Iterative Theory Research Articles

Iteration theory of noncommutative maps

Iteration theory of noncommutative maps

Computational Analysis of a Novel Iterative Scheme with an Application

The computational study of fixed-point problems in distance spaces is an active and important research area. The purpose of this paper is to construct a new iterative scheme in the setting of Banach space for approximating solutions of fixed-point problems. We first prove the strong convergence of the scheme for a general class of contractions under some appropriate assumptions on the domain and a parameter involved in our scheme. We then study the qualitative aspects of our scheme, such as the stability and order of convergence for the scheme. Some nonlinear problems are then considered and solved numerically by our new iterative scheme. The numerical simulations and graphical visualizations prove the high accuracy and stability of the new fixed-point scheme. Eventually, we solve a 2D nonlinear Volterra Integral Equation (VIE) via the application of our main outcome. Our results improve many related results in fixed-point iteration theory.

Generation of genetic codes with 2-adic codon algebra and adaptive dynamics

This is a brief review on modeling genetic codes with the aid of 2-adic dynamical systems. In this model amino acids are encoded by the attractors of such dynamical systems. Each genetic code is coupled to the special class of 2-adic dynamics. We consider the discrete dynamical systems, These are the iterations of a function F:Z2→Z2, where Z2 is the ring of 2-adic numbers (2-adic tree). A genetic code is characterized by the set of attractors of a function belonging to the code generating functional class. The main mathematical problem is to reduce degeneration of dynamic representation and select the optimal generating function. Here optimality can be treated in many ways. One possibility is to consider the Lipschitz functions playing the crucial role in general theory of iterations. Then we minimize the Lip-constant. The main issue is to find the proper biological interpretation of code-functions. One can speculate that the evolution of the genetic codes can be described in information space of the nucleotide-strings endowed with ultrametric (treelike) geometry. A code-function is a fitness function; the solutions of the genetic code optimization problem are attractors of the code-function. We illustrate this approach by generation of the standard nuclear and (vertebrate) mitochondrial genetics codes.

Combinatorial Estimations on Burnside Type Problems

The Burnside problem, formulated by W. Burnside in 1902, is one of the most well-known and important open questions in the field of Group Theory. Despite significant progress made in the past century towards solving this problem, its complete solution remains unknown. In this paper, we investigate one of the approaches to solving the Burnside problem based on the application of an iterative theory of small cancellations and canonical forms developed by E. Rips in recent years. We present a novel self-contained exposition of this theory and utilize it to obtain new estimates on the infiniteness of initial approximations of Burnside groups where only a finite number of periodic relations is used for relatively small odd exponents (n>120).

SOC estimation of lithium-ion batteries based on the improved Kalman filter

Nowadays, the environmental crisis and energy crisis have been known by people around the world as issues that have to be taken seriously. It is recognized that electric vehicles are more environmentally friendly than traditional fuel cars, but there are still many immature aspects of electric vehicle technology. Battery technology is one of them, and lithium battery is a common power storage tool for electric vehicles. Therefore, it is very important to monitor the status of the battery pack in real-time. Improving the practical capacity, safety, and service life of lithium batteries all have an important impact on the development of electric vehicles. In battery management systems (BMS), the accurate prediction of the battery state of charge (SOC) is particularly important. In this paper, the 18650 lithium battery was selected for the research experiment, and the circuit model of second-order RC was selected. Modeling by MATLAB, the SOC-OCV curve of the battery was determined through the data obtained from the charging and discharging experiment (HPPC), the original least squares method was improved, the forgetting factor was introduced, and the model was identified by the parameters based on the experimental data. The traditional SOC prediction method added noise adaptive rules and iterative theory in the extended Kalman filter(EKF), formed the adaptive Kalman filter (AIEKF) to estimate the SOC of lithium battery, and estimated the SOC using the American urban road cycle (UDDS) data. The results show that the AIEKF algorithm has fewer errors and advantages than the traditional EKF algorithm.

Research on Multichannel Marketing of Ostentatious Products Based on Consumption Iteration Theory -- Take Hermes as an Example

The iteration of China's consumer society has stimulated huge potential consumption momentum, and Chinese consumers' consumption focus has shifted from functional product consumption to ostentatious product consumption. With the deepening of economic globalization, international first-tier brands, which have the attribute of showing off, are increasingly entering the daily lives of Chinese consumers. Chinese consumers are regarded as the world's largest luxury consumer group with their strong purchasing power. If international luxury brands want to occupy a larger market share in China, they must conform to the characteristics of the Chinese market and meet the shopping habits of Chinese consumers. Therefore, based on the 4P marketing theory, this article analyzes the marketing background and current situation of the H brand, and conducts a PEST analysis of the H brand. It puts forward feasible suggestions on how to exploit the potential customers' consumption ability and obtain greater market share by taking advantage of the opportunities of sinking market and alternating generations of consumers after entering the Chinese market.

