and thus quaternion derivatives cannot be used. In this paper, we derive the quaternion gradient descent, approximated quaternion Gauss-Newton and quaternion Levenberg-Marquardt algorithms for feedforward QNNs based on the GHR calculus, which is suitable for analytic and non-analytic quaternion functions. Meanwhile, we solve a widely linear quaternion least squares problem in the derivation of quaternion Gauss-Newton algorithm, which is more general than the usual least squares probŹlem. A rigorous analysis of the convergence of the proposed algorithms is provided. Simulations on the prediction of benchmark signals support the approach.
Stability and convergence of the linear semi-implicit discrete duality finite volume (DDFV) numerical scheme in 2D for the solution of the regularized curvature driven level set equation is proved. Numerical experiments concerning comparison with exact solution and image filtering problem using proposed scheme are included.
The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability oThe main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation xn+1 = α0xn + α1xn−l + α2xn−k ⁄ β0xn + β1xn−l + β2xn−k , n = 0, 1, 2, . . . where the coefficients αi , βi ∈ (0,∞) for i = 0, 1, 2, and l, k are positive integers. The initial conditions x−k, . . . , x−l , . . . , x−1, x0 are arbitrary positive real numbers such that l < k. Some numerical experiments are presented.
The purpose of this paper is to introduce some new generalized double difference sequence spaces using summability with respect to a two valued measure and an Orlicz function in $2$-normed spaces which have unique non-linear structure and to examine some of their properties. This approach has not been used in any context before.
In this paper, following the methods of Connor \cite {connor}, we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely \cite {moe}) to $\mu $-statistical convergence and convergence in $\mu $-density using a two valued measure $\mu $. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure $\mu $ called the (APO$_2$) condition, inspired by the (APO) condition of Connor \cite {jc}. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure $\mu $ has the condition (APO$_2$).
The paper examines similarities between observer design as introduced in Automatic Control Theory and filter design as established in Signal Processing. It is shown in the paper that there are obvious connections between them in spite of different aims for their design. Therefore, it is prospective to make them be compatible from the structural point of view. Introduced error invariance and error convergence properties of both of them are unifying tools for their design. Lyapunov's stability theory, signal power, system energy and a power balance relation are other basic terms used in the paper.