Some Properties of the Basins of Attraction of the Newton’s Method for Simple Nonlinear Geodetic Systems

The research presented in this paper concerns the determination of the attraction basins of the Newton’s iterative method which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The authors considered simple observation systems corresponding to the intersections, or linear and angular resections, used in practice. The main goal was to investigate the properties of the sets of convergent in the initial points of the applied iterative method. An important issue regarding the possibility of automatic and quick selection of such points was also considered. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness or self-similarity were sought. The research also concerned the iterative structures of the basins, i.e. maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method, i.e. the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the essential results of the research is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises and their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points, i.e. located far from the solution points.

Iterative systems in sold structures and filigree structures / study in structural surface strategy

The relationship between the structure and the shape in contemporary architecture has different formatsaccording to design and structural requirements. The integration is important formulas among these relationship, asthey form one unit in architecture, where the integration is characteristic by the important property which is theiterative system. One of the strategies to find iterative systems in contemporary architecture over traditional is calledstructural surface. Previous knowledge has been differed in explaining the functional of systems and it’smechanisms, especially in the relationship among the form and structure, so the problem has surfaced “the necessityto know the difference of the properties and types of iterative systems among solid and perforated structures withinthe structural surface strategy, and it’s role between the form and the structure”. to achieve the research’s goal “whatare the repetition and iterative systems in contemporary architectural structure and it’s role to determine the shape ofthe relationship among the structure and the form for building of different heights “, which has depended ondescriptive analytical method in three stage after defining the repetition in general, iterative system in particular andprevious knowledge criticism. First stage has focused on building a theoretical framework (characteristics and typesof the iterative systems in contemporary architectural structure and structural surface strategy). Second stage hasfocused on knowing the levels of the relationship among the form and the structure, By studying selected sampleswithin building of different height In addition to determine the important basic assumption of the research ,which is(iterative systems is differ in architectural structure (solid and perforated) , through characteristics of the iterativesystems with the system and the relationship of the surrounding environment according to it’s height (high ,medium)and it’s formation method (orthogonal ,free) ) . Third stage has focused on analyzing the results and conclusions as inthe role of the iterative systems (structural surface strategy) in producing the solid structures by adopting therepetition of the structural elements and generative rules in perforated structures, and use it to achieve a fusionamong the form and the structure to produce structures with efficiency and aesthetic appearance and structures thatreflect movement and dynamism. The level of this relationship are :( first: the compositionl, through theorganizational depth of the architectural structure. Second: the expressive, by finding three types of the relationshipare (the merging, the discrete, and the hybrid).

Rotational periodic solutions for fractional iterative systems

<abstract><p>In this paper, we devoted to deal with the rotational periodic problem of some fractional iterative systems in the sense of Caputo fractional derivative. Under one sided-Lipschtiz condition on nonlinear term, the existence and uniqueness of solution for a fractional iterative equation is proved by applying the Leray-Schauder fixed point theorem and topological degree theory. Furthermore, the well posedness for a nonlinear control system with iteration term and a multivalued disturbance is established by using set-valued theory. The existence of solutions for a iterative neural network system is demonstrated at the end.</p></abstract>

On the Question of Asymptotic Integration of Singularly Perturbed Fractional-Order Problems

In this paper we consider an initial problem for systems of differential equations of fractional order with a small parameter for the derivative. Regularization problem is produced, and algorithm for normal and unique solubility of general iterative systems of differential equations with partial derivatives is given.Â

Existence results for positive solutions to iterative systems of four-point fractional-order boundary value problems in a Banach space

We investigate the eigenvalue intervals of [Formula: see text] for which the iterative system of four-point fractional-order boundary value problem has at least one positive solution by utilizing Guo–Krasnosel’skii fixed point theorem under suitable conditions. The obtained results in the paper are illustrated with an example for their feasibility.