On the convergence, stability and data dependence results of the JK iteration process in Banach spaces

Abstract This article analyzes the JK iteration process with the class of mappings that are essentially endowed with a condition called condition (E). The convergence of the iteration toward a fixed point of a specific mapping satisfying the condition (E) is obtained under some possible mild assumptions. It is worth mentioning that the iteration process JK converges better toward a fixed point compared to some prominent iteration processes in the literature. This fact is confirmed by a numerical example. Furthermore, it has been shown that the iterative scheme JK is stable in the setting of generalized contraction. The data dependence result is also established. Our results are new in the iteration theory and extend some recently announced results of the literature.

Modeling and prediction of surface topography and surface roughness in magnetic-field-enhanced shear-thickening polishing of SiC mold

To fill the gap in the research on the surface morphology formation theory of small aspheric magnetic-field-enhanced shear-thickening polishing (MESTP), a theoretical framework for surface morphology formation during the MESTP process was proposed for the first time. Based on the macroscopic computational fluid dynamics and convolution iteration theory, a multiscale surface morphology prediction model was established. This model considers the mechanical mechanism of the ductile-brittle transition of hard and brittle materials and effective cutting abrasive distribution law. Through a series of experiments and simulations, the maximum relative error between the measured Sa data and theoretical prediction data was 9.1%, and the accuracy of the model was analyzed based on RMSE. The proposed model can predict the microsurface morphology in MESTP.

Algorithms for computing basins of attraction associated with a rational self-map of the Hopf fibration based on Lyapunov exponents

The objective of this work is twofold. One, we develop a theory of iteration for complex rational functions so that the problems of overflow and indeterminacy caused by null, or almost null, denominators can be avoided when developing implementations. Second, we present easily implementable methods that allow the calculation of attracting cycles as well as the graphical representation of their basins of attraction.In order to deal with our first goal we work with homogeneous complex coordinates and we take the complex projective line as a model, which is expressed as a quotient of the 3-sphere through the Hopf fibration. An irreducible representation of a rational function can now be presented as a Hopf fibration endomorphism. As well as our second goal is concerned, we use Lyapunov functions and exponents to calculate the cycles associated with endomorphisms and to graphically represent the corresponding basins of attraction. We point out that our algorithms are based on the calculation of a finite set of non-negative real constants and their calculation does not depend on the previous determination of the fixed points or attracting cycles.

Evaluation, computation and coding of iterative function using recursive approach

Iteration and recursion are very pivotal concepts in understanding the logic and building blocks of all computer programs across all programming paradigms. Although the theory of iteration together with the development and implementation of iterative algorithm is easy to grasp, that of recursion remains elusive to programmers especially novice programmers. In this research, functions composition is applied in the explanation of iteration using recursion. The method demonstrates an easy and elaborate way of writing iterative programs using recursion by identifying the significant variables in both constructs. Function composition is used to write the recursive function, the recursive variable is identified as a variable that converges towards the base case, also the base case is also identified as being the terminating point of the function call else the function call runs and fill the stack memory causing stack overflow. The recursive part is the loop update and base case the termination condition in iterative programs. The results obtained simplifies how to write iterative codes using recursion.

Transformative learning theory: Where we are after 45 years

AbstractThis article provides an overview of the literature on transformative learning over the previous 45 years, describes its current condition as a relatively mature collection of theories, and calls for greater clarity, new iterations of theory, and productive and substantive steps forward. It then provides an overview of the contributions in this issue of New Directions for Adult and Continuing Education.

Multiple subharmonic solutions in Hamiltonian system with symmetries

In this paper, we prove the existence of infinitely many non-constant geometrically distinct symmetric subharmonic solutions and symmetric subharmonic brake orbits for symmetric Hamiltonian systems which are superquadratic at zero and infinity by combining iteration theory of Maslov-type indices under various boundary conditions, the Galerkin approximation arguments and link theorem of critical point theory. Furthermore, we prove the existence of multiple geometrically distinct symmetric subharmonic solutions and symmetric subharmonic brake orbits for the symmetric Hamiltonian systems.

Representation of exact trajectory solutions for chaotic one-dimensional maps in Schroder form

Purpose of the article is to illustrate the genesis, meaning and significance of the functional Schroder equation, introduced in the theory of iterations of rational functions, for the theory of deterministic chaos by analytical calculations of exact trajectory solutions, invariant densities and Lyapunov exponents of one-dimensional chaotic maps. We demonstrate the method for solving the functional Schroder equation for various chaotic maps by passing to a topologically conjugate mappings, for which finding the exact trajectory solution is a simpler mathematical procedure. Results of the analytical solution of the Schroder equation for 12 chaotic mappings of various types and the calculation of the corresponding expressions for exact trajectory solutions, invariant densities and Lyapunov exponents are presented. Conclusion is made about the methodological expediency of formulating and solving the Schroder equations by the study of the dynamics of one-dimensional chaotic mappings.

