Image Georeferencing using Artificial Neural Network Compared with Classical Methods

Georeferencing process is one of the most important prerequisites for various geomatics applications; for example, photogrammetry, laser scan analysis, remotely sensing, spatial and descriptive data collection, and others. Georeferencing mostly involves the transformation of coordinates obtained from images that are inhomogeneous due to accuracy differences. The georeferencing depends on image resolution and accuracy level of measurements of reference points ground coordinates. Accordingly, this study discusses the subject of coordinate’s transformation from the image to the global coordinates system (WGS84) to find a suitable method that provides more accurate results. In this study, the Artificial Neural Network (ANN) method was applied, in addition to several numerical methods, namely the Affine divided difference, Newton’s divided difference, and polynomial transformation. The four methods were modelled and coded using Matlab programming language based on an image captured from Google Earth. The image was used to determine reference points within the study area (University of Baghdad campus). The findings of this study showed that the ANN enhanced the results by about 50% in terms of accuracy and 90% in terms of homogeneity, compared with the other methods.

Extended Kung–Traub Methods for Solving Equations with Applications

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

Differential-difference method with approximation of the inverse operator

The problem of finding an approximate solution of a nonlinear equation with operator decomposition is considered. For equations of this type, a nonlinear operator can be represented as the sum of two operators – differentiable and nondifferentiable. For numerical solving such an equation, a differential-difference method, which contains the sum of the derivative of the differentiable part and the divided difference of the nondifferentiable part of the nonlinear operator, is proposed. Also, the proposed iterative process does not require finding the inverse operator. Instead of inverting the operator, its one-step approximation is used. The analysis of the local convergence of the method under the Lipschitz condition for the first-order divided differences and the bounded second derivative is carried out and the order of convergence is established.

Semilocal Convergence of the Extension of Chun’s Method

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

Some Remarks on Results Related to ∇-Convex Function

In the present article, we give new techniques for proving general identities of the Popoviciu type for discrete cases of sums for two dimensions using higher-order ∇-divided difference. Also, integral cases are deduced by different methods for differentiable functions of higher-order for two variables. These identities are a generalization of various previously established results. An application for the mean value theorem is also presented.

Some Remarks on Results Related to ∇-Convex Function

In the present article, we give new techniques for proving general identities of the Popoviciu type for discrete cases of sums for two dimensions using higher-order ∇-divided difference. Also, integral cases are deduced by different methods for differentiable functions of higher-order for two variables. These identities are a generalization of various previously established results. An application for the mean value theorem is also presented.

Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

A Family of Fifth and Sixth Convergence Order Methods for Nonlinear Models

We study the local convergence of a family of fifth and sixth convergence order derivative free methods for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the seventh derivative to show convergence. However, we only use the first divided difference of order one as well as the first derivative in our analysis. We also provide computable radius of convergence, error estimates, and uniqueness of the solution results not given in earlier studies. Hence, we expand the applicability of these methods. The dynamical analysis of the discussed family is also presented. Numerical experiments complete this article.