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified.

Integration with filters

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part, which can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations of real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.

Maximizing team synergy in AI-related interdisciplinary groups: an interdisciplinary-by-design iterative methodology

In this paper, we propose a methodology to maximize the benefits of interdisciplinary cooperation in AI research groups. Firstly, we build the case for the importance of interdisciplinarity in research groups as the best means to tackle the social implications brought about by AI systems, against the backdrop of the EU Commission proposal for an Artificial Intelligence Act. As we are an interdisciplinary group, we address the multi-faceted implications of the mass-scale diffusion of AI-driven technologies. The result of our exercise lead us to postulate the necessity of a behavioural theory that standardizes the interaction process of interdisciplinary groups. In light of this, we conduct a review of the existing approaches to interdisciplinary research on AI appliances, leading to the development of methodologies like ethics-by-design and value-sensitive design, evaluating their strengths and weaknesses. We then put forth an iterative process theory hinging on a narrative approach consisting of four phases: (i) definition of the hypothesis space, (ii) building-up of a common lexicon, (iii) scenario-building, (iv) interdisciplinary self-assessment. Finally, we identify the most relevant fields of application for such a methodology and discuss possible case studies.

Minimal [formula omitted]-symmetric periodic solutions of general Hamiltonian systems and delay differential equations

For a skew-symmetric non-degenerate 2n×2n matrix J∈L2n(R) and a J-symplectic matrix M with MTJ−1M=J−1, in this paper we redefine the Maslov-type (J,M)-index for J-symplectic path and develop the iteration theory of the Maslov-type (J,M)-index theory. As applications we study the minimal period problems for M-symmetric orbits of nonlinear autonomous supper-quadratic general Hamiltonian systems, where M is a J-symplectic matrix satisfying Mk=I2n. Also we study the minimal periodic problem for some super-linear delay systems as applications, and we give a positive answer to the Rabinowitz-type minimal period problem for a kind of super-linear delay equations with the forms studied by Kaplan and Yorke in [12].

A Music Curriculum Integration and Reconstruction Model Based on Advanced Iterative Reconstruction Algorithm

Music curriculum fusion and super-resolution reconstruction based on musical elements have gradually attracted the attention of researchers. The traditional music element fusion and reconstruction algorithm is based on the fusion and reconstruction of all the pixel information of the source music element, which has the problem of high time and space complexity. Based on the advanced iterative reconstruction theory, this paper uses the measurement matrix to measure the dimensionality of the music signal, compresses the music element data while acquiring the music elements, reduces the sampling frequency, reduces the sampling amount of the music element data, and greatly reduces the data. It solves the problem of spatial resolution reduction caused by degradation and the problem of music element feature extraction, classification, and identification and can obtain more music element features and detailed parameters. Aiming at the back-projection algorithm of musical symbol filtering, the advanced iterative reconstruction algorithm of musical symbol filtering of a triangular line array is simulated. The experimental results show that the feasibility analysis factor of the scheme reaches 0.917, and the running time of the reconstruction algorithm is reduced to 0.131 s, which promotes a large amount of data in music curriculum element fusion and super-resolution reconstruction.

#FreeBritney and the Pleasures of Conspiracy

#FreeBritney and the Pleasures of Conspiracy

Evaluating the item-level factor structure of anhedonia

BackgroundAnhedonia has long been theorized to be a multidimensional construct, focusing on domains of reward stimuli and temporal relationship to reward. However, little empirical work has directly examined whether there is support for this assertion. MethodsThe study used data from young adults from four independent samples (n = 2098). Participants completed multiple measures of anhedonia. ResultsWe used rigorous conducted exploratory and confirmatory factor analyses on items from six commonly used anhedonia measures to examine dimensions underlying anhedonia. Results suggested a four-factor solution with factors reflecting social reward, social disinterest, status/achievement, and physical/natural reward. The identified factors reflected broad content domains of pleasure, but not specific reward processes. The four factors were modestly associated with one another, suggesting a weak common underlying anhedonia trait that manifests across multiple dimensions. Factor scores were associated with personality measures, reward-related indices, and depression symptoms, supporting the validity of the factors. LimitationsParticipants were all young adults and we assessed anhedonia only at the level of self-report. ConclusionAnhedonia is a multidimensional construct. However, the dimensions of anhedonia only distinguish domains of, but not temporal processes of anhedonia. Future work should continue to refine the structures underlying the construct of anhedonia through iterative theory- and data-driven research and examine associations between anhedonia and clinical outcomes.